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inscribed 

TO 

CAPTAIN  JOHN  ERICSSON,  LL.D., 

AS  A SLIGHT  TRIBUTE  TO  HIS  GENIUS  AND  ATTAINMENTS, 
AND  IN  TESTIMONY  OF  THE  SINCERE  REGARD 
AND  ESTEEM  OF  HIS  FRIEND, 

THE  AUTHOR 


: it  > 


# 


PREFACE 

To  tlie  Forty-fifth.  Edition. 


The  First  Edition  of  this  work,  consisting  of  284  pages, 
was  submitted  to  the  Mechanics  and  Engineers  of  the  United 
States  by  one  of  their  number  in  1843,  who  designed  it  for 
a convenient  reference  to  Rules,  Results,  and  Tables  con- 
nected with  the  discharge  of  their  various  duties. 

The  Twenty-first  Edition  was  published  in  1867,  consisted 
of  664  pages,  and,  in  addition  to  the  original  design  of  the 
work,  it  was  essayed  to  embrace  some  general  information 
upon  Mechanical  and  Physical  subjects. 

The  Tables  of  Areas  and  Circumferences  of  Circles  have 
been  extended,  and  together  with  those  of  Weights  of  Metals, 
Balls,  Tubes,  Pipes,  etc.,  of  this  and  some  preceding  editions 
were  computed  and  verified  by  the  author. 

This  edition  is  a revision  and  an  entire  reconstruction  of 
all  preceding,  embracing  amended  and  much  new  matter,  as 
Masonry,  Strength  of  Girders,  Floor  Beams,  Logarithms,  etc., 
etc. 

To  the  young  Mechanic  and  Engineer  it  is  recommended 
to  cultivate  a knowledge  of  Physical  Laws  and  to  note  re- 
sults of  observations  and  of  practice,  without  which  eminence 
in  his  profession  can  never  be  attained ; and  if  this  work 
shall  assist  him  in  the  attainment  of  these  objects,  one  great 
purpose  of  the  author  will  be  well  accomplished. 


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- — - - — 


INDEX. 


A.  Page 

Abutments  and  Arches 604 

Acids 188 

Adulteration  in  Metals,  Proportion 

of, \ in  a Compound 216 

Aerodynamics 614 

Aerometry,  Course  of  Wind 675 

I ‘ Distance  of  Audible  Sounds  674 

“ Pneumatics 673-676 

“ Resistance  of  a Plane  Sur- 

“ face 675 

I I Resistance  to  a Steam  Vessel 

in  Air  or  Water 91 1 

‘ ‘ To  Compute  Height  of  a Col- 

umn of  Mercury  to  Induce 

an  Efflux  of  Air 675 

“ Velocity  and  Pressure  of 

“ Wind 674,  91 1,  921 

“ Volume  of  Air  discharged 
“ Through  an  Opening , 

“ etc 674-676 

“ Weight  of  Air 675 

Aerostatics 427-431 

“ Elevation  by  a Barometer  428 
“ “ by  a Thermometer  429 

“ Velocity  of  and  Sound. . . 428 

“ Velocity  of  Air  flowing 

into  a Vacuum 428 

Ages  of  Animals 192 

Air,  and  Steam 737 

“ Atmospheric 431-432 

“ Consumption  of. 432 

“ Decrease  of  Temperature  by  Al- 
titudes  522 

“ Expansion  of. 520 

“ Flow  of  in  Pipes 745,  746,  909 

“ Pressure  and  Resistance  of. ....  648 
u Resistance  of  different  Figures  in  646 
“ Velocity  Lost  by  a Projectile. .. . 648 
“ Volume  and  Weight  of  Vapor 

in 68,  69 

“ “0/  and  Gas  in  a Furnace.  760 

" " Pressure,  and  Density  of. . 521 

Pressure , Temperature , 

and  Density  of 522 

Required  per  Hour , etc. . 525 
194 


Alcohol.  . 


‘ ‘ Elastic  Force  of  Vapor  of. . 707 
“ Proportion  of  in  Liquors. . 204 

Ale  and  Beer  Measures 45 

Algebra,  Symbols  and  Formulas.. 22,  23 

Alimentary  Principles 200 

Alligation io6 

Alloys  and  Compositions 634-637 

Almanac,  Epacts  and  Dominical  Let- 
ters, 1800  to  1901 73 

Altitudes,  Decrease  of  Temperature 

by 522 

American  Gauge 118,  120 

Analysis  of  Organic  Substances..  190 

“ of  Foods  and  Fruits 201 

“ of  Meat , Fish , and  Vegetables  200 


Pag® 

Anchors  and  Kedges 174 

“ Cables , Chains , etc 173,  174 

“ Diameter  of  a Chain  Cable.  175 

“ Experiments  on 175 

“ Length  of  Chain  Cables. ...  175 

“ Number  and  Weight  of. . . . 174 

“ Resistance  to  Dragging 175 

Ancient  and  Scripture  Lineal  Meas- 
ures  53 

“ Weights 53 

Angle  and  T Iron,  Weight  of. 130 

Angles  and  Distances  Corresponding 

to  a Two  foot  Rule 160 

Angles,  To  Describe , etc 222 

Angles,  To  Plot  and  Compute  Chord  of  359 

Animal  and  Human  Sustenance 203 

Animal  Food 200-207 

u Power 432-440 

“ Birds  and  Insects 438 

“ “ Camel 438 

“ “ Crocodile 438 

“ “ Day's  Work. 434 

“ “ Dog, 438 

“ “ Horse . 435, 436, 437 , 439, 440 

“ “ Llama 438 

“ “ Men 433,434,438,439 

u “ Mule  and  Ass 437 

“ “ on  Street  Rails  or 

Tramways 435 

“ “ Ox 438 

Animals,  Proportion  of  Food  for 205 

Annuities no-in 

“ Amount  of m 

11  u at  Compound 

Interest m 

“ Present  Worth no 

11  Yearly  Amount  that  will 

Liquidate  a Debt no 

Anti-attrition  Metal 636 

Apartments,  Buildings,  Ventilation  of  524 

Appendix 913 

Aqueducts,  Roads , and  Railroads  ...  178 

Arc,  To  Describe 225,  227,  228 

Arches  and  Abutments 604 

Arches  and  Walls 602 

Areas  of  Circles 231-236 

“ by  Logarithms. . .236,  252 

‘ ‘ Wh  en  Composed  of  an 

Integer  and  a Frac- 
tion  236 

Area  of  a Circle,  When  Greater  than 

any  Contained  in  Tables 235,  252 

Areas  of  Segments  of  a Circle.  .267-269 
“ of  Circles , by  Birmingham 

W.G 236 

“ Greater  than  in 

Table 235 

“ of  Zones  of  a Circle 269-271 

“ of  Zones,  To  Compute 271 

“ and  Circumferences  of  Cir- 
cles by  10 ths  and  12 ths.  .243-257 


u 


INDEX. 


Page 

Apothecaries’  or  Fluid  Measure 46 

“ Weight 32 

Arithmetical  Progression 101 

Artesian  Well 179,  198 

Ash 482 

Asphalt 481,  689 

Asphalt  Composition 593 

11  Pavement 690 

Ass 437 

Atmosphere,  To  Compute  Volume  of 

Vapor  in ' 68 

Atmospheric  Air 431,  912 

“ Proportion  of  Oxygen 

and  Carbonic  Acid  at 

Various  Locations 432 

“ Carbonic  Acid  Exhaled 

by  Man 432 

Avoirdupois  Weight 32 

Axle,  Compound 627 

B. 

Babbitt’s  Anti-attrition  Metal 636 

Baking  of  Meats,  Loss  by. 206 

Balances,  Fraudulent 65 

Balloon,  Capacity  and  Diameter  of. . 218 

Balls,  Cast  Iron  and  Lead 153 

Balls,  Lead,  Weight  and  Dimension  of  501 
Barometer,  Elevations  by  Readings.  429 

Height  of 429 

11  Indications 429 

“ Weather  Glasses  430 

“ Weather  Indications  . . . 431 

Barrel,  Dimensions  of 30 

Beam  or  Girder  Trusses 823 

“ General  Deduction  824 

Beams,  Deflection  of. 77°~777 

‘ k Dimensions  of  which  a Struct- 
ure can  Bear 644 

“ Elements  of  Wrought  Iron , 

Rolled 807 

“ Elliptical  Sided 826 

“ Floor , Headers , Trimmers , 

etc. . . 835-838 

u Formula  of  Transverse  Stress  801 

11  of  Unsymmetrical  Section , 

Neutral  Axis  and  Strength 

of. 820 

“ or  Girders,  Moments  of. . .621,  622 

u Shearing  Stress 622 

Bearings  for  Propeller  Shafts 473 

Beet  Root  and  Beet  Root  Sugar 207 

Beeves  and  Beef,  Comparative  Weights 

of. 35 

Beils,  Weight  of. 180 

Belt,  Equivalent,  ahd  Wire  Rope 167 

Belting 9°7 

Belts  and  Belting 44!-443 

“ Width  of 44 

Bench  Marks 85 

Beton  or  Concrete 593 

Birds 44° 

‘ 1 and  Insects 196,  438 

Bissextile  or  Leap  Year 70 

Black  and  Galvanized  Sheet  Iron — 129 
Black  and  Galvanized  Sheet  Iron, 

Weight  of 129 

Blast  Furnace 529 


Page 

Blast  Furnace,  Pipe  of  a Locomotive.  907 

Blasting 443,  912,  913 

‘ ‘ Boring  Holes  in  Granite  . . 444 
“ Charge  of  Gunpowder  for. . 444 

“ Effects 444 

“ Weight  of  Explosive  Mate- 
rials in  Holes 444 

Blasts  and  Draughts,  Effects  of 746 

Blower  and  Exhausting  Force 898 

Blowers,  Fan 447 

Blowing  Engines 445,  898 

“ Memoranda 448 

“ Power  of  etc 446 

“ Pressure  of  Blast 447 

“ Root's  Rotary 449 

“ To  Compute  Dimensions  of 

a Driving  Engine . . . 446 

“ “ Elements  of. 447 

“ “ of  a Fan-Blower  448 

“ “ Power  of  a Centrifugal 

Fan 448 

“ “ Volume  of  Air  trans- 
mitted  447,  922 

Blowing  Off 726 

Board  and  Timber  Measure 61 

Boiler,  Steam 739 

“ and  Ship  Plates 828 

“ Areas  and  Ratio  of  Grate  and 

Heating  Surface , etc 741 

“ Draught 739,744,745 

“ “ and  Blasts , Compara- 
tive Effect  of. 746 

“ “ Velocity  of 746 

“ EvaporativeCapacity of  Tubes  742 

“ Evaporation,  Effects  of  for 

Different  Rates  of  Combus- 
tion  743 

“ Evaporation , Power  of. 757 

“ Fuel  that  may  be  Consumed..  742 

“ Heating  Surfaces 740 

I ‘ Loss  of  Pressure  by  Flow  of 

Air  in  Pipes 745 

‘ ‘ Minimum  Fuel  Consumed  per 

Square  Foot  of  Grate 740 

“ rower 760 

“ Rate  of  Combustion 760 

Relation  of  Grate , Heating 

Surface,  and  Fuel 741 

‘ ‘ Result  of  Experiments  with  a 

Steam  Jet 746 

II  Results  of  Operation  of 743 

“ “ of  Operation  of  Vari- 

ous  Designs  of  Boiler 744 

“ Riveting 755,  907 

“ Safety  Valves 746 

“ Steam 739—745,  829 

“ Steam  Heating 526 

“ Steam  Room 748 

“ Volume  of  Furnace  Gas  per 

Lb.  of  Coal 760 

“ Weights  of. 759 

Boiling  of  Meats,  Loss  by 206 

Boiling-Points 517 

Bolts  and  Nuts,  Dimensions  and 

Weights  of.....  156,  157 
“ English  Standard. . . 158 

“ French  Standard .. . 158 


INDEX. 


Ill 


Page 

Bolts  and  Nuts,  Square  Heads 159 

“ Tenacity  of 198 

“ Wrought-iron,  Experiments  on.  783 

“ and  Plates 749-75 7 

Boriug  aud  Turning  Metal 197 

Instruments,  Tempering  of. . . 197 

“ Wells 197 

Brain,  Weights  of. 192 

Brass,  Sheet,  Weight  of. 142 

Plates,  Weight  of 1 18,  1 19,  146 

Wire,  Weight  of 120,  121 

Brass 636 

Castings,  Weight  of. 155 

Tubes,  Weight  of 142 

* ‘ Weight  of. 136,  149 

Braziers’  and  Sheathing  Sheets 155 

Bread 207 

Breakwaters 181 

Breast- wheel 568 

Brick  Walls 603 

Brickwork 597,  801 

Brick  or  Compressed  Fuel 907 

Bricks .598,  599 

“ Crushing  Resistance  of 908 

“ Volume  of  and  Number  in  a 

Cube  Foot  of  Masonry 599 

Bridge,  Britannia  Tubular 178 

“ Highest 907 

“ Iron 178 

“ New  York , Erie , and  West- 
ern Railroad 178 

‘f  New  York  and  Brooklyn  ...  178 

“ Suspension 842 

Bridge  Plates  and  Rivets 830 

Bridges 178 

“ Lengths  and  Spans  of . 181 

“ Resistance  of. 645 

“ Suspension.  Length  of  Span 

°f- 199 

Suspension 178,842 

Bridles  or  Stirrups,  for  Beams 838 

British  and  Metric  Measures,  Com- 
mercial Equivalents  of 906 

Broccoli 207 

Bronze 637 

“ Malleable 907 

Browning  or  Bronzing  Liquid 874 

Builders’  Measure 46 

BuildingDepartment,.ReguiVeme/i!fso/;  907 
B uilding  Stones,  Expans  ion  and  Con- 
traction of ^4 

Buildings,  Walls  of. ^9 

“ Protection  of  907 

Buoyancy  of  Casks I92 

Burns  and  Stings,  Application  for . . . 196 

Buttress 696 

C. 

Cabbage 20? 

Cables,  Chain,  Weight  and  Strength  of  168 

“•  Chain,  Breaking  Strain  and 

Proof  of. 169 

“ Circumference  of I7I 

“ Galvanized  Steel ^3 

t <k  Strength  of. 168,170 

Cables,  Ropes,  Hawsers,  Anchors, 
and  Chains.  ....  ^75 


Pag® 

Calculus 24 

Calendar,  Ecclesiastical 70 

“ Gregorian  Or  N.  S 70,  71 

Caloric 504,  614 

Caloric  Engine,  Ericsson’s 903 

Canal,  Suez,  Via 912 

Canals 181 

“ Flow  of  Water  in 550 

“ Locks 183,  553-555 

“ Power  of  a Horse ~.  848 

“ Traction  on 848 

“ Transportation  of 193 

Candles,  Gas,  Light  of  etc 583,  584 

Cannon  Ball,  Flight  of 495 

Capillary  Tube 358 

Cargoes,  To  Ascertain  Weight  of.  176,  177 
ik  Units  for  Measurement  of  . . 176 

Carrot *. . . 207 

Cascades  and  Waterfalls 184 

Case  Hardening 644,  786 

Cask  Gauging 377 

Casks,  Buoyancy  of 192 

“ Ullage 378 

Cast  Iron 637,  765,  783,  784 

u and  Lead  Balls,  Weight  of.  153 
“ Balls,  Weight  and  Diam- 
eter of  . 153 

“ Bars,  Experiments  on 780 

“ . Crushing  Weight  of  Col- 
umns  768 

“ Columns,  Weight  Borne 

Safely 768 

“ Pipes , Weight  of. 132,  133 

“ Plates,  Weight  of. 146 

“ To  Compute  Weight  of  a 

Bar  or  Rod 131 

“ Weight  of.  , 136,155 

Castings,  Shrinkage  of. 218 

“ Weight  of , by  Pattern 217 

Catenary,  To  Describe 230 

Cathedral,  St.  Peter’s '. ..  I7Q 

Cattle  and  Horses,.  Transportation  of.  102 

“ Weight  of 35 

Cauliflower 207 

Cement ’ qG7 

Cements 589,  87.1-873 

Cements,  Limes,  Mortars,  and  Con- 
cretes   588-597 

Central  Forces 449-454 

Centre  op  Gravity 605-60^ 

u Of  a Vessel  and  Dis- 
placement  653,  658 

“ To  Ascertain  Centre 

u of. 605 

Centre  of  Gyration 609-611 

“ To  Compute  Ele- 
ments and  Cen- 

tre  of. 610,6x1 

Centres  of.  Oscillation  and  Per- 
cussion  612-614 

“ Centre  of.  in  Bodies  of  Va- 
rious Figures 613 

“ Centre  of  To  Compute.  .612-614 
Centrifugal  Fan,  Elements  and  Power 

of  a Fan  Blower , etc 448 

Centrifugal  Pump 579 

Chain,  To  Set  out  a Right  Angle  with.  69 


IV 


INDEX. 


Page 


Chain  Cables,  Breaking  Strain  and 

Proof  of 109 

Chaining  over  an  Elevation 69 

Chains  and  Ropes.. 457 

a Anchors,  etc.,  Diameter  and 
Length  for  a Given  Weight 

of  Anchor 175 

Characters  and  Symbols 21 

Charcoal 194?  48° 

“ Produce  of. 481 

Cheese,  Composition  of. 205 

Chemical  Composition  of  some  Com- 
pound Substances 461 

Chemical  Formula,  To  Convert 190 

Chimney  Draught 9°7 

Chimneys 179,  l8°,  9°4 

“ and  Smoke  Pipes 748,  749 

Chinese  Wall 179 

Chinese  Windlass 627 

“ or  Indian  Ink 903 

Chronological  Eras  and  Cycles.  . . 26 

Chronology 7°~74>  9*4 

u To  Ascertain  Years  of 

Coincidence 74 

Churches  and  Opera-Houses 180 

Circular  Arcs,  Length  of. 260-262 

Circular  Measure .113, 114 

“ Motion 618 

Circulating  Pumps 749 

Circles,  Areas  of 231-236 

“ “ by  Logarithms 236 

“ u by  Wire  Gauge .. . 236 

“ Sides  of  a Square  of. . . .258-259 

Circumferences  of  Circles 237-242 

“ “ by  Logarithms.  242 

“ “ by  Wire  Gauge  242 

Cisterns  and  Wells,  Excavation  of. . . 63 

“ u Capacity  of  63 

Civil  Day 37,  7° 

Civil  Year 7° 

Cloth  Measure 27 

Clouds 43° 

Coal,  Anthracite 480 

“ Average  Composition  of  Heat 
of  Combustion  and  Evapora- 

rative  Power  of. 486 

11  Bituminous 479 

“ “ Caking  Splint  or  Hard 

Cherry  or  Soft 479 

“ “ , Cannel 479 

“ “ Chemical  Composition , 

Varieties  of. 479 

“ Consumption  of  to  Heat  100 

Feet  of  Pipe 527 

“ Effective  Value  of 908 

“ Elements  of  Various 480 

“ Fields , Areas  of. 19 1 

‘‘  Japan 9°9 

“ Measure 46 

Coast  and  Bay  Service 9°8 

Cocks,  Composition,  and  Copper 

Pipes I5° 

Cohesion 614 

Coins,  British  Standards 38 

“ To  Convert  U.  S-  to  Bntish 

Currency 39 

“ U.  S.,  Weight  and  Fineness  of.  38 


Coins,  Values  of 39 

“ Weight , Fineness , and  Mint 

Value  of  Foreign 39 

Coke 480 

Cold,  Extremes  of  in  Various  Coun- 
tries  191 

“ Greatest . . . 908 

College,  Oxford 179 

Collision  or  Impact 580-582 

Color  Blindness 195 

Colors  for  Drawings 196,  913 

Colors,  Proportion  of  for  Paints 66 

Columns 180 

Combination 112 

Combustion 458-466 

Composition  and  Equiva- 
lents of  Gases 460 

Chemical  Composition  of 

some  Combustibles 461 

Evaporative  Power  of  1 
Lb.  of  a given  Combus- 
tible   462 

Heat  of 463 

Heating  Powers  of  Com- 
bustibles  461,  462 

Of  Fuel 463-465 

Rate  of 760 

Relative  Evaporation  of 

Combustibles. . . 465 
“ Volumes  of  Gases 
or  Products  of  per  Lb. 

of  Fuel 465 

Temperature  of. 462 

To  Compute  Consumption 

of  Fuel 446 

1 “ Volume,  of  Air 

Chemically  Consumed 
in  Complete  Combustion 

of  1 Lb.  of  Coal... 459 

‘ Volume  of  Air  Required . 465 
{ Weigh  t and  Sped fi  c Heats 

of  Products  of  Combus- 
tion, etc • 462 

Compass,  Degrees , etc.,  of  Each  Point  54 

Composition  and  Alloys 634-637 

Compound  Axle  or  Chinese  Windlass  627 

Compound  Interest 108 

Compound  Proportion 95 

Concretes,  Limes,  Mortars,  and 

Cements 588~597 

Concrete,  Cements,  and  Mortar  . .595,  914 

or  Belon 593 

Cones 353 

Conic  Sections 38o_384 

“ Conoid 38° 

“ Ellipse  or  Hyperbola , To  De- 
termine Parameter  of. . .380,  381 
“ Ellipse , To  Compute  Area  of 

Segment  of 382 

“ Hyperbola 383 

“ “ To  Compute  Abscissce . . 383 

“ “ Area  of 384 

“ “ Diameters 383>384 

“ “ Length  of  any  Arc  384 

Parabola 38z 

“ To  Compute  Area  of. . . 383 
“ “ Ordinate  or  Absdssa  382 


INDEX, 


Y 


Page 

Contractility 614 

Copper. . .. 750 

“ Braziers’  and  Sheathing 131 

‘ ‘ Plates , Weight  of. . . . 1 1 8,  1 1 9,  1 46 

1 ‘ Pipes 1 50 

“ Rods  or  Bolts , and  Pipes, 

Weight  of. i48 

“ Sheet,  Weight  of. ... 135 

“ Tubes,  Weight  of. 140,144 

“ Weight  of. 136,155 

“ Wire,  Weight  of. 120,121 

“ Wire  Cord .. 123 

Copying 2g 

Cord,  Copper  Wire I23 

Cordage,  Friction  and  Rigidity  of. 

Corn  Measure 

Corrosion  of  Iron  Steel 908 

Corrugated  Iron  Hoof  Plates,  Weight  of  131 

Co- SECANTS  AND  SECANTS 403-4 1 4 

“ . “ To  Compute,  etc.  414 

Cosines  and  Sines 390-402 

“ To  Compute,  etc.  401,  402 
Cost,  of  Family  of  Mechanics  in 

France.. 

Co-tangents  and  Tangents 416-426 

_ “ “ To  Compute,  etc.  426 

Cotton  Factories 800 

couple..,.,.... 

Couplings  of  Shafts 7Q6 

Coursing  and  Leaping 44o 

Crane,  Steam  Dredging 8qq 

“ Wood y 


Page 

• 37 

• 37 

37?  70 


Day,  Marine  or  Sea. . 

‘ ‘ Sidereal 

Solar  and  Civil. 

Day's  Work. 4_. 

Decimals * 

Deer  Park,  Copenhagen. 

Deflection 

Delta  Metal 

Departures,  Table  of... 

Desiccation 


Cranes 


Crank. 


900 

,•  • • • I79»  433,  455-457 

Machinery  of. 4^7 

To  Compute  Dimensions  of 

„ „,Post 456 

Stress  on 453 

“ Stress  upon  Strut  ....  45I 


“ Turning 433 

Cream,  Percentage  of,  in  Milk. 205 

Crocodile 43g 

Crops,  Mineral  Constituents  Absorbed 

189 

Croton  Aqueduct. I?8 

Crushing  Strength 76 4-760 

Cube  Measures 3o 

Cube  Root,  To  Extract . . ‘ [ gj 

Cubes,  Squares,  and  Roots 272-302 

“ To  Compute  and  to  As- 
certain, etc 300-302 

Cucumber 207 

Currency,  To  Convert  U.  S.  to  British  30 

Current  Wheel A/0 

Curvature- and  Refraction 

Cut  Nails,  Tacks.  Spikes,  etc k. 

Cutters  ( Yachts ) gq5 

Cycle,  Dominical  or  Sunday  Letter. . 70 

Lunar  or  Golden  Number 71 

Of  Sun 70 

Cycloid,  To  Describe 228 

Cyclones ’ g75 

Cylindrical  Flues  and  Tubes,  Hoilow  827 

D. 

Dams,  Embankments,  and  Walls. 700-703 
Day,  Astronomical, 


92-94 
...  179 

770-781 

384,  9X3 
• ••  54 

Detrusive  or  Shearing  Strength,  3 ^ 

_ _ . 782,  783 

Dew  Point,  and  To  Ascertain 68 

Diamond  Weight ’ * ‘ 32 

Diamonds,  Weight  of. * 

Diet , Daily,  of  a Alan. . 202  207 

Digestion  of  Food ’ 20g 

Dip  of  Horizon ’ gQ 

Discount  or  Rebate ’ * ’ ’ Ioq 

Displacement  of  a Vessel .’  653 

Distances,  Steaming 86 

“ Between  Cities  of  U.  S.  . . . . . 184 

u “ East  and  West 187 

“ “ Principal  Ports  of 

World.  87 

i “ ofU.S...  88 

“ Various  Ports  of  Eng- 
land, Canada,  and 

V.S. 86 

Distances  and  Angles  Corresponding 

to  a Two -foot  Rule jgo 

Distances  and  Geographic  Levelling.  '.  56 

_.  “ “ Measures..  54 

Distemper 

Distillation 

P°B '.'.'.'.'.438,440 

Domes  and  Towers.  I7g  Is0 

Dominical  Letters,  and  Epacts ’ 73 

“ or  Sunday  Letter 7Q 

Drainage  of  Lands 691 

Drains,  Diameter  and  Grade  of  to  ^ 

Discharge  Rainfall ’ qo5 

Draught 739/744,  746 

Drawing  and  Tracing  Paper 29 

Drawings,  Colors  for ’ ’ ' ' * \g§ 

“ Dimensions  of,  for  U.  S.  y 

Patents IQg 

Dredger  and  Hopper  Barge 8qq  qoo 

Dredging,  and  Cost  of \g7 

‘ ‘ Machines ’ gqo 

Drilling *-*  A/ 

Drowning  Persons,  Treatment  of. . ! 187 

Dry  Measure qo  J 

Dualin 

ia 

tk  Cellulose 444 

Dynamics.  .616-620 

‘ Circular  Motion 618 

“ Decomposition  of  Forces. . . 620 

“ Motion  on  an  Inclined 

Plane 619 

Uniform  Motion 617  618 

‘ ‘ Work  A ccum  ulated  in  Mov- 
ing Bodies 619 

By  Percussive  Force  . 620 


VI 


INDEX, 


E.  Pa&e 

Earth .188,  198 

“ Area  and  Population  of .... . 100 

“ Elements  of  Figure  of  61 

“ Motion  of 7° 

“ Pressure  of 695 

“ Weight  of  per  Cube  Yard 468 

“ Weights  of ;•  33 

Earthwork 4°7,  4°° 

“ Bulk  of  Rock 468 

‘ ‘ Js  umber  of  Loads  and  Vol- 
ume of  per  Day 908 

“ Shovelling 9°8 

“ Volume  of  Transported 

per  Day 9°8 

Easter  Day. . 71 

Ecclesiastical  Year 7° 

Egyptian  and  Hebrew  Measures 53 

Elastic  Fluids,  Specific  Gravity  of..  215 

Elasticity *95,  614 

“ Coefficients  of  761 

“ Modulus  of. 762 

“ Relative,  of  Metals 780 

Electric  and  Gas  Light 198 

“ Light , Candle-Power  of  ...  908 

Electrical  Weights 34 

Elementary  Bodies *9° 

Elevation  by  a Barometer 428 

Elevations  and  Heights  of  Various 

Places i83 

Elliptic  Arcs,  Length  of 263-266 

“ “ To  Ascertain  Length 

of. 266 

Ellipse,  To  Describe , etc 226,  380 

Embankment  and  Excavation 466 

‘ ‘ Walls  and  Dams . . 700-703 

“ Weight  of  a Cube  Foot 

of  Materials 694 

Endless  Ropes *67 

Engines  and  Machines 898 

“ Elements  of . 

“ and  Sugar -mills , Weights  of  908 
Ensigns , Pennants , and  Flags , U.  S. . 199 

Epacts,  and  Dominical  Letters 73 

Equation  of  Payments..  109 

Equilibrium,  Angles  of. 694 

“ Of  Forces 616 

Ericsson’s  Caloric 9°3 

Establishment  of  the  Port  for  Several 

Locations 8 5 

Ether,  Elastic  Force  of  Vapor 707 

Evaporation  of  Water 5J4 

Evaporative  Power  of  Tubes 5r3 

Evolution 9° 

Excavation  and  Embankment  of 

Earth  and  Rock 192 

Exhausting  Fan  and  Blower 898 

Expansion 614 

F. 

Fan  Blowers 447 

Fascines 690 

Fellowship 99 

Fence  Wire 

Fig i°l 

Filter  Beds 

Filtering  Stone 9°9 

Filters  for  Waterworks 184 


Page 

Fire  Bricks 600 

“ Clay ........ 597 

Fire-Engine,  Steam 904, 909 

Fish,  Meat , and  Vegetables , Analysis 

of 

Flags,  Ensigns,  and  Pennants,  U.S..  199 

Flax  Mill 47^ 

Floating  Bodies,  Velocity  of. 909 

Flood  Wave 912 

Floors  and  Loads,  Factor  of  Safety.  841 

Weight  of. 841 

Flour ... ........ 207 

“ Mills 9°° 

Flues  and  Tubes *. . . -747)  754?  827 

“ Corrugated 9°9 

“ or  Furnaces 754 

Fluid  Measure 3°,  46 

Fluids,  Impact  and  Resistance  of..  577 

“•  Lamp  and  Gas 584 

“ Percussion  of 579 

Flutter  Wheel 57* 

Fluxes  for  Soldering  or  Welding 636 

Fly  Wheel 451 

Flying  of  Birds 44° 

Food,  Animal 200-207 

Digestion  of 206,  914 

Nutritive  Constituents  and  Val- 
ues of. 202 

Nutritive  Equivalents  of. .... . 205 

Proportion  Expended  by  Ani- 
mals  205 

Foods,  Analysis  of .201,  203 

Nutritious  Properties  of. 204 

Relative  Values  of...  .202,  204,  912 
Thermometrical  Powers  and 

Mechanical  Energy  of 205 

Forces,  Composition  and  Resolu- 
tion of 615 

Division  of.  .- 614 

Equilibrium  of. 616 

Percussive 620 

Decomposition  of. 620 

Foreign  Measures  and  Weights.  .48-52 

Fortress  Monroe J79 

Foundation  Piles J98’  9°9 

Fractions 89-91 

Fraudulent  Balances 65 

Freeboard 666,  913 

Friction 469-478,  57 662 

“ and  Rigidity  of  Cordage. . . 472 

“ Application  of  Results 474 

“ Bearings  of  Propeller  Shafts  473 

Coefficien ts  of  Axle 47 1 

“ of  Motion 470 

“ of  of  Journals . 470 
To  Determine..  471 

47~« 
478 

478 


Frictional  Resistances 
Grain  Conveyers. . 

Launching  Vessels. 

Mechanical  Effect  of  To 

Compute 471 

of  Bottom  of  Vessels 9°9 

of  Pivots 472 

of  Planed  Surfaces 9°9 

of  Steam-Engines  and  Ma- 
chinery  475 

of  Walls  and  Earth 698 


INDEX. 


vii 


Page 

Friction,  Relative  Value  of  Angles. . 472 
“ Results  of  Experiments . 47 4,  475 

“ Rolling 473 

•*  Steam-engine 478 

k;  Steamers 478 

Tools 476 

“ Value  of  Unguents 471 

kk  Wood-sawing 477 

Frictional  Resistances 475 

of  Machinery,  Results  of 
Experiments  upon. . .475-478 

Fkigorific  Mixtures 193,  516 

Fruits,  Analysis  of 201 

kk  Proportion  of  Acid  and  Sugar  203 

Fuel v 479-487, 513 

kl  Area  of  Grate  and  Consump- 
tion of. 513 

Ash 482 

Average  Composition  of . . .485,  486 

Briclc  or  Compressed 90 

Elements  of. 486 

Lignite 481 

Liquid 484 

Miscellaneous 487 

Peat 482 

Produce  of  Charcoal. 481 

Relative  Values  of. 483 

Tan.. 482 

Values,  Weights,  and  Evapora- 
tive Power  of. 483,  91 

- 48! 

523,  754 

634 


“ Wo- l 

Furnaces 

Fusible  Compounds. 


G. 

Galvanized  Sheet  Iron,  Thickness 
and  IF eight  of. 

1 1 Charcoal  Iron. . , 

“ Iron  Wire  Rope, 

“ Steel  Cables 

Gas 


..  585- 

and  Electric  Light 

“ Atmospheric  Engine 587, 

“ Coal 

‘ 1 Candies , etc 

“ Engines 

“ Flow  of 

“ Mains,  Dimensions  of 

11  Pipes 138,160, 

‘ 1 Pipes , Thickness  of. 

“ Steam  and  Hot-air  Engines 

“ Temperature  of '. 

“ Tubing,  and  Number  of  Burners, 

Regulation  of. 

“ Volume  of 586, 

“ Volume  of  Furnace  per  Lb.  of 

Coal 

“ Weight  of  a Cube  Foot  of 

Gases,  Expansion  of 

Gauges , Wire 118- 

Gauging-  Cask 

Geographic  Levelling  by  Boiling-Point 
Geographic  Measures  and  Dis- 
tances  54i 

Geographical  and  Nautical  Meas- 
ures  

Geometrical  Progression 103- 


Page 

Geometry 219-230 

“ Angles 222 

“ Arcs 227 

Catenary 230 


Circles. . 


224 

Cycloid  and  Epicycloid  . . 228 

“ Ellipse 226 

‘ ‘ Hexagon 223 

“ Hyperbola 230 

u Involute 229 

“ Length  of  Elements 221 

“ Lines 221,  222 

u Octagon 223 

“ Parabola. 229 

“ Polygon 223 

“ Rectilineal  Figures 222 

“ Spiral 230 

Geostatics  and  Geodynamics 614 

Gestation,  Periods  of. 192 

Girder,  Beam,  etc.,  General  Deduction,  824 

Girders,  Beams,  Lintels,  etc 822 

Girders  and  Beams 805,  806 

“ “ Centre  of,  and  Ver- 

tical Distance  of 
Centres  of  Crush- 
ing and  Tensile 

Stress 819 

(i  ‘ £ Deflection  of. .... . 840 

“ “ Dimensions  and 

Load  of. 839,  840 

“ “ Factors  of  Safety. 

821,  841 

“ “ Moment  of  Stress 

of. 621,  623 

“ “ Trussed 823 

“ “ Tubular 775 

Glass  Globes  and  Cylinders 831 


124 

*97 

874 

119 

71 

207 

452 


Glass,  Window, 

Glazing 

Glues. 

Gold  Sheet , Thickness  of 

Golden  Number  or  Lunar  Cycle... 

Gooseberry 

Governors 

Grade,  Reduction  of,  to  Degrees. . . . 359 

Grain,  and  Roots,  Weights  of. 34 

“ Conveyers 478 

‘ ‘ Weights  per  Bushel 32 

Graphic  Delineation  of  Stress,  with 

a Uniform  Load,  etc 623 

“ Operation,  Solution  of  Ques- 
tions by 905 

Gravel 690 

or  Earth  Roads 688 

Gravitation..  487-496 

Accelerated  and  Retard- 


ed Motion . 


494 

Average  Velocity  of  a 

Moving  Body 495 

Formulas  to  Determine 
“ the  Various  Ele- 
ments   490 

“ “ of  Retarded 

Motion 492 

General  Formulas  for 
Accelerating  and  Re- 
tarding Forces 495 


INDEX. 


Vlll 


Page 


Gravitation,  Gravity  and  Motion  on 

an  Inclined  Plane . 492, 493 

“ Inclined  Plane 493,  494 

“ Miscell.  Illustrations .. . 496 

“ Promiscuous  Examples.  489 

“ Relation  of  Time,  Space, 

and  Velocities  . . . .488-491 
“ Retarded  Motion.. . .492,  496 

“ Space 489 

“ To  Compute  Action  of . . 488 

it  “ Velocity  of  a Fall- 

ing Stream  of 

Water 496 

tt  “ Action  of  by  a 

Body  Projected  Up- 
ward or  Downward. . 490 

“ Velocity 489 

it  “ due  to  a Given 

Height  of  Fall  and 
Height  due  to  Given 

Velocity 488 

Gravity  op  Bodies • • • • 208 

* ‘ at  Various  Locations  at  Level 

of  Sea 487 

“ Centre  of. 605 

“ Various  Formulas  for 488 

Grecian  Measures  and  Weights 53 

Gregorian  Calendar 7°?  7* 

Grindstones 47  8 

Grouting 593>  594?  598 

Gudgeons  and  Shafts 79°>  797 

Gun  Barrels , Length  of. 198 

Gun  Cotton 443 

Gun  Metal,  Weight  of 136,  146,  149 

Gunnery 49.7"5°3 

“ Charge , Range , and  Veloci- 
ty, To  Compute 497,  499 

‘ ‘ Comparison  of  Forces  oj  a 

Charge  in  Various  Arms . 502 
1 1 Experiments  with  Ordnance. 

498,  5 00 

“ Lead  Balls , Weight  and  Di- 
mensions of. 5QI 

“ Number  of  Percussion  Caps 

corresponding  to  B Gauge.  502 
“ Penetration  of  Lead  Balls 

in  Small  Arms 5°° 

“ Penetration  of  Shot  and 

Shell , etc 498»  5°° 

it  Proportion, Powder  to  Shot.  502 

“ Report  of  Board  of  Engi- 
neer s,U.S.  A.,  for  Fortifi- 
cations, etc • • 499 

‘ ‘ Velocity  and  Ranges  of  Shot 

and  Shells 49 8 

“ Windage  and  Waddings  ..  501 

Gunpowder 443>  5°2 

“ Charge  of. 444 

‘ ‘ Heat  and  Explosive  Pow- 
er of, , etc 5°3 

“ Proof  of -402 

tt  Properties  and  Results  of  503 

“ Relative  Strength  of  for 

Use  under  Water 503 

Gunter’s  Chain * 26 

Gyration,  Centre  of. 009-01 1 

t « Centre  of  of  a Water -wheel  6 1 1 


H.  Pag® 

Hammers  Steam 179 

Hancock  Inspirator 901 

Hawsers,  Wire,  and  Ropes  and  Ca- 
bles, Comparison  of. 169 

Hawsers,  Ropes,  and  Cables.  . . . 169-172 
“ u Strength  and  Cir- 
cumference of. . 171 

“ “ Weight  of. 172 

Hay  and  Straw 198 

Heat 504-529 

Available  Expended 909 

Capacity  for 5°5i  5°7 

Communication 5X5 

and  Transmission  of. ..  510 


522 


523 

513 

514 


Condensation • • 5 1 5 

Conduction  or  Convection  of...  5X4 
Congelation  and  Liquefaction. . 516 
Decrease  of  Temperature  by  Al- 
titudes   

Degrees  of  Fahrenheit  to  Reau- 
mur and  Centigrade , and  Con- 
trariwise   

Desiccation 

Distillation . 

Effect  upon  Various  Bodies 518 

Evaporation 5I2>  5*3>  5*4 

Expansion  and  Dilatation  of  a 

Bar  or  Prism. 5*9 

Expansion  of  Water 519,  520 

“ of  Liquids , Gases , and 

Air 52° 

Extremes  of  in  Various  Coun- 
tries   J9* 

Frigorific  Mixtures 193,  5l6 

Heating  and  Evaporating  Water 

by  Steam 511 

1 Latent , 508,509 

1 Latent,  of  Fusion 5°9 

‘ Latent,  of  Steam,  To  Compute..  707 
1 Mean  Temperatures  of  Various 

T ncnliti  •■»•»»•••*•*  IQ2 

‘ Mechanical  Equivalent  of  in 

Steam. 7°5 

1 Melting  and  Boiling  Points. .. . 517 

‘ Of  Sun *93 

• Perpetual  Congelation  or  Snow 

Line I92 

‘ Proper  Temperature  of  Enclosed 

Spaces 526 

it  Quantities  of,  Transmitted  from 

Water  to  Water  through  Plates, 

etc 511 

“ Radiation  of. 5°9 

‘ ‘ Radiating  and  Absorbing  Power 

of  Various  Bodies 5IQ 

“ Reduction  of  by  Surfaces 525 

“ Reflection 5 10 

“ Refrigerator,  To  Compute  Sur- 
face of ••••  512 

“ Relative  Power  of  Various  Sub- 
stances  5IQ 

“ Sensible 5°7 

“ Specific 5°5>  5°6>  5°7 

“ Temperature  by  Agitation 524 

“ Temperature  of  a Mixture  of 
Like  and  Unlike  Substances, 

To  Compute 5°6 


INDEX. 


IX 


Page 

Heat,  To  Compute  Volume  or  Pressure 
of  a Constant  Weight  of  Air 
or  other  Gas , etc.,  for  a Given 

Temperature 522 

“ To  Reduce  Degrees  of  Different 

Scales 523 

“ Transmission  of  through  Glass . 51 1 

“ Underground  Temperature 519 

“ Volume , Pressure , and  Density 
of  Air  at  Various  Tempera- 
tures  52 

Heating,  Air,  Length  of  Pipe  Required  525 

11  by  Hot  Water 524 

Hebrew  and  Egyptian  Measures  and 

Weights 53 

Height  and  Elevation  of  Various 

Places 183 

Heights,  Corresponding  to  Boiling 

Points  of  Water 519 

Hemp  and  Wire  Rope,  Circumference 
of  for  Rig- 
ging  172 

“ “ General  Notes  167 

“ “ Relative  Di- 

mensions of  172 
“ “ Weight  and 

Strength  of  172 
“ Weight  of  .. . 166 

Hemp  Rope,  Iron  and  Steel 164 

“ and  Iron  Wire 168 

“ Circumference  of 169 

“ Destructive  Strength  of.  171 

“ Iron  and  Steel , Relative 

Dimensions  of. 168  I 

High  Water,  Time  of. 74,  75  I 

Hills  or  Plants  in  an  Acre. . . * 193  i 

Hoggin 690  | 

Hoisting  Engines 901 

Honey 207 

Hoop  Iron,  Weight  of 129,  131 

Hopper  Barge  and  Dredger 899,  900 

Horizon,  Dip  of 60 

Horizontal  Wheels 572 

Horse  436 

Horse  Power 441,  733,  758,  914 

“ Tractive  Power  of ... . 436 

“ Transmission  of. 188 

Horses,  Age  of 186 

“ Labor  of  etc 435-437 

“ Performance  of. 439,  440 

“ Transportation  of  192 

“ Weight  of 35 

Horseshoe  Nails,  Length  of. i33 

Horseshoes  and  Spikes 152 

Hulls  of  Vessels,  Diameter  of  Rivets.  830 

Human  and  Animal  Sustenance 203 

Hydraulic  Radius  or  Mean  Depth. 

“ Ram 

“ Cement 

Hydrostatic  Press 561,  901 

Hydraulics 529,  557 

“ Canal  Lochs 553,  555 

“ Circular  Sluices , etc 537 

“ CoeffVts  of  Circular  Open- 
ings or  Sluices.  536 

“ In  Clack  or  Trap 

Valve  or  Cock. . 546 

A* 


552 

561 

591 


Page 

Hydraulics,  CoeffVts  of  Friction.  544-546 
“ u of  Resistance  in 

Bent  or  Angular 
Circular  Pipes , 
Valve  Gates , or 
Slide  Valves. . . 545 

“ Computation  of  Volume 

of  Discharge 533 

“ Curvatures,  Radii  of .. . 544 

“ Curves  and  Bends 545 

•“  •“  Coefficients  of  Re- 
sistance  545 

“ Cylindrical  Ajutage 549 

“ Discharge  from  a Notch.  541 

“ ‘‘  from  Conduits  or 

Pipes 530 

“ “ • from  Vessels  not 

Receiving  any 

Supply 538 

“ “ from  Vessels  of 

Communication  541 
■“  • “ from  Irregular - 

shaped  Vessels.  542 
‘‘  “ of  Water  in  Pipes 

for  any  Length 
and  Head , etc., 

547,  548 

“ “ or  Efflux  from  Va- 

rious Openings 
and  Apertures.  532 
“ “ under  Variable 

Pressures 540 

“ Experiments  on  Dis- 

charge  of  Fluids,  from 
Reservoirs , Conduits , 

or  Pipes 529,  531 

“ Flow  and  Velocity  in 

Rivers , Canals , and 

Streams 550,  552 

“ 11  in  Lined  Channels.  551 

“ “ of  Water  in  Beds. . 542 

“ Forms  of  Sections  of 

Canals 543 

“ Friction  in  Pipes  and 

Sewers 543,  544 

“ “ of  Liquids 531 

“ Heigh  t of  a Jet,  To  Com- 
pute  913 

“ Jets  d’Eau 550 

“ Miner's  and  Water  Inch  557 

“ Miscel.  Illustrations .556,  557 

“ Prismatic  Vessels 539 

‘ - Rectangular  Weir . . . 532-535 

“ Reservoirs  or  Cisterns . . 541 

“ Sluice  Weirs  or  Sluices..  535 

“ Submerged  or  Drowned 

Orifices  and  Weirs..  553 
u To  Compute  Depth  of  Flow 

over  a Sill , etc.  . . . 534 
“ Fall  of  a Canal  or 
Conduit  to  Conduct 
and  Discharge  a 
Given  Quantity  of 
Water  per  Second . 914 
u “ Head  and  Discharge 

of  Water  in  Pipes 
of  Great  Length. . . 914 


X 


INDEX. 


Page 

Hydraulics,  To  Compute  Head  or 
Height  of  Water 
from  Surface  of 
Supply  to  Centre 

of  Discharge 544 

<<  “ Vertical  Height  of  a 

Stream  Projected 

from  a Pipe 549 

u “ Volume  of  Water 

flowing  in  a Riv- 

543 


Page 

Hydrodynamics,  Water  -wheels,  Di- 
mensions of 

Arms 571 

“ Wldtelaw's  Wheel. . . 576 


“ Short  Tubes , Mouth 

pieces , and  Cylindri- 
cal Prolongations  or 

Ajutages 536~537 

“ Triangular , Trapezoid- 

al, Prismatic  Wedges , 

Sluices,  Slits,  etc 538 

u Variable  Motion 543 

« Velocity  of  Water  or  of 

Fluids S31 

“ Vena  Gontracta 52Q 

‘ < Weirs  or  Notches 5 39i  9 1 9 

Hydrodynamics . 558-580 

“ Barker's  Mill 5 77 

“ Boyden  Turbine 574 

“ Breast  Wheel 568-570 

“ Centrifugal  Pump . . 579 

“ Current  Wheel 570 

“ Flutter  Wheel 571 

“ Fontaine  Turbine. . . 574 

“ Friction  of  Journals 

or  Gudgeons 571 

u Horizontal  Wheels. . 572 

u Hydrostatic  Press. . . 561 

“ Hydraulic  Ram, 

561,  562 

n Impact  and  Reaction 

Wheels ••  576 

“ Impulse  and  Besist- 

ance  of  Iluids.  577,  578 

“ Jonval  Turbine 575 

“ Memoranda 571 

“ Overshot  Wheel . . 563-566 

“ Percussion  of  Fluids.  579 

“ Pipes,  Elements  of . . 561 

u “To  Compute 

Thickness  of  etc...  560 

“ PonceleV s Wheel 567 

u Pressure  and  Centre 

of 558-560 

“ Reaction  Wheel 576 

“ Tangential  Wheel. ..  576 

u Tremont  Turbine... . 576 

“ Turbine  and  Water 

Wheels , Compari- 
son Between 579 

“ Turbines 572~576 

“ Undershot  Wheel 566 

“ Victor  Turbine 576 

“ Water  Power 562 

“ Water- Pressure  En 

gine 579 

“ Water  Wheels 563 

a “ Diameter  and 

Journal  of  a 
Shaft , etc — 57 


Hydrometers 

“ To  Compute  Strength  of 

a Spirit 

Hygrometer 

“ To  Ascertain  Dew-point. 

“ To  Compute  Volume  of 

Vapor  in  Atmosphere. 
Hyperbola,  To  Describe \ 


I. 


Ice 


912 

and  Snow 849 

‘ ‘ Boats  and  Speed  of. 896,  909 

“ Strength  of. *95 

Impact  and  Reaction  Wheels 576 

“ or  Collision 580 

Impenetrability ^95 

Inclined  Plane 619,  628 

Incubation  of  Birds,  Periods  of 192 

Indicator 724 

Inertia,  Moment  of  a Revolving  Body, 

To  Compute 609 

“ Moment  of,  to  Ascertain  Ap- 
proximately  659 

“ of  a Revolving  Body,  To 

Compute 616 

“ of  a Solid  Beam 819 

Ink,  Chinese  or  India 9°7 

Inks 875 

Insects  and  Birds I9& 

Interest,  Simple  and  Compound.  107,  109 

Involute,  To  Describe 229 

Involution 96 

Iron -637-64o 

Bolts  in  Wood , Tenacity  of. 198 

Bridge , and  Iron  Pipe  Bridge. . 178 

Cast  Iron.. 

Pipes,  Weight  of 


637 

48 


Wrought  Iron 639 

Rope,  Hemp , Iron , and  Steel, . . 164 
* ‘ Hemp  and  Steel , U l timate 

Strength  and  Safe  Load 

of. «<>5 

“ Variable  Motion 543 

Steel  and  Hemp  Rope,  Relative 

Dimensions  of 168,  172 

Wire  and  Hemp  Rope 168 

‘ * Gauge , Weight  and  Length.  163 


“ “ Weight  of. 

Irregular  Bodies,  Volume  of 

.120,  124 

J. 

Jewish  Measures. 53 

.1  Ui  Kill  

Jumping,  Leaping,  etc 

K. 

Hedges  and  Anchors,  Weight 

and 

Khorassan,  or  Turkish  Mortar.. . 

INDEX. 


xi 


L.  Page 

LABOR 433,  434,  436,  468 

Lacquers 875 


Laitance  . 


593 

Lakes,  Areas,  Depth , and  Height 

of.  '. 181,  182 

Lamps,  Fluids,  and  Gas 584 

Land  Measure 29 

Larry ing 598 

Laths  603 

Latitude x98 

Latitude  and  Longitude 76-80 

Launching  Vessels,  Friction  of. 478 

Lead 640 

“ Encased  Pipe,  Weight  of  151 

“ Measure ... 32 

“ Pipe 831 

“ Pipe , Weight  of 139,  150 

“ Plates , Weight  of 146 

“ Weight  of 136,151 

Lead  and  Cast  - iron  Balls,  Weight 

“ “ and  Volume  of. 153 

“ “ To  Compute  Weight  of  155 

Leap  or  Bissextile  Year 7o 

Leaping,  Jumping,  etc 439,  440 

Leaves 207 

Lee -way  or  Drift  of  a Vessel 910 

Legal  Tenders 38 

Lenses  and  Mirrors 670 

Levelling,  Geographic 55,  56 

Lever 624,  626 

Lifting 430 

Ll?tHT  -- ;•••••; i95,  583-587 

Decomposition  of 583 

“ Gas,  Volumes  and  Temperature 

°f 585,  587 

‘ 1 Gas  and  Electric. ........... 

“ Intensity  of. ^5 

“ Loss  of  by  Use  of  Globes. .....  584 

11  Refraction • 584 

ii  Relative  Intensity , Consump- 
tion, and  Cost  of  Various 

Modes  of  Illumination 584 

“ Standard  of. OTo 

Lightning 9.  ° 

Lignite 479,  .8l 

Limes,  Cements,  Mortars,  and  Con- 
cretes  588-597 

“ Cements,  and  Mortars,  Ex- 
periments of  Gen.  Gillmore.  596 

“ and  Cements 394 

“ Asphalt  Composition 5g3 

“ Concrete  or  Beton 593 

“ Conclusions  from  Experiments  590 

‘ ‘ General  Deductions cg6 

“ Pozzuolana **  5gg 

“ Transverse  Strength sg6 

u Turkish  Plaster  or  Hydraulic 

Cement t-QI 

Lines,  To  Draw,  Bisect , etc. 221 

Liquid  Measure 30,31  46 

Liquids,  Expansion  of '$20 

“ Volume  of  at  Boiling-Points  518 
Liquors,  Proportion  of  Alcohol  in. ..  204 

_ . Proof  of  Spirituous 218 

Lithro-fracteur 443 


Locomotive  “Experiment” go2 

“ Axles gIO 

Locomotives,  Operation  of.  .681-685,  912 

“ Adhesion 681,685 

Tractive  Power 681 

“ Train  Resistances.  .682,  920 

Log  Lines 2- 

Logarithm  of  a Number 23 

Logarithms 305-310 

“ Hyperbolic 331-334 

“ . of  Numbers. ....... .311-330 

Longitude,  To  Reduce  to  Time 54 

“ and  Latitude 76 

“ Lengths  of  a Degree 60 

“ of  Observatories  in  Time.  80 

Luminous  Point IQ5 

Lunar  Cycle  or  Golden  Number. . . 1.  7I 
Lunar  Month 70 

M. 

Macadamized  Roads 687,  690 

Machines  and  Engines,  Elements  of  898 

Magnetic  Variation 57-59 

“ Bearing  of  N.Y. . 184 

Magnetism 614 

Malleable  Castings * ’ ’ 639 

Cast  Iron. ...............  785 

Manganese  Bronze ’ . 832 

Manures ’ [* " * jgg 

Marine  Day ’ ’ 37 

Marine  Steamers  and  Engines.  .886,  887 

“ Auxiliary  Freight 887 

“ Fire  boat 887 

‘ ‘ Freight  and  Passenger 886,  887 

“ Iron  Cruiser.. 886 

a u Freight  and  Passenger 

Propellers 886 

“ Steel  Launch 887 

Marine  Steam  Vessels  and  En- 

GINES 887-891 

Composite  Yachts. 888 

“ Ou tier ] 889 

“ Ferry  Boat 889 

“ Iron  Yachts ’ 888 

“ Light  Draught. ” .*  889 

“ side  Wheels 889,  890 

u Steel  Launch 887 

“ “ Yachts .’888,889 

“ Torpedo  Boats,  Iron,  Steel,  and 

Composite 889 

u Wood  Side  Wheels , Passengers 

and  Deck  Cargo . . . 890,  891 

“ 11  Propellers 891 

“ “ Towing 891 

Masonry 197,  597-605,  9x3 

Arch,  To  Compute  Depth  of . . 605 
“ Brick , Stone , and  Granite,  595-600 

“ . Designation  of .602-603 

“ Estimate  of  Materials  and 

Labor,  etc.  604 

“ Rubble 60j 

\\  %01}e .600,  603 

Technical  Terms cQ7  tQ8 

Mason’s  and  Dixon’s  Line 188 

Mastic Q 

Materials,  Strength  of 761-841 

Matter.. 


xii 


INDEX. 


Page 

Mean  Proportion 94 

Measures,  Ale  and  Beer 45 

“ Apothecaries' 47 

“ Avoirdupois 32 

“ Board  and  Timber 6i 

“ British  and  Metric , Com- 
mercial Equivalents  of  J 906 

“ Builder's 46 

“ Circular n3>  JI4 


Cloth 27 

“ Coal 33 

“ Copying 29 

“ Corn J98 

“ Cube 3° 

“ Dry 3° 

£ ‘ Foreign  of  Value 39"43 

u Foreign  Memoranda 43 

u Geographic,  and  Distances  54 

kC  Geographical  and  Nauti- 
cal  26 

u Grain 32 

“ Grecian 53 

“ Gunter's  Chain 26 

“ Hebrew  and  Egyptian. . . 53 

“ Jewish 53 

“ Land 29 

u Liquid 31 

u Men  and  Women 35 

“ Metric 27-33,  36,  44,  46,  47 

“ u Equivalent  Value , U.  S. , 

and  Old  and  New 

U.S 28,  30,  33 

u “ power  and  Work.  36 

u “ Temperatures 37 1 

u “ Velocities 37  I 

u “ Volumes 36546 

u “ Weights  and  Press- 
ures  36 

a Miscellaneous. . . .27,  29,  31,  46 

‘ ‘ Nautical 3° 

£k  Old  and  New  TJ.  S.  Ap- 

proximate Equivalents.  33 

“ of  Length 26,  44 

« of  Offal  in  a Beef  and  Sheep.  35 

“ of  Paper 29 

“ of  Surface 29,44 

it  of  Timber,  Local  Stand' ds  62 

“ of  Time 37 

tt  of  Value 38 

‘ ‘ of  Volume •'  • • • 3°^  45 

“ of  Weight.  32>  47 

“ Pendulum 27 

“ Roman  Long 53 

“ Ropes  and  Cables 26 

tt  Scripture  and  Ancient. . . 53 

“ . Shoemaker's 27 

« Timber , English 62 

“ Troy 47 

“ Vernier  Scale 27 

“ Wine  and  Spirit 45 

“ Wood 

Measures  and  Weights ..26-35 

a English  and  French,  44.  45 

“ Foreign 48"53 

Meat,  Analysis  of,  and  of  Ftsh  and 

Vegetables 2°9 

“ To  Preserve x96 


Mechanics.. 


Page 

Meats,  Roasting  of. • • 2°6 

Mechanical  Centres 605-614 

Centre  of  Gravity. . .605-608 
“ of  Gyration  of  a 

Water-wheel 61 1 

‘ ‘ of  Gyration . . . 609-61 1 
Elements  of  Gyration.  ..610 

Ratio  of  Gyration 609 

Mechanical  Powers 624-634 

Compound  Axle  or  Chi- 
nese Windlass 627 

Inclined  Plane 628 

Lever 624-626 

Pulley 632 

Rack  and  Pinion 628 

Screw 630 

“ Differential 632 

Wedge 630 

Wheel  and  Axle 626 

. m 4 .614-623 

Accumulated  Work 619 

Decomposition  of  Forces. . 620 

‘ ‘ Dynamics 616-620 

“ Moment 614 

“ Moments  of  Stress  on  Gir- 
ders  621-623 

*•  Motion  on  an  Inclined 

Plane 619 

a Solid,  Fluid,  and  Aeri- 
form Bodies 614 

11  Uniform  Motion 617 

tt  Work  by  Percussive  Force  620 

Melting-Points 5*7 

Memoranda • *9°7i  912 

“ Cast  and  Wrought  Iron 

and  Steel. 832 

Men., 433-435 

Mensuration  of  Areas,  Lines,  feUR- 

fages,  and  Volumes,  335"378 
Any  Figure  of  Revolution, 

3585  376 

“ Plane  Figure 359 

Arc  and  Chord,  etc.,  of  a 

Circle ..^.343-345 

Area  Bounded  by  a Curve.  342 

Capillary  Tube 358 

Cask  Gauging 377 

Chord  of  an  Angle 359 

Circle 

“ Section  of 34® 

“ Segment  of. 346 

Circular  Zone 349 

Cones 363 

Cube  and  ParaUelopipedon  360 

Cycloids 352 

Cylinder 35°,  363 

“ Sections 357 

Ellipsoid,  Paraboloid,  and 
Hyperboloid  of  Revolu- 
tion   357 > 375 

Helix  {Screw) 354 

Irregular  Bodies 377 

“ Figures 341 

Links • • *3535  37° 

Lune 352 

of  Areas,  Lines,  and  Sur- 
faces   335~359 


INDEX. 


xm 


Page 

Mensuration  op  Areas,  Lines,  Sur- 
faces, and  Volumes. 360-378 

“ Parallelograms 335 

“ Polygons 338,343 

“ Polyhedrons 362 

“ Prism . 350,360 

“ Prismoid 351,  361 

“ Pyramids 354,  365 

“ Reduction  of  Ascending  or 
Descending  Line  to  Hori- 
zontal Measurement 359 

“ Regular  Bodies,  340, 341,362, 364 

“ Rings 353  368 

“ Sphere 347,367. 

‘ ‘ Spheroids  or  Ellipsoids . 348,  368 

‘ ‘ Spindles 355,  370-374 

Spirals 355 

‘ To  Ascertain  Area  of  any 

Plane  Figure 359 

“ To  Compute,  Chord  of  an 

Angle 359 

“ To  Plot  Angles 359 

“ Trapezoid 338 

“ Triangles 335-338 

“ Ungulas 35  x 

“ Useful  Factors  343 

“ Wedge 350,  361 

Mercurial  Gauge 9IO 

Meta- Centre  of  Hull  of  a Vessel , To 

Compute 659,  913 

Metal  Products  ofU.S. 910 

Metals,  Alloys  and  Compositions  . . 634 

“ Comparative  Quality  of  Va- 
rious  821 

“ Lustre  of X94 

“ To  Compute  Weight  of 131 

“ To  Compute  Weight  of  by 

Pattern 217 

Milk,  Nutritive  Values  and  Constitu- 
ents of 202 

“ Percentage  of  Cream 205 

“ Relative  Richness  of  of  Several 

Animals 207 

“ To  Detect  Starch  in 196 

Mineral  Constituents  from  an  Acre 

of  Soil... 189 

Mineral  Waters,  Analysis  of  etc.. 850,  851 

Minerals,  Hardness  of..... X93 

Miner’s  Inch cc7 

lining ] 445 

Mining  Engines  and  Boilers 901 

Mining  Ropes  x65 

Mirage 195,  669 

Mirrors  and  Lenses 670 

Miscellaneous  Elements 188-198,  906 

Mixtures 871-879 

Operations  and  Illus- 
trations   879-885 

Mississippi  River,  Silt  in 910 

Models,  Strength  of 

Molasses 207 

Molding  and  Planing a76 

Molecules,  Velocity , Weight , and  Vol- 
ume of I94 

Momentum !.!!!!].  105 

Monoliths 179 


Page 

Month,  Lunar 70 

Months,  Numbers  of 74 

Moon’s  Age,  To  Compute 74 

Mortar 590,  592,  595 

Mortars,  Limes,  Cements,  and  Con- 
cretes  588-597 

Motion,  Accelerated  and  Retard- 

ED * 494,  495 

Motion  of  Bodies  in  Fluids 645 

“ Pressure , Velocity,  Time,  etc., 

To  Compute 648 

i 1 Resistances  of  Areas  and  Dif- 
ferent Fig' sin  Water  or  Air.  646 
Mountains  and  Passes,  Heights  of. . . 182 

Mowing 433 

“ Machine 910 

Mule 437 

Mural  Efflorescence 593 

N. 

Nails,  Length  and  Number  of ...  153,  154 
“ . and  Spikes,  Retentiveness  of. ..  159 

National  Road x7g 

Natural  Powers X98 

Nautical  Measure 3Q 

Naval  Architecture 649-667 

“ Angles  of  Courses  and  Sails. . 665 
‘ 4 Heel  and  of  Steady 

Heel 664,  665 

“ Area  of  Sails,  etc 663,  664 

“ Centre  of  Effort,  To  Compute 

Location  of. 659 

“ “ of  Gravity  of  Hull,  En- 
gines, etc 658 

“ Centres  of  Lateral  Resistance 

and  Effort. 658 

“ Course  and  Apparent  Course 

of  Wind 666 

ut  Curve  and  Coefficients  of  Dis- 
placement.  657 

“ Displacement,  and  its  Centre 

of  Gravity 653,  655 

“ Elements  of  a Vessel 653,  660 

“ of  Capacity  and  Speed 
of  Several  Types  of 

Steamers 660 

“ Experiments  upon  Forms  of 

Vessels 649 

1 1 Freeboard 666 

‘ 1 Lee-way ‘ 666 

“ Masts,  Location  of 66  4 

“ Memoranda 667 

‘ ‘ Meta-Centre  of  Hull  of  a Vessel, 

To  Compute 659,  913 

“ Moment  of  Inertia  and  Meta- 

Centre  659 

“ . Pitch  of  Propeller  and  Slip  of 

Side  Wheels 662 

‘ ‘ Plating  Hulls 667 

“ Power  Utilized  in  a Steam- 

vessel  662 

“ Relative  Positions  of  Lateral 
Resistance  and  Centre  of 

Effort. 659 

Resistance  of  Bottoms  of  Hulls,  663 

“ “ to  Wet  Surface  of  Hull, 

To  Compute 653 


XIV 


INDEX. 


Page 


Naval  Architecture,  Rudder  Head . 667 
“ Sailing,  Ratio  of  Effective 

Area  of  Sails , eic., 

to  Wind 663 

“ “ Power  of  a Vessel 665 

“ Sails,  Area  and  Trimming  of 

664,  665 

‘ ‘ Surface , Bottom , and  Immersed 

Hull , To  Compute 653 

44  Stability 649,  650 

44  “ Results  of  Experiments 

upon 649,  650 

“ “ To  Compute  Statical 

and  Dynamical 651 

“ 11  To  Determine  Mean  of 

of  Hull  of  a Vessel..  650 
“ To  Compute  Displacement,  Ap- 

proximately, and  Co- 
efficients of 655 

“ “ Elements  of  Power  Re- 

quired to  Careen  a 

Vessel 652 

“ 44  Power  Required  in  a 

Steam  - vessel.  Speed 
and  IP,  and  Coeffi- 
cient of. 661 

“ Wind,  Effective  Impulse  of. . . 665 

Needle,  Magnetic  Variation  of. 57 

44  Decennial  Variation  of 58 

44  Variation  of  it  in  U.  S.  and 

Canada 59 

Neutral  Axis  of  a Beam 820 

New  and  Old  Style 37'  7° 

Niagara,  Falls  of. *98 

Nitro-Glycerine 443 

Non-conductibility  of  Materials 911 

Notation 25 

Number  of  Direction ••  7* 

Numbers,  Properties  and  Powers  of.  % 98 
“ 4 th  and  5 th  Powers  of. . 303,  304 

44  To  Compute  4 th,  $th,  and 
6th  Power,  and  4th  and 
5thRootof 304 


Organic  Substances,  Analysis  of  by 

Weight 1 9° 

Orthography  of  Technical  Words 

and  Terms * 021-928 

Oscillation  and  Percussion,  Centres 

of..., 612-614 

44  Centre  of  in  Bodies  of 

Various  Figures 613 

4 4 To  Compute  Centre  of,  6 1 2 , 6 1 4 

Overshot  Wheel 563 


Nutritive  Equivalents,tfuma«  Milk,  1 205 


0. 


179 


Obelisks 

Observatories,  Latitude  and  Longi 

tude 

Oceans  and  Seas,  Depths  and  Areas. . 182 
Offal,  Weight  of,  in  a Beef  and  Sheep . 35 

Oil , Yield  of , from  Seeds 189 

“ Calce  and  Vegetables,  Nutritious 

Properties  of. 2°4 

44  Proportions  of , in  Air-dry  Seeds.  203 

Oils,  Petroleum,  Schist,  and  Pine- 

Wood 484 

Old  and  New  Style 37?  7° 

Onion 2°7 

Opera-Glasses 671 


Opera-Houses. . 


Optics -668-671 

44  Elements  of  Mirrors  and  Lenses  670 

44  Refraction 668,669 

44  To  Compute  Dimensions  or  Vol- 
ume of  an  Image 668 

Ordnance,  Energy  of. 9JO 


Painting 66 

Parabola,  To  Describe 229 

Park,  Deer *79 

Parsnips 2°7 

Passages  of  Steamboats 896 

44  Steamer  and  Sailing  Vessels.  897 
Passes,  Mountains,  and  Volcanoes. . . 182 

Pavement,  Asphalt 690 

4 4 Block  Stone 689,  690 

44  Granite 690 

44  Rubble  Stone. . .1.. 689 

44  Telford 688 

44  Wood 689,690 

Pavements,  Roads,  and  Streets. . .686-690 

Payments,  Equation  of 109 

Peat 482 

Pendulum  Measure 27 

Pendulums 452 

44  Centre  of  Gravity  of....  453 

44  Lengths  and  Number  of 

Vibrations  of. 453,  454 

44  Time  of  Vibration. . 454 

Pennants,  Ensigns,  and  Flags,  U.S.. . 199 
Percussion  and  Oscillation,  Centres 

of 612-614 

Performances  of  Men,  Horses,  etc.  438 

Perimeter  of  a Figure 912 

Permutation 100 

Perpetuities 112 

Petroleum,  Elastic  Force  of  Vapor. . 707 
44  Evaporative  Effects  of. . . 910 

Pile  Driving 433i  671-673,  902 

44  Coefficient  of  Resistance  of 

Earth,  To  Compute 672 

44  Pneumatic 672 

44  Ringing  Engine 672 

44  Sheet  Piling 672 

44  Sinking 673 

44  To  Compute  Weight  of  Ram 672 

Piles,  Foundation 198,  909,  912 

44  Extreme  Load  a Pile  will  Bear  913 

44  Retaining  Walls  of  Iron 196 

Piling  of  Shot  and  Shells 65 

Pillar  at  Delphi *79 

Pipes 747 

44  Lead  and  Tin,  Weight  of. 139 

4 4 Riveted  Iron  and  Copper,  Weight 

of • H8 

44  Steam,  Gas,  and  Water 138 

44  To  Compute  Thickness  of. 560 

a 44  Weight  of  Metal. . . 147 

4 4 Tin , Weight  of. 

44  Lead  Encased 



Pivots,  Friction  of..... 


80 


151 

151 

593 

472 


INDEX. 


XV 


Position  . 
Potato  . . . 


Page 

Planing  Cast  Iron  and  Molding 476 

Plank  Roads 688 

Plants  or  Hills,  in  an  Acre 193 

Plaster,  Turkish 591 

Plastering 197, 604 

Plate  Bending 476 

Plates,  Test  of  and  Bolts.. 749,  753 

“ Thickness  of,  by  Wire  Gauges  121 
“ To  Compute  Thickness  of. . . . 751 

“ Wrought- Iron  Shell ....  750 

Plating  Iron  Hulls 667 

Ploughing 433 

Pneumatics. — Aerometry 673-676 

Pointing 598 

Poisons,  Antidotes  and  Treatment  of.  185 

Poles  and  Spars 62 

Poncelet’s  Wheel 567 

Population,  Comparative  Demity  of 
and  Number  of  Persons 
in  a House  in  Different 

Cities 910 

“ of  Earth 188 

of  Principal  Cities 187 

98>  99 

207 

POWER  AND  WORK,  METRIC 36 

“ and  Mechanical  Energy  of  io 
Grains  of  Various  Substances 
when  Oxidized  in  Human 

Body 205 

Motive 910 

Movers  and  Transmitters  of..  797 
of  a Quantity , To  Ascertain 

Value  of. 359 

“ Required  to  Draw  a Vessel  up 

an  Inclined  Plane 910 

“ To  Sustain  a Vehicle  on  an  In- 
clined Road 845 

“ Transmission  of. 167 

Powers,  Natural I98 

“ of  first  9 Numbers 98 

“ of  4 th  and  $th  Numbers. 303 

“ of  6th  Number , etc 304 

Probability 114-117 

Progression . 101-105 

Proof  of  Spirituous  Liquors 218 

Propellers 73o,  886-891 

Propeller  Steamers,  Ordinary  Distri- 
bution of  Power  in gn 

Properties  of  Numbers g8 

Proportion 94_?6 

Pulley 

433 

Pumping  Engines 738  902-903 

Pumps,  Fire 738  Centrifugal..  01  x 

••  Worthington , 738  Circulating , 740 

Pushing  or  Drawing 

Pyramids,  Statues,  etc 178 

Q. 

Quartermasters,  Service  Train  of. . . . 198 

R. 

Race-Courses,  English,  Lenqth  of....  c2 

Rack  and  Pinion 628 

Railroads ...].!!!  178 

Rails,  Iron  and  Steel * * gi2 


„ . Par* 

Rails,  Tangential  Angles  for  Chords 

and  Curves 677,  678 

“ To  Define  a Curve 677 

* ‘ To  Determine  Elevation  of  Out- 
er Rail 4 679 

Railways 677-685 

“ Curves  by  Offsets 678 

‘ ‘ Elevation  of  Rail 678,  679 

Operation  of  Locomotives . 

681-685 

“ Points  and  Crossings 678 

“ Radii  of  Curves 679 

“ Rise  per  Mile  and  Resist- 
ance to  Gravity 679 

Sidings 677 

“ To  Compute  Weight  of  Rail  679 

‘ ‘ Load  a Locomotive 
will  Draw  up  an 

Inclination 680 

“ “ Maximum  Load 

Drawn  up  the 
Maximum  Grade 
it  can  Attain. . . 680 
“ “ Resistance  of  Grav- 
ity  679 

‘ ‘ Traction , A dhesion , 

etc 680 

“ Velocity  of  Trains 680 

Railway  Trains gn 

Rainfall,  Volume  of. 850 

Reaction  and  Impact  Wheel 576 

Reaping..... 433 

Rebate  or  Discount 109 

Reciprocals 3q4 

Rectilineal  Figures,  To  Describe. . . . . 222 

Refraction. 668 

“ of  Light 584 

Refrigerator,  Surface  of 5I2 

Rendering 598 

Resilience  of  Woods 763 

Retaining  Walls ! ! ] 695 

“ of  Iron  Piles 196 

Revetment  Walls 694-700 

“ Surcharged 699 

“ To  Compute  Elements  of  696 

Ringing  Engine 672 

River  Steamboat gj3 

River  Steamboats  and  Engines.  .892,  914 

“ Traction  on  848 

Rivers,  Current  of Xg3 

“ Descent  of  Western 188 

‘ ‘ Flow  of  Water  in c c0,  c c 1 

“ Lengths  of ...183 

Obstruction  in 55! 

Riveted  Joints,  Comparative  Strength 

of  etc. . .751,  752,  755, 829,  829 
Experiments  on 783 

RiVET,!iG 755-757 

_ . 829,  830 

Diameter  of  etc 756,  829 

Memoranda 830 

Roads I7g 

Streets,  and  Pavements,  686-690 
690 
689 


Rivets. 


Bituminous . 

Concrete 

Construction. 
Corduroy. . . . 


686 

688 


XVI 


INDEX. 


Page 


Roads,  Getters , Fillers , and  Wheelers , 
Proportion  of, \ in  Different 

Soils £88 

“ Macadamized 687,  690 

“ Materials , etc 689 

“ Metalling 690 

“ Miscellaneous  Notes 690 

“ Planlc 688 

“ Rolling,  Sprinkling,  etc 690 

“ Ruts 687 

“ Sweeping,  Watering, and  Wash- 
ing  690 

“ . Telford . 688 

Roadway,  Construction  of 687 

Rock • 467 

Rock  and  Earth,  Excavation  and 

Embankment  of T92 

Rocks.  Bulk  of 468 

“ ' Weight  of,  per  Cube  Yard 468 

Roman  Calendar 71 

“ Indiction , To  Compute 71 

“ Long  Measures 53 

Roof  Plates,  Corrugated,  Weight  of...  131 

Roofs  of  Buildings *79 

“ Wooden i89 

Root,  To  Compute,  of  an  Even  Power.  98 

“ . To  Extract  any • 97 

Roots  and  Grains,  Weights  of 34 

Roots,  Square  and  Cube 272-302 

To  Compute  4 tli  and  6th 301 

Ropes 782 

‘ and  Cables 26 

and  Chains  of  Equal  Strength.  165 

Cables,  Chains,  etc 163-175 

“ Endless. ..... . *67 

“ Equivalent,  and  Belts 167 

‘ ‘ Hawsers  and  Cables 169-172 

“ 1 ‘ To  Compute  Strain  0/170,  1 7 1 

u u u Circumference  of  171 

“ “ “ Weight  of. 172 

“ Hemp  and  Wire 167 

“ “ Iron,  and  Steel 164,168 

“ “ Weight  and  Strength  of . 172 

“ Mining. *65 

‘ To  Compute  Stress,  Tension , and 

Deflection 166 

u “ Circumference  of, 

etc 169 

“ Transmission  of  Power. 167 

“ Wire 161-172 

“ u Experiments  on 169 

Rowing 433 

Rubble  Stone  Pavement 689 

Rule  of  Three 95 

Running.. 43s?  44° 


Safety  Valves 746,  912 

Sago..... 2°7 

Sailing •••  ° • * 

“ . Vessels,  Iron 894,  095 

Sails,  Propulsion  and  Area  of. 663 

“ . . Trimming  of. 665 

Saline  Saturation 726 

“ Matter  in  Sea  Water 727 


Page 

Sandstones *93 

Saw  Mill 9°4 

Sawing  Stone  and  Wood 196 

Saws,  Circular i97j477>911 

“ Vertical  and  Band 477 

Scale  or  Sediment  in  Boilers 726 

Scales,  To  Divide  a Line , etc 221 

“ Weighing  without 66 

Scarfs,  Resistance  of. 841 

Screw 630 

Cutting 477 

Differential 632 

Propeller,  Pitch  and  Speed  of . 662 
Screw  Propeller,  Friction  of  Engines  663 
Scripture  and  Ancient  Measures..  53 

Sea  Depth l84 

Seas,  Depth  and  Area 182 

Secants  and  Cosecants 4°3“4I4 

“ “ To  Compute  Degrees , 

Minutes,  etc 414 

Seeds,  Humber  of,  in  a Bushel,  and 

per  Sq.  Foot  per  Acre 193 

Proportion  of  Oil  in  Air -dry . 203 
Segments  of  a Circle,  Area  of.  .267,  268 
44  ‘‘  To  Compute  Area 

of 268 

Sewers • • • *6^1’  692 

‘■4  Drains,  Diameter  and  Grade 

of,  to  Discharge  Rainfall. . 906 

“ Pipes  and  Sewage 692 

Shaft,  Bearings  for  Propeller 473 

Shafts 778>  793?  794>  796,  797 

“ and  Gudgeons 79° 

Deflection  of. 77° 

“ Supports  of 796 

Shearing  or  Detrusive  Strength, 

782, 783 

Sheathing  and  Braziers’  Sheets 155 

and  Braziers'  Copper 131 

Nails,  Weight  of 
Sheet  Iron,  Galvanized  — 

Sheet  Piling 

Shingles..,.. 

Ship  and  Boiler  Plates. . . . 

Shoemaker’s  Measure, 

Shot  and  Shell,  Piling  of 05 

Shot,  Chilled  and  Drop 90b 

“ No.,  Diameter,  and  Numbers 

Of 9°6 

Shrinkage  of  Castings. 2i« 

Shrouds,  Hemp  and  Wire 173 

Side  Wheels,  Area  of  Blades 662  # 

“ Friction  of  Engines..  662 

“ Slip  of. 662  1 

Sides  of  Equal  Squares 258,259  \ 

Silt,  in  Mississippi  River 910 

Silver  Sheet,  Thickness  of. IX9  < 

Simple  Interest io7  . 

Simpson’s  Rule,  To  Compute  Area.. . 344  . 

“ Volume  of  an  Irreg- 
ular Body 87° 

Sines  and  Cosines 39°"402 

“ To  Compute 4®1 

u “ Number  of  Degrees, 

etc 4°2 

Sixth  Power  of  a Number,  To  Com 


i35 
.124, 129 
....  672 
. . . . . 63 

828 


Sand. , 


599 


pule. 


3°4 


INDEX. 


xvii 


Pape 

Skating 439 

Slackwater,  Canal , etc.,  Traction  on.  848 

Slaking 594 

Slate,  To  Compute  Surface  of. 64 

Slates  and  Slating 64 

Slates,  English 64 

“ Weight  per  1000  and  Number 

Required  to  Cover  a Square  64 

Slide  Valves 731,  733 

Smelting  of  Ore 445 

Slotting 477 

Smoke  Pipes  and  Chimneys 748 

Snow-flakes 195 

“ Line  of  Perpetual  Congelation.  192 

“ Melted 195 

Snow  and  Ice 849 

Solar  Day  and  Year 70 

Solders 634,  636 

Soldering 875 

Sound 195 

“ To  Compute  Velocity  of 428 

Soundings,  To  Reduce  to  Low  Water.  60 

Spars  and  Poles 62 

Specific  Gravity,  To  Ascertain,  of  a 
Body  Heavier  or  Lighter 

than  Water 209 

“ of  a Body  Soluble  in  Water  209 

“ of  a Fluid 209 

Specific  Gravity  and  Weight.  .208-215 
‘ ‘ To  Compute  Weight  of  a Body  215 
“ “ Proportions  of  Two  In- 

gredients in  a Com- 
pound, etc 216 

“ Weights  and  Volumes  of  Va- 
rious Substances  in  Ordi- 
nary Use 216 

Spikes  and  Nails,  Retentiveness  of. . . 159 
Spikes,  Ship,  Boat,  and  Railroad. 

i52i  i54 

and  Horseshoes 152 

“ and  Nails 159 

“ Wrought  - Iron  Nails  and 


Tacks. 


154. 

Spiral,  To  Describe % §3^ 

Spires v . . x%>' 

2l8( 

191 

7.79 


Spirituous  Liquors,  Dilution  Per  Cent] 

“ Proof  of ...  %VN 

“ Proportion  of  Alcohol. 

Springs,  Deflection  of. v. ....... . 

Square  and  Cube  Root,  To  Compute^ 
of  a Higher  Number  than 

cop  tainted  in  fable. 

“ Ta  Ascertain,  of  a Number 
consisting  of  Inte 
gers  and  Decimals 

“ of  Decimals v , , 

Square  PvOOT,  To  Extract.... . . 

Square,  To  Ascertain  One  that  has 

Same  Area  as.  a given  Circle. 

Squares,  Cubes,  and  Square  and 
Cube  Rgqts  , .272- 
M , To  Compute  Square  and 
Cube  Roots  of  Roots,  of 
Whole  Numbers,  and  of 
Integers  and,  Pecimals. . . 
Squads,  Sides  of  Equal,  in  Area  to  a 
Circle 258, 


Pape 

Stability 693-703 

“ Dynamical 651 

“ Dynamical  Surface 651 

“ of  Earth... 695 

“ of  Models 649 

“ To  Ascertain , of  a Body . . 693 

“ To  Careen  a Vessel 652 

“ To  Compute  Statical 651 

“ Equilibrium  of  Walts  701 

“ “ Stability 701 

“ To  Determine  Measure  of 

Hull  of  a Vessel , etc 650 

Staging,  Coach 44o 

Staining 876 

Starch,  Proportion  of  in  Vegetables. . 205 

Stars,  Velocity  of. I98 

Statics 6x5-617 

“ Composition  and  Resolution 

of  Forces. 615 

“ Equilibrium  of  Force 616 

Statues,  Pyramids,  etc 178 

Stay  Bolts,  Diameter , etc 754 

Steam 704-727 

“ and  Air.  Mixture  of. 737 

“ Average  Pressure , etc 71 1,  7I2 

“ Blowing  off  of  Saturated  Water  726 

“ Boilers 829 

“ “ Zinc  Foil  in nI2 

“ Clearance,  Effect  of 7^ 

‘ ‘ Compound  Expansion 720-7 24 

“ Conclusions  on  Actual  Effi- 
ciency of. 724 

“ Condensing  Surface , Experi- 
ments on qjj 

“ Density  or  Specific  Gravity  of  706 

“ Expansion 7IO 

“ Effects  of. . 713,718,719 

‘ ' Points  of. 7 1 2 

“ Feed  Water , Gain  in,  at  High 

Temperatures,  To  Compute. . 719 

“ Gain  in  Fuel,  To  Compute 725 

u Gaseous  and  Total  Heat  of. ..  710 

“ Ifammers i79 

“•  Heat  of  Saturated 705 

“•  Heating  Co.  of  N.  Y. 904 

Indicator ... 724 

“•  Injector , . . . . . . 736 

“ Latent  Heat  of. 70.7 

“ Mean  Pressure,  To  Compute . . 713 
“ Mechanical  Equivalent  of. .. . 705, 

“ - Pipes , Gas,  etc , 138 

‘ ‘ Poin  t .of  Cutting  off,  and  Press- 
ure at 7,I(t> 

Pressure  q/.,  * 705,  710,  7/11 

u Weight , and  Temper- 
ature  ......  705 

“ Properties  of,  of  Maximum 

Density 7^7- 

Ratio  of  Expansion ^q, 

u Relative  Effect  of  Equal  Vol- 
ume Of. 7x4; 

u - Saline  Saturation  in  Boilers:. 

‘ * Saturated  704,  708,  7 1 C 

u • Scale  or  Sediment * Removal  off  72S 
“ Specific  Gravity. of. , , . , . .704,  706* 

“•  Superheated 7^, 

“ Temperature  of. 703; 


XV111 


INDEX. 


Page 


Pagt 


Steam,  Temperature  of  Water  m a 

Condenser 7°7 

tt  To  Compute  Volume  of  Water 

Evaporated  per  Lb.  of  Coal  725 
tt  “ Consumption  of  Fuel ... . 725 

“ Total  Effect  of  1 Lb.  of 7*4 

it  “ Heat  of  Saturated 707 

t<  Velocity  of  to  Compute 7IO>  9*3 

“ Volume  of  a Cube  Foot  of  Wa- 
ter in 7°6 

« tt  of  Cylinder,  to  Compute.  715 
“ “ of  to  Raise  a Given  Vol- 
ume of  Water 7°6 

« t.  of  Water  Contained  in..  706 

a “ of  Water  to  Raise  or  Re- 

duce to  any  Required 

Temperature 7°7 

“ Weight  of. 7°5 

“ Wire  Drawing 7l8 

“ Woolf  Engine  . 722 

Steamboats,  River,  and  Engines  . 892, 893 

‘ 1 Passages  of 896 

“ Wood  Side  Wheels 892 

“ “ Ferry  Boat 893 

“ “ Passenger  and  Light 

Freight 893,894 

a “ Stern  Wheels  .. : 893,894 

■Steam-Engine 727-760 

‘ ‘ and  Boilers , Cost  of  per  Day.  904 

“ Area  of  Feed  Pump 736 

a a injection  Pipe 735 

“ Boiler 73^-745 

a a Evaporative  Power  oj . 757 

a a Weights  of 759 

a Circulating  Pumps 749 

“ Condensing • 727 

“ Elements  and  Capacity  of 

Steam  Pumps 738 

a Evaporation '• 747 

a Fire 

a Flues  and  Tubes. 747 

a Friction  of  Side  Lever 478 

a General  Rules 728,  730 

a IP,  to  Compute , etc 733,  734 

a Injection  Pipe , to  Compute 

Volume  of  Flow 735 

a Eon-condensing 728 

a Plates  and  Bolts 749~753 

a Practical  Efficiency  of 737 

a Propeller , to  Compute  Thrust 

of 73i 

a Propellers 73°j  731 

a Relative  Cost  of 757 

a Results  of  Operation  of. . .737,  921 

a Riveting -753-757 

a a General  Formulas. . 757 

a Safety  Valves 746,  912 

a Smoke  Pipes  and  Chimneys . 

748,  749 

a Stay  Bolts , Rods , etc 753,  754 

a Steam  Room 748 

“ Slide  Valves 73* 

tt  a To  Compute  and  ^4s- 
certain  Lap,  Breadth 
of  Ports,  Portion  of 
Stroke,  Lead , etc. 731-733 
a Volume  of  Circulating  Water  735 


Steam-Engine,  Volume  of  Injection 

and  Feed  Water 735,  736 

a Volume  of  Water  Required  to 

be  Evaporated 734 

a Water  Surface 748 

Water-  Wheels 730 

Weights  of 758,  759,  9” 

Steam  Fire-Engine 9°4 

Steam  Pumps,  Elements  and  Capaci- 
ties of. 738 

Steam- Vessel,  Power  Utilized  in 662 

Steam- Vessels,  Resistance  to,  in  Air 

and  Water  911 

Steamer  u Great  Eastern” 179 

Steamers,  Friction  of  Screws 478 

Steaming  Distances 86 

Steel 640-643,  750,  783,  787,  827 

a Cables,  Galvanized ... 163 

a Columns , Crushing  Weight  of . 768 
< ‘ Hemp  and  Iron  Ropes,  Relative 

Dimensions  of 168 

a Hemp,  and  Iron  Wire  Rope. . . 164 

a Locomotive  Tubes 138 

a Manufacture  of. 642 

“ Plate 83o 

a Plates,  Weight  of. 118,  119,  146 

“ Relative  Dimensions  of 172 

‘ ‘ Rolled  and  Bar,  Weight  of. . 1 34,  1 35 
a Rope , Hemp  and  Iron,  Round 

and  Flat 164 

a weight  of. 136.149 

t ‘ Wire , Weight  of 20,  1 2 1 

Sterling,  Pound 38 

Stings  and  Burns,  Application  for. . . 196 

Stirling’s  Mixed  Iron 785 

Stirrups  or  Bridles,  for  Beams 838 

Stone  and  Ore  Breakers 9°3 

Stone,  Expansion  and  Contraction 

Of- 

a Hauling 4°  8 

a Masonry 595 

a Resistance  of , to  Freezing 184 

a Sawing 196,904? 

a Voids  in  a Cube  Yard 690  . 

Stones,  Cements,  etc 766 

Streams,  Flow  of  Water  in 55° 

Street  Rails  or  Tramways 4351  9°2 

Streets,  Roads,  and  Pavements.. 

686,  690 

Strength  of  Materials 761-841 

Crushing  Strength 764-769 

“ Comparative  Value  of  Long 

Solid  Columns 769  : 

a of  Cements,  Stones,  etc 596  ; 

a of  Columns . to  Compute  } 

Weight  of. 768 

“ of  i-inch  Cubes 767^ 

1 ‘ of  Various  Materials . . 765,  766. 

a Resistance  of  Rivets 9°8  - 

1 a Riveted  Joints 828,829 

a Safe  Load  of  Columns , 
Arches , Chords , etc.,  of 

Cast  Iron 766 

a Weight  borne  with  Safety 

by  Cast-Iron  Column. . . 768 
a Woods,  Destmictive  Weight 

of  Column  of. 769 


INDEX. 


xix 


Page 

Strength  of  Materials,  Crushing 
Strength  of  Woods , Rel- 
ative Value  of  Various  . 769 
“ Wrought- Iron  Cylinder  and 

Rectangular  Tubes 767 

Deflection 770-781 

“ and  Distributed  Weight  for 

Limits  of  to  Compute. . . 778 
‘ ‘ Cast-Iron  Bars  and  Beams, 

777,  778 

“ Continuous  Girders  or 

Beams  of  Wood 772 

“ Formulas  for  Beams. . .771,  772 

“ General  Deductions 779 

“ Mill  and  Factory  Shafts. . . 779 
“ of  a Shaft  from  its  Weight 

alone 778 

“ of  Bars,  Beams,  Girders, 

etc 770-782 

“ of  Rectangular  Bars  or 

Beams  of  Cast  Iron 777 

“ of  Wrought- Iron  Bars 773 

“ of  Wrought -Iron  Rolled 

Beams 774 

“ Rails 776 

“ Relative  Elasticity  of  Vari- 
ous Materials 780 

“ Results  of  Experiments. . . . 780 

“ Shafts  of  Wrought  Iron 778 

“ To  Compute,  and  Compar- 
ative Strength  of  Cast- 
Iron  Flanged  Beams. . 778 
“ “ and  Weight  that  may  be 

Borne  by  a Rectan- 
gular Bar  or  Beam 

of  Cast  Iron 777 

u 11  Maximum  Load  that 
may  be  Borne  by  a 
Rectangular  Beam.  773 
“ 11  of  and  Weight  Borne  by 

a Rectangular  Bar 

or  Beam 773,774 

u “ of  Cast-Iron  Flanged 

Beams 777 

“ Woods 772 

“ Wrought  Iron  and  Woods. . 772 

“ Wrought  - Iron  Bars  or 

Beams .773,  774 

“ Wrought  - Iron  Riveted 

Beams 774,  775 

“ Wrought  - Iron  Tubular 

Girders 775 

Detrusive  or  Shearing  Strength  .782, 783 
“ Comparison  between  it  and 

Transverse 782 

“ of  Woods 782 

Results  of  Experiments.  782,783 
“ Riveted  Joints,  Cast  Iron , 

Treenails,  and  Woods. . . 783 

“ Shearing 783 

‘ ‘ To  Compute  Length  of  Sur- 
face of  Resistance  of 
Wood  to  Horizontal 

Thrust 782 

“ Wrought  and  Cast  Iron 
Riv'd  Joints , Steel,  Tree- 
nails, and  Woods 783 


Page 

Strength  of  Materials,  Elasticity 

and  Strength 761-764 

“ Coefficients  of 761 

“ Comparative  Resilience  of 

Woods 763 

I ‘ Extension  of  Cast-Iron  Bar  762 

“ Modulus  of  Cohesion 763 

“ “of  Elasticity  ....  .762,  763 

“ “ of  Elasticity  and 

Weight 763 

“ of  Elasticity,  Height 

of,  to  Compute. . . 763 

u “ of,  to  Compute 762 

“ Weight  a Material  will 
Bear  without  Permanent 

Alteration 763 

Tensile  Strength 784-789 

“ Elements  Connected  with 
Resistance  of  Various 

Bodies 786 

“ Malleable  Iron 785 

“ Manganese  Bronze 832 

“ of  Cast  and  Wrought  Iron, 

78 4,  785 

“ of  Tie-rods 787 

II  of  Wrought  Iron 785 

‘ ‘ Ratio  of  Ductility  and  Mal- 
leability of  Metals 787 

‘ ‘ Steel , Bars  and  Plates. . 787,  788 

“ Stirling's  Mixed  Iron 785 

“ Various  Materials 788-790 

Torsional  Strength 790-797 

“ Couplings 796 

“ Hollow  Shafts 792,  794 

“ Journals  of  Shafts,  etc 796 

“ Metals  and  Woods 793 

“ Mill  and  Factory  Shafts . . 797 
“ Minimum  and  Maximum 
Diameter  of  Shafts,  For- 
mulas for 796 

1 1 of  Various  Materials 793 

“ Shafts  and  Gudgeons. . . 790-795 

‘ 1 To  Compute , of  Shafts 794 

“ “ Diameter  of  a Shaft  to 

Resist  Lateral 

Stress 791 

“ “ “ of  Shafts  of  Oak 

or  Pine 793 

“ “ “ of  a Centre  Shaft.  794 

“ “ “ of Solid  and  Hollow 

Shafts.  .791,  792,  794 

Transverse  Strength 799-841 

“ Bars,  Beams,  Cylinders, 

etc 801-805 

“ “ Girders,  or  Tubes, 

Comparative  Value 

of. 824 

“ Bow-string  Girder 812 

“ Brick-work 801 

“ Bridge  Plates  and  Rivets . . 830 

“ Cast-Iron 814-817 

“ “ and  Woods .'...  798 

“ “ Girders  and  Beams.  813 

“ Channel  and  Deck  Beams , 

and  Strut  Bars 808 

“ Comparative  Qualities  of 

Various  Metals 8q,. 


XX 


INDEX. 


Page  | 


Strength  of  Materials,  Transverse , 
Comparative  Strength  and 
Deflection  of  Cast-Iron 
Beams 8o9 

“ Cylinders, Flues,  and  Tubes,  ^ 

u . Cylindrical  and  Elliptical 

Beams  or  Tubes 810 

“ Dimensions  and  Propor- 
tions of  Wrought -Iron 

Flanged  Beams 809 

“ Elastic  StrengthofWrought- 

Iron  Bars 808 

“ Elements  of  Rolled  Beams.  807 
“ Flanged  Beams , Compara- 
tive Strength  and  Deflec- 
tion of. 809 

“ Flanged  Hollow  or  Annu- 
lar Beams  of  Symmetri- 
cal Section 815 

a Form  and  Dimensions  of  a 
Symmetrical  Beam  or 

Girder 825 

“ General  Deductions 824 

“ General  Formulas  for  De- 
structive Weight  of  Solid 
Beams  of  Symmetrical 
and  Unsymmetrical  Sec- 
tion   816, 817 

“ Girders  and  Beams  of  Un- 
symmetrical Section  810 
“ « Beams , Lintels , etc., 

822-826 

11  Iron  and  Steel  Rails 812 

“ Memoranda ••  83° 

“ “ Cast  and  Wrought 

Iron 832 


Pag? 

Strength  of  Materials,  Transverse , 

To  Compute  Section  of 
Flange  of  a Girder 
or  Shaft  of  Cast 

Iron 817,  818 

“ “ Ultimate  Strength  of 

Homogeneous  Beams  820 
“ T missed  Beams  or  Girders , 

823,  824 

“ Unequally  Loaded  Beams. . 810 
“ Wrought  - Iron  Inclined 


Beams,  etc..  81 1 
« “ Plate  Girders  81 1 

u “ Rectangular 

Girders  or 
Tubes 809 

Working  Strength  and  Factors  of 

Safety 7Sl 

Strength  of  Models ; • °44 

u To  Compute  Dimensions  oj  a 

Beam,  etc 644 

« “ Resistance  of  a Bridge 

from  a Model 645 

Stress,  Moment  of. 621-623 

Stucco : 591 

Suez  Canal,  Via 

Sugar  Cane,  and  Beet  Root 207 

Sugar-Mill  Rollers 911 

Sugar  Mills ••••••'•*  9°3 

Sulpliuret  of  Carbon,  Elastic  Force 

of  Vapor  of 7°7 

Sun 168 

“ Heat  of J93 

Sunday  Cycle  or  Cycle  of  the  Sun — 70 

“ or  Dominical  Letter 7° 

69 


Sun-dial,  To  Set - 

Tron  ....  032  ! Surcharged  Revetments  699 

" S"  **::  lit1 

596  Sustenance, 

ing-man 2°7 

Sweet  Potato 2°7 

Swimming * * 439 

Symbols,  Algebraic,  and  Formulas.  22 


of  Various  Figures  of  Cast 

Iron 800 

‘ 1 Materials. . . 799-801 

“ Metals 799 

Rectangular , Diagonal , or 
Circular  Beam  or  Shaft.  817 
Relative  Stiffness  of  Mate- 
rials  798 

Rivets  and  Plates. . ..  .828,  830 

Solid  and  Hollow  Cylinders 

of  Various  Materials 801 

Steel  Bars 8x7 

u plates 83° 

To  Compute  Centres  of  Grav- 
ity and  of  Crushing 
and  Tensile  Strength 
of  a Girder  or  Beam  819 
“ Destructive  Weight  or 
Loads,  Borne  by 
Rolled  Beams  or  Gir- 
ders or  Riveted 

Tubes 805-807 

‘ “ Inertia,  Moment  of  of 

a Solid  Beam 819 

» “ Neutral  Axis  of  a Beam 

of  Unsymmetrical 
Section 


Tacks,  Nails , Spikes,  etc. 
Iron 


820 


Wrought 
754 

Tan 482 

Tangential  Wheel • 57° 

Tangents  and  Co  tangents 410-426 

41  “ To  Compute.  426 

Tannin,  Quantity  of  in  Substances. . . 190 
Tee  and  Angle  Iron,  Weight  of  ...... . 13° 

Teeth  of  Wheels 859-861 

“ Involute 8 59 

Telegraph  Wire,  Span  of £79 

Telescopes,  Opera-Glasses,  etc • * • 071 

Telford  Roads 688>  69° 

Temperature *95 

“ by  Agitation . 524 

“ Decrease  of  .by  Altitude.  522 

“ of  Enclosed  Spaces 526 

u 0f  Various  Localities.. . 192 

“ To  Reduce  Degrees  of  ■ 

Different  Scales 523 


INDEX. 


XXI 


Page 

Temperature,  Underground 519 

Temperatures,  Metric 37 

Tempering  Boring  Instruments 197 

Tenacity  of  Iron  Bolts  in  Woods 198 

Tensile  Strength 784-700 

Terne  Plates 124 

Terra  Cotta 602 

Test  of  Plates  of  Iron 749 

Theatres  and  Opera-Houses 180 

Thermometers 523 

Throwing  Weights 439 

Thrust,  To  Compute  Weight  of  a Body, 

To  Sustain 693 

Tidal  Phenomena 75 

Tide  Table,  Coast  of  U.  S. 84 

Tides igs 

“ of  A tlantic  and  Pacific 19 1 , 198 

‘ ‘ of  Pacific  Coast 85 

“ Rise  and  Fall  of  Gulf  of  Mexico  85 

‘ { Time  of  High  Water. 74,75 

Tie-rods .787 

Time,  after  Apparent  Soon,  before 

Moon  next  passes  Meridian.  75 

“ Difference  of 81-83 

u Measures  of. . . . 37 

“ New  Style 37 

“ Sidereal  and  Solar 37 

“ To  Compute  Difference  of,  be- 

tween New  York  and  Green- 
wich  83 

“ To  Reduce  to  Longitude 54 

Timber,  Comparative  Weight  of  Green,  217 
“ Measure,  and  to  Compute 

Volume  of ...61,  62 

“ Waste  in  Hewing  or  Sawing,  62 

“ and  Board  Measure 61 

“ and  Woods 865-870 

“ Impregnation  of 868 

Seasoning  and  Preserving. . . 865 

Strength  of 870 

644 

and  Lead  Pipes,  Weight  of. 139 

“ Plate,  Marks  and  Weights 139 

Tolerance,  of  Coins 38 

Tonite 443 

Tonn age,  Approximate  Rule 176 

“ Builder"1  s Measurement. .. . 176 

1 1 Corinthian  and  New  Thames 

Yacht  Club jyy 

“ Freight  or  Measurement. . . 177 

“ of  Suez  Canal I77 

“ of  Vessels 175-177 

“ Royal  Thames  Yacht  Club.  177 

“ To  Compute x73 

“ Units  for  Measurement, 

and  Dead  Weight  Cargoes  176 

‘ ‘ Weight  of  Cargo 1 7 7 

Tools 476 

Torsional  Strength 790-797 

Towers  and  Domes x8o 

Towing,  Erie  Canal  and  Hudson 
River .................... ....  103 

Traction 843-849 

“ Ascending  or  Descending 


Page 

Traction,  Friction  of  Roads 847 

“ Grade 847 

“ Omnibus 844 

u on  Common  Roads 843,844 

“ Resistance  of  a Car 849 

“ of  a Stage  Coach  848 
“ of  Gravity  and 

Grade 847 

“ on  Common  Roads 

843-845 

“ Results  of  Experiments  on . 843 

“ To  Compute  Power  neces- 

sary to  Sustain  a Vehicle 
on  an  Inclined  Road.  845,  846 
“ Various  Roads  and  Vehi- 


845 

845 

198 

9i5 

i93 


Tin . 


cles 

Wagon 

Train,  Service , of  a Quartermaster. . 
Tramways  or  Street  Rails.  ..435  84* 

Transportation,  Canal 

“ of  Horses  and  Cattle.  192 

Transverse  Strength 708-841 

Treadmill 433 

Treenails 783 

Trees,  Large,  in  California 184 

Trigonometry,  Plane 385-389 

Trim,  Change  of,  in  a Vessel 65s 

Tripolith 41 

Trotting 439 

Troy  Measure 32 

Truss,  Iron 178 

Tubes,  and  Flues 747,  827 

“ and  Pipes,  To  Compute  Weight 

of 147 

“ Brass , Weight  of. 142 

“ Copper  Drawn,  Weight  of.  140,  144 

“ Lap  Welded  Iron  Boiler 137 

“ or  Girders 809 

“ Seamless  Copper 140,  144 

u Steel  Locomotive. . . / 138. 

“ Wrought-iron 143,  i4«t 

Tubular  Bridge x-g 

Tunnels,  Lengths  of. x jg, 

Turbines ’ ^ 

Boy  den 574 

“ CompaHson  with  Water- 


wheels. . 


5791 

Downward-flow 574 

Fontaine 574 

Fourneyron 37^ 

High  - Pi'essure,  Operaticm 

°f- 

Inward-flow 

Jonval 

Low-pressure 

Outward-flow 


Poncelet . 


574- 

57S 

57S 

575 

575 


574- 


Ratio  of  Effect  to  Power 577 


an  Elevation 

Canal,  Slackwater , 
River. 


Swain 
Tremont, 

Victor. 

Turkish  Plaster  and  Mortar 591  592, 

Turning 

“ and  Boring  Metal 197 


575 

576 

576 


and 


846  Turnips 2D7 

Turpentine,  Elastic  Force  of  Vapor 
°f' * JO? 


XXII 


INDEX. 


U.  Page 

Underground  Temperature 5*9 

Undershot-Wheel 5°6 

Unguents,  Value  of 471 

Uniform  Motion 


6x7 


V. 


Value  of  Coins : — * • : • * • 

“ and  Weight  of  Foreign  Coins  . 
Vapor  in  Atmosphere,  Volume  of... 

“ Weight  of •••••• 

‘ « Elastic  Force  of  A Icohol , Ether , 
JSulphuret  of  Carbon , Petroleum , 

and  Turpentine 7°7 

Variable  Motion 6l7 

Variation  of  Magnetic  Needle 57 

“ Decennial,  of  Needle 50 

* ‘ of  in  U.  S.  and  Canada ...  59 

Varnishes 876 

Vegetable  Marrow 207 

Vegetables,  Analysis  of  Meat  and  Fish , 200 
« and  Oil -cake,  Nutritious 


Properties  of 204 

“ Proportion  of  Starch  in. . 205 

“ Tubers 2°7 

Vegetation,  Limit  of *92 

Velocities,  Metric 37 

Ventilation 524 

•“  of  Mines 449 

Vernier  Scale 27 

Vessels,  Elements  of 653 

“ Hulls  of 830 

Veterinary * 

Volcanoes,  Height  of ........... 

“ Power  of 

Volume  and  Weight  of  Various  Sub- 
stances  

Volumes 


Page 

Washington  Aqueduct *78 

Water 849-852 

Boiling-Points  of 851 

Density  of 52o 

Deposits  of 852 

Expansion  of 5*9 

Fresh  and  Sea 849,  851 

Inch 557 

Motors , Ratio  of  Effective 

Power 503 

Power 502 

Pressure  Engines 579 

Rainfall  and  Volume  of 850 

Resistance  of,  to  an  Area  of 

One  Sq . Foot 646 

Velocity  of  a Falling  Stream 

of. f6 

Volumes  of. °49 

Weight  of 852,  920 

Wheels  • • • *563j  73° 

“ Compared  with  Tur 

bines 

“ Overshot 

‘ ‘ Undershot 

Waterfalls  and  Cascades 

Watermelon 

Water  Pipes,  Cast-Iron 

“ Dimensions,  etc.  ..  .138, 139 
Water-Wheel,  Centre  of  Gyration. . 611 
Water-Wheels,  Diameter  and  Journal 

of  a Shaft,*  tc 581 

“ Dimensions  of  Arms 57 1 

Waves  of  the  Sea 852,  853 


579 

563 

566 

184 

207 

H7 


W. 


Walking • -433,  438 

Wall,  Chinese *79 

Walls  and  Arches 602 

“ Centre  of  Gravity  of  702 

“ Dams  and  Embankments 700 

“ Elements  of 

‘ 1 Friction  of. 698 

“ Moment  of 7°* 

a “ of  Pressure 098 

“ of  Buildings i89 

“ Retaining,  of  Iron  Piles 190 

“ Revetment 694 

“ Stability  of. 7°2 

Warehouses,  Brick  Walls  for 003 

Warming  Buildings 527-528 

1.  By  Hot-air  Furnaces  or 

Stoves 528 

“ By  Hot  Water.. 524 

“ By  Steam 527 

u Coal  Consumed  per  Hour  . 527 

“ Furnaces 528 

« Illustrations  <>f  Heating. . . 527 

Open  Fires 528 

M Volume  of  Air  Heated  by 

Radiators , Consumption 
of  Coal , Areas  of  Grate 
and  Heating  Surface,  of 
Boiler,  etc..  • 528 


Tidal. 

Velocity  of 

Weather-Foretelling  Plants 

Glasses 

Indications 

Wedge 

Weighing  without  Scales. . 
Weight , Avoirdupois 


853 
853 
. 185 

• 430 

• 43i 
. 630 
. 66 

A 32,47 

and  Diameter  of  Cast-Iron 

Balls J53 

and  Dimensions  of  Lead 

Balls 5oi 

and  Dimensions  of  Wr ought- 
iron  Bolts  and  Nuts  . .156-158 
and  Fineness  of  U.  S.  Coins.  38 
and  Marks  of  Tin  Plates. . . 139 
and  Mint  Value  of  Foreign 

Coins 4° 

and  Strength  of  Hemp  and 

Wire  Ropes *72 

and  Strength  of  Iron  Wire , 

etc I24 

and  Strength  of  Stud -link 

Chain  Cables 168 

Angle  and  T Iron 125, 130 

Apothecaries' 32>  47 

Bells 180 

Brain J92 

Brass 136} 1 49 

“ Plates M6 

“ Sheet  and  Tubes  cor- 
responding to  Iron . 142 

“ Wire 20, 121 

Cast  Iron *49 


INDEX. 


XX111 


Page 

Weight,  Cast-Iron  and  Lead  Balls . . 153 

“ “ Bar  or  Rod 131 

“ “ Pipes  or  Cylinders  132 

“ 44  Plates 146 

4 4 Composition  S heath  ing  Nails  135 

44  Copper 136 

44  “ Rods  or  Bolls 148 

44  44  /Seamless  Tubes 144 

• 4 4 4 Sheet 135 

44  Conjugated  Roof  Plates 13 

44  Cube  Foot  of  Embankments, 

Walls , etc 694 

44  Diamond,  and  Diamonds . 32,  193 

44  Electrical 34 

44  Flat  Rolled  Iron 126-128 

44  Foods , to  Furnish  Nitroge- 
nous Matter 202 

44  Galvanized  Sheet  Iron 124 

“ Grain 32 

44  44  and  Roots 34 

44  Green  and  Seasoned  Timber  217 

44  Gun  Metal 149 

44  Hemp  and  Wire  Rope 166 

44  Hexagonal,  Octagonal,  and 

Oval 135 

44  Horses 35 

44  Ingredients , that  of  Com- 
pound being  given 218 

4 4 Iron,  Steel,  Copper , and  Brass 

Plates.  1 1 8,  1 19 

44  4 4 44  Wire. . 1 20-1 2 1 

44  Lead 32 

“ 44  Encased  Tin  Pipes .. . 151 

“ “ Pipes 139,150 

4 4 4 4 Plates: 146 

“ 44  Sheet 151 

44  Men  and  Women 35 

44  Metal  by  Weight  of  Pattern.  217 

44  Metals  of  a Given  Sectional 

Area 149 

44  Molecules,  Weight , etc 194 

44  of  Articles  of  Food  Consumed 

in  Human  System  to  De- 
velop Power  of  Raising  140 
Lbs.  to  a Height  of  10000  Ft.  204 

44  of  Beef  and  Cattle 35 

44  of  Cast  and  Wrought  Iron. . 155 

44  44  4 4 44  Steel , 

Copper , and  Brass.  136 

44  of  Earths 33 

“ of  Offal 35 

“ of  Sq.  Foot  of  Slating 64 

44  Rocks , Earth , etc 468 

44  Rolled  and  Bar  Steel 134,  135 

44  Round  Rolled  Iron 126 

44  Sheet  Iron 129 

4‘  Steam ■ 7°5 

44  Steel 136,  149 

4 4 4 4 Plates 134,  146 

44  Tin  Pipe 139,  i5I 

44  To  Ascertain , of  a Solid  or 

Liquid  Substance 217 

44  To  Compute,  of  an  Elastic 

Fluid 217 

44  Various  Materials 155 

44  Substances  in  Bulk  217 

“ 44  per  Cube  Foot  217 


Page 

Weight,  Wire  and  Hemp  Rope 166 

44  Wood 33 

44  Wrought  and  Cast  Iron 155 

4 4 4 4 Iron.  .125,  126,  136,  149 

4 4 4 4 Plates 146 

44  Sheet  and  Hoop. . . 129 

4 4 4 4 Tubes 143-145 

44  4 4 Tubes  and  Plates, 

145, 146 

44  Zinc  Sheets 123,  146, 151 

4 4 4 4 4 4 and  Dimensions 

°f  - *r< I51 

Weights  and  Measures 26-35 

44  and  Volumes  of  Various 

Substances 216 

4 4 English  and  French 44 

44  Foreign 48 

44  Grecian 53 

44  Hebrew  and  Egyptian 53 

44  Measures  of 32,  47 

“ Metric 33,36 

4 4 Miscellaneous 33 

44  of  Steam-Engines.  .758,  759,  91 1 

44  Roman 53 

Well,  Artesian 179,  198 

44  Boring i97 

Wells  or  Cisterns,  Excavation  of, 

etc 63 

Welding 786 

44  Cast  Steel,  Composition  for. . 634 

44  Fluxes  for 636 

Wheel  and  Axle 626 

44  and  Pinion,  Combinations  or 

Complex  Wheel  Work 628 

Wheel  Gearing.  854-861 

44  Circumference  of 857 

“ General  Illustrations 858 

44  Pitch , Diameter , Number  of 

Teeth , Velocity,  etc 855,  857 

44  Revolutions  of  858 

44  Spur  Gear 91  x 

4‘  Teeth  of 859 

44  To  Compute  Diameter  of. . . . 857 

4 4 4 4 W of  a Tooth 861 

4 4 4 4 Velocities  of. 856,857 

Wheels,  Proportions  of 862 

44  Teeth  of. 859-862 

Whitewash  or  Grouting 594 

Wind,  Course  of. 675 

44  Effective  Impulse  of 665 

4 4 Force  of. 674 

44  Pressure  of. 91  x 

44  Velocity  and  Pressure 674 

Winding  Engines 476,  862,  863 

44  To  Compute  Diameter  of  a 

Drum 862 

4 4 4 4 Number  of  Revo- 
lutions  863 

Windlass 433 

44  Chinese 627 

Windmills  • 863-865 

44  Results  of  Operation  of  865,  921 

44  To  Compute  Elements  of  . . 864 

Window  Glass I24 

Wine  and  Spirit  Measures 45 

Wire  Gauge,  French 123 

“ Standard  of  Great  Britain  122 


XXIV 


INDEX. 


Page 

Wire  Gauges 122 

“ Iron  Gauge,  Weight  and  Length 

of 163 

Wire,  Length  of. 124 

“ Rope 161,473 

“ and  Equivalent  Belt 167 

“ and  Hemp,  General  Notes 167 

“ Cables,  Galvanized  Steel 163 

“ Endless 167 

“ Fence , Weight  and  Strength  of.  164 
“ Results  of  an  Experiment  with 

Galvanized 161 

“ “ of  Experiments  on,  at 

U.  S.  Navy  Yard 169 

“ Transmission  of  Power  of ... . 167 
“ Ropes,  Hemp,  Iron,  and  Steel, 

Relative  Dimensions  of. ... . 168 
1 ‘ and  Hemp  Rope,  Iron  and,  Steel, 

Relative  Dimensions  of 172 

“ and  Hemp  Ropes,  Weight  and 

Strength  of. 172 

u and  Tarred  Hemp  Rope,  Haw- 
sers, Cables , Comparison  of. . 169 
“ Rope , Circumference  of,  to  Com- 
pute  169 

“ “ for  Standing  Rigging, 

Circumference  of,  to  Compute  172 

“ Shrouds 173 

Woods 481,  765,  769,  782,  783 

“ Bituminous  or  Lignite 479 

“ Coefficients  for  Safety 835 

“ Detrusive  Strength  of. 782 

“ Floor  Beams 835 

“ Measure 47 

“ Pavements 689,690 

u Relative  Value  of  their  Crush- 

ing Strength  and  Stiffness 

combined 769 

“ Safe  Statical  Loads  for 834 

“ Sawing 196 

“ Weights  of. 33 

Wood  and  Timber 865-870 

“ Creosoting,  Effects  of. 869 

“ Decrease  by  Seasoning ... . 869 


Page 

Wood  and  Timber,  Defects  of 866 

“ Durability  of  Various 869 

‘ ‘ Impregnation  of. 868 

‘ ‘ Proportion  of  Water  in. . . 869 
“ Seasoning  and  Preserving , 

866,  868 

“ Selection  of  Trees 865 

“ Strength  of. 833,  870 

“ Transverse  Strength  of,  to 

Compute 833 

“ Weight  of  Oak  and  Yellow 

Pine  per  Cube  Foot 870 

Work 432 

“ Accumulated  in  Moving  Bod- 
ies, etc 619 

“ and  Power , Metric 36 

Works  of  Magnitude 178,  179 

Wrought  Iron 639,  765,  768,  773,  785 

“ Crushing  Weight  of 

Columns 768 

“ Deflection  of  Bars , 

Beams,  etc.  773-775 

“ “ of  Rails 776 

“ Plates  and  Bolts 749 

“ Plates,  Weight  of.  118,  119 

“ To  Compute  Weight 

for 125 

“ Weight  of. 155 

“ Wire,  Weight  of..  120,121 


Yam 207 

Year,  Bissextile  or  Leap 70 

“ Civil 70 

“ Solar 70 

Years  of  Coincidence 74 


Zinc 644 

“ Plates,  Weight  of 146 

“ Sheets,  Thickness  and  Weight  of, 

123,  151,  152 

Zinc  Foil  in  Steam  Boilers 912 

Zones  of  a Circle,  Areas  of 269-271 


Addenda. 


Page 

Asbestos 9*3 

Asphalt,  Mortar  and  Concrete. .....  913 

Blowing  Engine,  Friction  of  Air  in 

Pipes 921 

Boilers,  Cylindrical  Shells 751,  752 

“ Fiued , Arched,  or  Circular 

Furnaces 754 

“ Girders 754 

“ Plates , Straps,  and  Stays. . 753 

Chain  Cables,  Stowage  of 913 

Concrete,  CoigneVs 914 

Distances,  Velocities,  and  Accel- 
erations  91** 

Flexible  Paint  for  Canvas 915 

Gas,  Natural 9*3 

Gauges,  Steam,  Vacuum,  and  Hy- 
drostatic  91^ 

Hose,  Delivery  and  Friction  of. 919 


Page 

Ice  or  Cold  Producing  Machines..  922 

Jarrah  Wood 913 

Light,  Penetration  of,  in  Water 915 

Locomotive,  Brakes 920 

Materials,  Non-conducting 914 

Ocean.  Depths  of. 912 

Pumps,  Direct  Acting 738 

Safety  Valve,  Adjustable  Pop 918 

Shafts 914 

Siphon.  Steam 918 

Steam  Heating 913 

Steel  Guns 913 

Temperature,  Conductivity  of 914 

Tramways  or  Steel  Railroads 915 

Troops,  Marine  Transportation  of  . . 914 

Water,  Friction  of,  in  Pipes 922 

Weirs,  Gauges  of 919 

Windmills 921 


EXPLANATIONS  OF  CHARACTERS  AND  SYMBOLS 
Used  in  Formulas , Computations , etc.,  etc. 


= Equal  to,  signifies  equality ; as  12  inches  = 1 foot,  or  8 X 8 = 16  x 4. 

-f  Plus , or  More , signifies  addition  ; as  4 + 6 -f-  5 = 15. 

— Minus , or  Less,  signifies  subtraction ; as  15  — 5 = 10. 

X Multiplied  by,  or  Into,  signifies  multiplication;  as  8 x 9=72.  a X d, 
a.d,  or  ad,  also  signify  that  a is  to  be  multiplied  by  d. 

-r - Divided  by,  signifies  division  ; as  72  9 = 8. 

: Is  to,  ::  /So  is, : 7b,  signifies  Proportion,  as  2 4 ::  8 : 16;  that  is,  as  2 zs 
/o  4,  50  is  8 to  16. 

signifies  Therefore  or  Hence,  and  v Because. 

Vinculum,  or  Bar,  signifies  that  numbers,  etc.,  over  which  it  is 
placed,  are  to  be  taken  together ; as  8 — 2 -j-  6 = 12,  or  3 x 5 + 3 = 24. 

. Decimal  point , signifies,  when  prefixed  to  a number,  that  that  number 
lias  some  power  of  10  for  its  denominator;  as  .1  is  — , .is  is  — etc. 

Difference , signifies,  when  placed  between  two  quantities,  that  their 
difference  is  to  be  taken,  it  being  unknown  which  is  greater. 

V Radical  sign,  which,  prefixed  to  any  number  or  symbol,  signifies  that 
square  root  of  that  number,  etc.,  is  required ; as  Vg,  or  Va+b.  The  degree 
of  the  root  is  indicated  by  number  placed  over  the  sign,  which  is  termed 
index  of  the  root  or  radical ; as  V , V , etc. 

> H , < L signify  Inequality , or  greater,  or  less  than,  and  are  put  between 
two  quantities ; as  a r|  b reads  a greater  than  b,  and  a L b reads  a less  than  b. 

()[]  Parentheses  and  Brackets  signify  that  all  figures,  etc.,  within  them 
are  to  be  operated  upon  as  if  they  were  only  one ; thus,  (3  + 2)  x 5 = 25 ; 
[8  — 2]  x 5 = 30.  " ’ 


± =F  signify  that  the  formula  is  to  be  adapted  to  two  distinct  cases,  as 
c ^v==c,  either  diminished  or  increased  by  Here  there  are  expressed 
two  values : first,  the  difference  between  c and  v ; second,  the  sum  of  c and  v. 
In  this  and  like  expressions,  the  upper  symbol  takes  preference  of  the  lower. 
p or  7r  is  used  to  express  ratio  of  circumference  of  a circle  to  its  diameter 
= 3.1416;  L p = .785 4, and  ^p  = .523 6. 

signify  Degrees,  Minutes,  Seconds,  and  Thirds. 

U\lnlhlpri°r  t0  a figUre  °r  figUreS’  signify> in  tienoting  dimensions,  Feet 


a'  a"  a'"  signify  a prime,  a second,  a third,  etc. 

1,  2,  added  to  or  set  inferior  to  a symbol,  reads  sub  1 or  sub  2 and  is  used 
to  designate  corresponding  values  of  the  same  element,  as  h,  hz,h2,  etc. 

b i;  ^dt®dK0r  set  *uPf‘°r  to  a number  or  symbol,  signify  that  that  mini- 

pft  bv  1“  l\S7iX’£i  'tS 

2X5- “ b “ i,y  ,h; 


22 


ALGEBRAIC  SYMBOLS  AND  FORMULAS. 


i1  etc.,  set  superior  to  a number,  signify  square^  or  cube  root,  etc.,  of  the 

number;  as  2^  signifies  square  root  of  2;  also  3,  a,  3,  3,  etc.,  set  superior 
to  a number,  signify  two  thirds  power,  etc.,  or  cube  root  of  square,  or  square 

or  cube  root  of  4th  power,  or  cube  root  of  sixth  power;  as  8*=  VW  or 

= (v/8)2. 

1.7  3.6^  etc.,  set  superior  to  a number,  signify  tenth  root  of  17th  power,  etc. 

.02  .059  set  superior  to  a number,  signify  hundredth  root  of  2d  power,  or 
thousandth  root  of  59th  power,  the  numerator  indicating  power  to  which 
quantity  is  to  be  raised,  and  denominator  indicating  root  which  is  to  be  ex- 
tracted. 

CO  signifies  Infinite,  as  - or  a quantity  greater  than  any  assignable  quan- 
tity. Thus,  - = oo  signifies  that  o is  contained  in  any  finite  quantity  an  in- 
0 „ . a a , 

finite  number  of  times : - — a%  — = 10a,  etc. 

ac  signifies  Varies  as.  Thus,  M oc  D X,V  signifies  that  mass  of  a body  in- 
creases  or  diminishes  in  same  ratio  as  product  of  its  density  and  \olume,  or 
S oc  t2,  signifies  S varies  as  t2 . 

/ signifies  Angle.  -L  Perpendicular.  A Triangle.  □ Square , as  D 
inches ; "and  g|  cube,  as  cube  inches. 

Notes.— Degrees  of  temperature  used  are  those  of  Fahrenheit. 

g is  common  expression  for  gravity  = 32. 166,  2 g = 64. 33,  v 2 g — 8.02  jeet. 

0 signifies  Dead  Flat , denoting  dimensions  or  greatest  amidship  section 
of  hull  of  a vessel. 


ALGEBRAIC  SYMBOLS  AND  FORMULAS. 

I representing  length,  h'  representing  h prime,  v representing  versed  sine,  , 

b “ breadth , c “ chord,  h ‘ h sub, 

d “ depth,  a “ area,  sin.  ‘ sine, 

h “ height , r “ radius , g gravity. 

= sum  of  length  and  breadth  divided  by  depth. 

^ = product  of  length  and  breadth  divided  by  depth. 

— = difference  of  length  and  breadth  divided  by  depth. 

d 

p b 3 = product  of  square  of  length  and  cube  of  breadth. 

VJl  = square  root  of  length  divided  by  cube  root  of  breadth. 

V b 

= square  root  of  sum  of  length  and  breadth  divided  by  depth. 

d 


<j- 


yWcoft,  _ cube  roQt  of  difference  of  h prime  and  h sub,  divided  by 
V2g 
square  root  of  2 g. 

VaA-(c-rY  = x.  Add  square  of  difference  between  the  chord  and  ra- 
dh)s  to  the  area,  and  extract  the  square  root ; the  result  will  be  equal  to  x. 

Note. -It  is  frequently  advantageous  to  begin  interpretation  of  a formula  at  its 
right  hand,  as  in  the  above  case. 


ALGEBRAIC  SYMBOLS  AND  FORMULAS. 


23 


‘V 


{x+yY 


— i=zz.  Divide  square  of  sum  of  x and  y by  square  of  y ; 

subtract  unity  from  quotient ; extract  square  root  of  result ; multiply  it  by 
length,  and  product  will  be  equal  to  z. 

2 (sin  7c0)2 

iV  . 3^.  Divide  twice  square  of  sine  of  the  angle  of  750  by  square 

i-r^sin.  75  ) 


of  sine  of  the  angle  of  75 0 added  to  unity. 
■ \ SV  2 a (\Zh  — 


] SV  2 g ( Vh-Vh  ) +2.303  c.  log.  [■  = t.  Multiply 

(bv  2 g)  l SV 2 yh—b) 

S by  the  V of  2 g,  and  this  product  by  difference  between  square  roots  of  h 
and  h prime ; add  this  to  2.303  times  common  logarithm  of  quotient  arising 
from  dividing  product  of  S into  V 2 gli  diminished  by  b by  product  of  S 
into  Vr2yh  prime  diminished  by  b,  and  multiply  this  sum  by  the  quotient 
of  2 a divided  by  square  of  product  of  S into  V"^  which  will  be  equal  to  t. 

2rt  + 3 cos.  98°  = 2 a — 3 cos.  82°  = twice  a diminished  by  three  times 
cosine  of  82°. 


SV  2 gh 


zgh—b  \_t 
2,qh — b ) 


Cosine  of  any  angle  greater  than  900  and  less  than  2700  is  always  — or  negative, 
but  is  numerically  equal  to  cosine  of  its  supplement,  i.  e.,  remainder  after  subtract- 
ing angle  from  1800. 

39.127  — .09982  cos.  2L  = /.  Assuming  L less  than  450,  as  420,  this  equation 
becomes  39.127  — .09982  cos.  (2  X 420  = 84°)  = £ ; and  also,  L greater  than  450,  as 
500,  it  becomes  39.127+  .099  82  cos.  (1800—  2 X 50°  = 80 °)  = l. 

m L — io°  N = L + 10°  S,  as  a negative  result  furnished  by  a formula  in- 
dicates a positive  result  in  an  opposite  direction. 


(B -b)  v 2 BV 
B + b 


Minus,  the  fraction  B minus  b,  times  v,  plus 


2 times  BV,  divided  by  B plus  b , is  equal  to  y. 


Sin..  1 x , tan. -I  x , cos.  _I  x , signifies  the  arc,  the  sine,  tangent  or  cosine 
of  which  is  x.  Thus,  if  x = .5,  this  is  30°,  as  30°  is  the  arc,  the  sine  of 
which  is  .5. 


(Sin.  x)  1 


sin.  x 


Raise  r to  rath  powder,  i.  e.,  multiply  r by  itself  and  this  result  by  r,  and  so 
on,  until  r appears  in  result  as  a factor,  as  many  times  as  there  are  units 
in  n.  Multiply  this  result  by  /,  diminish  this  by  Z;  divide  remainder  by  r 
raised  to  the  nth.  power,  diminished  by  r raised  to  a power  whose  exponent 
is  n diminished  by  1,  and  quotient  = or  is  value  of  S. 


Divide  l by  a and  extract  that  root  of  the  quotient,  index 


of  which  is  n diminished  by  1,  and  this  root  is  = or  value  of  r. 


Logarithm  of  a Number  is  exponent  of  the  power  to  which  a particular 
constant  quantity  must  be  raised  in  order  to  produce  that  number. 

Constant  Quantity  is  termed  the  base  of  the  system. 

Common  (or  Brigg's ) Log.  is  the  logarithm  the  base  of  which  is  10. 


Hyperbolic  Log.  is  the  logarithm  the  base  of  which  is  2.718  28. 

Com.  IjOg.  = Hyp.  log.  X -434  294. 

Hyp.  Log.  = Com.  log.  x 2.302  585  052  994,  ordinarily  2.303  or  2.3026. 


2Af  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


= a giyen  figure  or  number. 


Illustration.  — When  a number,  hyp.  log.  of  which 
is  required.  . . 

Multiply  figure  or  number  (hyp.  log.)  by  ,434294  (modulus  of  com.  log.)  _ com.  log. 

°f  Thus, Required  the  number,  hyp.  log.  of  which  = .02.  .02  X -434  294  = -°°  868  588, 

com.  log.,  and  1.0202 — number. 

L °oo.0S9=  059  x log.  of  ioo  = .o59  X 2 = .ii8;  the  number  correspond, ng  to 
log  ,18  is  ,.3122:  hence,  ,oo-°»  = 1.31*2.  That  is,  if  100  is  raised  to  59^  l»wcr, 
and  the  1000th  root  is  extracted,  the  result  will  be  1.3122. 


In  Equation , u = 3 x2  - 2 x,  u is  termed  a function  of  *.  If  it  is  desired 
to  indicate  the  fact  that  u thus  depends  for  its  value  upon  the  value  of  x 
without  expressing  exact  value  of  u in  terms  of  x,  following  notation  is 
used:  «=/(*),  u = F(x ),  or  u = <j>  (x). 

Each  of  these  notations  is  read,  a is  a function  of  x.  If  in  such  function 
of  x the  value  of  * is  assumed  to  commence  with  o and  to  increase  uniformly 
the  notation  indicating  rate  of  increase  is  dx,  and  is  read  the  differential 
of  X.” 


Differentiation,  d is  its  symbol,  and  it  is  the  process  of  ascertaining  the 
ratio  existing  between  the  rate  of  increase  or  decrease  of  a function  of  a 
variable  and  the  rate  of  increase  or  decrease  of  the  variable  itself.  If 
V = 3Z2,V  or  its  equal  3X2  is  the  function  of  *,  and  * is  the  independent 
variable  while  the  exponent  of  the  variable  or  the  primitive  exponent  is  2. 

By  the  operation  of  Calculus,  such  expressions  are  differentiated  by  ui- 
minishing  the  exponent  of  the  variable  by  unity,  multiplying  by  the  pnm- 

iti This  indicates  *¥  fla‘ionwbetween 

the  differential  of  y,  the  function  of  x and  the  differential  of  x itself. 

Assume  that  a-  increasing  at  rate  of  3 per  second  becomes  4;  that  1^*  - 4, 
QriH  j r/.  — ^ . hence  dv  — 6 X 4 X 3 — 72.  That  is,  if  x is  increasing  at 
rate  of  3 per  second,  at  the  time  that  x = 4 , the  function  itself  is  increasing 
at  rate  of  72  per  second.  # e 

To  differentiate  an  expression  of  two  or  more  terms,  it  ls  ne^^sar^.t? 
differentiate  them  separately  and  connect  the  results  with  the  signs  with 
which  the  terms  are  connected.  __  , 

Thus,  differentiating  u — 3 x2  - 2 x,  we  have  du  — d (3  x - 2 x)—Sxdx 

Assuming  a = 4 and  d x= 3,  we  have  d u — (6  X 4 “ 2)  X 3 = ^ ^is 
indicates  that  when  x = 4,  and  is  increasing  at  rate  of  3 per  second,  the  func- 
t“or  ^ - a *,  is  at4kme  instant  incasing  at  rate  of  66  per  second. 


Intearation.  Its  symbol  / was  originally  letter  S,  initial  of  sum,  the 
symbol  of  an  operation  the  reverse  of  differentiation;  and  when  the  oper- 
ation of  integration  is  to  be  performed  twice,  thrice,  or  more  times,  it 
written  f f , f f f 5 etc.  _ ..  , 

Bv  the  operation  of  Calculus,  expressions  are  integrated  by  increasing  the ; 
exponent  of  the  variable  by  unity,  dividing  by  the  new  exponent,  and  do 

ta Hence, integrating  the  differential  6 x dx, we  have  / 6 at d x = 3 x . Thi= 
result  is  the  function,  the  differential  of  which  is  6xdx^  j 

To  integrate  an  expression  of  two  or  more  terms,  it  is  necessary  , 

grate  the  terms  separately  and  connect  the  results  with  the  signs  with  which 

the  terms  are  connected.  . , r (A-jr—zdx) 

Thus  integrating  (6x— 2)  dx,wo,  have  f (6 x— 2)  dx—  f {ox d x 

is  the  function  the  differential  of  which  is  (6.x 

— 2)  dx  or  (6a;  — 2 x° ) d x. 

Note. -A  quantity  with  the  exponent  °,  as  *°  or  30,  is  equal  to  unity. 


NOTATION. 


25 


The  operation  of  summation  may  also  be  illustrated  in  use  of  the  sym- 
bol / . Assuming  x = 4,  the  former  of  the  preceding  results  becomes 
/ 6 x d x = 3 x2  = 48,  the  latter  / (6  x — 2)  d x = 3 x2  — 2 x = 40. 

Here  2;  is  assumed  to  commence  at  o and  to  continue  to  increase  by  in- 
finitely small  increments  of  d x until  it  becomes  4.  The  summation  is  the 
addition  of  all  these  values  of  x from  o to  4. 

Arithmetically . — The  first  formula  may  be  written 

6 (x  + x + x"  -f-  etc.)  dx.  If  then  x is  to  advance  from  o to  4 by  in- 
crements of  1,  we  have  6 (o  + 1 + 2 + 3 -f-  4)  x 1 = 60,  which  exceeds  48. 
H dx  is  assumed  to  be  .5,  the  result  is  54.  The  correct  result  is  obtained 
only  when  d x is  taken  infinitely  small.  By  Arithmetic  this  is  approximated, 
but  it  is  reached  by  the  operations  of  Calculus  alone. 

The  second  formula  may  be  written 

(6  \x  + x"  -f-  x"‘  -}-  etc.]  — 2 [ x° ' -f-  x°"  + x°'"  etc.] ) dx.  Assuming  x = 
4,  and  dx  = i,  we  have  (6  [1  4-  2 4-  3 4-4J  - 2 [1  4-  1 4-  1 4-  1])  x 1 = 52, 
which  exceeds  40.  If  d x=  .25,  the  result  would  be  43,  and  if  .125  it  would 
be  41.5,  ever  approaching  but  never  reaching  40,  so  long  as  a finite  value  is 
assigned  to  d x. 

A,  Delta , when  put  before  a quantity,  signifies  an  absolute  and  finite  in- 
crement of  that  quantity,  and  not  simply  the  rate  of  increase. 

2,  Sigma,  signifies  the  summation  of  finite  differences  or  quantities.  Thus, 
2y2  Ax  = (, y '2  -f-  y"2  4-  y"'2  4-  etc.)  A x.  Assume  y = 6,  y“  = 8,  y " = 4,  and 
A x the  common  increment  of  x = 5,  then  2y2  A x = (36  -j-  64  4-  16)  X 5 = 


NOTATION. 


1 = 1. 

20  = XX. 

1 000  = M,  or  CIO. 

2 = II. 

30  = XXX. 

2 000  = MM. 

3 = HI. 

40  = XL. 

5 000  = V,  or  100. 

4 = iv. 

50  = L. 

6 000  = VI. 

5 = V. 

60  = LX. 

10  000  = X,  or  CCIOO. 

6 = VI. 

70  = LXX. 

50  000  = L,  or  IOOO. 

7 = VII. 

80  = LXXX. 

60  000  = LX. 

8 = VIII. 

90  = xc. 

100  000  = C,  or  CCCIOOO. 

9 = ix. 

100  = c. 

1 000  000  = M,  or  CCCCIOOOO. 

10  = X. 

500  = D,  or  10. 

2 000  000  = MM. 

as ^C<~“^2oo^  a character  is  rePeate^}  so  many  times  is  its  value  repeated, 

A less  character  before  a greater  diminishes  its  value,  as  IV  = V — I. 

A less  character  after  a greater  increases  its  value,  as  XI  = X -f- 1. 

For  every  0 annexed  to  ID  the  sum  as  500  is  increased  10  times. 

number'is 'doubted!*  ^ ^ °f  1 “ "‘any  timeS  as  0 is  on  the  ri8ht>  the 

A bar,  thus  over  any  number,  increases  it  1000  times. 

Illustration  i.— 1880,  MDCCCLXXX.  18  560,  XVmDLX. 

rp2’  „ ^ ^°°‘  ~ 5 00  X 2 = IOOO.  IOD  =r  500  X IO  =:  5000. 

_ 1JJ  — 5000  X 2=10000.  1000  = 500  x iox  10  = 50000.  CCCI000 

— 50000  x 2 = IOOOOO. 


26  CHRONOLOGICAL  ERAS. — MEASURES  AND  WEIGHTS. 


CHRONOLOGICAL  ERAS  AND  CYCLES  FOR  1884. 

The  year  1884,  or  the  109  th  year  of  the  Independence  of  the  United  States  of  America , 
corresponds  to 

The  year  7392—93  of the  Byzantine  Era) 

“ 6597  of  the  Julian  Period; 

“ 5644-45  of  the  Jewish  Era ; 

“ 2660  of  the  Olympiads,  or  the  last  year  of  the  665th  Olympiad,  commenc- 

ing in  July  (1884),  the  era  of  the  Olympiads  being  placed  at  775-5 
years  before  Christ,  or  near  the  beginning  of  July  of  the  3938th 
year  of  the  Julian  Period; 

“ 2637  since  the  foundation  of  Rome,  according  to  Varro; 

n 2196  of  the  Grecian  Era,  or  the  Era  of  the  Seleucidae; 

“ 1600  of  the  Era  of  Diocletian. 

The  year  1301  of  the  Mohammedan  Era,  or  the  Era  of  the  Hegira,  begins  on  the 
7th  of  February,  1884.  ' , ^ 

The  first  day  of  January  of  the  year  1884  is  the  2,409,178th  day  since  the  com- 
mencement of  the  J ulian  Period. 

Dominical  Letters F,  E I Lunar  Cycle  or  Golden  Number 4 

Epact 3 I Solar  Cycle 17 

Homan  Indiction  was  a period  of  15  years,  in  use  by  the  Romans.  The  precise 
time  of  its  adoption  is  not  known  beyond  the  fact  that  the  year  313  A-D.  was  a first 
year  of  a Cycle  of  Indiction. 

Julian  Period  is  a cycle  of  7980  years,  product  of  the  Lunar  and  Solar  Cycles  and 
the  Indiction  (19  X 28  X 15),  and  it  commences  at  4714  years  B.C. 

6^3  (given  year  — 1800)  = year  of  Julian  Period,  extending  to  3267. 

Note. — if  year  of  Julian  Period  is  divided  by  19,  28, 15,  or  32,  the  remainders  will 
respectively  give  the  Lunar  and  Solar  Cycles , the  Indiction , and  the  Year  qf  the 
Dionysian. 


MEASURES  OF  LENGTH. 

Standard  of  measure  is  a brass  scale  82  inches  in  length,  and  the 
yard  is  measured  between  the  27th  and  63d  inches  of  it,  which,  at  tem- 
perature of  62°,  is  standard  yard. 


Lineal 

12  inches  = 1 foot. 

3 feet  = 1 yard. 

5.5  yards  = 1 rod. 

40  rods  = 1 furlong. 

8 furlongs  = 1 mile. 

Inch  is  sometimes  divided  into  3 barleycorns , or  12  lines. 
A hair’s  breadth  is  .02083  (48th  part)  of  an  inch. 

1 yard  = .000  568,  and  1 inch  = .000015  8 of  a mile. 


Inches.  Feet.  Yards.  Rods.  Furl. 

36=  3- 

198=  16.5=  5-5- 

7 92O  = 660  = 220  = 40. 

63  360  = 5 280  = I 760  = 320  — 8. 


G-unter’s  Chain. 


7.92  inches  = 1 link. 


100  links  = 1 chain,  4 rods,  or  22  yards. 
80  chains  — 1 mile. 


Ropes 
1 fathom  = 6 feet. 


and  Cables. 

I 1 cable’s  length  =120  fathoms. 


Greographical  and  Nautical. 

1 degree,  assuming  the  Equatorial  radius  at  6974532.34  yards,  as  given 
by  Bessel,  = 69.043  statute  miles  = 364  556  feet. 

1 mile  = 2028.81  yards  or  6086.44  feet. 

1 league  = 3 nautical  miles. 


MEASURES  AND  WEIGHTS. 


27 


Log  Lines. 

Estimating  a mile  at  6086.43  feet,  and  using  a 30"  glass, 

1 knot  = 50  feet  8.64  inches.  | 1 fathom  = 5 feet  .864  inches. 

If  a 28"  glass  is  used,  and  8 divisions,  then 

1 knot  = 47  feet  4 inches.  | 1 fathom  = 5 feet  1 1 inches. 

The  line  should  be  about  150  fathoms  long,  having  10  fathoms  between  chip  and 
first  knot  for  stray  line. 

Note.  — This  estimate  of  a mile  or  knot  is  that  of  U.  S.  Coast  Survey,  assuming 
equatorial  radius  of  Earth  to  be  6974532.34  yards  and  a meter  to  be  39.36850535 
inches  of  the  Troughton  scale  at  62°. 

Cloth.. 

1 nail  = 2.25  inches.  | 1 quarter  = 4 nails.  | 5 quarters  = 1 ell. 
Pen  cl  11  lmxL . 

6 points  = 1 line.  J 12  lines  = 1 inch. 
Shoemakers’. 

No.  1 is  4.125  inches,  and  every  succeeding  number  is  .333  of  an  inch. 
There  are  28  numbers  or  divisions,  in  two  series  or  numbers— viz.,  from  1 
to  13,  and  1 to  15. 

IVI  iscellaneous. 

12  lines  or  72  points  = 1 inch.  I 1 hand  = 4 inches. 

1 palm  = 3 inches.  | 1 span  = 9 inches. 

1 cubit  = 18  inches. 

■Vernier  Scale. 

V ernier  Scale  is  divided  into  10  equal  parts ; so  that  it  divides  a scale 

of  ioths  into  iooths  when  two  lines  of  the  two  scales  meet. 


Metric,  by  _A.ct  of  Congress  of  Jnly  28,  1866. 
Unit  of  Measurement  is  the  Meter,  which  by  this  Act  is  declared  to  be  39.37  ins. 


Denominations. 

Meters. 

Inches. 

Feet. 

Yards. 

Miles. 

Millimeter 

. IOO 

•°394 

•3937 

3-937 

39-37 

393-7 

Centimeter 

— 

Decimeter 

i° 

.328083 
3.28083 
32.80833 
328.083  33 
3280.83333 

— 

Meter 

' 

1.093  61 
10.936  11 
109.361  11 
1093.611  11 

— 

Dekameter 

IO 

— 

Hektameter 

IOO 

— 

Kilometer 

IOOO* 

— 

Myriameter 

I OOOO* 

.62137 

6.2137 

In  Metric  system,  values  of  the  base  of  each  measure — viz.,  Meter  Liter  Stere 
Are,  and  Gramme— are  decreased  or  increased  by  following  prefix.  Thus,  ’ ’ 


Milli,  1000th  part  or  .001. 
Centi,  100th  “ ,01. 


Deci,  10th  part  or  .1. 

I Deka,  10  times  value. 
Myria,  10000  times  value. 


Hekto,  100  times  value. 
Kilo,  1000  “ 


Note  —The  Meter,  as  adopted  by  England,  France,  Belgium  Prussia  and  Russia 
of  WortT  ^inad  ?aPo4; R Clarke>  RE.,  F.R.S.,  1866,  Which  at  32° in  terms 

^,IlPPienf1uSta'xrdar^  at  6z°  18  39-37° 432  inches  or  1.09362311  yards  its  leo-al 

France6111  ^ MetnC  Act  of  1864  being  39-37°8  inches,  the  same  as  adopted  °i  11 


Coro?1  wa?  comparison  and  the  one  formerly  adopted  by  the  U.  S.  Ordnance 
T 39‘ 370  797 J mches ’ or  3-280899  76  feet,  and  the  one  adopted  bv  the 
U.  S.  Coast  Survey,  as  above  noted,  is  = 39. 368  505  35  inches.  P tne 


28  MEASURES  AND  WEIGHTS. 

Equivalent  Values  in  Metric  Denominations  of  UJ.  S. 


Denominations. 

Value  in  Meters.  1 

1 Denominations. 

Values  in  Meters. 

.0254 
. 304  800  6 
.9144018  [ 

Rod 

5.029  209  9 

XT’ 

Furlong 

201. 168  396 

Yard 

1 Mile 

1609.347  168 

i Chain = 20  meters . 

1 Furlong  . . . = 200  . “ 

5 Furlongs  . . . = 1 kilometer . 


Approximate  Equivalents  of  Old  and  Metric  XJ.  S. 
Measures  of  Eengtli. 

i Kilometer  . . . . = .625  mile. 

1 Mile  = 1.6  kilometers. 

1 Pole  or  Perch  . = 5 meters. 

x Root =3  decimeters  or  30  centimeters. 

1 Metre =3.2808 33  feet  = 3 feet  3 ***•  and  3 eighths. 

n Meters =12  yards.  | 1 Decimeter ...  =4  inches. 

1 Millimeter  . . = 1 thirty-second  of  an  inch. 

To  Convert  Meters  into  Inches.— Multiply  by  40;  and  to  Convert  Inches 
into  Meters. — Divide  by  40. 

Approximate  rule  for  Converting  Meters  or  parts,  into  Yards—  Add  one 
eleventh  or  .0909. 

Inches  Decimally  — Millimeters. 

Milli- 


Inches. 

Milli- 

meters. 

Inches. 

Milli-  I 
meters. 

Inches. 

Milli-  I 

meters.  ] 

Inches.  1 

Milli- 

meters. 

.01 

•25 

.2 

5.08 

.48 

12.2  ' 

.76 

l9.3 

.02 

• 51 

.22 

5-59 

. -5 

12.7 

•78 

19.8 

.03 

.76 

.24 

6. 1 

•52 

13.2 

.8 

20.3 

.04 

1.02 

.26 

6.6 

•54 

13-7 

.82 

20.8 

•°5 

1.27 

.28 

7-11 

•56 

14.2 

.84 

21.3 

.06 

1.52 

•3 

7.62 

•58 

14.7 

.86 

21.8 

.07 

1.78 

•32 

8-13 

.6 

15.2 

.88 

22.4 

.08 

2.03 

•34 

8.64 

.62 

15-7 

•9 

22.9 

.09 

2.29 

•36 

9.14 

.64 

16.3 

.92 

23-4 

. 1 

2-54 

.38 

9-65  | 

.66 

16.8 

.94 

23-9 

.12 

3-°5 

•4 

10.2  i 

.68 

17-3 

.96 

24.4 

.14 

3-56 

.42 

10.7 

•7 

17.8 

.98 

24.9 

.16 

4.06 

•44 

11. 2 

.72 

18.3 

1. 

25-4 

.18 

4-57 

.46 

n-7  1 

•74 

18.8 

127 

152.4 

177.8 
203.2 
228.6 
254 
2794 

304.8 
foot. 


Inches  in  Fractions  = Millimeters. 


Eighths. 

Six- 

teenths. 

Thirty- 

SecOnds. 

Milli- 

meters. 

Eighths. 

Six- 

teenths. 

Thirty- 

seconds. 

Milli- 

meters. 

J3 

w 

Six- 

teenths. 

t e 
.-  0 
JS  « 
E-1  as 

Milli- 

meters. 

J Eighths. 

ST| 

' 

1 

•79 

9 

7.!4 

17 

13- 5 

13 

1 



i-59 

5 

— 

7-94 

9 

14-3 

3 

2.38 

11 

8-73 

*9 

15- 1 

1 



3*7 

3 

— 

— 

9-52 

5 

— 

— 

!5.9 

7 

5 

3-97 

13 

10.32 

21 

16.7 

i5 

3 

4.76 

7 

— 

11. 11 

11 

— 

I7-5 

2 

7 

5- 56 

6- 35 

4 

— 

15 

11. 91 
12.7 

6 

- 

23 

18.3 

*9 

8 

- 

H S 


Bv  means  of  preceding  tables  equivalent  values  ot  incnes  ana  mumneie  s, 
equivalent  values  of  inches  in  centimeters,  decimeters,  and  meters,  may  be 
ascertained  by  altering  position  of  decimal  point. 

Illustration  —Take  1 millimeter,  and  remove  decimal  point  successively  by  one 
figure  to  the  right;  the  values  of  a centimeter,  decimeter,  and  meter  become 


1 millimeter. . . . 

1 centimeter. . . . 


Ins. 

I 1 decimeter 3.94 

1 meter 39-4 


.32  inch  = 8.13  millimeters. 
3.2  inches  = 81. 3 


MEASURES  AND  WEIGHTS. 


29 


MEASURES  OF  SURFACE. 

144  square  inches  = 1 square  foot.  | 9 square  feet  = 1 square  yard. 
Architect's  Measure , 100  square  feet  = 1 square. 

Land. 


30.25  square  yards  = 
40  square  rods  = 

4 square  roods  ) 

10  square  chains  J 

640  acres  = 


1 square  rod. 

1 square  rood. 

1 acre. 

1 square  mile. 


Yards.  Rods.  Roods. 

1210. 

4840  = l6o. 

3 097  600  = 102  400  = 2560. 


208.710326  feet,  69.570109  yards  square,  or  220  by  198  feet  square  = 1 Acre. 


Paper. 


24  sheets  = 1 quire.  | 20  quires  = 1 ream.  | 21.5  quires  = 1 printer’s  ream. 
2 reams  = 1 bundle.  | 5 bundles  = 1 bale. 


Drawing. 


Cap  

, . 13  X 16  inches. 

Demy 

Medium  . . . . , 

u • 

Royal 

ll 

Super-royal  . . 

. 19  x 27 

n 

Imperial . . . . . 

u 

Elephant  . . . . 

Peerless 

'll 

Columbier  .... 

23 

X 

34 

inches. 

Atlas  ........ 

26 

X 

34 

li 

Theorem 

28 

X 

34 

u 

Doub.  Elephant, 

27 

X 

40 

11 

Antiquarian  . . . 

31 

X 

53 

u 

Emperor 

40 

X 

60 

u 

Uncle  Sam  .... 

48 

X 

120 

ll 

18  x 52  inches. 

Tracing. 


Double  Crown 20  X 30  inches. 

Double  D.  Crown  . . 30  X 40  “ 

Double  D.  D.  Crown,  40  X 60 


Mounted  on  cloth,  38  ins.  in  width. 


Grand  Royal 18  X 24  inches. 

Grand  Aigle 27  X 40  “ 

Vellum  Writing,  18  to  28  ins.  in  -width. 


Miscellaneous. 


1 sheet  = 4 pages. 
1 quarto  =8  “ 

1 octavo  = 16  “ 


1 duodecimo  = 24  pages. 
1 eighteenmo  = 36  “ 

1 bundle  = 2 reams. 


1 piece  wall-paper,  20  ins.  by  12  yards. 

1 “ “ “ French,  4.5  sq.  yards. 

Roll  of  Parchment  = 60  sheets. 


Copying. 

100  Words  = 1 Folio. 

Metric,  by  ^Act  of  Congress  of  Jnly  28,  1866. 
Unit  of  Surface  is  Are  or  Square  Delcameter. 

A square  meter  (39.37?)  = 1549.9969  sq.  ins.,  but  by  this  Act  is  declared  to  be 
1550  sq.  ins. 


Denominations. 

Sq.  Meters. 

Sq.  Inches. 

Sq.  Feet. 

Sq.  Yards. 

Acres. 

Centimeter 

.0001 

.01 

1. 

100. 

JO  000. 

•155 

15-50 

1550. 

.107638 
10.763  888 
1076.388  88 

1.196 

119.6 
II  960. 

.02471 

2.471, 

Decimeter 

Centare  or  ) 

Square  Meter  j 

Are 

Hectare 

30 


measures  and  weights. 


Denominations. 


Sq.  Inch . 
Foot 
Yard 
Rod 


Sq.  Meters'.  1 1 Denominations. 


.00064516  Sq.  Chain  . 
.09290323  “ Rood.. 

.83612907  ^ Acre.. 

25.292904  II  u Mile..!. 


Sq.  Meters. 

Sq.  Hectares. 

Sq.  Ares. 

404.686  47 

1011.716  175 

4046.864699 

V f — 

. 404  686 
258.999  34 

4.046  865 
10. 117  162 
40.468647 

25  899-934  °74 

Approximate  ^ ^ 

5. 5 square  centimeters  %**  inch.^  j « ? 


measures  of  volume. 

C ,1  „a»nn  measures  231  cube  ins.,  and  contains  8.338  882  2 
Standard  g Xroy  grains  of  distilled  water,  at  temper- 

Standard  bushel  is  the  Winchester,  which  contains  215042 

“ -r"?  "Set'S 

,„X“”hS  s,r.h« »».  "»« <■>  - -»  ^ “»"■ 

equal  2747.715  cube  ins.  for  a true  cone. 

\ struck  bushel  contains  1.24445  cube  feet 
Liqnid. 


4 gills  = 1 pint. 

2 pints  = 1 qnart. 
4 quarts  = i gallon. 


2 pints  = 1 quart. 
4 quarts  = i gallon. 
2 gallons  = 1 peck. 

4 pecks  = 1 bushel. 


Dry. 


Cube  Ins. 

28.875 

57-75 

231. 

Cube  Ins. 
67.2006 
268.8025 
537-6°5 
2150.42 


Gills.  Pints. 

8. 

32  = 8. 


Pints.  Quarts.  Galls. 

8. 

16  = 8. 

64  — 32  = 8. 


Cnloe. 


Inches. 

46656 


1728  cube  inches  = i foot.  I 

27  cube  feet  = i yard.  | . . 

Note-A  cube  foot  contains  2200  cylindrical  inches,  or  3300  spherical  inches. 

Din  id. 


60  minims  = i dram. 

8 drams  = i ounce. 
16  ounces  = i pint. 

8 pints  = 1 gallon. 


Minims.  Drams.  Ounces. 

480. 

•7  680  = 128. 

6l  24O  = 1024  = 128. 


ISTantical. 

j. =r  ii;  cube  feet. 

1 ton  displacement  in  salt  water __  ^ « u 

1 “ registered  internal  capacity 

Dimensions  of  a Barrel. 

Diameter  of  head,  ,7  ins. ; hung,  ,9  ins. ; length,  28  ins. ; volume,  7689  cube  ina 
= 3.5756  bushels. 


MEASURES  AND  WEIGHTS. 


31 


Miscellaneous. 

I?ub,e*oot 74805  gallons. 

1 bushel  9.309 18  gallons. 

1 chaldron  _ 36  bushels,  or 57-244  cube  feet. 

I “IdlofiW”0d: c»be  feet. 

24.75  cube  feet. 

1 load  hay  or  straw  = 36  trusses. 


i quarter  = 

8 bushels. 

1 Barrel 

Galls. 

1 Tierce 

Butt  of  Sherry 

Pipe  of  Port 

Pipe  of  Teneriff'e 

•••35X50. .. 

. 108 

• 115 

Butt  of  Malaga 

•••33X53*  •• 

• 105 

Puncheon  of  Scotch  Whisky. . no  to  %o 
Puncheon  of  Brandy  34X52.  .no  to  120 

1 uncheon  of  Rum t0  no 

Hogshead  of  Brandy  28X40..  55  to  60 

Pipe  of  Madeira Q2 

Hogshead  of  Claret 46 

oft  ®p?St,ior°Pun\hf^QUarter  cask  iS  0De  f0Urth’  and  an  0ctave  is  eighth 

Tt  Act  of  Congress  of  July  28,  1S66 

brut  or  Base  of  Measurement  is  a cube  Decimeter  or  Liter,  which  is  declared  to  be 
01.022  cube  ms. 


Cube  Measures. 


Denominations. 

Values. 

Cube  Inches. 

Cube  Centimeter 

“ Decimeter 

.001  cube  milliliter 
1 cube  liter  . 

.061  022 
61.022 

“ Meter 

Kiloliter  or  stere. . 

Cube  Feet.  I Cube  Yards. 


Denominations. 


Milliliter 

Centiliter 

Deciliter 

Liter 

Dekaliter 

Hektoliter 

Kiloliter  ) 
or  Stere  j * * * 


Dry  IVTeasnres, 

| Cube  Ins.  Quarts. 


cube  centimeter. 
1 “ decimeter 


meter. 


* Or  .227  gallon. 


.061 

.6102 

6.1022 

61.022 


.908* 

9.08 


•035  3i3  657  — 

35-3I3  657  I 1.308 


Bushels.  [Cube  Yards. 


•1135 
c*  *35 
[*35 


•283  75 
2-8375! 

28.375 

+ 3-531  365  7 ^be  feet. 


.1308 

1.308 


to  cc,  is  used  instead  of 

in.  Metric  Denominations  of  TJ.  S. 
Dry  Measures. 


Equivalent  Values 


Denominations. 


Inch . . . 
Pint... 
Quart . . 
Gallon  . 
Peck. . . 
Bushel. 


Centiliters. 


Deciliters. 


.0881 

•3524 


.110  125 
.4405 
.881 
3-524 


Denominations. 


Milliliter . . . 
Centiliter. . . 

Deciliter 

Liter 

Dekaliter. . . 
Hektoliter. . 
Kiloliter  ) 
or  Stere  j 


Liters. 


Dicfuid.  3VIeasu.res. 


Liters. 


1.10125 

4-405 

8.81 

35-24 


Dekaliters. 


11. 0125 
44-05 


352.4 


.001 

.01 


100 

1000 


•27 

2-7 

27 


Ounces. 


•338 

3-38 

33-8 


•21134 

2.1134 

21.134 


Quarts. 


1.0567 

10.567 


Gallons. 


.26417 

2.6417 

26.417 

264.17 


32 


measures  and  weights. 


Approximate  E^“™u*^S0f  Volume 


of  Old  and  Metric  TJ.  S. 


; liters. 


i cube  foot . 


. = 28. 3 liters. 


’ cube  meter.... ,.^.33^^ 

\ u kiloliter  = 2240  lbs.  nearly  of  water. 


measures  OE  WEIGHT. 

Standard 

tilled  water  weighed  in  air,  at  v39-  j > 

A cube  inch  of  such  water  weighs  252.693?  g™“S. 

^voircTupois. 


16  drams  = i ounce. 
16  ounces  = i pound. 
1 12  pounds  = 1 cwt. 
20  cwt.  = 1 t°n* 


Drams.  Ounces.  Pounds. 

256. 

28672=  I792- 
573  440  = 35  840  — 2240. 


\ dram  = M 3-343  75  grains  Troy,  or  53-5  *"»■* 
i stone  =14  pounds. 

Troy. 

I Grains.  Dwt. 


24  grains  = i dwt. 

20  dwt.  = 1 ounce. 

12  ounces  = i pound. 

7000  Troy  grains 
437.5  “ “ 

27.343  75  Troy  grains 
!75  Troy  pounds 
175  u ounces 
•r  “ ounce 
pound 


480. 

5760  = 240, 

— 1 lb.  avoirdupois. 

a 

— I OZ. 

= i dram  u 

= 144  ibs.  ;; 

= 192  OZ. 

= 48ogrs.  “ 

.822  857  lb. 


?.  Troy . 


20  grains 
3 scruples 
8 drams 
12  ounces 
45  drops 


Grains.  Scruples.  Drams. 

60. 

480  = 24. 

3760  = 288  — 96. 


avoirdupois  pound  = t.215  270 

.A.potliecaries 
= 1 scruple. 

= 1 dram.  . 

— 1 ounce. 

= 1 pound. 

45  arops  = 1 teaspoonful  or  a fluid  dram. 

The  SulTouhct^ata  are  the  same  as  in  Troy  weight. 

Diamond. 

. __  I 4 grains  = 3.2  grains. 

1 gram- i6 parts  j carat  =4  grams. 

16  parts  = .0  gram.  \ 

15.5  carats  = i Troy  ounce. 

Dead. 

Shlutdfo}S.58  tots  feet  in  width  and  from  30  to  35  feet  to  length. 
G-rain. 

Standard  Weights  per  Bushel.  ^ 

Lbs.  I n * ^nd  c8  I Rve  ...  56 ' Oats 32  1 Barley ....  48 

Wheat....  60  1 Corn 56  and  58  l nye 5 


MEASURES  AND  WEIGHTS, 


33 


Miscellaneous. 

COAL. 

Anthracite i cube  foot  = 1.75  broken. 

“ 50  to  55  lbs.  per  cube  foot. 

44  41  to  45  cube  feet  = 1 ton  broken. 

Bituminous 70  to  78  lbs.  per  heaped  bushel. 

a 40  to  50  lbs.  per  cube  foot. 

“ Cumberland 53  “ “ “ “ 

44  Cannel 50.3  lbs.  per  cube  foot. 

44  Welsh 43  cube  feet  = 1 ton. 

44  Lancashire 44  “ “ = 1 “ 

44  Newcastle 45  “ “ =1  “ 

44  Scotch 43  “ 44  = 1 44 

44  R.  N.  allowance 48  “ “ = 1 “ 

Charcoal,  hardwood  18.5  lbs.  per  cube  foot. 

44  pine  wood  . 18  44  44  “ “ 

WOOD. 

Virginia  pine  . . 2700  lbs.  = 1 cord.  | Southern  pine  . 3300  lbs.  = 1 cord. 

EARTH. 

River  sand  . . 21  cube  feet  ~ 1 ton.  I Marl  or  Clay,  28  cube  feet  — 1 ton. 
Coarse  gravel,  23  44  44  = 1 44  | Mold 33“  44  = 1 44 

Metric,  L>y  J^ct  of  Congress  of  July  SB,  1866. 


Unit  of  Weight  is  the  Gram,  which  is  weight  of  one  cube  centimeter  of  pure  water 
weighed  in  vacuo  at  temperature  of  40  C.,  or  39. 20  F.,  which  is  about  its  tem. 
perature  of  maximum  density  = 15.432  grains. 


Denominations. 

Values. 

Grains.  | 

Ounces.  | 

Lbs. 

Ton. 

Milligram 

1 cube  millimeter 

_ 

— 

— 

Centigram 

10  44  4 4 

•154  32 

— 

— 

— 

Decigram 

.1  44  centimeter 

z-543  2 

— : 

— 

— 

Gram 

1 4 4 4 4 

15-432 

•035  27 

— 

— 

Dekagram 

Hektogram 

10  4 4 4 4 

1 deciliter 



•3527 

3-527 

.220  46 

Kilogram  or  Kilo . . 

x liter 

— 

35-27 

2.2046 

— 

Myriagram 

10  44  

— 

— 

22.046 

— 

Quintal  

1 hektoliter 

— 

— 

220.46 

.098  419 

Millier  or  Tonneau. 

1 cube  meter 

— 

— 

2204.6 

.984  196 

Kilogram  — 2.679  *7  Troy , or  2 lbs.  8 oz.  3 dwts.  .3072  grain. 


Equivalent  Values  in  NEetric  Denominations  of  XT.  S. 


Denominations. 

Grams. 

Dekagrams. 

Denominations.  1 

Grams. 

Kilograms. 

Grain 

.0648 

1.296 

1-5552 

1.77187 

3.888 

Ounce 

28.3502 

31.1042 

453.6028 

373.2504 

.028  35 
.031  1 
•4536 
•373-25 
1016.057  28 

Scruple 

Pennyweight 

Drachm 

44  (Apoth.) 

17.7187 

38.88 

44  Troy 

Pound 

4 4 Troy 

Ton 

Approximate  Equivalents  of  Old.  and  New  IT.  S. 
Measures  of  Weight. 

The  ton  and  the  gram  are  at  nearly  equal  distances  above  and  below  the 
kilogram.  Thus, 

1 ton  . . . . — 1 016057.28  grams.  | 1 kilogram = 1000  grams. 

1 gram  is  nearly  15.5  grains  (about  .5  per  cent.  less). 

1 kilogram  about  2.2  pounds  avoirdupois  (about  .25  per  cent.  more). 

1000  kilograms,  or  a metric  ton,  nearly  1 Engl,  ton  (about  1.5  per  cent.  less). 


34 


MEASURES  AND  WEIGHTS. 


Electrical.  ( British  Association.') 

T?^«iQtanoe  —Unit  of  resistance  is  termed  an  Ohm , which  represents  resist- 
ant of  a column ‘of  mercury  of  i sq.  millimeter  in  section  and  1.0486  meters  m 
length,  at  temp.  o°  C.  Equivalent  to  resistance  of  a wire  4 millimeters  in  diameter 
1 and  100  meters  in  length. 

1 000  000  Microhms . . . . . . . . .—  absolute  electro-magnetic  units. 

T ohm  10000000  “ “ ‘ 

1 ooo  000  Ohms  .* : = i Megohm  or  id*  “ 

Electro-motive  Eorce.-Unit  of  tension  or  difference  of  potentials  is 

1 .e<xo  000^ Microvolts. = I Volt  or . I of  an  absolute  electro-magnetic  unit. 

i Volt = 100000  “ 

1 Megavolt = 1000000  'Volts. 

r.irrent  —Unit  of  current  is  equal  to  1 Weber  per  second,  or  the  current  in  a 
ciroiX  has  an  electro -motive  force  of  one  Volt  and  a resistance  of  an  Ohm. 

Volume. — Unit  of  volume  is  termed  an  Ampere , and  represents  that  volume 
of  electricity  which  flows  through  a circuit  having  an  electro-motive  force * of  one  Fo« 
and  a resistance  of  one  Ohm  in  a second,  or  it  represents  a Volt  diminished  by  an  Ohm. 

1 000000  Microvolts  or  100  absolute  units  of  volume = 1 

1 000000  Amperes 1 ® 

Capacity.— Unit  of  capacity  is  termed  a Farad. 

1 000  000  Microfarads  or  iooooqoo  absolute  units  of  capacity ==  1 ^™farad 

Heat.— Unit  of  heat  is  quantity  required  to  raise  one  gram  of  water  to  i°  C. 
of  temperature. 

Weiglits  of  Grrain  and.  Boots. 

Following  weights  have  been  fixed  by  statute  in  many  of  the  States;  and 
these  weights  govern  in  buying  and  selling,  unless  a specific  agreement  to 
the  contrary  has  been  made. 

Bonnds  in  a Bnsliel. 


ARTICLES. 


Barley 

Beans 

Blue  Grass  Seed. 

Buckwheat 

Castor  Beans 

Clover  Seed 

Dried  Apples — 
Dried  Peaches  . . 

Flaxseed 

Hemp  Seed 

Corn 

Corn  in  ear 

Corn  Meal 

Coal 

Oats 

Onions  

Pease 

Potatoes 

Rye 

Rye  Meal 

Salt 

Timothy  Seed. . . 

Wheat 

Wheat  Bran 


California. 

Connecticut. 

Delaware. 

Illinois. 

Indiana. 

Iowa. 

| Kentucky. 

| Louisiana. 

1 Maine. 

j Massachusetts. 

I 

la 

§ 

| Minnesota. 
1 IVTioormri 

5° 

— 4« 

3 48 

48 

48 

32 

46 

48 

48  4 

— 6. 

D 60 

60 

60 

— 

— 

— 

— 6 

— I. 

4 14 

14 

14 

— 

— 

— 

— 

— 1 

4° 

45 

— 4' 

0 50 

52 

52 

— 

— 

46 

42 

42  5 

— 4( 

6 46 

46 

— 

— 

— 

— 

*— 

— 4 



— 6 

0 60 

60 

60 

— 

— 

— 

60 

60  6 





— 2 

4 ! 25 

24 

— 

— 

— 

— 

28 

28  2 





— 3 

3133 

33 

— 

— 

— 

28 

28  3 

— 

— 

— 5 

6 56 

56 

56 

— 

— 

— 



— 

— 4 

4 44 

44 

44 

56 

56 

56  5 

52 

56 

56  5 

,2  56 

56 

50 

— 

— 7 

0 68 

68 

1 

— 

— 

— 4 

£ 50 

— 

5o 

50 

50 

— 

— 

— 8 

5o  70 

32 

28 

l2  32 

35 

33* 

• 32 

3C 

» 30 

32 

32 : 

— 

57  48 

57 

57 

— 

■ 52 





I 

— 

, 

— 

’ 

60 

— ( 

So  60 

1 60 

1 60 

6c 

> — 

54 

56 

54  56 

' 56 

’ 56 

60 

■ 56 

» 5^ 

» 56 

. 

• — 

• 5< 

> 5C 

» — 

— 

. — 

— 5C 

> 5C 

> 5o 

45  45 

i 45 

i 45 

— 

* — 

6c 

> 5^ 

i 60  1 

60  6c 

> 6c 

> 60 

6c 

i — 

-6c 

) 6c 

> 60 

-1- 

■ — 

20|  — 

- 2C 

> 20 

1 

3° 


60 


48 


60 


47 


60 


56 


34 


60 


46 


48 


50 


60 


56 


60 


28 


50 


60 


56 


60 


MEASUKES  AND  WEIGHTS. 


35 


XV  eight  of  Men.  and.  Women. 

Average  weight  of  20000  men  and  women,  weighed  in  Boston,  1864,  was 
— men,  141.5  lbs. ; women,  124.5  lbs.  Average  of  men,  women,  and  chil- 
dren, 105.5  lbs. 


XVeight  of  Horses.— (XT.  S.) 
Weight  of  horses  ranges  from  800  to  1200  lbs. 


WEIGHT  OF  CATTLE. 


To  Compute  Dressed  XVeight  of  Cattle. 

Rule. — Measure  as  follows  in  feet: 

1.  Girth  close  behind  shoulders,  that  is,  over  crop  and  under  plate, 
immediately  behind  elbow. 

2.  Length  from  point  between  neck  and  body,  or  vertically  above 
junction  of  cervical  and  dorsal  processes  of  spine,  along  back  to  bone  at 
tail,  and  in  a vertical  line  with  rump. 


Then  multiply  square  of  girth  in  feet  by  length,  and  multiply  product 
by  factors  in  following  table,  and  quotient  will  give  dressed  weight  of 
quarters. 


Condition. 

Heifer,  Steer, 
or  Bullock. 

Bull. 

Condition. 

Heifer,  Steer, 
or  Bullock. 

Bull. 

Half  fat 

3*15 

3-36 

Very  prime  fat . . . 

3-64 

3.85 

Moderate  fat 

Prime  fat 

3-36 

3-5 

3-5 

3-64 

Extra  fat 

3-78 

4.06 

Illustration.— Girth  of  a prime  fat  bullock  is  7 feet  2 ins.,  and  length  measured 
as  above  4 feet  5 ins. 


7'  2"  = 7.i7,  and  7.172  51.4,  which  x 4'  5"  and  by  3.5  — 794-5  lbs.  Exact 

weight  was  799  lbs. 

Note. — 1.  Quarters  of  a beef  exceed  by  a little,  half  weight  of  living  animal. 

2.  Hide  weighs  about  eighteenth  part,  and  tallow  twelfth  part  of  animal. 


Comparative  'Weigh. ts  of  Live  Beeves  and.  of  Beef. 


Lbs. 

Per  cent. 

Lbs. 

Bullocks 

2800 

72  to  78 

Rnl  looks 

Heifers 

2600 

Heifers 

I55° 

Bullocks 

2600 

| 70  to  76 

Ru  Hocks 

I55° 

1260 

1200 

Heifers 

2400 

2400 

2100 

| 66  to  70 

Heifers 

Bullocks 

Rnl locks  . 

Heifers 

| 64  to  68 

Heifers 

Bullocks 

2100 

Rnl  locks 

1050 

980 

950 

Heifers 

1800 

63  to  66 

Heifers 

Per  cent. 
61  to  64 
58  to  61 
57  to  58 
50  to  56 


"Weight  of  Offal  in  a Beef  and.  Sheep. 


BEEF. 

Lbs. 

Hide  and  Hair 56  to  98 

Tallow 42  “ 140 

Head  and  Tongue  . 28  “ 40 

Feet.  " y 


SHEEP. 

Lbs. 

8 to  16* 


BEEF. 

Lbs. 


SHEEP. 

Lbs. 


Kidneys, Heart,)  . , , . 

Liver,  etc....’! 31  to  62  6 to. 

Stomach,  Entrails, etc.,  126  u 196 
Blood 42  u 56 


32 


36  MEASURES,  WEIGHTS,  PRESSURES,  ETC. 

To  Compute  Equivalents  of  Old.  and.  New  TJ . S.  and 
of  Metric  Denominations. 

By  Act  of  Congress , July  28,  1866. 

Rule.  — Divide  fourth  term  by  second,  multiply  quotient  by  first 
term,  and  divide  product  by  third  term. 

Or,  Ascertain  relative  ratio  of  first  and  second  terms,  and  multiply 
result  by  ratio  of  third  and  fourth  terms. 

Note.  — When  result  is  required  in  French  or  other  Metric  denominations  than 
those  of  U.S.,  use  exact  denominations,  as,  61.025  387  for  61.022,  39.370432  for  39.37? 
etc. 

Example  1.— If  one  gallon  (1st),  per  sq.  foot,  yard,  acre,  etc.  (2d) ; how  many  liters, 
(3d),  per  sq.  foot,  yard,  acre,  etc.  (4th)  ? 


_L  x 231  -4-  61.022 .=3.7851  liters  or  3.7848  litres. 

1 

Or,  — = 1.604,  and  =2.3598;  hence,  1.604  X 2.3598  = 3.7851  liters- 

5 144  61.022 

Note.— In  computing  ratios,  first  term  is  to  be  dividedby  second,  and  fourthby  third. 
Example  2.— If  one  ton  per  cube  foot,  how  many  kilograms  per  cube  decimeter? 

6l,?1 2 * * *?-X  2240-4-2.2046  = 35.881  liters , or  35.882  litres. 

1728 


MEASURES. 

By  Act  of  Congress  of  U.  S.  By  Metric  Computation. 

1 Liter  per  sq.  foot,  etc.  = .2642  Gallon  per  sq.foot , or  .264  2 gallon. 

1 Liter  per  sq.  meter  . = .0245  Gallon  per  sq.foot , or  .024  5 gallon. 

1 Gallon  per  sq.  foot  . = 40.746  Liters  per  sq.  meter , or  40.745  4 litres. 

1 1 Sq  foot  per  acre  . . . = .2296  Sq.  meters  per  hectare , or  2.29609  metres . 


WEIGHTS  AND  PRESSURES. 


By  Act  of  Congress  of  U.  S. 
Per  sq.  inch.  Lev  sq.  inch. 

— .1929  Lb. 

= 6.6679  Kilograms, 

z=  2.54  Centimeters, 
= 453.6029  Grams, 


1 Centimeter 
1 Atmosphere  . 

1 Inch  mercury 
1 Pound 

1 Kilogram  . J — o-#- -r— t * ~ - 

Note  — -*o  ins.  of  mercury  at  62°  = 14  7 lbs-  Per  inch  >'  hcnce> 1 j6 * * * * **  = 2-  °4°8  ins. , 

and  a centimeter  of  mercury  = 30  = . 3937  for  U.  S.  computation,  and  30= . 393  704  32 
for  French  or  Metric. 


By  Metric  Computation . 

• .19292  lb. 

6.667  8 kilogrammes. 
_r  2.54  centimetres. 
or  453.592  6 grammes. 


— ^ ^ "T  7 J 

= 317.4624  Lbs.  per  sq.foot , or  317.465 


POWER  AND  WORK. 

1 Horse  - power  = Cheval  or  Cheval  - vapeur  = 4500  k X 772  = 33  000-4- 
(4500  X 2.2046  X 39.37  -4-  I2)  — 1.01388  chevaux. 

1 Cheval  or  Cheval-vapeur  (75  kxm  per  second)  = horse-power. 

(4500  X 2.2046  X 39-37  -M2)  -4-  33  000  = .9863  horse-power. 

By  A ct  of  Congress  of  TJ.  S.  By  Metric  Computation . 

1 Foot-pound  = Kilogrammeter  kxm  — 7.233  foot-lbs. ; hence, 

1— (2.2046x3.280833)  = .13826  Kilogrammeter , or  .13825  kilogrammeire . 

1 Cube  foot  per  IP = .0279  Cube  meter  per  cheval . or  .0279  cheval. 

1 Pound  “ “ . . = .447  38  Kilogram  per  cheval.  or  .447  38  kilogramme. 
1 Cube  meter  per  cheval  = 35-8038  Cube  feet  per  IP,  or  35.8058  IP. 


I 

1 

i 

<. 

i 

i 

\ 


PRESSURES,  ETC. — MEASURES  OF  TIME. 


37 


TEMPERATURES. 

i Caloric  or  French  unit  = 3.968  Heat-units , and  1 heat-unit  = 1 -r-  3.968 
= .252  c alone. 

1 U.  S.  Mechanical  equivalent  ( 772  foot -lbs. ) = 772  ■—  7.233  = 106.733 
Kilo gr ammeters  and  106.733  kilogrammetres. 

1 French  Mechanical  equivalent  (423.55  k X m)  = 3.280833  X 2.2046  X 
423.55  = 3063.505  foot-lbs.,  or  3063.566 foot-lbs.  Metric. 

1 Heat-unit  per  pound  = .5556  Kilogram , or  .5556  kilogramme. 

1 Heat-unit  per  sq.  foot  = .2715  Caloric  per  sq.  meter , or  .271 %per  sq.  metre. 


VELOCITIES. 

1 Foot  per  second,  minute,  etc.  = .3047  Meter  per  second,  or  .3047  metres. 
1 Mile  per  hour =1 .447  “ u u or  .447  “ 


MEASURES  OF  TIME. 

60  thirds  = 1 second.  I 60  minutes  = 1 degree. 

60  seconds  = 1 minute.  | 30  degrees  = 1 sign. 

360  degrees  = 1 circle. 

True  or  apparent  time  is  that  deduced  from  observations  of  the  Sun, 
and  is  same  as  that  shown  by  a properly  adjusted  sun-dial. 

Mean  Solar  time  is  deduced  from  time  in  which  the  Earth  revolves 
on  its  axis,  as  compared  with  the  Sun ; assumed  to  move  at  a mean 
rate  in  its  orbit,  and  to  make  365.242  218  revolutions  in  a mean  Solar 
or  Gregorian  year. 

Sidereal  time  is  period  which  elapses  between  time  of  a fixed  star 
being  in  meridian  of  a place  and  time  of  its  return  to  that  place. 

Standard  unit  of  time  is  the  sidereal  day. 

Sidereal  day  = 23  h.  56  m.  4.092  sec.  in  solar  or  mean  time. 

Sidereal  year , or  revolution  of  the  earth,  365  d.  5 h.  48  m.  47.6  sec.  in  solar 
or  mean  time  = 365.242  24  solar  days. 

Solar  day,  mean  = 24  h.  3 m.  56.555  sec.  in  sidereal  time. 

Solar  year  (Equinoctial,  Calendar,  Civil  or  Tropical)  =365.242  218  solar 
days,  or  365  d.  5 h . 48  m.  47.6  sec . 

Civil  day  commences  at  midnight.  Astronomical  day  commences  at 
noon  of  the  civil  day,  having  same  designation,  that  is,  12  hours  later 
than  the  civil  day. 

Marine  or  sea  day  commences  12  hours  before  civil  time  or  1 day 
before  astronomical  time. 

New  Style  was  introduced  in  England  in  1752. 

Note  —In  Russia  days  are  reckoned  by  Old  Style,  and  are  consequently  12  days 
behind  Gregorian  record. 


D 


38 


MEASURES  OF  VALUE. 


32 


til 


MEASURES  OF  VALUE. 

10  mills  = 1 cent.  I 10  dimes  = 1 dollar. 

10  cents  = 1 dime.  | 10  dollars  = 1 eagle. 

Standard  of  gold  and  silver  is  900  parts  of  pure  metal  and  100  of 
alloy  in  1000  parts  of  coin. 

Fineness  expresses  quantity  of  pure  metal  in  1000  parts. 

Remedy  of  the  Mint  is  allowance  for  deviation  from  exact  standard 
fineness  and  weight  of  coins. 

Nickel  cent  (old)  contained  88  parts  of  copper  and  12  of  nickel. 
Bronze  cent  contains  95  parts  of  copper  and  5 of  tin  and  zinc. 

Pure  Gold  23.22  grains  = $1 00.  Hence  value  of  an  ounce  is 
$20.67.183+. 

Standard  Gold,  $18.60.465+  per  ounce. 

» 

WEIGHT,  FINENESS,  ETC.,  OF  U.  S.  COINS. 

GrOld.. 


Denomination. 

Weigh 

of  Coin. 

t 

of  Pure 
Metal. 

Denomination. 

Weigh 

of  Coin. 

t 

of  Pure 
Metal. 

Dollar 

Oz. 

•053  75 
•134  375 
.161  25 

Grs. 

25.8 

64-5 

77-4 

Grs. 

23.22 

58-05 

j 69.66 

Half  Eagle 

Eagle 

Oz. 

.268  75 
•537  5 

1-075 

Grs. 

129 

258 

5x6 

Grs. 

1 16. 1 

232.2 
464.4 

Quarter  Eagle . . 
Three  Dollar . . . 

Double  Eagle. . . 

Silver. 

r)ime  1 .08037=;  ! 38.58  I 34.722  II  Half  Dollar ! .401  875 

20  Cent .16075  77.16  69.444  1 Trade  Dollar. .875 

Quarter  Dollar  . j .200937  5 | 96.45  | 86.805  ||  Silver  Dollar  . . . | -859375 


192.9 

420 

412.5 


173.61 

378 

37i-25 


Weight. 

Copper. 

Tin  and 
Zinc. 

Weight. 

Copper. 

Tin  and 
Zinc. 

One  Cent 

Two  Cents . . . 

Grains. 

48 

96 

Per  cent. 

95 

95 

Per  cent. 

5 

5 

Three  Cents. 
Five  Cents. . 

Grains. 

30  * 
77.16 

| Per  cent. 

75 
1 75 

Per  cent. 
25 
25 

rrolerance. — Gold,  Dollar  to  Halt  Eagle,  .25  grams,  regies, 

—Silver,  1.5  grains  for  all  denominations.  — Copper,  1 to  3 cents,  2 grains, 
5 cents,  3 grains.  „ , 

LeJal  Tenders. — Gold,  unlimited.  - Silver.  Dollars  of  412.5  grams 
u+mited;  for  subdivisions  of  dollar,  *10.  (Trade  dollars  [420 grams]  are 
not  legal  tender.) — Copper  or  cents,  25  cents. 

Note. -Weight  of  dollar  up  to  1837  was  416  grains,  thence  to  1873,  412.5-  Weight 
of  $1000,  @ 4x2.5  Sr-  =859.375  OK. 

British  standards  are:  Gold , fg  of  a pound,*  equal  to  11  parts  pure  gold 
and  1 of  alloy  ; Silver,  fit  of  * pound,  or  37  parts  pure  silver  and  3 of  alloy 

=A9TroyCounce  of  standard  gold  is  coined  into  £3  i7»; 
ounce  of  tandard  silver  into  5..  6 d.  1 lb.  silver  is  coined  into  66  shillings. 
Copper  is  coined  in  proportion  of  2 shillings  to  pound  avoirdupois. 

£ Sterling  (1880)  $486.65;  hence  ^ of  this — value  of  1 penny 

2.027  708  33  cents.  __ 

* A pound  is  assumed  to  be  divided  into  a4  equal  parts  or  carats,  hence  the  pro- 
portion  is  equal  to  22  carats. 


FOREIGN  MEASURES  OF  VALUE. 


39 


To  Compute  "Value  of  Coins. 


Rule. —Divide  product  of  weight  in  grains  and  fineness,  by  480 
(grains  in  an  ounce),  and  multiply  result  by  value  of  pure  metal  per 
ounce. 

Or,  Multiply  weight  in  ounces  by  fineness  and  by  value  of  pure  metal 
per  ounce. 

Example  i. — When  fine  gold  is  $20.67.183-}-  per  oz.,  what  is  value  of  a British 
sovereign? 


By  following  tables,  p.  40,  Sovereign  weighs  .2567  oz.,  and  .2567  X 480  = 12^  216 
grains,  and  has  a fineness  of  .9165.  H J 


Hence, 


123.216  x 916  5 
480 


X 20. 67. 1 83+  = $4.86.34. 


Example  2.— When  fine  silver  is  $1. 15. 5 per  oz.,  what  is  value  of  U.  S.  Trade  dollar? 
By  table,  p.  40,  Dollar  weighs  .875  oz.  and  has  a fineness  of  .900. 

Hence,  .875  x -900  X 1-15-5  = 90-95625  cents. 

nr5Xr^P-LE  3-  —A  4-Florin  (Austrian)  we:ghs  49.92  grains  and  has  a fineness  of  .000. 
What  is  its  value  ? y 


49.92x.900_  „ „ , ■ 

—■ — X 20.67.183+=  $1.93.49. 


To  Convert  LJ.  S.  to  Britisli  Currency  and.  Contrari- 
wise. 

Rule  i.— Divide  Cents  by  2.027  7i~  (2.027  708  33),  or,  Multiply  by 
.493  12-  (.493  1 18  26),  and  result  is  Pence. 

2.  Multiply  Pence  by  2.02771— , or  divide  by  .49312—,  and  result 
is  Cents. 

Example. — What  are  100  cents  in  pence? 

100  X -493  12 — — 49-3i2 — pence  = 45.  1.312 d. 

2.  What  is  a Pound  sterling  in  cents? 

20  X 12  = 240 pence,  which  x 2.027  71—  — $4  86.65. 


FOREIGN  MEASURES  OF  VALUE. 


Weight,  Fineness,  and  Mint  Values  of  Foreign 
Silver  and.  CL  old  Coins. 

By  Laws  of  Congress,  Regulations  of  the  Mint,  and  Reports  of  its  Directors. 

# Current  Value  of  silver  coins  is  necessarily  omitted,  as  the  value  of 
silver  is  a variable  element.  Hence,  in  order  to  compute  current  value 
of  a silver  coin,  the  price  of  fine  or  a given  standard  of  silver  bein" 
known,  0 

Proceed  as  per  above  rule  to  compute  value  of  coins. 

The  price  of  silver  should  be  taken  as  that  of  the  London  market  for 
British  standard  (925  fine),  it  being  recognized  as  the  standard  value 
and  governing  rates  in  all  countries. 


Example.— If  it  is  required  to  determine  value  of  a Mexican  dollar  in  cents. 

Weight  867.  5 oz.  .903  fine.  Value  of  Silver  in  London  52.75  pence  per  ounce  - 
100.9616+  cents. 

Then  — - 5X9  3 = .846 867—  and  106.9616  x .846  867  = 90.5822  cents. 


925 


40 


foreign  measures  of  value. 


Weight  and  Mint  Values  of  Foreign  Coins. 
Countries  given  in  Italics  have  not  a National  Coinage. 


Country  and  Denomination.  Weight. 


Fine- 

ness. 


Oz.  Thous’s. 


Arabia. 

Piastre  or  Mocha  Dollar. . 
Argentine  Republic. 

Dollar  = ioo  Centisimos 

(Employs  South  American  and 
Foreign  Coins.) 
Australasia. 

Same  as  British. 

Australia. 

Sovereign,  1855 

Pound,  1852 

Austria. 

Kreutzer  (copper) 

Florin,  new 

Dollar,  “ 

4 Florins 

Ducat 

Souverain 

Belgium. 

Same  as  France. 

Bolivia. 

Centena 

Dollar,  new 

Doubloon,  1827-36 

Brazil. 

Rei. 

Milreis . 

Double  Milreis 

20  Milreis,  1854-56  . 

Moidore,  4000  Reis . 

Canada. 

Mil,  sterling. . 

Cent 

20  Cent,  currency  . 

25  u “ 

Penny  “ 

Shilling  “ 

Dollar,  sterling 
4 “ =20  shillings,  currency 

Pound  “ 

Cape  of  Good  Hope. 

Same  as  British. 

Central  America. 

4 Reals 

Dollar 

2 Escudos 

Doubloon  ante  1834 

Chili. 

Centaro 

Dollar,  new 

10  Pesos. - 

Doubloon 

China. 

Cash,  Lc 

10  Cents,  Leang 

Dollar 

Cochin  China. 

Mas,  60  Sapeks 

10  Mas,  1 Quan 


Pure  I 

Silver  Current 
or  or 

Gold.  Nominal. 

Grains.  | Cents. 

83.14 
— 50.69 


Value. 

Gold. 

U.  S.  British. 


.256.5 

.281 


•397 

•596 

.104 

.112 

•363 


916 

916.5 


900 

900 

900 

986 

900 


£ 8.  d. 


.801 

.867 

.028.8 

.82 

•575 

.261 


900 

870 


i7J-47 

257-47 


•75 

346.03 
362.06  — 


4- 85-7 

5- 32-37 


1.93.49 
2. 28. 3 
6-75-4 


19  11.5 
: 1 10. 5 


711  . 
9 4-6 
7 9-* 


i5-59‘3  3 4 1 


— — -547  — 

916.66  12.67  — 


9i8-5 

9I7-5 

914 


•54-59 
393.6  — — 

10.90.6 
4.92 


— — — 1. 01  — 


•i5 
.187  5 


925 

925 


66.6 

83-25 


/27 

26.92 

2 4 9.84 
1 o 2.63 

•05 

•5 


— — I — 1-52 


.027 

.866 

.209 

.869 


.801 

•492 

.867 


.087 

.866 


l75 

850 

853-x 

833 


900.5 

900 

870 


901 

901 


ii-34 

353-33 


346.22 


37-98 

374-63 


6.75 

67.52 


3-97-43 

3-99-97 


3.68.8 

14.96.39 


9-I5-4 

15-59-3 


•75 

4 2 
16  4 
16  5.25 


15  1.88 
3 1 5-97 

-45 

1 17  7-45 
3 4 1 

.07 


3-33 

9-33 


FOREIGN  MEASURES  OF  VALUE. 


41 


Weigh,  t and.  Mint  Values. 


Country  and  Denomination. 

Weight. 

Fine- 

ness. 

Pure 

Silver 

or 

Gold. 

Current 

or 

Nominal. 

V A V V I 
G 

U.  S. 

E. 

old. 

British. 

Cuba. 

Same  as  Spain. 

Colombia. 

Centaro 

Oz. 

Thous,s. 

Grains. 

Cents. 
1. 01 

$ c. 

£ 9.  d. 

Peso,  new 

.801 

900 

844 

870 

346-03 

'•5 

4 Escudos 

7-55-5 

15-59-3 

1 11  0.58 

Doubloon,  old 

.867 

Costa  Rica. ' 

Same  as  Mexico. 

Denmark. 

Mark,  16  Skilling 

8.94 

3 4 1 
4-39 

Crown 7 

900 

877 

895 

26.8 

2 Risrsdaler 

.927 

390.23 

13-  22 

10  Thaler 

7.90 

1 12  5.6 

East  Indies. 

See  Hindostan  and  Japan. 

Ecuador. 

Centaro 

1.  Ol 

Peso 

.801 

900 

346.03 

•5 

England. 

Penny 

2,  Q2  j 

Groat 

26.82 

80.99 

79-°3 

201.8 

161.44 

1 

Shilling,  new. . 

.182.5 

.178 

# AC  A.  C 

925 

‘ ‘ average 

924-5 

925 

925 

925 

916.5 

916.5 

Half  Crown 

Florin 

•363-6 

.256.7 

.256.2 

Sovereign  or  Pound,  new  . . . 
“ “ average. 

Egypt. 

Piastre,  40  Paras 

- 

4.86.65 

4-85-1 

04.9 

100 

100 

7 c t 

14-5 

Guinea,  Bedidlik 

. 27  £ 

/DD 

875 

875 

875 

10  6. 84 

Pound 

. 27  C 

5-  0.52 

Purse,  5 Guineas 

• ^ J O 

i-375 

.032 

4.97.4 
25.  2.6 

1 0 5-3 

France. 

Centime 

5 2 10. 2 

Sou,  5 Centimes 

» l6l 

.2 

. 1 

Franc,  100  Centimes 

.l6l 

.804 

.207.5 

900 

1. 01 

•5 

5 Francs 

°9-55 

347-76 

20  Francs,  Napoleon,  new . . . 
25  Francs  20  centimes =£1  Stg. 
Germany. 

Groschen,  10  Pfenning 

899 

2.38 

3.85.8 

15  10.26 

Mark,  10  Groschen 

.012.8 
.128 
. CO  c 

900 

900 

23.8 

2.38.24 

I-I75 

10  Marks 

11.74 

Thaler 

257.04 

9 9-5 

Ducat 

•0  yo 

986 

2.28.38 

i9-3 

9 4-63 
9-5 

Greece  and  Ionian  Islands.. 

Same  as  France. 

Drachma,  100  Lepta 

5 Drachmas 

.010.4 

900 

900 

900 



I 

20  Drachmas 

.719 

.185 

310.61 

■ 

— 

Pound 

44.2 

14  i-75 
1 0 9.6 

Guatemala. 

Same  as  Mexico. 

Guiana , British,  French,  and 
Dutch. 

Same  as  that  of  their  Countries. 
Hanse  Towns. 

Mark 

.012.8 

900 

5-  6. 11 

Holland. 

Cent 

23.8 

11.74 



•4 

— 

.2 

* 2.027  71  cents. 

D* 


42 


FOREIGN  MEASURES  OF  VALUE, 


Wei glit  and  Mint  Values. 


Country  and  Denomination. 


Holland. 

Florin  or  Guilder,  ioo  cents. 

io  Guilders 

Hindostan. 

Rupee 

Honduras. 

Same  as  Mexico. 

Italy. 

Same  as  France. 

Lira,  ioo  Centimes 

Scudo 

Indian  Empire. 

Pic,  nominal 

Anna  “ 

Rupee,*  16  Annas 

io  Rupees,  and  4 Annas 

Moliur,  15  Rupees 

Japan. 

Sen 

Itzebu,  new 

Yen,  100  Sen 


Weight. 


.021.6 

.215 


•374 


.16 

.864 


•375 

•375 


Thous’s, 

900 

899 

916.  s 


835 

900 


9i6-5 

9i6-5 


.279 

.866.7 

.053-6 


377-^7 

372.98 


336-25 


Cobang,  old -2^9  57^ 

' new 362  568 

20  Yen 1.072  900 

Java. 

Same  as  Holland. 

Liberia. 

U.  S.  Currency. 

Mciltci 

12  Scudi  = 1 Sovereign 
Mexico. 

Peso,  new 

“ Maximilian 

Doubloon,  new 

20  Pesos,  Republic 

Morocco. 

Ounce,  4 Blankeels .... 

10  Ounces,  Mitkeel 

Naples. 

Scudo 

6 Ducati 

Netherlands. 

Same  as  Holland. 

New  Brunswick. 

Same  as  Canada. 

Newfoundland. 

Same  as  Canada. 

New  Granada. 

Dollar,  1857 803 

Doubloon,  Popayan -867 

Norway. 

Alike  to  Denmark. 

Mark,  24  Skillingen  . . . . 

Nova  Scotia. 

Same  as  Canada. 

Persia. 

Keran,  20  Shahis 

10  Keran,  Toman 

Paraguay.  Foreign  coins.  

* .092  76  of  a X Stg.,  nominal  value  = 2 shillings  sterling. 


.867. 
.861 
.867. 
1. 081 


Fine- 

ness. 


Pure 

Silver 


Current 

or 

Nominal. 


164.53 


65. 12 
373-24 


900 

900 


165 


119.19 

374-4 


9°3 

902.5 

870.5 
873 


830 

996 


896 

858 


•25 

3-03 


Value. 

Gold. 


$ c. 


40.49 

3-99-7 


4.86.65 
6. 84.36 


99.72 

3-57-6 

4.44 

19.94.4 


21.63 


22.81 


1 8 

16  5.11 


1 10.5 


4.86.65 


15.  6.1 
I9-5I-5 


5-  4-4 


15-37-8 


.125 

i-5 


I-5 

•5 


4 1. 18 
14  8.35 
18  2.96 
. 1 11. 6 


4 

3 4 

4 o 


1.88 

2.4 


1 o 8.75 


3 3 3-39 


10.66 


11.25 


FOREIGN  MEASURES  OF  VALUE, 


43 


Weight  and.  Mint  Values. 


Country  and  Denomination. 

Weight. 

Fine- 

ness. 

Pure 

Silver 

or 

Gold. 

Current 

or 

Nominal. 

Value 
G < 

U.  S. 

Peru. 

Dollar,  1858 

Oz. 

.766 

Thous?s. 

900 

900 

868 

912 

912 

835 

Grains. 

341.01 

346.46 

Cents. 

$ c. 

Sol...'...'! 

Doubloon,  old 

.867 

.308 

•095 

15-55-7 

5. 80. 66 
10.8 

Portugal. 

Coroa,  1838,  10000  Reis 

— 

— 

Roumania. 

2 Lei 

129.06 

Russia. 

Copek 

•77 

100  Copek,  Rouble 

.667 

875 

916.6 

835 

277-73 

5 Roubles 

3-97-6 

Sandwich  Islands. 

U.  S.  Currency. 

Sardinia. 

Lira 

.16 

65.12 

Spain. 

Centimo 

.19 

100  Centimo,  Peseta 

. 16 

837 

900 

64.13 

345-6 

Dollar,  5 Peseta 

.8 

100  Reals 

.268 

.270.8 

4.96.4 

10  Escudos 

896 

896 

750 

75o 

900 

20  Reals  vellons=i  U.S.  Dollar. 
Sweden. 

Riksdaler,  100  Ore 

28 

5-  I-5 

Rixdollar 

•273 

1.092 

.104 

393.12 

• 

Carolin,  10  Francs 

1-93-5 

Switzerland. 

Same  as  France. 

St.  Domingo. 

Gomdes,  100  Cents 

6-33 

11.83 

Tunis. 

Piastre,  16  Karubs 

5 Piastre 

Ctt 

220.38 

25  Piastre 

O A A 
. l6l 

090.5 

900 

2.99.5 

Turkey. 

Piastre,  40  Paras 

4-39 

20  Piastre 

•77 

9QT 

830 

915 

900 

306.77 

100  Piastre,  Medjidie 

4.36.9 

Tuscany. 

Zecchino,  Sequin 

Tripoli. 

20  Piastres,  Mahbub 

74-3 

2-3I-3 

Uruguay. 

Dollar,  100  Centimes 

West  Indies,  British. 

Same  as  England. 

Venezuela. 

Centaro 

Bolivar,  1 Franc 

— 

— 

_ 

7 



Memoranda. 

France.— Bronze  coins  9.5  copper,  4 tin,  and  1 zinc. 

Hanse  Towns. —Monetary  system  same  as  that  of  German  Empire. 
Switzerland.— The  Centime  is  termed  a Rappe. 

Spain.— 25  Peseta  piece  is  19.5.  9.5 d.  Stg. ; Real  vellon  was  2.5 d.  Stg. 
Italy. — All  coins  same  weight  and  fineness  as  those  of  France. 
Malta.— 7 Tari  and  4 Grani ■=  1 Shilling  Sterling. 

Egypt.— A Para  — .061  5 d.  Sterling,  and  97.22  Piastres  = 1 Sovereign. 
Indian  Empire.— 1 Lac  Rupees=:£ioooo  Sterling.  In  Ceylon,  Rupees 


3 3 ii-22 

2 4 5-5 


.38 


16  4.8 


•095 


1 o 4.8 

I o 7.32 


7 11.42 

3-125 
_5-83 
12  3-7 
2.16 
18  o 
9 6.1 
3 0.89 


: 100  Cents. 


44  ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS. 


ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS. 
MEASURES  OF  LENGTH. 

English. — Imperial  standard  yard  is  referred  to  a natural  standard, 
which  is  a pendulum  39.1393  ins. in  length  vibrating  seconds  in  vacuo 
in  London,  at  level  of  sea ; measured  between  two  marks  on  a brass 
rod,  at  temperature  of  62°. 

Note.  — In  consequence  of  destruction  of  standard  by  fire  in  1834,  and  difficulty 
of  replacing  it  by  measurement  of  a pendulum,  the  present  standard  is  held  to  be 
about  1 part  in  17  230  less  than  that  of  U.  S.,  equal  to  3.67  ins.  in  a mile. 

Miscellaneous. 

Land  —Woodland  pole  or  perch  or  Fen = iS  feet. 

Forest  pole — 21 

Irish  mile = 2240  yards.  | Scotch  mile  ...*.  = 1984  yards. 

Sea. — IO  cables,  or  1000  fathoms,  or  6086.44  feet,  or  1.152  8 statute  miles 
= 1 Admiralty  or  Nautical  mile  or  knot. 

3 miles  = 1 league.  60  Nautical  or  69.168  Statute  miles  or  20  Leagues  — 

1 Mean  length  of  a minute  of  latitude  at  mean  level  of  the  sea  =1.1508 

statute  miles.  . A ^ 

Nautical  mile  is  taken  as  length  of  a minute  at  the  Equator. 

Nautical  fathom  is  1000th  part  of  a nautical  mile,  and  averages  about 
.0125  longer  than  the  common  fathom. 

French.— Standard  Metre  or  unit  of  measurement  is  defined  as  the 
ten  millionth  part  of  the  terrestrial  meridian,  or  the  distance  from  the 
Equator  to  the  Pole,  passing  through  Paris.  .Actual  standard  is  a plat- 
inum metre,  deposited  in  the  Palais  des  Archives,  Paris. 


Metric  Length,  in.  Inches,  Feet,  etc. 
Denomination.  Metres.  Inches.  Feet.  Yards. 


100 
1 ooo 
10000 


.039  37 
•393  7 
3-937  °4 
39- 370  43 


3. 280  87 
32.808  69 
328.086  9 
3280. 869 


1 Millimetre. . . 

1 Centimetre  . . 

1 Decimetre . . . 

1 Metre 

1 Dekametre  . . 

1 Hektometre. . 

1 Kilometre  . . 

1 Myriametre. . 

Note.— For  length  of  metre  see  p.  27. 

Old.  Measure. 

1 Terrestrial  league  = 4.444  kilometres. 
1 Nautical  league  . = 5.555  . “ 

1 Arpent 900  sq.  toises. 


1.093  62 
10.93623 
109.36231 
1 093.623 1 
10936.231 


Miles. 


.621  38 
6.21377 


i Toise = 1.949  metres. 

1 Mille = 1.949  kilometres. 

1 Nocud  (knot).  = 1.855  “ 


MEASURES  OF  SURFACE. 

English.— Same  as  that  of  United  States  of  America. 

Miscellaneous. 

Builders.  i superficial  part = 1 ^uare  inch- 

12  parts — 1 2WC"* 

12  inches ==  square  foot. 

Boards. — Boards  7 inches  in  width  are  termed  battens,  9 inches  deals,  and 
12  inches  planks. 


ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS.  45 


French. 

IMetric  Surfaces  in.  Square  Inches,  Feet,  etc. 


Denomination. 

Sq.  Inches. 

Sq.  Feet. 

Sq.  Yards. 

Sq.  Acres. 

i Square  millimetre 

.001  55 
•155  003 

I K.  ^OO  300 

1 “ centimetre 

1 “ decimetre 

.107  641 
10.764  104 
1076.410358 

1 “ Metre  or  Centiare 

1 ‘ ‘ dekametre  or  are 

1 “ hektometre  or  hectare 

1 u kilometre 

1550.0309x6 

1. 19601 
119.601  15 
1 1 960. 1 1 5 09 

.024  711 
2.471  098 
247. 109  816 
24  710.981  6 

1 “ myriametre* 

— 

— 

— 

* Equal  38.610  908  sq.  miles. 


Old.  System. 

1 square  inch  = 1.135  87  inches. 

1 toise  = 6.394  6 feet. 

1 arpent  (Paris)  = 900  square  toises  = 4089  square  yai'ds. 

1 arpent  (woodland)  = 100  square  royal  perches  = 6108.24  square  yards. 


MEASURES  OF  VOLUME. 

Imperial  gallon  measures  277.123  cube  ins.,  but  by  Act  of  Parliament 
1825  its  volume  is  277.274  cube  ins.,  equal  to  10  lbs.  avoirdupois  of 
distilled  water,  weighed  in  air,  at  temperature  of  62°,  barometer  at  30 
inches.  6.2355  gallons  in  a cube  foot. 

Imperial  bushel , 18.5  ins.  internal  diameter,  19.5  external,  and  8.25 
in  depth,  contains  2218.192  cube  ins.,  and  when  heaped  in  form  of  a 
right  cone,  at  least  .75  depth  of  the  measure,  must  contain  2815.4872 
cube  ins.  or  1.6293  cube  feet. 

Grain. — 1 quarter  = 8 bushels  or  10.2694  cube  feet. 

Vessels.  — 1 ton  displacement  = 35  cube  feet ; 1 ton  freight  by  measure- 
ment = 40  cube  feet. 

1 ton  internal  capacity  = 100  cube  feet , and  1 ton  ship  - builders  = 94 
cube  feet. 

English  standard  No.  5 is  .008  grain  heavier  than  the  pound,  and  U.  S.  pound  is 
.001  grain  lighter  than  English. 


Wine  and  Spirit  Measures. 

4 quarts  (231  cube  ins.) = .8333  Imperial  gallon. 

10  gallons = 1 anchor. 


18 
31*5 
42 

63 
84 
126 

2 pipes  or 

3 puncheons 


(15  imperial)  , 
26.25 
35 

52.5 

70 

105 


= 1 runlet. 

= 1 barrel. 

= 1 tierce. 

— 1 hogshead. 

= 1 puncheon. 
= 1 pipe  or  butt . 

= 1 tun. 


4 quarts  (28c  cube  ins.)  . . = 1.017 

9 gallons  = 1 firkin = 9.153 

2 firkins  = 1 kilderkin  . . . = 18.306 


-Ade  and  Beer  Measures. 
Imp’l  gall’s. 


Imp’l  gall's. 

2 kilderkins  = 1 barrel  = 36.612 
54  gallons  = 1 hogshead  = 54.918 
108  “ = 1 butt  . . . . = 109.836 


46  ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS. 

Apothecaries’  or  Fluid.  Measures. 

T ,irfm  = 1 grain.  I 4 drachms = 1 tablespoon. 

^ol's  ; ; ; ; — 1 drachm.  I 2 ounces  (875  grains)  = 1 wineglass. 

Coal  Measures. 

1 sacks = 1 chaldron . 


50  pounds  . . . . = 1 cube  foot. 

88  “ = 1 bushel, 

g bushels  . . . . = 1 vat. 

, ( 1 London  or 

80  or  84  pounds  ^ Newcastle  bushel. 
go  or  94  tk  = 1 Cornish 
93  pounds  . . . . — i Welsh  bushel. 

3 heaped  bush.  = i sack. 

10  sacks = 1 ton. 


1 chaldron = 58.6548  cube  ft. 

5.25  chaldrons  . . = i room. 

1 London  chaldron  =26.5  cwts. 

1 Newcastle  “ = 53  “ 

i ton =44-5  cube  feet. 

i room = 7 ions. 

21  chaldrons = i score. 

i barge  or  keel . . = 21.2  tons. 


1 last  corn 

1 ton  water 

1 dicker  hides  .... 

1 last  hides 

1 barrel  tar  ...... 

6 bushels  wheat  . . 

1 clove 

1 score •'* 

1 sack  flour 

1 truss  straw 


NLiscellaneoms. 

1 truss  old  hay = 50  pounds. 

1 “ new  “ = 60  “ 

1 bushel  oats = 4°  u 

1 “ barley  . . . . = 47  “ 

1 “ wheat = 60  “ 

1 cube  vard  new  hay  = 84  “ 

1 “ “ old  “ = 126  “ 

1 quintal  = 100 

1 boll = 140  “ 

1 sack  wool  = 364  ‘ 


= 80  bushels. 

= 35-9  cube  feet. 
= 10  skins. 
z=z  20  dickers. 

— 26.5  gallons. 
=z  i sack  flour. 
= 7 pounds. 

= 20  “ 

= 28.2  “ 

= 36 


35.9  cube  feet  = 1 ton  water. 
Liquid. 


1 wine  gallon  = 231  cube  ins. 

1 beer  “ =282  “ “ 

1 litre = .220  09  gallon. 

1 gallon = 4-544 

1 cube  foot . . = 6.2321  gallons. 

1 anker  . . . . = 8.333 


1 hogshead  wine  . . = 52-5  gallons . 
1 “ beer . . . = 54-9l8  “ 

1 puncheon  wine  . . = 70 
1 pipe  or  butt  wine  = 105 
1 u “ “ beer  = 109.836  “ 

1 tun  

ton  water  62°  = 224  gallons. 


Builders. 


1 solid  part = 12  cube  ins 

12  “ parts = 1 “ inch. 

12  “inches  ” • . = 1 cube  foot. 

1 load  timber,  rough  = 40  “ feet. 
x « u hewn  = 50 
1 “ lime  .......  = 32  bushels. 

1 “ sand  . 36  “ 


1 square sq.  feet. 


bundle  laths 
1 rod  brickwork  . 
1 rood  masonry  . 
Batten,  in  section 
Deal,  ? u 

Plank,  “ “ 


= 120  laths. 

— 306  cube  feet. 
= 648  “ “ 

= 7 X 2.5  ins. 
= 9X3  “ 

= 11X3 


Metric  Volumes  in.  Cube  Indies,  Feet,  etc. 


Denominations. 

Litres. 

Gills. 

Pints. 

Quarts. 

Gallons. 

Bushels. 

Quarters 

Cent  i 1 i t.rp. 

.01 

.0704 

.0176 

— 

— 

— 

— 

Decilitre 

. 1 

•7°43 

. 1761 

— 

— 

Litre * 

1 

7.0429 

1.7607 

.8804 

.2201 

Dekalitre 

10 

— 

8. 8036 

2.2009 

.275 11 

•3439 

3-43^9 

Hectolitre 

100 

— 

— 

— 

22.OO9I 

220.0908 

2.751  J3 

Kilolitre 

1000 

— 

— 

— 

27-5II35 

* Equal  61.025  24  cube  ins. 


ENGLISH  AND  FRENCH  MEASURES  AND  WEIGHTS.  4.7 


Wood  NTeasnre. 

i Stere  or  cube  metre  = 35.3150  cube  feet  or  1.308  cube  yards. 

1 Voie  de  bois  (Paris)  = 70.6312  cube  feet ; 1 voie  de  charbon  (charcoal) 
= 7.063  cube  feet ; 1 corde  = 4 cube  metres  = 141.26  cube  feet. 


MEASURES  OF  WEIGHT. 


British. — 1 Troy  grain  = .003  961  cube  inches  of  distilled  water. 

1 Troy  pound  =22.815  689  cube  inches  of  water. 

1 Avoir,  drachm  = 27.343  75  Troy  grains. 


16  drachms,  or 
437.5  grains 
16  ounces,  or 
7000  grains 


-Avoirdupois. 

8 pounds 


, = 1 ounce. 


14 

28 


. = 1 ton 


= 1 stone  (for  meat). 
= 1 stone. 

= 1 quarter. 

= 1 cwt. 


= 1 pound. 

20  hundredweights  . 

The  gram,  of  which  there  are  7000  to  the  pound  avoirdupois,  is  same  as 
Troy  grain,  of  which  there  are  by  the  revised  table  7000  to  the  Troy  pound. 
Hence  Troy  pound  is  equal  with  the  Avoirdupois  pound. 

In  Wales,  the  iron  ton  is  20  cwt.  of  120  lbs.  each. 


Troy. 


24  grains = 1 dwt. 

20  pennyweights,  or\  

437-5  grams  j • • - 1 <Hmce- 


16  ounces =i  pound. 

25  pounds = 1 quarter. 

4 quarters,  or  100  pounds  = i cwt. 


By  this  are  weighed  gold,  silver,  jewels,  and  such  liquors  as  are  sold  by 
weight. 

The  old  Troy  ounce  to  the  Avoirdupois  ounce  was  as  480  grains,  the 
weight  of  the  former,  to  437.5  grains,  weight  of  the  latter ; or,  as  1 to  .9115. 

.Apotliecaries.* 

437-5  grains  = 1 ounce.  | 16  ounces  = 1 pound. 

French. 

NLetric  Weights  in  Avoirdupois. 

Denominations.  Grammes.  Grains.  Ounces.  Pounds. 


Milligramme  . . 
Centigramme. . 
Decigramme  .. 

Gramme 

Dekagram  me. . 
Hektogramme. 
Kilogramme^ . . 
Myriagramme  . 

Quintal 

Millier  or  Ton. . 


10 
100 
1 000 
10000 
100  000 
1 000000 
f Kilogramme 


*015  43 
•154  32 
1 543  23 
I5-432  35 
154.32349 
1 543.23487 
15432.348  74 


•3527 

3-5274 

35-2739 


.22046 
2.204  62 
22.046  21 
220.462  12 
2204.621  25 


= 2 lbs.  3 os.  4 drachms,  10.4734  grains. 

Note.— For  the  values  of  the  prefixes,  as  Milli,  Centi,  etc.,  see  p.  27. 

Old.  System. 


.9842 


1 grain . 

. = 0.8188  grains  Troy. 

I 1 ounce  = 1.0780  oz.  Avoirdupois. 

1 gross  . 

. = 58.9548  “ 

j 1 livre  = 1.0780  lbs. 

* As  by  revised  Pharmacopoeia. 


48 


foreign  measures  and  weights. 


foreign  measures  and  weights. 


Tt  beino-  wholly  impracticable  to  give  all  the  denominations  of  measures 
and  weights  of  ah  countries,  the  f oUowing  cases  are  selected  as  essential  and 

“flTpatnt  countries,  as  England,  France,  etc  their  denominations  ex- 
South  America,  and  those  of  France  to  a part  of  the  West  Indies,  Algiers,  e c. 


Abyssinia. 

Pic,  Stambouili 26.8  ins. 

“ geometrical 3°-37  ‘ , 

Madega 3466bu?h. 

Ardeb 34-66  ^ 

“ Musuah 83-i84. 

Wakea 

Mocha 1 

Rottolo , • 

Also , same  as  in  Egypt  ana  Cairo. 

Africa,  Alexandria,  Cairo, 
an. cl  Egypt. 

Cubit 20.65  ins. 

Derail 25.49  u 

Pic,  cloth ^ 

“ geometrical 29.53 

Kassaba,  4.73  Pics H* 

Mile...!:.. 2146  yds 

Feddan  al-risach 552  48  acre. 

Roobak r.684  galls. 

Ardeb 4.9  bush. 

Rottol 9821  lb. 

Distances  are  measured  by  time. 

A Maragha  = 15  Dereghe  or  1 hour. 
Aleppo  and  Syria. 

Dra  Mesrour 2*-845  ’ «s' 

Pic 

Road  Measures  are  computed  by  time. 

Algeria. 

Rob,  Turkish 3.11  ins. 

pic  “ 24.92 

“ ’ Arabic i8,89 

Also  Decimal  System. 


Alicante. 

^f38  '“s- 

Vara 35-632 

Amsterdam. 

ins- 

El ‘ 

Faden 5-57  ll- 

Lieue 6.  383  y^s- 

Maat 1.6728  acres. 

Morgen 2-°°95 

yat  ® 40  cub.  ft. 

Also  Decimal  System. 


Arabia,  Bassora,  and 
Mocha. 

Foot,  Arabic 1.0502  ft 

Covid,  Mocha ins. 

Guz,  “ 2 5 

Kassaba I2-3  jt 

Mile,  6000  feet 2146  yds. 

Baryd,  4 farsakh 21 120  ‘ 

Feddan 57  &>o  sq.  ft. 

Noosfia,  Arabic J38  culJ- ins- 

Gudda. ^ galls. 

3 *6f- 

Tomand 

Other  Measures  like  those  of  Egypt. 

Argentine  Confederation, 
Paraguay,  and  TJ ruguay. 

Fanega 

Arroba 25-35  16s- 

Quintal 101,4 

Also  Decimal  System  in  Argentine  Con- 
federation and  Paraguay. 

Australasia. 

Land  Section 80  acres- 

Other  Measures  same  as  English. 
Austria. 

Zoll 1 *037 1 ins. 

Fuss 1 0371  it 

Meile 24  000  It. 

Klafter,  quadrat 35-854  sq.  yds. 

Cube  Fuss ft' 

Achtel 1.692  galls. 

Eimer u 

Mptyp  1.6918  bush. 

Unze  8642  grains. 

Pfund  (1853, 500  grammes),  1.2347  lbs. 

Centner 123.47 

Also  Decimal  System. 


Antwerp. 

Fuss h\s. 

Corde 24.494  cub  ft. 

Bonnier 3.2507  acres. 

Also  Decimal  System. 


, 18.205  ins. 


Babylon 

Pachys  Metrios 

Baden. 

Fuss 

Klafter 5-9°55  ft- 

Stundcn 488°  5 “s- 

Morgen ^ galls. 

. 1 1268  bush. 

Malter ; ; ; ; ; ; ; ; 1. 1023  ibs. 

Also  Decimal  System. 


Pfund  . 


foreign  measures  and  weights. 


49 


Bagdad. 

Guz 31-665  ins. 

Barbary  States. 

Pic,  Tunis  linen 18.62  ins. 

“ “ cloth 26.49  “ 

“ Tripoli 21.75  “ 


Batavia. 


Foot . . 
Covid . 
El 


12.357  ms. 

27  “ 

27.75  “ 

Bavaria. 

Fuss ii-49  ins- 

Klafter 5-745  36  It. 

Ruthe 3.1918  yds. 

Meile 8060 

Ruthe,  quadrat 10. 1876  sq. yds. 

Morgen  orTagwerk 8416  acre. 

Klafter,  cube 4. 097  cub.  yds. 

Eimer 15.058  56  galls. 

Soheffel 6.119  “ 

Metze 1.0196  bush. 

Pfund 8642  grains. 

Also  Decimal  System. 
Belgium. 

Meile 2.132  yds. 

Also  Decimal  System. 
Benares. 

Yard,  Tailor’s 33  ins. 

Bengal,  Bombay,  and  Cal- 
cntta. 

Moot 3 ins. 

Span 9 “ 

Ady,  Malabar 10.46  ins. 

Hath 18  “ 

Guz,  Bombay 27  “ 

“ Bengal 36  “ 

Corah,  minimum 3.417  ft. 

Coss,  Bengal 1.136  miles. 

“ Calcutta 1-2273  “ 

Kutty. 9.8175  sq.  yds. 

Biggah,  Bengal 3306  acre. 

Bombay 8114  “ 

Seer,  Factory 68  cub.  ins. 

Covit,  Bombay 12.704  cub.  ft. 

Seer,  Bombay 1.234  pints. 

Parah 4. 4802  galls. 

Mooda 1 12.0045  u 

Liquids  and  Grain  measured  by  weight. 
Bohemia. 

Foot,  Prague 11.88  ins. 

“ Imperial 12.45  “ 

Also  same  as  Austria. 
Bolivia,  CHili,  and  Bern. 


• 33 


Vara 

Fanegada, 

Gallon. . . . 

Fanega. . . 

Libra 

Arroba 25. 

Originally  as  in  Spain;  now  Decimal 
System  in  Chili  and  Peru. 


333  ms. 
.5888  acres. 
-74  gall 
•572  “ 

.014  lbs. 

■ 36 


.927  inch. 
. 11.128  ins. 


Brazil. 

Palmo,  Bahia 8.5592  ins. 

Vara 3. 566  ft. 

Braca 7-132  “ 

Geora 1.448  acres. 

Also  same  as  Portugal , and  sometimes 
as  in  England. 

Buenos  Ayres. 

Vara 2. 84  ft. 

Legua 3.226  miles. 

Suertes  de  Estancia ....  27  000  sq.  varas. 
Also  same  as  Spain * 
Burmah. 

Paulgat 1 men. 

Dain 4.277  yds. 

Viss 3.6  lbs. 

Taim 5.5  “ 

Saading 22  u 

Also  same  as  England. 

Canary  Isles. 

Onza 

Pic,  Castilian 

Almude 0416  acre. 

Fanegada 5 “ 

Libra 1.0148  lbs. 

Also  same  as  Spain. 

Cape  of  Good  Hope. 

Foot n. 616  ins. 

Morgen 2. 1 16  54  acres. 

Also  same  as  in  England. 
Ceylon. 

Seer 1 quart. 

Pariah 5.62  galls. 

Also  same  as  in  England. 
China. 

Li 486  inch. 

Chih,  Engineer’s 12.71  ins. 

“ or  Covid 13-125  “ 

“ legal 14. 1 “ 

Chang 131-25  “ 

“ legal 141  “ 

P« 4-°5  ft- 

Chang,  fathom 10.0275  ft. 

Li 486  yds 

Pu  or  Rung 3.32  sq.  yds. 

King,  100  Mau 16.485  acres. 

Tau 1. 1 3 galls. 

Tael i-333  oz. 

Catty 1.333  lbs. 

Cochin  China. 

Thuoc  or  Cubit 19.2  ins. 

Sao 64  sq.  yds. 

Mao 1.32  acres. 

Hao 6. 222  galls. 

Shita 12.444  u 

Nen 8594  lb. 

Colombia  and  Venezuela. 

Libra 1. 102  lbs. 

Oncha 25  “ 

Also  Decimal  System. 


50 


foreign  measures  and  weights. 


Hentnarli,*  Greenland,  Ice- 
land, and  Norway. 

Tomme 1-0297  ins. 

Fod  — 

Favn,  3 Alen 


1.0297 


ft. 


6.1783  “ 

Mil'"?. “7"  4-6&> 55  mites. 

“ nautical o6lOJ2«ik 

Anker  8.0709  galls. 

"nKtr .a78  bush. 

Skeppe y o u 

^ingkar;...;.....,. 

Lispund i7'367  u 

Centner 

* Also  Decimal  System. 


Hungary. 

Fuss...:.. 

30-67  “ 

9.139  yds- 

Also  as  in  Vienna. 


Elle. 


Meile  . 


Guz 

Cowrie 


Indian.  Empire. 

27.125  ins. 

i sq.  yd. 

Sen  ... . . • ' • ' 61025  39  <^  ms 

u 2.204737  IDS. 

Uniform  standard  of  multiples  of  the  Sen 
adopted  in  1871. 


Ecuador. 
Decimal  System. 


Genoa,  Sardinia,  and 
Turin. 

Palmo. 9-8076  ins. 

Piede,  Manual,  8 oncie. . . 13- 488  u 
“ Lipraudo,  12  •“  ...20.23 
Trabuco  orTesa IO,IrL  miles 

Starello 98o4  ac™- 

Giomaba 9394 


Italy. 

IMilan  and  "Venice. 
Decimal  System. 

The  Metre  is  termed  Metra ; the  Are,  Ara ; 
the  Stere,  Stero;  the  Litre,  Litro;  the 
Gramme,  Gramma,  and  the  Tonneau, 
Tonnelata  de  Mare. 

Naples  and  Two  Sicilies. 
Palmo 

mi 


Germany. 

The  old  measures  of  the  different  States 
differ  very  materially ; generally , how- 
ever, 

Foot.  Rhineland 12. 357 

Meile 4.603  miles. 

Decimal  System  made  compulsory  in  1872. 


Greece. 

Stadium 6155  niile. 

Also  Decimal  System. 


,r.  ,•  i.mo6  miles. 

MSltego:::::::::::'.::'.::  .7467 acre. 

Moggia ° « 

Pezza,  Roman 

Roman  States. 

Old  Measure. 

Foot to* 

“ Architect’s.. n-73 

Braccio.. 30-73  u 

Palmo 8-347 

Miglio 1628  yds. 

Quarta 1. 1414  acres. 

Lucca  and  Tuscany. 


Guinea. 


Jachtan  . 


, 12  ft. 


Hamburg. 

Fuss 11.2788  ins. 

Klafter  5-6413  “-•  - - 

Morgen 2.386  acres. 

Cube  Fuss -831 1 CUV;  ft’ 


Pie 

Palmo 

Braccio 

Passetto 

Passo 

Miglio 

Quadrato 

Saccato  


Cube  Fuss. 

Tehr 99-73  ,7 

Viertel i-594  7 galls- 

Pfund  (500  grammes) . . . 1-102  32  lbs. 

Ton 21 35- 3 lbs. 

Also  Decimal  System. 


.6476  acre. 


Hanover 

Fuss i1-5,5' 

Morgen 

Hindostan. 

Borrel '“f- 

2.387  u 



Kobe 29-o65 

Coss 3-65  f - 

Tuda 1.184  cub.  ft. 

Candy ..,.14.209 


. 11.94  ms. 
• XI-49  “ 


. 3.829  ft. 

• 5-74  “ 

. 1.0277  miles. 
. .8413  acre. 

. i-324  u 


Sun, 


1. 193*  ins. 
,5*  ins. 


Japan 
303  03  -Metre. . . 

Shaku,  3.0303  Metres. . . . n-93°5* 

Jo,  30.303  ‘ ••••  9-9421*  ft. 

R,e,nA88o5  • “ 8%| tmnek- 

Kai-ri 6080  feet  t 

Hiro •••••  4-97**  4 


1*  feet. 


Momme : . 3-756  521  7 g™™™^ 


Hiyaku-me o'8qSi7 

Kwam-me 8.28171  ^ 

Hiyak-kin - 132-507  32  u 

Man’s  load S7-972  u 

Koku ^‘q68!1  “ 

Hiyak-koku 33126.8308 

* These  are  as  equivalent  as  they  are  prac  1 
cable  of  reduction. 

+ Admiralty  knot. 


FOREIGN  MEASURES  AND  WEIGHTS. 


51 


Java. 

Duim 1.3  ins. 

Ell 27.08 

Rjong 7.015  acres. 

Ran 328  galls. 

Tael 593-6  grains. 

Sach 61.034  lbs. 

Pecul 122.068  “ 

Catty... 1.356  u 

IVTadras. 

-Ady — ... 10.46  ins. 

Covid 18.6  “ 


Guz. , 


■ 33 


Culy 20.92  ft. 

League 3472  yds. 

338  galls. 

M areal 2.704  “ 

Tola 180  grains. 

Seer 625  lbs. 

Viss 3.086  “ 

Maund .....24.686  u 

Malabar. 

10.46  ins. 

iVlalaeoa. 

Hasta  or  Covid 

Depa 

Orlong 


. 18.125  ms. 
. 6 ft. 

. 80  yds. 


Palmo. 
Pie. . . . 
Canna. , 
Salma. , 


NXalta. 


Foot 

Kot,  silk. , 
Fathom  . 


Also  as  in  Sicily. 
^Moldavia. 


. 10.3125  ins. 
. 11.167  “ 

• 82.5  “ 

4.44  acres. 


. 8 ms. 

. 24.86  ins. 
. 8 ft. 


Molucca  Islands. 

Covid 18.333  ins. 

Morocco. 

Tom  in 
Cadee. 

Cubit. 

Muhd. . 


. 2.81025  ms. 
. 20.34  ins. 


Kma, on.:: :::::: :::::::  ll%35e^ 

Rotal  or  Artal I-I2  ibs. 

Liquids  other  than  oil  are  sold  by  weight. 

Mysore. 

Angle 

Haut 

Guz 



Netherlands. 

Elle*--v 39-370  432  ins. 

Decimal  System  since  1817. 


2.12  ms. 

19. 1 11 

38.2 


Gereh . 


JPersia. 


Gueza,  common ’ 2t- 

“ Monkelrcr 37.5 


375  ms. 


Archin,  Schah  ...........  31.55  ins. 

“ Arish 38.27  “ * 

Parasang 6076  yds. 

Chenica 80. 26  cub.  ins. 

Artaba 1 . 809  b u sh. 

Mi  seal grains. 

Eatei 2.1136  lbs. 

Batman  Maund 6.49  “ 

Liquids  are  measured  by  weight. 


Trewice  . 


IPoland. 


. 14.03  ms. 


Precikow jns> 

Pretow 4.7245  yds. 

Mile,  short 6075  yds 

Morgen 1.3843  acres. 

Portugal  and  Mozambique. 

foot ins. 

M]iila 1.2788  miles. 

Almude 3.7  galls. 

ffD£a 1.488  bush. 

Alguieri 3.5  « 

Libra..: 1. 012  lbs. 

Also  Decimal  System. 

Prussia. 

12.358  ins. 

Rathe.v . 4.  n92  yds. 

Meile 24  ooo  feet. 

Quadrat  Fuss 1.0603  sq.  ft. 

MR1^ 631 03  acre. 

Cobe  FuSS 1.092  cub.  ft. 

Scheffel 1.5121  bush. 

p"fnertr 7-559.galls. 

iouna..  7217  grams. 

Zollpfund 1.1023  lbs. 

Centner 113-43  tbs. 

K-nssia. 

Vershok 1.75  ins. 

f o°t  12  ins. 

Arschine u 

Rhein  Fuss. 


1.03  ft. 


Sajene  ...  £ 

^.rst 3500  - 

Dessatiila 5-5574  miles. 

uessatina ; . 2.4954  acres. 

Vedro 2. 7049  galls. 

1.4426  bush. 

Tschetwert 5.7704  “ 

E™?d 63i7  grains. 

90285  lbs. 

Decimal  System  adopted  in  1872. 

Siam. 

S^R 9-75  ins. 

Ken 39  u 

dod; 09848  mile. 

Roeneng 2.462  miles. 

Silesia. 

I™? ins. 

S 4.7238  yds. 

Meile 7086  yds. 

MorSen 1.3825  acres. 


52 


FOREIGN  MEASURES  AND  WEIGHTS. 


Singapore. 

Hasta  or  Cubit 18  ms. 

Dessa “• 

Orlong 80  yds- 

Smyrna. 

pjc  26.48  ins. 

Indise  24-648  “ 

Berri 1S28  yds. 

Spain,  Cuba,  Malaga,  Ma- 
nilla, Guatemala,  Hondu- 
ras, and.  Mexico. 

Pie 11.128  ins. 

Vara 33-384  “ 

Milla ...  .865  mile. 


Legua,  8000  varas 

Fauegada 

Vara,  cubo 

Cuartilla 

Arroba,  Castile 

Fanega 

Libra 

Tonelada 

Also  Decimal  System. 


4. 2151  miles. 
. 1.6374  acres. 
. 21.531  cub.  ft. 
..  .888  gall. 

• • 3-554  galls- 

. . 1.5077  bush. 

. . 1. 0144  lbs. 
2028.2  lbs. 


Stettin. 


Fuss 

Foot,  Rhineland. 

Elle 

Morgen 


, 11. 12  ms. 

• 12.357  “ 

. 25.6  ins. 

. 1.5729  acres. 


Sumatra. 

Jankal  or  Span 9 ins. 

Elle - *8  ‘ 

Hailoh 36 

Fathom ° 

Tung 4 yds. 

Surat. 

Tussoo,  cloth 1.161  ins. 

Guz,  “ 27.864  1 

Hath 20.9 

Covid . . .. i8-5 

Biggah acre. 

Sweden. 

Fot 11.6928  ins. 

Ref  ’. yds- 

Faden 5.845  ft. 

League 3-  35^4  miles. 

Meile 6-64i7 


Tunnland 1.2198  acres. 

Anker 8.641  galls. 

Spann 1.962  bush. 

Centner 1 12.05  lbs. 

Also  Decimal  System. 
Switzerland. 

Fuss,  Berne n.52  ms. 

“ n-54  “ 

Vaud 1 1- 81  “ 

Klafter 5-77  .. 

Meile 4.8568  miles. 

Juchart,  Berne 85  acre. 

Maas 2.6412  pints. 

Eimer S^^galls 

M alter 4.1268  bush. 

Ffund 1 -1023  lbs. 

Also  Decimal  System. 
Tripoli. 

Pik,  3 palmi 26.42  ins. 

Almud 4 cu>>.  ins. 

Killow 2°23 

Barile 14.267  galls. 

Temer 7383  bush. 

Kottol 768°  §»'“*• 

Turkey. 

Pic,  great 27.9  ins. 

“ small .....27.06  ‘ 

Berri I-828  yd.®- 

Alma 1. 1 54  galls. 

Also  Decimal  System. 

W urtemberg 


Fuss 

FUp.  

Meile 

Morgen 

8146.25  yds. 

7703  acre. 

Cube  Fuss. . . . 

. .83045  cub.  ft 

Eimer 

Scheflfal  . - - - 

4.878  bush. 

Pound  

Fuss 

Zurich. 

FUft  

Klpft^r 

^.0062  ft. 

4.8568  miles. 

Jachart. ..... 

808  acre. 

1 Cube  Klafter. 

144  cub.  ft. 

LENGTHS 
Course.  Miles. 


OF  ENGLISH  RACE-COURSES. 


NEWMARKET. 

Across  the  Flat 

Beacon 

Cambridgeshire 

Cesare  witch 

Round  

Rowley  Mile 

Summer  Course 

Two-year  old,  new . . 
Yearling 


1.292 
4.206 
1. 136 
2.266 
3-579 
1.009 
2 

.702 

.277 


Course. 


DONCASTER. 

Circular 

Fitzwilliam 

Red  House 

St.  Leger 

Cup  Course 

EPSOM. 

Craven 

Derby  and  Oaks. . 
i Metropolitan 


I-9I5 

1 

•711 

1.825 

2.634 

1.25 
i-5 

2.25 


Course. 


• goodwood. 
Cup  Course 

LIVERPOOL. 

New  Course 

New  Castle  — 
Oxford 

YORK. 

Stakes  Course. . 
Two-mile 


Miles. 


2-5 

1.5 

1.796 


x-75 

1.923 


SCRIPTURE  MEASURES. ANCIENT  WEIGHTS. 


53 


SCRIPTURE  AND  ANCIENT  LINEAR  MEASURES. 
Scripture. 

912  inch.  I Span,  3 palms IO  inc: 

Palm,  40  digits 3.648  ins.  | Cubit,  2 spans u * 

Fathom,  4 cubits. . 7 feet  3 .552  ins. 

Hebrew  and.  Egyptian. 

Nahud  cubit i.475  feet.  I Babylonian  foot T T .n  fppt 

Royal  “ 1 . 721 6 “ Hebrew  “ 1,140  leet- 

Egyptian  finger 06145  “ I “ cubit.  ..!!*!!!  1 

Hebrew  sacred  cubit 2.002  feet. 


1. 212 
i.8i7 


Digit 

Pons  (foot) 

Cubit 

Pythic  or  natural  foot . 
Attic  or  Olympic  “ . 


Grrecian. 
7554  inch. 
oo73  feet. 

1332  “ 

814  foot. 

009  feet. 


Ancient  Greek  foot  ) 

(16  Egyptian  fingers) ) •••••  -9841  foot. 

Arabian  foot 1.095  feet. 

Stadium 604.0^5  “ 

Olympic  stadium 606.29  “ 


Mile,  8 stadium 48^  feet 

Alexandrian  or  Phileterian  stadium  (600  Phil.  feet)  = 7o8  65  feet 
Volume.  Keramion  or  Metretes 3.488  gallons.  ' 


Jewish. 

Cubit.... 1.824  feet. 

Sabbath  day’s  journey 3648  “ 


I Mile,  4000  cubits ?2g6  feet 

I Day’s  journey 33.i64  mfles. 

Roman.  Long  Measures. 


Digit 723  73  ins. 

Uncia  (inch) q67  “• 

Pes  (foot) 11.604  “ 


Passes "I505  *!** 

r assus 4*835  tf 

Mile,  milliarium 4842  “ 


ANCIENT  WEIGHTS. 
Hebrew  and  Egyptian. 


Attic  obolus 

Troy  grains. 
{ «•’? 

Denarius, 

Roman 

Troy  grains. 
j 51-9* 

“ drachma 

X 9-*t 

(51-9* 

U 

Shekel. . . 

Nero 

( 62. 5f 

Lesser  mina 

(69 1 

Ounce  . . . 

(4I5-I* 

Greater  mina 

Egyptian  mina 

Drachm. . 

(43i-2$ 

Ptolemaic  “ . 

Libra 

Alexandrian  “ 

y D 

• • • 

Pound. . . 

. . 12  Roman  ounces. 

Obolus 

Talub 

Talent  (60  minae) 


Obolus,  ancient . 


Gramme ......! 

Drachma. 50 

u great .’! !!.’!!!  69 


Grrecian 

Troy  grains. 

" 33 
57 
i5 


. 56  lbs.  avoirdupois. 


Mina 

‘ ‘ great . . 

Talent 

“ Attic . 


47  I 


Troy  ounces. 
. ..  10.41 

...  14.472 

. . 625.19 
. . 868.32 


Roman. 

0unce 416.82  grains.  | Pound. 


10.41  ounces. 


* Christian!. 


+ Arbuthnot. 

E* 


X Paucton. 


54 


GEOGRAPHIC  MEASURES  AND  DISTANCES. 


GEOGRAPHIC  measures  and  distances. 

rp  oil  nee  Longitude  into  Time. 

• and  seconds  by  4,  and  product  is 

th.6  time.  , . . o ‘31^  v 4 = fh.  22 w.  4®* 

Example.— Required  time  corresponding  to  50  31  • 50  31  X 4 3 

To  Reduce  Time  into  Longitude. 

Rule Reduce  hours  to  minutes  and  seconds,  divide  by  4,  and  quo- 

tient  is  the  longitude.  Or,  Multiply  them  by  15. 

EXAMPLE. -Required  longitude  corresponding  tc 

5h.  8 m.  11. 2 s.  = 308m.  11.2s.,  which  . 4 — 77  45-5  • 

Or,  multiplying  by  15:  Sh-  8w*  I1*2S*  X 15  = 77°  2 45-5  • 


Table  of  Departures  for 
— * ' Course. 


Course. 

Departure. 

3.5  points. 
4 “ 

•773 

•7°7 

Distance  run  of  1 IVtile. 
Departure.  II  Course.  | Departure. 


4.5  points. 


5. 5 points. 
6 


Thus  if  a vessel  holds  a course  of  4 points  that  is  without  leeway,  for  distance 
0fo“a  ™sseeiIaUi“galeN7E7  upo^a'  c'oursTof  6 points  for  100  miles  will  make  38.3 

froox  083)  miles  of  longitude.  _ . ^ 

tioo  x r l point  of  tiie 

Miinntes,  and  Seconds  of  eacn  ro 
Compass  with.  Meridian. 

« • " Sin.  A.* 


Degrees 

North. 


South. 


N.N.E 

N.N.W 


N.E.  by  N.  . 
N.  W.  by  N. . 


Points. 


N.E. . 
N.W. 


N.E.  by  E.  .. 
N.W. by  W. . . 


E.N.E 

W.N.W 


E.  by  N 

W.  by  N 

East  or  West. 


S.S.E 

s.s.w 


S.E.  by  S.  ...  . 
S.W.  by  S.... 


S.E 

S.W 


S.E.  by  E.... 
S.W.  by  W... 


E.S.E.  .. 
W.S.W.  . 


E.  by  S.. 

W.  by  S.. 

East  or  West. 


2 48  45 
5 37  30 
8 26  15 

11  i5 
14  3 45 
16  52  30 
19  41  *5 
22  30 
25  18  45 
27  7 30 
30  56  15 

33  45 
36  33  45 
39  22  30 
42  11  15 

45 

47  48  45 
50  37  3° 
53  26  *5 
56  15 
59  3 45 
61  52  30 
64  41  15 
67  30 
70  18  45 
73  7 3° 
75  56  i5 

78  45 
81  33  45 
84  22  30 

87  11  15 

90 


| Cos.  A.* 

Tan.  A.* 

.9988 

.0491 

.9952 

.0985 

.9891 

.1484 

.9808 

.1989 

•97 

.2504 

.9569 

•3034 

.9415 

•3578 

.9239 

.4142 

•9°4 

•4729 

.8819 

•5345 

•8577 

•5994 

• 8315 

.6682 

.8032 

.7416 

•773 

.8207 

.7409 

.9063 

.7071 

1 

•6715 

1-103 

•6344 

1.218 

•5957 

1.348 

•5556 

1.497 

•5141 

1.668 

.4714 

1.871 

•4275 

2.114 

.3827 

2.414 

•3368 

2-795 

.2903 

3.296 

.2429 

3-94i 

.195 

5.027 

.1467 

6.741 

.098 

10.153 

.0489 

20.555 

.0000 

03 

* A.  re 


■presenting  course  or  points  from  the  meridian. 


GEOGRAPHIC  LEVELLING. 


55 


GEOGRAPHIC  LEVELLING. 

Curvature  and  Refraction. 

Correction  for  Curvature  of  Earth,  to  be  subtracted  from  reading  of 
a levelling-staff,  is  determined  as  follows  : 

Divide  square  of  distance  in  feet  from  level  to  staff,  by  Earth’s  Equa- 
torial diameter — viz.,  41  852  124  feet. 

Or,  Two  thirds  of  square  of  distance  in  statute  miles  equal  the  cur- 
vature in  feet. 

Correction  for  Refraction  is  to  be  added  to  reading,  and  as  a mean 
may  be  taken  at  about  one  sixth  of  that  for  curvature. 

Correction  for  Curvature  and  Refraction  combined,  is  to  be  subtracted 
from  reading  on  staff. 

Formulas  of  Capt.  T.  J.  Lee , U.  8.  Engineers . 

D2  . D2 

— = correction  for  curvature , — m = correction  for  refraction,  and 
D2 

(r  — 2 m)  — = correction  for  curvature  and  refraction.  D representing 

distance , R radius  of  earth , and  m a coefficient  of  refraction  = 075  all 
in  feet.  /J’ 

Illustration.  A distance  is  3 statute  miles,  what  is  correction  for  curvature 
and  refraction? 


5280  X s 

(1  — 2 x .075)  4I  g52 124  = • 8s  X 5-996  = 5-09 7 feet 


Approximately , — D2  = curvature  in  feet. 
3 


Revelling  Toy  Roiling  Boint  of  Water. 

To  Compute  Height  Above  or  Below  Bevel  of  Sea. 

517  (2i2°-T)  + (2120  - T Y = Height. 

Illustration.— What  is  height  of  an  elevation,  when  boiling  point  of  water  is  182°  ? 
517  X 212°  182°  + 2I2°-i82°2-  5I7  X 30+  302  = 1 6 410  feet. 

Corrections  for  Temperature  to  he  made  in  Connection  with  Formula. 

Temp.j  Temp.  Temp.  C^T  Temp.  Temp.  C?"n?  Temp. 


.972 

.976 

.98 

.984 

.988 

.992 

.996 

r.  004 


1.008 
1. 012 
1. 016 
1.02 
1.024 
1.028 
1.032 
1.036 
.041 


54 

56 

58 

60 

62 

64 

66 

68 

70  | 


1.046 
1.05 
1.054 
1.058 
1.062 
1.066 
1. 071 
I-°75 
•°79 


1.083 
1.087 
r.091 
[.096 
c.  1 
[.  104 
[.  108 
[.  112 
[.  116 


90 

92 

94 

9o 

98 

100 

102 

104 

106 


1. 124 
1. 128 
1. 132 
1.136 
1. 14 
r- 144 
1.148 
1. 152 


' ' 7 I 1UU 

Illustration.  Ass u m e temperature  in  preceding  illustration  to  have  been  8o-. 
Then  16410X  1.1  = 18051/eet 


and  Differences  Let  ween.  True 


56  GEOGRAPHIC  LEVELLING  AND  DISTANCES. 


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GEOGRAPHIC  LEVELLING. — MAGNETIC  VARIATION.  57 

Illustration.  — Curvature  of  Earth  independent  of  refraction  is  computed  at 
.667  foot  = 8.004  ins.  for  1 geographical  mile,  and  as  refraction  on  land  is  taken  as 
.104  foot  or  1.248  ms.,  and  on  ocean  at  .099  foot  or  1.188  ins.,  relative  visible  dis- 
tances of  an  object,  including  curvature  and  refraction,  for  an  elevation  of 
.667  foot  is  1.09  miles  on  land,  and  1.08  miles  at  sea. 

I “ “ I.33  “ “ “ “ j o2  “ a 

9 feet  “ 4 “ “ “ “ 3 8 u « u 

1 mile  “ 104.03  “ “ “ “ 103.54  “ “ “ 

Difference  between  two  levels  in  feet  is  as  square  of  their  distance  in 
miles. 

Illustration  —At  what  elevation  can  an  object  be  seen,  at  surface  of  ocean  when 
it  is  2 miles  distant  ? * 

I2  : 22  ::  .568  : 2.272  feet  '==  2 feet  3.25+  ins. 

. 5^erence  between  two  distances  in  miles  is  as  square  root  of  their  heights 
m feet.  0 

Illustration  i.  — At  an  elevation  of  9 feet  above  level  of  sea,  at  what  distance 
can  an  object  be  seen  upon  its  surface?  ’ bianco 

V- 568  = .754::  1 : : if  9 ; 3. 98  miles. 

2 ATIf  a I5an,at  the  ^re-topgallant  mast-head  of  a vessel,  100  feet  from  water  sees 
another  and  a large  vessel  “hull  to,”  how  far  are  the  vessels  apart?  ’ 

A large  vessel’s  bulwarks  are  at  least  20  feet  from  water 
Then,  by  table,  100  feet = 11.27 

20  “ 5-93 

Distance 19.20  miles. 

When  an  observation  for  distance  is  taken  from  an  elevation,  as  from 
a light-house,  a vessel’s  mast,  etc.,  of  an  object  that  intervenes  between 
observer  and  horizon,  or  contrariwise,  observer  being  at  a horizon  to 
elevated  object,  distance  of  observer  from  intervening  object  can  be 
determined  by  ascertaining  or  estimating  its  elevation  from  horizon  and 
subtracting  its  distance  from  whole  distance  between  observed  and 
pomt  from  which  observation  is  taken,  and  remainder  will  give  distance 
of  object  from  observer.  & 

fa —Top  of  smoke-pipe  of  a steamer,  assumed  to  be  50  feet  above  sur- 

fr0m  aQ  100  ^et;  whatTs  ffi 

100  feet = 13  27 

50  “ ■ 9*  38 

Distance 3.89  miles. 

D Refraction  = -S«  ^ for  land  and  .563  D=  for  sea 


MAGNETIC  VARIATION  OF  NEEDLE. 

-Needle  reached  a Westerly  maximum  in  1660  and  then 
varied  to  East  until  1800,  when  it  reversed  to  West.  ’ 

Easf trf o ’ w"~ frT  t0  1815  variation  ranged  from  n°  15' 

24  27  West,  when  it  receded  gradually  to  210  in  186^. 

Jamaica  (W.  I.). — No  variation  from  year  1660. 

Diurnal  Variation.— There  is  a small  diurnal  variation  bein-  greatest 

3 r; zw&ztnzr*'  «• 


58 


MAGNETIC  VARIATION  OF  NEEDLE. 


Variation  in  U.  &— Professor  Loomis  concludes  that  the  Westerly 
variation  is  increasing  and  Easterly  diminishing  in  every  part  of  United 
States-  that  this  change  occurred  between  1793  and  1819,  and  that 
present  annual  change  is  about  2'  in  Southern  and  Western  States,  from 
3'  to  4'  in  Middle  States,  and  5'  to  7' m Eastern  States. 

Rules  for  computation  of  variation  are  empirical,  except  in  each 
particular  locality,  as  the  annual  and  diurnal  variations  of  the  needle, 
added  to  local  attraction,  render  it  altogether  unreliable. 

Decennial  Variation  of  Needle. 

Mr.  Schott , U.  S.  Coast  and  Geodetic  Survey. 

From  January  i,  1790,  to  January  1, 1880. 


Location. 


Halifax,  N.  S 

Quebec,  Can 

Portland,  Me 

Burlington,  Vt. . . . 
Newburyport,M’s. 
Portsmouth,  N.  H. 

Rutland,  Vt 

Salem,  Mass 

Boston,  Mass 

Cambridge,  Mass. . 


1 790.  | 1 800.  | 1810. 

wT 


w. 


w. 


i5-9 


1820. 


W. 


Hartford,  Conn.  . . 

5-2- 

New  Haven,  Conn. 

4.8 

New  York,  N.  Y.  . 

4.29 

Philadelphia,  Pa. . 

2.4 

Baltimore,  Md. . . . 

— 

Albany,  N.  Y 

— 

Buffalo,  N.  Y 

.14 

E. 

Erie,  Pa. 

•03 

Cleveland,  0 

2.2 

Detroit,  Mich 

- 

Washington,  D.  C. 

.1 

Acapulco,  Mex.. . . 

7.2 

Charleston,  S.  C.. . 

Havana,  Cuba 

— 

Kingston,  W.  I — 

6-3 

San  Diego,  Cal 

11 

Savannah,  Ga 

— 

Mobile,  Ala 

— 

Key  West,  Fla 

. — 

Monterey,  Cal.  ... 

11. 4 

Mexico,  Mex 

7.1 

New  Orleans,  La. . 

■ 7 

San  Bias,  Mex 

■ 7-41 

San  Francisco, Cal 

. 12.8 

Sitka,  Alaska 

. — 

Vera  Cruz,  Mex. . 

• 8.37 

E. 


•35 


1830. 


1840.]  1850.  | i860. 


W. 


7- 1 


W. 


w. 


w. 


1870. 


w. 


8.95  I 9-32 


IV 

8 

8.64 

9-33 

10.03  i' 

6-73 

7-43 

8.31 

9.09 

5. 46 
5 

5-8 

5-43 

6.24 

5-99 

6.77 

6.67 

4-47 

4.91 

5-59 

6-34 

2.28 

2.71 

3-33 

4. 11 

.8 

1.2 

x-7 

2.4 

5-79 

6.32 

6.97 

7-7 

• 3 

•74 

i-33 

2.05 

E. 

E. 

•83 

•43 

•17 

•25 

E. 

E. 

i-5 

1.05 

.6 

'I4! 

2.9 

2-55 

2.09 

1.56 

W. 

W. 

W. 

w. 

.6 

1 

1.49 

1.99 

E. 

E. 

E. 

E. 

8.68 

8.88 

8.91 

8.79 

4.04 

3-44 

2.78 

2.12 

5 6.22 

6. 12 

5-94 

5-71 

5-4 

5 

4.6 

4.2 

11. 6 

11. 9 

12.2 

12. 54 

4.8 

4-5 

4.14 

3-65 

7-3 

7.2 

7-1 

7 

6.9 

6.52 

6.03 

5-47 

13-3 

13-9 

14.44 

14-95 

8.6 

8.8 

8.9 

8.76 

8.1 

8.2 

8.14 

7-94 

8 8.61 

8.84 

8.97 

9.9 

1 14.42 

14.92 

x5-38 

15-78 

1 27. 89 

28.48 

1 28.88 

29.08 
. 8.66 1 

12  9.48 

9.42  1 9.14 

i3- 15 
11.97 


11.49 

12.8 


i-5 


7-99 

8.18 


2.23 


1.07 


3.08 

8.8 
4.86 

15.42 
8.48 
7.61 
8.91 
16. 11 
29.08 


16.36 

28.88 

7-i5 


16. 52 
28.5 


For  variation  in  other  locations  in  unirea  oiawb  *uu 
see  treatises  of  J.  B.  Stone,  C.E.,  New  York,  and  Heller  and  Brightly, 
Philadelphia,  1878. 


MAGNETIC  VARIATION  OF  NEEDLE. 

Table  for  Reducing  Observed.  Daily  Variation,  of  Needle 
to  Mean  Variation  of*  the  Day. 

* TJ.  S.  Coast  and  Geodetic  Survey,  1878. 


Needle  East  of  Mean  Mag- 

Needle  West  of  Mean  ft 

Season. 

netic  Meridian. 

Meridian. 

A.M. 

A.  M. 

A.M. 

A.M. 

A.M. 

A.M. 

NOON. 

P.M. 

P.M. 

P.  M. 

h. 

h. 

h. 

h. 

h. 

h. 

h. 

h. 

~hT 

6 

t 

7 

1 

8 

/ 

9 

f 

10 

II 

Noon. 

1 

2 

3 

Spring 

3 

4 

4 

3 

I 

I 

4 

5 

5 

4 

Summer. 

4 

5 

5 

4 

I 

2 

4 

6 

5 

4 

Autumn 

2 

3 

3 

2 

— 

2 

3 

4 

3 

2 

Winter 

1 

i 

2 

2 

I 

— 

2 

3 

5 

2 

Variation  of*  Needle 
U.S. 

Location. 


Astoria,  TV.  T 

Augusta,  Ga 

Austin,  Tex 

Bismarck,  Dak 

Chicago,  111 

Cincinnati,  0 

Colorado  Springs,  Col. , 

Columbia,  S.  C 

Columbus,  0 

Deadwood,  Dak 

Denver,  Col 

Detroit,  Mich 

Duluth,  Min 

Galveston,  Tex 

Green  Bay,  TVis 

Houston,  Tex 

Indianapolis,  Ind 

Jackson,  Miss 

Jacksonville,  Fla 

Kansas,  Kan 

Keokuk,  la 

Little  Rock,  Ark 

Louisville,  Ky 

Milwaukee,  \75s. 


at  Locations  in  United  States  and 
Canada,  1S775. 

Coast  and  Geodetic  Survey. 

EAST. 

Variation. 


30 

28 

15 

6 

55 

18 

45 


38 


Location. 


Augusta,  Me 

Bangor,  Me 

Batavia,  N.  Y, 

Belfast,  Ale 

Bridgeport,  Conn. . 

Calais,  Me 

Concord,  N.  H 

Dover,  Del 

Fall  River,  Mass. . . 

Hamilton,  Can 

Harrisburg,  Pa.. . . . 

Hudson,  N.  Y. 

Lewiston,  Me 

Lowell,  Mass 

Montpelier,  Vt 

Montreal,  Can 

New  Bedford,  Mass, 
New  London,  Conn. 
Newark,  N.  J. 


48 
WEST. 


Montgomery,  Ala 

Natchez,  Miss 

Nebraska,  Neb 

New  Orleans,  La 

Olympia,  W.  T 

Omaha,  Neb 

Oregon  City,  Or 

Paducah,  Kan 

Portland,  Or 

Port  Townsend,  TV.  T.. 

Sacramento,  Cal 

Salt  Lake  City,  Utah. . 

San  Antonio,  Tex 

Santa  Barbara,  “ ... 

Santa  Fe,  N.  Mex 

Springfield,  111 

St.  Augustine,  Fla.  . . . 

St.  Louis,  Mo. 

St.  Paul,  Minn 

Tallahassee,  Fla .* 

Toledo,  0 

Topeka,  Kan 

Vincennes,  Ind 

Yazoo,  Miss 


14 

16 

34 

4 

40 

i5 

22 

8 

18 

12 

11 

42 

4 

12 

10 

30 

2 

55 

4 

18 

8 

14 

48 

11 

15 

12 

5 

12 

20 

10 

30 

9 

15 

7 

18 

Newburgh,  N.  Y 

Newport,  R.  I 

Norfolk,  Va 

Ogdensburgh,  N.  Y. . 

Oswego,  N.  Y 

Ottawa,  Can 

Pittsburgh,  Pa 

Raleigh,  N.  C 

Richmond,  Va 

Rochester,  N.  Y. 

Saratoga,  N.  Y 

Stamford,  Conn.  . . . 

Syracuse,  N.  Y 

Toronto,  Can 

Trenton,  N.  J 

Troy,  N.  Y 

Utica,  N.  Y. 

Wilmington,  Del 

Wilmington,  N.  C. . . 


Variation. 


5 2 

7 26 

11  20 

6 


55 


23 

*7  4 

*7 

9 17 
J4  58 
13  18 
6 3 

2 55 
6 30 

10  30 

4 14 


4 

35 

25 

8 

38 

28 

24 

48 


60  GEOGRAPHIC  LEVELLING. — BASE  LINE. — SOUNDINGS. 


Dip 


of  Horizon. 

Approximate,  57.4  V H =.  dip  in  seconds,  varying  with  temperature  of 
air.  H representing  height  of  observer's  eye  in  feet. 

.667 «2  = H : ' .498 s"  = H : 1.42  VH  = *:  1.23  VH  = n. 

n representing  distance  in  geographical  miles  and  s in  statute. 


Multi- 

plier. 

Angle. 

Multi- 

plier. 

Angle. 

Multi- 

plier. 

Angle. 

Multi- 

plier. 

Angle. 

Multi- 

plier. 

Angle. 

1 

45  0 

2.5 

68  11 

4 

75  58 

5-5 

79  42 

8 

82  52 

1.5 

56  18 

3 

7i  34 

4-5 

77  29 

6 

Bo  32 

9 

83  40 

2 

63  26 

3-5 

74  4 

5 

78  41 

7 

81  52 

10 

84  17 

uptruuLuri.  — oci  dvakcj-lu  vw  — o — ; « 

multiplied  by  number  opposite  to  it. 

Illustration.  — When  sextant  is  set  at  8o°  32',  and  horizontal  distance  from  ob- 
ject in  a vertical  line  is  100  feet,  what  is  its  height? 

100  X 6 = 600  feet. 

By  Trigonometry:  1 : 100  *.*.  5.997  (tan.  angle)  : 599-7  feet- 


\{ 


To  Reduce  a Sounding  to  Dow  NV'ater. 

180 1'\ 


1 zp  cos>  2zrJLj  = h'.  h representing  vertical  rise  of  tide,  and  h!  sound U 


ina  or  depth  at  low  water,  both  in  feet;  t time  betiveen  high  and  low  water,  and 

J L _ 180 1 0 

t>  time  from  time  of  sounding  to  low  water,  in  hours.  — cos.  when  -j-  <90  7 

and  4*  cos.  when  >90°. 

Illustration.  — Low  water  occurring  at  3.45,  and  high  water  at  10.15  p.m;,  a 
sounding  taken  at  5.30  p.m.  was  18.25  feet;  what  was  depth  at  low  water,  vertical 
rise  being  10  feet? 

h = 10  feet ; V — 5 h.  30 m.  — 3 h.  45™.  = ih.  45™-  = 1.75  hours. 
t=z  ioh.  15 m.  — 3 h.  45 m.  = 6h.  30WI.  — 6.5  hours. 

10/  180X1.75' 

Then  “ ( 1 4- cos- 


6-5 


3/1.  45WI.  = On.  30/rt.  _ O.  5 iivui  a. 

— 5(1  — 48°  27'  24")  = 5 X (1—  .663 186)  = 1.684 07  feet 


Sounding  18.25 /eei  — Reduction  1.68407/^=16.56593/^. 


Ijeii"tlis  of  a Degree  of  Longitude  on  parallels  of  Dati- 
tvTde,  for  each,  of  its  Degrees  from  Equator  to  Dole. 


Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

i° 

59-99 

1 6° 

57- 67 

3i° 

5i-43 

46° 

41.68 

6i° 

29.09 

76° 

14.52 

2 

59.96 

17 

57-38 

32 

50.88 

47 

40.92 

62 

28.17 

77 

x3-5 

3 

59.92 

18 

57.06 

33 

50.32 

48 

40.15 

63 

27.74 

78 

12.48 

4 

59-85 

x9 

56.73 

34 

49-74 

49 

39-36 

64 

26.3 

79 

n-45 

5 

59-77 

20 

56.38 

35 

49- 1 5 

50 

38.57 

65 

25-36 

80 

10.42 

6 

59.67 

21 

56.01 

36 

48.54 

5i 

37-76 

66 

24.4 

81 

9-38 

7 

59.55 

22 

55-63 

37 

47.92 

52 

36.94 

67 

23-44 

82 

8.35 

8 

59. 42 

23 

55-23 

38 

47.28 

53 

36.11 

68 

22.48 

83 

7-31 

9 

59.26 

24 

54.81 

39 

46.63 

54 

35-27 

69 

21.5 

84 

0.27 

10 

59-  09 

25 

54-38 

40 

45.96 

55 

34-4i 

70 

20.52 

85 

5-23 

1 1 

58.89 

26 

53-93 

41 

45.28 

56 

33-45 

71 

x9-53 

86 

4- 1° 

12 

58. 69 

27 

53-46 

42 

44-59 

57 

32.68 

72 

18.54 

87 

3-J4 

13 

58.46 

28 

52-97 

43 

43.88 

58 

3i-79 

73 

x7-54 

88 

2 

14 

58.22 

29 

52.48 

44 

43.16 

59 

30.9 

74 

16.54 

89 

1.05 

15 

57-95 

30 

51.96 

45 

42-43 

60 

30 

75 

15-53 

90 

.oo 

Note.  — Degrees  of  longitude  are  to  each  other  in  length  as  Cosines  of  their 
latitudes. 


FIGURE  OF  EARTH. — BOARD  AND  TIMBER  MEASURE.  6 1 


Elements  of*  F’igu.re  of*  tlie  Eartla. 
Capt.  A.  R.  Clarke , 1866. 


Feet. 

Major  semi-axis  of  Equator  (longitude  150  34'  E.) 20926  350 

Minor  “ “ “ ( “ 1050  34'  E.) 20919972 

Polar  “ “ 20853429 

Equatorial  semi-axis 20  926  062 


Diameter, 


Miles. 
3963.324. 
3 962. 1 15. 
3 949-513- 
3963.269. 
24  898.562. 
7916. 


BOARD  AND  TIMBER  MEASURE. 

BOARD  MEASURE. 

In  Board  Measure,  all  boards  are  assumed  to  be  1 inch  in  thickness. 
To  Compute  Measure  or  Surface. 

When  all  Dimensions  are  in  Feet. 

Rule. — Multiply  length  by  breadth,  and  product  will  give  surface  in 
square  feet. 

When  either  of  Dimensions  are  in  Inches . 

Rule. — Multiply  as  above,  and  divide  product  by  12. 

When  all  Dimensiom  are  in  Inches . 

Rule. — Multiply  as  before,  and  divide  product  by  144. 

Example.  — What  are  number  of  square  feet  in  a board  15  feet  in  length  and  16 
inches  in  width  ? 

15X16  = 240,  and  240  -r- 12  = 20  feet 


TIMBER  MEASURE. 

To  Compute  "Volume  of  Round  Timber. 

When  all  Dimensions  are  in  Feet. 

Rule. — Add  together  squares  of  diameters  of  greater  and  lesser  ends, 
and  product  of  the  two  diameters ; multiply  sum  bv  .7854,  and  product 
by  one  third  of  length. 

Or,  a + «'  + a"  X ; = V,  and  c2  -f  c'2  -f  c x c'  X .07958  X - = Y.  a and 
, . 3 3 

a representing  areas  of  ends , a'  area  of  mean  proportional,  l length  and  c 
and  c circumference  of  ends. 

Note.— Mean  proportional  is  square  root  of  product  of  areas  of  both  ends. 

Illustration. — Diameters  of  a log  are  2 and  1.5  feet,  and  length  15  feet. 

22+  i-  52— :4  ~f~  2-  25  4“  2 X 1.5  = 9.25,  which  x .7854  and  ^=36.32  cube  feet. 

3 

When  Length  in  Feet,  and  A.reas  or  Circumferences  in  Inches. 

Rule. — Proceed  as  above,  and  divide  by  144. 


When  all  Dimensions  are  in  Inches. 

Rule. — Proceed  as  before,  and  divide  by  1728. 

Note.  — Ordinary  rule  of  Hutton,  Ordnance  Manuat  of  U.  S.,  and  Molesworth,  of 

4’  ?'VGu  a result  of  ab°ut  .25  less  than  exact  volume,  or  what  it  would  b( 
11  me  log  was  hewn  or  sawed  to  a square,  c representing  mean  circumferences. 

F 


62 


BOARD  AND  TIMBER  MEASURE. 


To  Compute  Yolume  of  Squared  Tiirfber. 
When  all  Dimensions  are  in  Feet. 

Rule. — Multiply  product  of  breadth  by  depth,  by  length,  and  product 
will  give  volume  in  cube  feet. 

When  either  Dimension  is  in  Inches. 

Rule. — Multiply  as  above,  and  divide  product  by  12. 


When  any  two  Dimensions  are  in  Inches. 

Rule. — Multiply  as  before,  and  divide  by  144. 

Example.— A piece  of  timber  is  15  inches  square,  and  20  feet  in  length;  required 
15  X 15  X 20 


its  volume  in  cube  feet. 


144 


- = 31.25  cube  fed. 


Allowance  Is  to  be  made  for  bark,  by  deducting  from  each  girth  from 
.5  inch  in  logs  with  thin  bark,  to  2 inches  in  logs  with  thick  bark. 


ALeasnres  of  Tircfber. — [English.) 


100  superficial  feet » are. 

of  planking  ) u 
120  deals = 1 hundred. 


50  cube  feet  of  squared  ) __  joa(j 
timber  J 

40  feet  of  unhewn  timber  = 1 load. 


600  superficial  feet  of  inch  planking  = 1 load. 


Deals. 

Deals.  — Boards  exceeding  7 ins.  in  width,  and  if  less  than  6 feet  in 
length,  are  termed  deal  ends. 

Battens  are  similar  to  deals,  but  only  7 inches  in  width. 

Balk. — Roughly  squared  log  or  trunk  of  a tree. 

Blanks  are  boards  12  ins.  in  width. 


JLocal  Standards. 


Country. 

Long. 

Broad. 

Thick. 

Volume. 

Country. 

Long.  ! 

Broad. 

Thick. 

Volume. 

Ft. 

Ins. 

Ins. 

Cub.  ft. 

Ft. 

Ins. 

Ins. 

Cub.  ft. 

Russia  and 

Norway . . 

12 

9 

3 

2.25 

Prussia . . 

12 

II 

i-5 

i-375 

Christiana 

II 

9 

1.25 

•859 

Sweden . . . 

14 

9 

3 

2.625 

Quebec. . . 

12 

11 

2-5 

2.292 

100  Petersburg!!  standard  deals  equal  60  Quebec  deals. 


SPARS  AND  POLES. 

Pine  and  Spruce  Spars , from  10  to  4.5  inches  in  diameter  inclusive, 
are  to  be  measured  by  taking,  their  diameter,  clear  of  bark,  at  one  third 
of  their  length  from  abut  or  large  end. 

Spars  are  usually  purchased  by  the  inch  diameter ; all  under  4 inches  [ 
are  termed  Poles. 

Spars  of  7 inches  and  less  should  have  5 feet  in  length  for  every 
inch  of  diameter,  and  those  above  7 inches  should  have  4 feet  in  length  * 
for  every  inch  of  diameter. 


IjOss  or  "Waste 


Oak,  English . . 
“ African.. 
“ Dantzic  . . 
“ American 


in  Herwing  or  Sawing  of  Timber. 
[C.  Mackrow.) 


200  per  cent. 
100  “ “ 

50  “ “ 

10  “ “ 


Yellow  Pino  from  planks. . 10  per  cent. 

Teak *5  “ ‘‘ 

Elm,  English 200  “ 

u American 15  ‘ 


CISTERNS. SHINGLES. 


63 


CISTERNS. 

Capacity  of  Cisterns  in  Cube  IFeet  and.  GJ-allons. 
For  each  10  Inches  in  Depth. 


Diam. 

[ Cub.  ft. 

| Gallons. 

Diam. 

Cub.  ft. 

| Gallons. 

| Diam. 

| Cub.  ft. 

Feet. 

2 

2.618 

19.58 

Feet. 

9-5 

59.068 

441.8 

Feet. 

17 

189.15 

2-5 

4.091 

30.6 

10 

65.449 

489.6 

I7-5 

200.432 

3 

5-89 

44.07 

10.5 

72.158 

539-78 

18 

212.056 

3-5 

8.018 

59-97 

11 

79.194 

592.4 

*9 

236.274 

4 

10.472 

78.33 

n-5 

86.558 

647-5 

20 

261.797 

4-5 

13-254 

99.14 

12 

94.248 

705 

21 

288.632 

5 

16.362 

122.4 

12.5 

102.265 

764.99 

22 

316.776 

5-5 

19.798 

148.1 

13 

1 10.61 

827.4 

23 

346.23 

6 

23.562 

176.24 

13-5 

119.282 

892.29 

24 

376.992 

6-5 

27.652 

206.84 

14 

128.281 

959-6 

25 

409,062 

7 

32.07 

239.88 

14-5 

137.608 

1029.38 

26 

442.44 

7-5 

36.816 

275-4 

i5 

147.262 

1101.6 

27 

471-!  3 

8 

41.888 

3I3-33 

15-5 

I57-243 

1176.26 

28 

513.126 

8-5 

47.288 

353-72 

16 

167.552 

I253-37 

29 

550.432 

9 

53014 

396-55 

16.5 

178.187 

J332-93  !l 

30 

589.048 

Gallons. 


1414.94 

1 499 -33 
1586.28 
1767,45 

1958.3 

2159.11 

2369.64 

2589.97 

2820.09 

3059-8 

3309-67 

3569-17 

3838.44 

4ii7-5i 

4406.08 


Excavation  and  Lining  of  Wells  or  Cisterns 
For  each  10  Inches  in  Depth. 


Feet. 

3 

3- 5 

4 

4- 5 

5 

5- 5 

6 

6- 5 
7 

7- 5 

3 


c 

.2 

Bricks. 

Masonry. 

J 

.2 

Bricks. 

Num- 

Laid 

8 inches 

1 foot 

I 

g 

g 

Num- 

Laid 

1 W 

ber. 

dry. 

thick. 

thick. 

s 

W 

ber. 

dry. 

Cub.  ft. 

126 

Cub.  ft. 

Cub.  ft. 

Cub.  ft. 

Feet. 

Cub.  ft. 

Cub.  ft. 

12. 29 

5-24 

6.4 

IO-47 

8-5 

63.29 

356 

14.83 

I5-29 

18.62 

J47 

168 

6. 11 
6.98 

7.27 
8. 14 

11.78 

13.09 

9 

9-5 

69. 89 
76.81 

377 

398 

15-71 

16.58 

22.27 

26.25 

188 

209 

7-85 

8.73 

9.02 

9.89 

14.4 

I5-7I 

10 

10.5 

84.07 

9i-65 

419 

440 

17-45 

18.33 

3°.  56 

230 

9.6 

10.76 

17.02 

11 

99-56 

461 

19^2 

35-2 

251 

10.47 

11.64 

i8-33 

12 

116.36 

503 

20.94 

40. 16 

272 

n-34 

12.51 

19.63 

13 

134.46 

545 

22. 69 

45-45 

293 

12.22 

13-38 

20.94 

14 

153-88 

586 

24-43 

51-07 

3i4 

13.09 

14-25 

22.25 

15 

174.61 

628 

26.18 

57.02 

335 

13.96 

15-13 

23-56 

16 

196.64 

670 

27-92  1 

8 inches 
thick. 


Masonry. 


Cub.  ft 
16 

16.87 

I7-75 

18.62 

19.49 

20.36 

22. 11 

23-85 

25.6 

27-34 

29. 09 


Cub. ft. 
24.87 
26. 18 
27.49 
28.8 
30.11 
31.4a 
34-03 
36-65 
39-27 
41.89 
44-51 


brick vi 7 « h v d • curD  ?re. taken  at  dimensions  of  ordinary 

oricK  \iz.,  8 by  4 by  2.25  ins.  = 72  cube  ins.  J 

hnlrliS0T?Uting  ™mlrer.  of  bricks  required,  an  addition  of  5 per  cent  should 

**  *-  ' - excavation 


SHINGLES. 

» 1*"»h  - . <•  7 

abut.  ’ t0  llght  ‘25  inch  ^ point  and  .3125  inch  at 

than'^V^rshinHpT863  Hc°uraes  of  about  8 inches-  so  that  less 

quired3p^s;:aoV:ffi?ed  * ^ shingles  are  re- 

Shingles,  alike  to  Slates,  are  laid  upon  boards  or  battens. 


6 4 


SLATES  AND  SLATING. 


SLATES  AND  SLATING. 

A Square  of  Slate  or  Slating  is  ioo  superficial  feet. 

Gauge  is  distance  between  the  courses  of  the  slates. 

Lap  is  distance  which  each  slate  overlaps  the  slate  lengthwise  next 
but  one  below  it,  and  it  varies  from  2 to  4 inches.  Standard  is  assumed 

to  be  3 inches.  # . 

Margin  is  width  of  course  exposed  or  distance  between  tails  01  the 

slates.  , 

Pitch  of  a slate  roof  should  not  be  less  than  1 in  height  to  4 of  length. 

rp0  Compute  Surface  of  a Slate  'when  laid,  and  Num- 
ber of  Squares  of  Slating. 

Rule.  — Subtract  lap  from  length*  of  slate,  and  half  remainder  will 
give  length  of  surface  exposed,  which,  when  multiplied  by  width  of 
slate,  will  give  surface  required. 

Divide  14  400  (area  of  a square  in  inches)  by  surface  thus  obtained, 
and  quotient  will  give  number  of  slates  required  for  a square. 

Example.  — A slate  is  24  X 12  inches,  and  lap  is  3 inches;  what  will  be  number 
required  for  a square  ? 

24  _ 3 = 21,  and  21-4-2  = 10.5,  which  X 12  = 126  inches;  and  14400-4-126  = 


xi  4. 29  slates. 


Dimensions  of  Slates. 


Ins. 

Ins. 

i 

Ins.  | 

American. 

Ins. 

] 

Ins. 

Ins.  [ 

Ins. 

14  X 7 
14  X 8 

14  x 9 

14  X IO 
16  X 8 

16  x 9 

16  X 10 

18  X 9 

18  X 10 

| 18  X II 
l8  X 12 
1 20  X IO 

20  X II 
20  X 12 
22  X II 

22  X 12 
22  X 13 
24  X 12 

24  x 13 

24  X 14 
24  X 16 

English. 


Ins. 

Ins. 

Ins. 

Doubles 

13  X 10 

12X  8 

Marchioness  . . 

22X22 

u 

Small  doubles  . 

u 

13X  7 
11X  6 
10X  5 
12x10 
13  x IO 
18X  10 

Ladies - 

14X  8 
14X  12 
15  x 8 

Duchess 

Imperial 

Rags 

24  X 12 

30X24 

36X24 

Plantations . . j 
Viscountess  . . . 

Countess  .... 

i6x  8 

16  X 10 
20  X 10 

Queens 

Empress 

Princess 

36X24 

26X15 

24XI4 

Thickness  of  slates  ranges  from  .125  to  .3125  of  an  inch,  and  their  weight 
varies  from  2 to  4.53  lbs.  per  sq.  foot. 

Weight  of  One  Square  Foot  of  Slating. 

.125  in.  thick  on  laths 4 75  lbs.  -25  in.  thick  on  laths. . • • • • • • 9-25  lbs* 

“ “ “ “ 1 in.  boards..  6.75  “ “ ‘‘  u 1 m boards. . 11.25 

.1875  in.  thick  on  laths 7 “ -3I25  in-  thmk  on  laths. . ....  zz.  5 

V5  u u a , in.  boards.  9 “ “ “ “ “ 1 in.  boards,  14.10 

Slate  weighs  from  167  to  181  lbs.  per  cube  foot,  and  in  consequence  of 

laps,  it  requires  an  average  of  nearly  2.5  square  feet  of  slate  to  make  one  01 

slating. 

Weights  per  1000  and  Number  Required  to  Cover  a Square. 

I Lb*.  | No.  I]  I Lbs< 

Doubles 13  X 6 1680  480  j|  Countess. . .20X10  6720 

Ladies 15  X 8 I 2800  I 240  1 1 Duchess  . . . 24  X 12  1 44«° 

* Length  of  a slate  is  taken  from  nail-hole  to  tail. 


SHOT  AND  SHELLS. FRAUDULENT  BALANCES.  65 

PILING  OF  SHOT  AND  SHELLS. 

To  Compute  NTxiinUer  of  Sliot. 

. Triangular  Pile.  Rule.— Multiply  continually  together,  number  of  shot 
m one  side  of  bottom  course,  and  that  number  increased  by  1 and  a«-ain  bv 
2,  and  one  sixth  of  product  will  give  number.  0 

ing  30  shotrWhat  iS  nUmbGr  °f  Sh0t  in  a trian£ularPil€V  each  side  of  base  contain- 


To  Detect  Tliem.— After  an  equilibrium  has  been  established  between 
weight  and  article  weighed,  transpose  them,  and  weight  will  preponder- 
ate if  article  weighed  is  lighter  than  weight,  and  contrariwise  if  it  is 
heavier. 

To  Ascertain  True  Weight.  Rule. — Ascertain  weight  which  will  produce 
equilibrium  after  article  to  be  weighed  and  weight  have  been  transposed  • 
reduce  these  weights  to  same  denomination,  multiply  them  together  and 
square  root  of  their  product  will  give  true  weight. 


24  lbs.  8 oz.  =24.5  lbs. 

Then  32  X 24.5  — 784,  and  ^784  = 28  lbs. 

Or,  when  a represents  longest  arm , I A greatest  weight,  and 

0 lt  shortest  arm,  | B least  weight. 

or  W^L^B^ndV- VAb"6*’  multiplying  these  two  e<luations,  W2a&  = ABa&, 

Illustration.  —A  = 32 ; B = 24. 5 ; W = 28.  Assume  length  of  longest  arm  = IO, 


30X30+1X30  + 2 20760 

— — — — ^ — • = 4960  shot. 


Example.  How  many  shells  are  there  in  a square  pile  of  30  courses? 


3°  X 3°+  1 X 3°  X 2 + 1 56  730 


— 5 — = 9455  shells. 


6 


will  nu^ber?^1  7 U1CaUU1’  lilcreasea  I>  ^ one  sixth  of  product 
be?ngAf6>and+qUired  number  of  shells  in  an  oblon«  Pile,  numbers  in  base  course 


being  16  and  7? 


16X3-7-1X7X7  + 1 _ 2352 


— 392  shells. 


6 


' y^yMuiuwa, 


FRAUDULENT  BALANCES. 


Hence,  a _ 10,  6 = 8. 75,  or  282  = 32  x 24. 5,  and  V32  X 24.5  = 28. 


Then  32  : 28  ;;  10  : 8.75. 


3 


66  WEIGHING  WITHOUT  SCALES. PAINTING. 


NV'eiglxing  witliovit  Scales. 

To  Ascertain  Weiglit  of  a Bar,  Beam,  etc.,  by  Aid  of* 
a,  known.  "Weiglit. 

Operation.— Balance  bar,  etc.,  over  a fulcrum,  and  note  distance  between 
it  and  end  of  its  longest  arm.  Suspend  a known  weight  from  longest  arm, 
and  move  bar,  etc.,  upon  fulcrum,  so  that  bar  with  attached  weight  will  be 
in  equilibrio : subtract  distance  between  the  two  positions  of  fulcrum  trom 
longest  arm  first  obtained ; multiply  this  remainder  by  weight  suspended, 
divide  product  by  distance  between  f ulcrums,  and  quotient  will  give  weight. 

Example. — A piece  of  tapered  timber  24  feet  in  length  is  balanced  over  a fulcrum 
when  n feet  from  less  end;  but  when  the  body  of  a man  weighing  210 * lbs-  's  sus- 
pended from  extreme  of  longest  arm,  the  piece  and  weight  are  balanced  when  ful- 
crum is  12  feet  from  this  end.  What  is  weight  of  the  timber? 

I3_I2_Ij  and  13  — 1 = 12  feet  Then  12  X 210-^1  = 2520  lbs. 


PAINTING. 

1 pound  of  paint  will  cover  about  4 square  yards  for  a first  coat  and  about 
6 yards  for  each  additional  coat. 


Colors. 

White 

Lead. 

Red 

Lead. 

Red 

Ochre. 

>•  u 

I Spanish 
1 Brown. 

Colors. 

White 

Lead. 

Lamp- 

black. 

| Red 
1 Lead. 

White 

100 

100 

- 

- 

- 

Lead 

Red 

98 

2 

50 

Green 

25  1 

- 

- 

75 

- 

Chocolate. . 

— 

4 

> fe 


50 


96 


These  are  the  colors  alone,  to  whicn  Doueci  nnseeu  on,  nmuge,  .japan  , 

and  spirits  turpentine  are  to  be  added  according  to  the  application  of  the  paint. 
Lamp-black  and  litharge  are  ground  separately  with  oil,  then  stirred  into  the 

^Thus  for  black  paint:  Lamp-black  25  parts,  litharge  1,  Japan  varnish  1,  boiled  lin- 
seed oil  72,  and  spirits  turpentine  1. 


Tar  Baint. — Coal  tar  9 gallons,  slaked  lime  13  lbs.,  turpentine  or  naphtha  2 
or  3 quarts. 

Superficial 

A Gallon  of  Paint  will  cover  feet. 


On  stone  or  brick,  about . 
On  composite,  etc.,  from  . 
On  wood,  from  . 


■go  to  225 
300  “ 375 
375  “ 525 


A Gallon  of  Paint  will  cover 


On  well-painted  surface  or  iron 

One  gallon  tar,  first  coat 

“ “ “ second  coat .. . 


Superficial 

feet. 


600 

90 

160 


Boiled.  Oil Raw  linseed  oil  91  parts,  copperas  3,  and  litharge  6. 

Put  litharge  and  copperas  in  a cloth  bag  and  suspend  in  middle  of  a kettle  Boil 
oil  four  hours  and  a half  over  a slow  fire,  then  let  it  stand  and  deposit  the  sediment. 

Wliite  Baint. 

Inside  work.  Outside  work.  I . , Inside  work.  Outside  work. 

White  lead  in  oil . . 80  80  Raw  oil 9 

Boiled  oa  .’.  . . . .4.5  9 I Spirits  torpentme  8 4 

New  wood- work  requires  i lb.  to  square  yard  for  three  coats. 

Coats  for  ioo  Square  Yards  New  White  Pine. 


Inside. 

White 

lead. 

Raw 

oil.- 

Turpen- 

tine. 

Drier. 

Lbs. 

Pts. 

Pts. 

Lbs. 

Priming 

16 

— 

6 

•25 

2d  coat 

15 

3-5 

I*5 

•25 

3d  “ 

1 13 

2-5 

1-5 

•25 

Outside. 


Priming — 
2d  and  3d  \ 
coats  j | 


White 

lead. 

Lbs. 

18.5 


Raw 

oil. 


Boiled 

oil. 


Turpen- 

tine. 


lb.  of  drier  with  priming  and  coating  for  outs.de. 


HYDROMETERS. 


67 


HYDROMETERS. 


IT.  S.  Hydrometer  (Tralle’s)  ranges  from  o (water)  to  100  (pure  spirit) ; 
it  has  not  any  subdivision  or  standard  termed  “Proof,”  but  50,  upon 
stem  of  instrument,  at  a temperature  of  6o°,  is  basis  upon  which  com- 
putations of  duties  are  made. 

In  connection  with  this  instrument,  a Table  of  Corrections,  for  differences  in  tem- 
perature of  spirits,  becomes  necessary;  and  one  is  furnished  by  the  Treasury  De- 
partment, from  which  all  computations  of  value  of  a spirit  are  made. 

Illustration.  — A cask  contains  100  gallons  of  whiskey  at  700,  and  hydrometer 
sinks  in  the  spirit  to  25  upon  its  stem. 

Then,  by  table,  under  700,  and  opposite  to  25,  is  22.99,  showing  that  there  are  22  qq 
gallons  of  pure  spirit  in  the  100. 


Commercial  Hydrometer  (Gendar’s)  has  a “Proof”  at  6o°,  which  is 
equal  to  50  upon  U.  S.  Instrument  and  its  gradations,  run  up  to  100 
with  it,  and  down  to  10  below  proof,  at  o upon  U.  S.  Instrument ; or  o 
of  the  Commercial  Instrument  is  at  50  upon  U.  S.  Instrument,’ from 
which  it  progresses  numerically  each  way,  each  of  its  divisions  being 
equal  to  two  of  latter. 

In  testing  spirits,  Commercial  standard  of  value  is  fixed  at  proof; 
hence  any  difference,  whether  higher  or  lower,  is  added  or  subtracted’ 
as  case  may  be,  to  or  from  value  assigned  to  proof. 

. A scale  0*'  Corrections  for  temperature  being  necessary,  one  is  fur- 
nished with  a Thermometer. 

Application  of  Thermometer.—  Elevation  of  the  mercury  indicates  correction  to 
be  added  or  subtracted,  to  or  from  indication  upon  stem  of  hydrometer. 

When  elevation  is  above  6o°,  subtract  correction ; and  when  below,  add  it. 

Illustration.— A hydrometer  in  a spirit  indicates  upon  its  stem  50  below  proof 
and  thermometer  indicates  4 above  6o°  in  appropriate  column. 

Then  50  — 4 = 46  = strength  below  proof 

To  Compute  Strength,  of  a Spirit,  or  Volume  of  its  Pure 
Spirit,  by  Commercial  Hydrometer,  and.  Convert  it  to 
Indication  of  a TJ.  S.  Hydrometer. 

TFTien.  Spirit  is  above  Proof.  Rule. — Add  100  to  indication,  and  divide  sum  by  2. 

re mai nder^byl  **  below  Proof  Rule.  — Subtract  indication  from  100,  and  divide 

pofTon  of^iure^pWt'does'it  contain f°°^ ^ * CommerciaI  Hydrometer;  what  pro- 
11  -f- 100  — 2 = 55.5  per  cent. 

To  Compute  Strength,  etc.,  by  a TJ.  S.  Hydrometer. 
When  Spirit  is  above  Proof  Rule. -Multiply  indication  by  2,  and  subtract  100. 

U bd0W  Pr°°f'  RCLE'  ~ MultipI^  indication  by  2,  and  subtract  it 

Example.— A spirit  is  55.5;  what  is  its  per  centage  above  proof? 

55-  5 X 2 — 100  ==  1 r per  cent 

Commercial  practice  of  reducing  indications  of  a hydrometer  is  as  follows: 

nuP1V.er.P^  gMlons  of  spirit  by  per  centage  or  number  of  degrees  above 

2 X7e.and  qU°tient  Wi“  *">  “»“**  of  ganonlThl  £22 

Illustration.— 50  gallons  of  whiskey  are  n per  cent,  above  proof. 

Then  50  X iz  zoo  = 5. 5,  which  added  to  50  = 55. 5 gallons. 


68 


HYGROMETER. 


HYGROMETER. 

D e w-p  oint .—When  air  is  gradually  lowered  in  its  temperature  at  a 
constant  pressure,  its  density  increases,  and  ratio  of  increase  is  sensibly 
same  for  the  vapor  as  for  the  air  with  which  it  is  combined,  until  a point  is 
reached  at  which  the  density  of  the  vapor  becomes  equal  to  the  maximum 
densitv  corresponding  to  the  temperature,  ^ „ , 

This  temperature  is  termed  dew-point  of  given  mass,  and  any  further  re- 
duction of  it  will  induce  the  condensation  of  a portion  of  the  vapor  m form 
of  dew,  rain,  snow,  or  frost,  according  as  temperature  of  surface  is  above  or 
below  freezing  point. 

Mason’s  or  like  Hygrometer. 

To  Ascertain  Dew-point. 

Rule.  — Subtract  absolute  dryness  from  temperature  of  air,  and  remainder  is 
dew-point. 

Example.— Temperature  of  air  570,  and  absolute  dryness  70. 

Hence  57 0 — 70  = 500  dew-point. 

To  Ascertain  Absolute  Existing  Dryness. 

Rule  —Subtract  temperature  of  wet  bulb  from  temperature  of  air,  as  indicated 
by  a dry  bulb,  add  excess  of  dryness  from  following  table,  multiply  sum  by  2,  and 
product  will  give  absolute  dryness  in  degrees. 

Example.— Temperature  of  air  57 °,  wet  bulb  54°- 

Then  570  — 540  = 30,  and  30  -f  • 5°  (from  table)  X 2 = 70  absolute  dryness. 


Observed  jExcess  of  j 
Dryness.  | Dryness,  j 

Observed 

Dryness. 

Excess  of 
Dryness. 

Observed] Excess  of! 
Dryness,  j Dryness,  j 

• 5 

.083 

5 

•833 

9-5 

1-583 

I 

. 166 

5-5 

.9165 

10 

1.666 

1-5 

.2495 

6 

1 

10.5 

1-7495 

2 

•333 

6-5 

1.083 

11 

1-833 

2-5  * 

•4i65 

7 

I. 166 

n-5 

1.9165 

3 

•5 

7-5 

1.2495 

12 

2 

3.5 

. 583 

8 

i-333 

12.5 

2.083 

4 

.666 

8-5 

1.4165 

13 

2. 166 

4-5 

•7495  1 

9 

i-5 

I3-5 

2.2495 

Observed 

Dryness. 

Excess  of  1 
Dryness. 

Observed 

Dryness. 

Excess  of 
Dryness. 

14 

8-333 

18.5 

, 3-o83 

14-5 

2.4165 

19 

3.166 

15 

2-5 

19-5 

3-2495 

15-5 

2.583 

20 

3-333 

16 

2.666 

20.5 

3-4i65 

16,5 

2-7495 

21 

3-5 

17 

2.833 

21-5 

3-583 

17-5 

2.9165 

22 

3.666 

18 

3 

22.5 

3-7495 

To  Compute  Volume  of  Vapor  in  Atmosphere. 
By  a Hygrometer. 

When  temperature  of  atmosphere  in  shade , and  of  dew-point  are  given.  If  temper- 
ature of  air  mid  dew-point  correspond,  which  is  the  case  when 
are  alike  and  air  consequently  saturated  with  moisture,  then  m table  opposite  to 
temperature  will  be  found  corresponding  weight  of  a cube  foot  of  vapor  in  grams. 

Illustration. -Assume  temperature  of  air  and  dew-point  70°  Then  opposite 
tomnerature  weight  of  a cube  foot  of  vapor  = 8. 392  grains. 

But  if  temperature  of  air  is  different  from  dew-point,  a correction  is  necessary  to 

obtain  exact  wTeight. 

Illustration.— Assume  dew-point  7o°  as  before,  but  temperature .o^1^  s^ade 
80°  then  the  vapor  has  suffered  an  expansion  due  to  an  excess  of  10  , which  re 

^InLble^rcm^ections  for  10°  is  1.0208.  Then  divide  8.392  grains  at  dew-point- 
viz.,  700  by  correction  corresponding  to  degrees  of  absolute  dryness— viz.,  10  . 

8,392  =8.221  grains  of  existing  vapor , which,  subtracted  from  weight  of  vapor 

col-responding  to  temperature  of  80°,  will  give  number  of  grains  required  for  satu- 
ration  at  that  temperature.  . 

n.333  grains  at  temperature  of  8o°— 8.221  contained  in  the  air  = 3.112  required 

for  saturation.  - . 

* For  tabic,  see  Mason’s  as  published  by  Pike  & Sons,  New  York,  and  compared  with  Sir  John 
Leslie’s  and  l’rofcssor  Daniel  ?s. 


HYGROMETER. SUN-DIAL. — CHAINING. 


69 


To  ascertain  relations  of  these  conditions  on  natural  scale  of  humidity  (complete 
saturation  being  1000),  divide  weight  of  vapor  at  dew-point  by  weight  at  tempera- 
ture of  air,  and  quotient  will  give  degrees  of  saturation. 

Illustration.— Dew-point  = 700,  weights 8.392. 

Then  8.392-^-11.333  (at  8o°)  = .7405  degrees  of  humidity;  saturation  ~ 1000. 

To  Compute  Weight  of  Vapor  in  a Cube  Foot  of  Air. 

See  Pressures,  Temperatures,  Volumes,  and  Density  of  Steam,  p.  708. 

Thus,  Required  weight  of  vapor  in  a cube  foot  of  saturated  air  at  21 20. 

At  a temperature  of  2120  density  or  weight  of  1 cube  foot  of  air  = .038  lb. 

If  density  is  required  for  any  temperatures  not  in  table,  see  rule,  p.  706. 

Humidity.— Condition  of  air  in  respect  to  its  moisture  involves  amount  of 
vapor  present  in  air  and  ratio  of  it  to  amount  which  would  saturate  it  at  its 
temperature,  and  it  is  this  element  which  is  denoted  by  term  humidity,  and 
it  is  expressed  as  a per  centage;  thus,  if  weight  of  vapor  present  is  .7  of  that 
required  for  saturation,  the  humidity  is  70. 

Dry  Air  is  air,  humidity  of  which  is  below  zero,  but  it  is  customary  to 
term  it  dry  when  its  humidity  is  below  the  average  proportion. 

Note. —Air  in  a highly  heated  space  contains  as  much  vapor  (when  weight  of  it 
is  equal)  as  a like  volume  of  external  air,  but  it  is  drier  as  its  capacity  for  vapor 
is  greater. 


SUN  - DIAL. 

To  Set  a Su.11-d.ial. 

Set  column  on  which  dial  is  to  be  placed  perpendicular  to  horizon.  Ascertain  by 
spirit  level  that  upper  surface  is  perfectly  horizontal ; screw  on  plate  loosely  by  means 
of  centre  screw,  and  bring  gnomon  as  nearly  as  practicable  to  its  proper  direction 
wn  a bright  day  set  dial  at  9 a.m.  and  3 f.m.  exactly,  with  a correctly  regulated 
watch;  observe  difference  between  them,  and  correct  dial  to  half  difference.  Pro- 
ceed in  same  manner  till  watch  and  dial  are  found  to  agree  perfectly.  Then  fix 
plate  firmly  in  that  situation,  and  dial  will  be  correctly  set. 

This  is  obvious;  for,  if  there  were  any  defects,  the  Sun’s  shadow  would  not  agree 
w;th  time  indicated  by  watch,  both  before  and  after  he  passed  meridian.  Take 
care,  however,  to  allow  for  equation  of  time,  or  you  may  set  dial  wrong.  Best  day 
in  the  year  to  set  a dial  is  15th  of  June,  as  there  is  no  equation  to  allow  for  and  no 
error  can  arise  from  change  of  declination.  A dial  may  be  set  without  a watch  by 
drawing  a circle  around  centre,  and  marking  spot  where  top  of  shadow  of  an  upright 
pm  or  piece  of  wire,  placed  in  centre,  just  touches  circle  in  a.m.,  and  again  in  p m 
A une  should  be  drawn  from  one  spot  to  the  other,  and  bisected  exactly;  then  a 
line  drawn  from  centre  of  dial  through  that  bisection  will  be  a true  meridian  line 
on  which  the  XII  hours’  mark  should  be  set.  ’ 


CHAINING  OVER  AN  ELEVATION. 

I C = L,  and  C = cos.  angle. 

I representing  length  of  line  chained , C cos.  angle  of  elevation  with  horizon, 
and  L Length  of  line  reduced  to  horizontal. 

horizSXanTjf  “Sth  °f  “ eIeVati°n  at  ““  angle  of  3°°  V is  100  f<*‘;  "hat  is 
By  Table  of  Cosines,  30°  17' = .863  54.  Hence,  100  X .863  54  = 86. 354  feet. 

To  set  out  a Right  Angle  with  a Chain,  Tape-line,  etc. 

and^nfor  Chain  0r  fefu  0f  Iine  for  base’  30  links  or  feet  for  perpendicular, 

and  50  for  hypothenuse,  or  in  this  ratio  for  any  length  or  distance. 


Useful  Numbers  in  Surveying. 


For  Converting 

Multiplier. 

Converse. 

For  Converting 

Multiplier. 

Converse. 

Feet  into  links.. 
Yards  “ “ 

I-5I5 

4-545 

.66 

.22 

Square  feet  into  acres. . 
Square  yards  “ “ .. 

.000022  9 
. 000  206  6 

43  56o 
4 840 

3 


70 


CHRONOLOGY. 


CHRONOLOGY. 

Solar  day  is  measured  by  rotation  of  the  Earth  upon  its  axis  with  respect 

to  the  Sun.  _ . . „ , , 

Motion  of  the  Earth,  on  account  of  ellipticity  of  its  orbit,  and  o±  perturba- 
tions produced  by  the  planets,  is  subject  to  an  acceleration  and  retardation. 
To  correct  this  fluctuation,  timepieces  are  adjusted  to  an  average  or  mean 
solar  day  ( mean  time ),  which  is  divided  into  hours,  minutes,  and  seconds. 

In  Civil  computations  day  commences  at  midnight,  or  A.M.,  and  is  divided  into 
two  portions  of  12  hours  each. 

In  Astronomical  computations  and  in  Nautical  time  day  commences  at  M.,  or 
12  hours  later  than  the  civil  day,  and  it  is  counted  throughout  the  24  hours. 

Solar  Year,  termed  also  Equinoctial , Tropical , Civil , or  Calendar  Y ear,  is  the 
time  in  which  the  Sun  returns  from  one  Vernal  Equinox  to  another;  and  its  average 
time,  termed  a Mean  Solar  Year , is  365.242218  solar  days,  or  365  days , 5 hours , 48 
minutes , and  47. 6 seconds. 

Year  is  divided  into  12  Calendar  months,  varying  from  28  to  31  days. 

Mean  Lunar  Month , or  lunation  of  the  Moon,  is  29  days,  12  hours,  44  minutes, 

2 seconds,  and  5.24  thirds.* 

Bissextile  or  Leap  Year  consists  of  366  days;  correction  of  one  year  in  four  is 
termed  Julian  ; hence  a mean  Julian  year  is  365.25  days.  ^ 

In  vear  1*82  error  of  Julian  computation  of  a year  had  amounted  to  a period  ot 
10  days,  which,  by  order  of  Pope  Gregory  VIII.,  was  suppressed  in  the  Calendar,  and 
5th  of  October  reckoned  as  15th. 

Error  of  Julian  computation,  .00776  days,  is  about  1 day  in  ia8.7oyeMS,  and  adop- 
tion of  this  period  as  a basis  of  intercalation  is  termed  Gregorian  Calendar j or  JSew 
Style, t Julian  Calendar  being  termed  Old  Style. 

Error  of  Gregorian  year  (365.2425  days)  amounts  to  1 day  in  3571.4286  years. 

New  Style  was  adopted  in  England  in  1752  by  reckoning  3d  of  September  as  14th. 
Bv  an  English  law,  the  years  1900,  2100,  2200,  etc.,  and  any  other  100th  year,  ex- 
cepting only8 every  400th  year,  commencing  at  2000,  are  not  to  be  reckoned  bissex- 
tile years. 

Dominical  or  Sunday  Letter  is  one  of  the  first  seven  letters  of and  is 
used  for  purpose  of  determining  day  of  week  corresponding  to  an>  g^ven  date  In 
Ecclesiastical  Calendar  letter  A is  placed  opposite  to  1st  day  ^f^no’ 

B to  second-  and  so  on  through  the  seven  letters;  then  the  lettei  which  falls  oppo- 
site to  first  Sunday  in  year  will  also  fall  opposite  to  every  following  Sunday  in  tha 

^ Note.— In  bissextile  years  two  Dominical  letters  are  used,  one  before  and  the  other 
after  the  intercalary  day. 

In  Ecclesiastical  Year  the  intercalary  day  is  reckoned  upon  24th  February; 
hence  24th  and  25th  days  are  denoted  by  same  letter,  the  dominical  letter  beine  se 
back  one  place. 

In  Civil  Year  the  intercalary  day  is  added  at  end  of  February,  the  change  of  letter 
taking  place  at  1st  of  March. 

Dominical  Cycle  is  a period  of  400  years,  when  the  same  order  of  dominical  letters 
and  days  of  the  week  will  return. 

Cycle  of  the  Sun,  or  Sunday  Cycle , is  the  28  years  before  same  onjei ot 
letters  return  to  same  days  of  month,  and  it  is  considered  as  ha\mg  commenced  9 
years  before  the  era  of  Julian  Calendar. 

To  Compute  Cycle  of  tlie  Sxxn. 

Rule.— Add  9 to  given  year;  divide  sum  by  28;  quotient  is  number  of  cycles  that 
have  elapsed,  and  remainder  is  number  or  years  of  cycle. 

Note  —Use  of  this  computation  is  determination  of  dominical  letter  for  any  given 
year  of  Julian  Calendar  for  each  of  the  28  years  of  a cycle. 


* Ferguson. 


t Now  adopted  in  every  Christian  country  except  Russia  and  Greece. 


CHRONOLOGY. 


7 I 


By  adoption  of  Gregorian  Calendar , order  of  the  letters  is  necessarily  interrupted 
by  suppression  of  the  century  bissextile  years  in  1900,  2100,  etc.,  and  a table  of  Do- 
minical letters  must  necessarily  be  reconstructed  for  following  century. 

Lunar  Cycle , or  Golden  Number , is  a period  of  19  years,  after  which  the  new 
moons  fall  on  same  days  of  the  month  of  Julian  year,  within  1.5  hours. 

Year  of  birth  of  Jesus  Christ  is  reckoned  first  of  the  Lunar  Cycle. 

To  Compute  Lunar  Cycle,  or  Grolden  Number. 
Rule.— Add  1 to  given  year;  divide  sum' by  19,  and  remainder  is  Golden  Number. 
Note. — If  o remain,  it  is  19. 

Example.— What  is  Golden  Number  for  1879  ? 

i879+  1 -L 19  = 98?  aud  remainder  = 18  = Golden  Number. 

Epact  for  any  year  is  a number  designed  to  represent  age  of  the  moon  on  1st  dav 
of  January  of  that  year.  See  table,  p.  73.  * 

To  Compute  tlie  Roman  Indiction. 

Rule.— Add  3 to  given  year;  divide  sum  by  15,  and  remainder  is  Indiction. 

Note. — If  o remain,  Indiction  is  15. 

March”667*  °f Direction  is  the  nlimber  of  days  that  Easter-day  occurs  after  21st  of 

Easter-day  is  first  Sunday  after  first  full  moon  which  occurs  upon  or  next  after 
21st  of  March;  and  if  full  moon  occurs  upon  a Sunday,  then  Easter-day  is  Sundav 
after  and  it  is  ascertained  by  adding  number  of  direction  to  21st  of  March  It  is 
therefore  March  N 21,  or  April  N — 10. 

Illustration.  — If  Number  of  Direction  is  19,  then  for  March,  i04 _2I  =4D  and 
40  — 31=9  = 9 th  of  April;  ; y 1 

again  for  April,  19  — 10  = 9 — 9 th  of  April. 

Note.  Moon  upon  which  Easter  immediately  depends  is  termed  Paschal  Moon. 
Full  Moon  is  14th  day  of  moon,  that  is,  13  days  after  preceding  day  of  new  moon. 

Days  of*  tlie  Roman  Calendar. 

Calends  were  the  first  6 days  of  a month,  Nones  following  9 days,  and  Ides  remain 
ing  days.  ’ 

In  March,  May,  July,  and  October,  Ides  fell  upon  15th  and  Nones  began  upon  7th 
In  other  months  Ides  commenced  upon  13th  and  Nones  upon  5th. 

For  Roman  Indiction  and  Julian  Period  see  p.  26. 

Chronology. 

Creation  of  World  (according  to  Julius  Africanus,  Sept.  1,  5508-  Samaritan 
Pentateuch,  4700;  Septuagint,  5872 ; Josephus,  4658 ; Talmudists  Sea 
liger,  395o;  Petavius,  3984;  Hales,  5411).  d 

576.  Money  coined  at  Rome. 

562.  First  Comedy  performed  at  Athens. 
480.  First  recorded  Map  by  Aristagoras 
420.  First  Theatre  built  at  Athens. 

336.  Calippus  calculates  the  revolution  of 
Eclipses. 

320.  Aristotle  writes  first  work  on  Me- 
chanics. 

310.  Aqueducts  and  Baths  introduced  in 
Rome. 

306.  First  Light  house  in  Alexandria. 

289.  First  Sun-dial. 

267.  Ptolemy  constructs  a Canal  from  the 
Nile  to  the  Red  Sea. 

224.  Archimedes  demonstrates  the  Prop- 
erties of  Mechanical  Powers  and 
the  Art  of  measuring  Surfaces,  Sol- 
ids, and  Sections. 

219.  Hannibal  crossed  the  Alps. 

219.  Surveying  first  introduced. 

202.  Printing  introduced  in  China. 


B.  C. 
4004. 


2348 

2247. 

2203. 

2090. 

1920. 

1891. 

1822. 

1490. 

1240. 


1180. 

1120. 

753- 

640. 

605. 


Deluge  (according  to  Hales,  3154). 

Bricks  made  and  Cement  first  used. 
Tower  of  Babel  finished. 

Chinese  Monarchy. 

First  Egyptian  Pyramid  and  Canal. 

Gold  and  Silver  Money  first  intro- 
duced. 

Letters  first  used  in  Egypt. 

Memnon  invents  the  Egyptian  Al- 
phabet. 

Crockery  introduced. 

Axe,  Wedge,  Wimble,  Lever,  Masts 
and  Sails  invented  by  Daedalus 
of  Athens. 

Troy  destroyed. 

Mariner's  Compass  discovered  in 
China. 

Foundation  of  Rome. 

Thales  asserts  Earth  to  be  spherical. 

Geometrjr,  Maps,  etc.,  first  intro- 
duced. 


7 2 


CHRONOLOGY. 


B.C. 

198.  Books  with  leaves  of  vellum  first 
introduced  by  Attalus. 

170.  Paper  invented  in  China. 

168.  An  eclipse  of  the  Moon  which  was 
predicted  by  Q.  S.  Gallus. 

162.  Hipparchus  locates  the  first  degree 
of  Longitude  and  the  Latitude  at 
Ferro. 

A.D. 

69.  Destruction  of  Jerusalem. 

79.  Destruction  of  Herculaneum  and 
Pompeii. 

214.  Grist-mills  introduced. 

622.  Year  of  Hegira,  commencing  16th 
July ; Glazed  windows  first  intro- 
duced into  England  in  thiscent’y. 

667.  Glass  discovered. 

670.  Stone  buildings  introduced  into  Eng- 
land. 

842.  Lands  first  enclosed  in  England. 

933.  Printing  said  to  have  been  invented  1 
by  the  Chinese. 

991.  Arabic  Numerals  introduced. 

1066.  Battle  of  Hastings. 

mi.  Mariner’s  Compass  discovered. 

1180.  Destruction  of  Troy.  Mariner’s 
Compass  introduced  in  Europe. 

1368.  Chimneys  first  introduced  into 
Rome  from  Padua. 

1383.  Cannon  introduced. 

1390.  Woollens  first  made. 

1434.  Printing  invented  at  Mayence. 

1460.  Wood-engraving  invented  and  First 
Almanac. 

1471.  Printing  in  England  by  Caxton. 

1477.  Watches  first  introduced  at  Nurem- 
berg. 

1492.  America  discovered. 

1497.  Vasco  de  Gama  discovers  passage 
to  India. 

1500.  Variation  of  Mariner’s  Compass  ob- 
served. 

1522.  F.  de  Magellan  circumnavigates  the 
Globe. 

1530.  Incas  conquered  by  Pizarro. 

1545.  Needles  first  introduced. 

1586.  Potato  introduced  into  Ireland  from 
America, 

1590.  Telescopes  invented  by  Jansen  and 
used  in  London  in  1608. 

1616.  Tobacco  first  introduced  into  Vir- 
ginia. 

1620.  Thermometer  invented  by  Drebel. 

1627.  Barometer  invented. 

1629.  First  Printing-press  in  America. 

1639.  First  Printing-office  in  America  at 
Cambridge. 

1647.  Otto  Van  Gueriche  constructed  first 
electric  machine. 

1650.  Railroads  with  wooden  rails  intro- 
duced near  Newcastle. 

1652.  First  Newspaper  Advertisement. 

1704.  First  Newspaper  in  America. 

1705.  Blankets  first  made  at  Bristol,  Eng- 

land. 


159.  Clepsydra,  or  Water- clock,  invent- 
ed. 

146.  Carthage  destroyed. 

70.  First  Water-mill  described. 

51.  Caesar  invaded  Britain. 

45.  First  Julian  Year  by  Caesar. 

8.  Augustus  corrects  the  Calendar. 


A.V. 

1752.  Benjamin  Franklin  demonstrated 
identity  of  the  electric  spark  and 
lightning,  by  aid  of  a kite. 

1752.  New  Style,  introduced  into  Britain; 

Sept. *3  reckoned  Sept.  14. 

1753-  First  Steam-engine  in  America. 

1769.  James  Watt— First  design  and  pat- 
ent of  a Steam-engine  with  sepa- 
rate vessel  of  condensation. 

1772.  Oliver  Evans— Designed  the  Non- 
condensing Engine.  1792.  Ap- 
plied for  a patent  for  it.  1801. 
Constructed  and  operated  it. 

1774-  Spinning-jenny  invented  by  Robert 
Arkwright, 

1776.  Iron  Railway  at  Sheffield,  England. 
1783.  First  Balloon  ascension,  and  Vessel’s 
bottoms  coppered. 

1700  Water-lines  first  introduced  in  mod- 
els of  Vessels  in  the  U.  S. 

1797.  John  Fitch— Propelled  a yawi-boat 
by  application  of  Steam  to  side- 
wheels,  and  also  to  a screw'-propel-  - 
ler,  upon  Collect  Pond,  New  York. 
1807.  Robert  Fulton  — First  Passenger 
Steamboat. 

1824.  Compound  marine  steam-engines : 

first  introduced  by  James  P.  Al-t 
lan,  New  York. 

1825.  Introduction  of  steam  towing  by 

Mowratt,  Bros.  & Co.,  of  New  York, 
by  steam  boat  “ Henry  Eckford,”, 
New  York  to  Albany.* 

1826.  Voltaic  Battery  discovered  by  Alex.  > 

Volta,  and  First  Horse-railroad. 

1827.  First  Railroad  in  U.  S.,  from  Quincy 

to  Neponset. 

1829.  First  Lucifer  Match  and  first  Loco- 

motive in  America. 

1830.  Liverpool  and  Manchester  Railroad. 

opened.  First  Steel  Pen  and  first 
Iron  Steamer. 

1832.  S.  F.  B.  Morse  invents  the  Magnetic 
Telegraph.  { 

1836.  Robert  L.  Stevens  first  burned  An  ' 
thracite  Coal  in  furnace  of  boiler, 
of  steamboat  “Passaic.” 

1840.  First  steam-boiler  constructed  for 
burning  Anthracite  Coal  in  steam- 
boat “North  America,”  N.  Y. 
1844.  Telegraph  line  from  Washington  to 
Baltimore,  Md. 

1846.  First  complete  Sewing-machine. 

Elias  Howre,  inventor. 

1866.  Submarine  Telegraph  laid  from 
Valencia  to  Newfoundland,  N.S. 


* Witnessed  by  author. 


CHRONOLOGY. 


73 


Dates  oT  Day-  of  Week,  corresponding  to  Day  deter- 
mined "by  following  Table. 


February, 

March, 

November. 

February,* 

August. 

May. 

January, 

October. 

January,* 

April, 

July. 

September, 

December. 

June. 

z 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

3i 

• Thus,  if  Monday  is  the  day  determined  by  the  year  given,  the  following  dates  are 
the  Mondays  in  that  year. 


Epacts,  Dominical  Letters,  and  an  Almanac,  from 
1SOO  to  1901. 

Use  of  Table.  — To  ascertain  day  of  the  week  on  which  any  given  day  of  the 
month  falls  in  any  year  from  1800  to  1901. 

Illustration. — The  great  fire  occurred  in  New  York  on  16th  of  December,  1835; 
what  was  day  of  the  week  ? 

Opposite  1835  is  Sunday;  and  by  following  table,  under  December,  it  is  ascertained 
that  13th  was  Sunday;  consequently , 16th  was  Wednesday. 


Years. 

Days. 

Dom 

Let- 

ters. 

l w 

Years 

Days. 

Dom. 

Let- 

ters. 

cl 
• W 

Years. 

Days. 

Dom, 

Let- 

ters. 

4 

e* 

1 w 

1800 

Saturday. 

E 

.4 

1834 

Saturday. 

E 

20 

1868 

Sunday.* 

ED 

6 

1801 

Sunday. 

D 

15 

1835 

Sunday. 

D 

I 

1869 

Monday. 

C 

17 

1802 

Monday. 

C 

26 

1836 

Tuesday.* 

CB 

12 

1870 

Tuesday. 

B 

28 

1803 

Tuesday. 

B 

7 

1837 

Wednesd. 

A 

23 

1871 

Wednesd. 

A 

9 

1804 

Thursday.* 

AG 

18 

1838 

Thursday. 

G 

4 

1872 

Friday.* 

GF 

20 

1805 

Friday. 

F 

29 

1839 

Friday. 

F 

15 

1873 

Saturday. 

E 

1 

1806 

Saturday. 

E 

11 

1840 

Sunday.* 

ED 

26 

1874 

Sunday. 

D 

12 

1807 

Sunday. 

D 

22 

1841 

Monday. 

C 

7 

1875 

Monday. 

C 

23 

CO 

00 

Tuesday.  * 

CB 

3 

1842 

Tuesday. 

B 

18 

1876 

Wednesd.* 

BA 

4 

1809 

Wednesd. 

A 

14 

i843 

Wednesd. 

A 

29 

1877 

Thursday. 

G 

15 

1810 

; Thursday. 

G 

25 

i844 

Friday.  * 

GF 

11 

1878 

Friday. 

F 

26 

1811 

{ Friday.  * 

F 

6 

1845 

Saturday. 

E 

22 

1879 

Saturday. 

E 

7 

1812 

Sunday.* 

ED 

*7 

1846 

Sunday. 

D 

3 

1880 

Monday.* 

DC 

18 

1813  Monday. 

C 

28 

1847 

Monday. 

C 

14 

1881 

Tuesday. 

B 

29 

1814 

i Tuesday. 

B 

9 

1848 

Wednesd.* 

BA 

25 

1882 

Wednesd. 

A 

11 

1815 

i Wednesd. 

A 

20 

1849 

Thursday. 

G 

6 

1883 

Thursday. 

G 

22 

1816 

Friday.* 

GF 

1 

1850 

Friday. 

F 

17 

1884 

Saturday.* 

FE 

3 

1817 

Saturday. 

E 

12 

1851 

Saturday. 

E 

28 

1885 

Sunday. 

D 

14 

1818 

Sunday.' 

D 

23 

1852 

Monday.* 

DC 

9 

1886 

Monday. 

G 

25 

1819 

Monday. 

C 

4 

i853 

Tuesday. 

B 

20 

1887 

Tuesday. 

B 

6 

1820 

Wednesd.* 

BA 

*5 

i854 

Wednesd. 

A 

1 

1888 

Thursday.* 

AG 

17 

1821 

Thursday. 

G 

26 

i855 

Thursday. 

G 

12 

1889 

Friday. 

F 

28 

1822 

Friday. 

F 

7 

1856 

Saturday.* 

FE 

23 

1890 

Saturday. 

E 

9 

1823 

Saturday. 

E 

18 

i857 

Sunday. 

D 

4 

1891 

Sunday. 

D 

20 

1824 

Monday.* 

DC 

29 

j 1858 

Monday. 

C 

15 

1892 

Tuesday.  * 

CB 

j 

1825 

Tuesday. 

B 

11 

1859 

Tuesday. 

B 

26 

1893 

Wednesd. 

A 

12 

1826 

Wednesd. 

A 

22  I 

i860 

Thursday.* 

AG 

7 

1894 

Thursday. 

G 

23 

1827 

Thursday. 

G 

3 1 

1861 

Friday. 

F 

18 

1895 

Friday. 

F 

4 

1828 

Saturday.* 

FE 

H 1 

1862 

Saturday. 

E 

29 

1896 

Sunday.* 

ED 

15 

1829 

Sunday. 

D 

25  j 

1863 

Sunday. 

D 

11 

1897 

Monday. 

c 

26 

1830 

Monday. 

C 

6 

1864 

Tuesday.  * 

CB 

22 

1898 

Tuesday. 

B 

7 

1831 

Tuesday. 

B 

17 

1865 

Wednesd. 

A 

3 

1899 

Wednesd. 

A 

18 

l832  j 

Thursday.* 

AG 

28 

1866 

Thursday. 

G 

14 

1900 

Thursday. 

29 

i833  1 

Friday. 

F 

9 I 

1867 

Friday.  | 

F 1 

25 

1901  j 

Friday. 

F j 

11 

In  leap-year,  January  and  February  must  be  taken  in  columns  marked  *. 


CHRONOLOGY.— '-MOON’S  AGE. — TIDES. 


74 


To  ^Ascertain  Year  or  Years  of  Coincidences  of  a given 
Day  of  the  Week  with,  a given  Day  of  a Month. 

Look  in  preceding  table  and  ascertain  day  of  week  opposite  to  year  of 
occurrence,  and  every  year  in  which  same  day  is  given  will  be  year  of  coin- 
cidences required. 

Illustration.— If  a child  was  born  on  Saturday,  19th  Sept.  1829,  when  could  and 
can  his  birthdays  be  celebrated,  that  occurred  or  are  to  occur  on  same  day  of  week 
and  date  of  month  ? 

Opposite  to  1829  is  Sunday,  and  in  preceding  table  the  Sundays  for  September  of 
that  vear  were  6th,  13th,  20th",  hence,  if  20th  was  Sunday,  the  19th  was  Saturday. 

Hence,  every  year  in  table  opposite  to  which  is  Sunday  are  the  years  of  the  coin- 
cidence required,  as  1835, 1840, 1846, 1857, 1863, 1868, 1874, 1885,  etc. 


moon’s  age. 

To  Compute  Moon’s  -A_ge. 

Rule. — To  day  of  month  add  Epact  and  Number  of  month;  from  sum 
subtract  29  days,  12  hours,  44  min.  and  2 sec.,  as  often  as  sum  exceeds  this 
period,  and  result  will  give  Moon’s  age  approximately  at  6 o’clock  a.m.  in 
United  States,  east  of  Mississippi  River. 

Numbers  of  the  Months. 


d.  h. 


January 5 

February 1 22 

March 9 


d.  h. 

April 1 21 

May 2 8 

June.. 3 19 


d.  h. 

July 4 7 

August 5 18 

September ...  7 5 


d.  h. 

October 7 16 

November 9 4 

December ....  9 15 


Example.— Required  age  of  Moon  on  25th  February,  1877? 

Given  day  25  + epact  15 -{-number  of  month  1.22  = 41  d.  22  li.  —29  d.  12  h.  44  m. 
2 sec.  = 12  d.  9 h.  15  min.  58  sec. 


In  Leap-years  add  1 day  to  result  after  28th  February. 


To  Compute  Age  of  Nfoon  at  Mean  .Noon,  at  any  other  r 
Location  than  that  Griven. 

Rule. — Ascertain  age,  and  add  or  subtract  difference  of  longitude  or  time, 
according  as  place  may  be  West  or  East  of  it,  to  or  from  time  given. 

Or , when  time  of  new  Moon  is  ascertained  for  a location,  and  it  is  required 
to  ascertain  it  for  any  other,  add  difference  of  longitude  or  time  of  the  place, 
if  East,  and  subtract  it  if  it  is  West  of  it. 

Moon’s  Southing,  as  usually  given  in  United  States  Almanacs,  both  Civil  and  Nau- 
tical, is  computed  for  Washington. 


To  Compute  Tiine  of  High-water  by  jAid  of  A.xnerican 
N autical  Almanac. 

Rule. — Ascertain  time  of  transit  of  Moon  for  Greenwich,  preceding  time 
of  the  high-water  required.  ( 

For  any  other  location  (west  of  Greenwich),  multiply  the  time  in  column 
“ diff.  for  one  hour”  by  longitude  of  location  west  of  Greenwich,  expressed 
in  hours,  and  add  product  to  time  of  transit. 

Note.— It  is  frequently  Necessary  to  take  the  transit  for  preceding  astronomical 
day,  as  the  latter  does  not  end  until  noon  of  day  under  computation. 

Example. — Required  time  of  liigh-water  at  New  York  on  25th  of  August,  1864. 

Longitude  of  New  York  from  Greenwich  = 4 Ji.  56  m.  1.65  sec.,  which,  multiplied 
by  2. 17  min. , the  difference  for  1 hour  = 10.71  min.  for  correction  to  be  added  to  time 
of  transit,  to  obtain  time  of  transit  at  New  York. 


TIDES. MOON’S  SOUTHING. 

Time  of  transit,  18  h.  38.8  m. ; then  18  h.  38.8  m.  -f  10.71  m.  = 
Time  of  transit  at  New  York,  24  d.  18  h.  50  m. 
Establishment  of  the  Port,  8 13 


7 5 

18  hours  49.51  min. 


Note.  - 


25  d.  3 h.  3 m.  = time  of  high-water. 

p.H.  Civil  mine0  °f  25th  at  3 A'  3 Astron<>mical  computation  = 25th  at  3 h.  3 m. 

To  Compute  Time  of  High-water  at  Full  and  Change 
of  Moon. 

Time  of  High-water  and  Age  of  Moon  on  any  Day  being  given. 

Rule.  Note  age  of  Moon,  and  opposite  to  it,  in  last  column  of  following 
tabie,  take  time,  which  subtract  from  time  of  high-water  at  this  ara  of 
Moon  added  to  12  h.  26  m .,  or  24  h.  52  m.,  as  case  may  require  (when  sum  to 
be  subtracted  is  greatest),  and  remainder  is  time  required. 

Example.— What  is  time  of  high-water  at  full  and  change  of  Moon  at  New  York? 
Time  of  high-water  at  Governor’s  Island  on  25th  of  Jan.  1864,  was  q h 20  m a m 
civil  time.  Age  of  Moon  at  12  m.  on  that  day  was  16  d.  8 h.  59  m.  9 

nn2pP°fe  ‘°  16  in  following  table,  is  13  h.  28  m.,  and  difference  between  16  d 
Th  12  ^ (- 1 r I ? or  I3-5*3  ~ l3* 28)  is  25  ^ hence?  ^ « h. = 25  m. , 16  a. 

8h.j9  m.  — 16  d.  _ 8 h.  59  m.  = 18.71  or  19  w.,  which,  added  to  13  h.  28  w.  = 13  h. 
™en  9 12  \ 26  (aS  Su?>  t0  be  subtracted  is  greater  than  time)  _ 13  h 

47  m.  = 21  n.  46  w.  — 13  h.  47  m.  = 7 7i.  59  m.  7 J 

This  is  a difference  of  but  13  minutes  from  Establishment  of  Port. 

Time  after  apparent  Noon  before  Moon  next 
passes  Meridian,  Age  at  Noon  being  given. 


Days 


1-5 


Moon  at 
Meridian. 

Age  of 
Moon. 

Moon  at 
Meridian. 

Age  of 
Moon. 

Moon  at 
Meridian. 

H.  M. 

P.  M. 

Days. 

H.  M. 

P.  M. 

Days. 

H.  M. 

P.  M. 

0 

6 

5 3 

12 

10  6 

25 

6-5 

5 28 

12.5 

10  31 

50 

7 

5 53 

13 

10  56 

1 16 

7-5 

6 19 

J3-5 

11  21 

I 41 

8 

6 44 

H 

11  47 
A.  M. 

2 6 

8.5 

7 9 

*4-5 

12  12 

2 31 

9 

7 34 

15 

12  37 

2 57 

9-5 

7 59 

i5-5 

13  2 

3 22 

10 

8 25 

16 

13  28 

3 47 

10.5 

8 50 

16.5 

*3  53 

4 12 

11 

9 i5 

17 

14  28 

4 38  1 

II*5 

9 40  J 

1 W-5 

H 43  1 

Days. 

18 

18.5 

J9 

i9-5 


20.5 

21 

21.5 

22 

22.5 

23 

23-5 


15  8 
*5  34 
*5  59 

16  24 

16  49 

W i5 

17  40 

5 

18  30 

18  56 

19  21 
19  46 


Age  of 
Moon. 

Moon  at 
Meridian. 

Days. 

H. 

M. 

A. 

M. 

24 

20 

II 

24-5 

20 

37 

25 

21 

2 

25-5 

21 

27 

26 

21 

52 

26.5 

22 

17 

27 

22 

43 

27-5 

23 

8 

28 

23 

33 

28.5 

23 

58 

29 

24 

24 

29-5  I 

24 

48 

Tidal  IPlienomena. 

tJK6  elevati°n  of  a tidal  wave  towards  the  Moon  slightlv  exceeds  tint  nf  the 
P The  Sun  & 1f.tens.,ty  of  diminishes  from  Equator  to  the  Poles 

the 1 actiTo^th^Cn^urwith1 TTSE"  «“  sea  lowing 

on  boZiwTlrnEmh  C°mbinCd aCti°n  °f  the Sun  aDd Moon  wh™  they  are 

•J&S  StST.g’ iMKasiS  Sr s* » «» 


;6 


LATITUDE  AND  LONGITUDE, 


LATITUDE  AND  LONGITUDE. 

Latitude  and  Longitude  of  Principal  Locations 
and  Observatories. 

Compiled  from  Records  ofU.  S.  Coast  and  Geodetic  Survey  and  Topograph- 
ical  Engineer  Corps , Imperial  Gazetteer , and  Bowditclis  Navigator. 

Longitude  computed  from  Meridian  of  Greenwich. 

A represents  Academy;  Az.,  Azimuth ; A.  S.,  Astronomical  Station;  C.,  College; 
Cap  , Ctopitol;  Ch. , Church ; C.  YL  , City  Hall ; C.  S.,  Coast  Survey ; Cl  Court-house; 
Cy .'Chimney;  F.S.,  Flagstaff;  G.S Geodetic  Station ; Hos.  . Hospital ; L.  Light- 
house ; Obs  ..Observatory;  S.H.,  State-house;  Sp.  Spire ; Sq.,  Square; 

Station  • T Telearanh  ; T.  H. , Toten  JTaM ; U. , University;  L n. , I mow  ; B. , Baptist , 
Co Episcopal ; P.',  rSesby. ; and  M.Ch,  JfcO.  U/mrc^S. 


Location. 

Latitude.  1 

NORTH  AND  SOUTH 

N. 

AMERICA. 

0 t n 

Acapulco 

Mex. 

16  50  19 

Albany,  P.Ch 

N.Y. 

42  39  3 

Ann  Arbor Mich. 

42  16  48 

Annapolis 

. Md. 

38  58  42 

Apalachicola,  F.S.  Fla. 

29  43  30 

Astoria,  F.S 

46  11  19 

Atlanta,  C.  H 

..Ga. 

33  44  57 

Auburn 

N.Y. 

42  5.5 

Augusta 

..Ga. 

33  28 

Augusta,  B.Ch.. . 

. Me. 

44  18  52 

Austin 

.Tex. 

30  16  21 

Balize 

..La. 

29  8 5 

Baltimore,  Mon’t 

. Md. 

39  i7  48 

Bangor, Tho’s  Hill.  Me. 

44  48  23 

Barbadoes,  S.Pt. 

. W.I. 

13  3 

Barnegat,  L 

.N.  J. 

39  46 

Bath,  W.S.Ch... 

..Me. 

43  54  55 

Baton  Rouge 

30  26 

Beaufort,  Ct 

.N.C. 

34  43  5 

Beaufort,  E.Ch. . 

. S.C. 

32  26  2 

Belfast,  M.Ch... 

. . Me. 

44  25  29 

Benicia,  Ch 

. Cal. 

38  3 5 

Benington 

...Vt, 

42  40 

Bismarck,  S.  S . . 

.Neb. 

46  48 

Boston,  L 

Mass. 

42  19  6 

Boston,  S.H 

“ 

42  21  30 

Brazos  Santiago. 

.Tex. 

26  6 

Bridgeport 

Conn. 

41  10  3° 

Bristol 

.R.  I. 

41  40  11 

Brooklyn,  C. H. . 

.N.Y. 

40  41  31 

Brownsville,  S.S. 

..Tex. 

26 

Brunswick,  C.Sp.  .Me. 

Buffalo,  L N.Y. 

Burlington N.  J. 

Burlington,  C Vt. 

B u rl  i n gton,  P ub.  Sq.,  Ia. 

Bushncll Neb. 

Cairo Til. 

Calais,  C.S.  Obs. . . Me. 

Callao,  F.S Peru 

Cambridge,  Obs.  ‘Mass. 

Camden S.C. 

Campeachy . .Yucatan 


Location. 


3i 

43  54  29 
42  50 
40  4 52 

44  28  52 

40  48  22 

41  i3  54 
36  59  48 

45  11  5 
S. 


42  2.2  52 
34  17 
19  49 


W. 

99  49  9 

73  45  24 

83  43 
76  29 

84  59 
123  49  42 

84  23  22 
76  28 
81  54 
69  46  37 
97  44'  I2 
89  1 4 
76  36  59 

68  46  59 
59  37 

74  6 

69  49 
91  18 

76  39  48 

80  40  27 

69  19 
122  9 23 

73  18 

100  38 

70  53  6 

71  3 3° 
97  12 
73  11  4 
71  16  5 

73  59  27 
97  30  r 

81  29  26 
69  57  24 
78  59 

74  52  37 
78  10 

91  6 25 
103  52  57 
89  11  14 
67  16  5 

77  !3 

71  7 43 
80  33 
9°  33 


Latitude,  j Longitude. 


NORTH  AND  SOUTH  N. 

AMERICA. 

Canandaigua N.Y. 

Cape  Ann,  S.  L.  .Mass. 

Cape  Breton Va. 

Cape  Canaveral. . .Fla. 

Cape  Cod,  L.  P.  L. . . Ms. 

Cape  Fear N.C. 

Cape  Flattery, L..W.T. 

Cape  Florida,  L. . . Fla. 

Cape  Hancock,  Colo.  R. 

Cape  Hatteras,  L. , N.C. 

Cape  Henlopen,  L. , Del. 

Cape  Henry,  L Va. 

Cape  Horn,  S.  Pt.,  Her- 
mit’s Island 


Cape  May,  L N.  J. 

Cape  Race N.  S 

Cape  Sable N.S. 

Cape  Sable,  C.S. . . Fla. 

Cape  St.  Roque,  Brazil 


Carthagena. . . . . .N.G. 

Castine Me. 

Cedar  Keys,  Depot  Isl. 

Chagres N.G. 

Charleston , C.  Ch. , S.  C. 
Charlestown,  Mon. , Ms. 
Cheboygan,  L . . . Mich. 

Chicago,  C.Ch 111. 

Chickasaw Miss. 

Cincinnati,  Obs 0. 

Cleveland,  Hos “ 

Colorado  Springs.. Col. 

Columbia,  S.H S.C. 

Columbus,  Cap O. 

Concord,  S.  H N.  H. 

Corpus  Christi. . .Tex. 
Council  Bluffs.  .Neb. T. 
Crescent  City,  L.  .Cal. 

Cumberland Md. 

Darien.  W.H Ga. 

Davenport,  S.  S la. 

Dayton 0.  „ 

Deadwood.  S.  S. . . Dak.  ',44  22 
Decatur,  S.  S Tex.  133  10 


42  54  9 
42  38  11 

45  57 

28  27  30 

42  2 
33  48 
48  23 
25  39  54 

46  16  35 

35  i5  2 
38  46  6 

36  55  3° 
S. 

53  $ 

38  55  48 
46  39  24 

43  24 

25  6 53 
S. 

5 28 
N. 

10  26 

44  22  30 
29  7 3° 

9 20 

32  46  44 

42  22  36 

45  40  9 
41  53  48 
35  53  30 

39  6 26 

41  30  25 

38  50 

33  59  58 

39  57  40 

43  12  2 
27  47  1 

3° 

44  34 
39  39  i4 
31  21  54 
41  32 
39  44 


W. 

0 / n 

77  17 
70  34  10 
59  48  5 
80  33 
70  9 48 
77  57 
124  43  54 
80  9 2 
124  1 45 
75  3°  54 
4 7 


67  16 

74  57  18 
53  4 3 

65  36 
81  15 

35  17 

75  38 

68  45 

83  2 45 

80  1 21 
79  55  39 
71  3 18 

84  24  37 

87  37  47 

88  6 .25 
84  29  45 

81  40  30 
104  49  8 

8123 

82  59  40 

71  29 
97  27  2 
95  48 
124  11  22 
78  45  25 
81  25  39 
90  38 
84  11 
103  34 
97  30 


LATITUDE  AND  LONGITUDE. 


77 


Latitude  and  Longitude — Continued. 

Location.  | Latitude.  Longitude.)  Location.  Latitude.  Longitude. 


NORTH  AND  SOUTH 
AMERICA. 

Denver,  S.  H.Sp. . Col. 
Des  Moines,  C.  H . . . la. 
Detroit,  St.  P.  Ch. , M ich. 

Dover  Del. 

Dover N.  H. 

Dubuque Ia. 

Duluth,  S.  S Min. 

Eastport,  Con.Ch. . Me. 
Eden  ton,  C.H....N.  C. 
Elizabeth  City,  Ct.  “ 

Erie,  L Penn. 

Eureka,  M.  Ch Cal. 

Falls  St.  Anth’y..Minn. 
Fernandina,  A.  S. , Fla. 

Florence Ala. 

Fort  Gibson Ind.  T. 

Fort  Henry Tenn. 

Fort  Laramie.  .Neb.T. 
Fort  Leavenworth,  Ks. 

Frankfort Ky. 

Frederick Md. 

Fredericksburg,  E.Ch., 
Ya. 

Fredericton N.B. 

Galveston.Cath’l..Tex. 

Georgetown Ber. 

Georgetown,  E.Ch., S.C. 
Gloucester, U. Ch. . . Ms. 
Grand  Haven,  S.  S., 
Mich. 

Halifax,  Obs N.S. 

Harrisburg Penn. 

Hartford,  S. H. . .Conn. 
Havana,  Moro . . .Cuba 
Hole  in  the  Wall,  L. 

Bahamas 
Holmes’s  Hole, Ch., Ms. 

Hudson N.Y. 

Huntsville Ala. 

Indianapolis Ind. 

Indianola,G.S Tex. 

Jackson Miss. 

Jacksonville,  M.  Ch. 

Fla! 

Jalapa Mex. 

Jefferson  City Mo. 

Jersey  City,  Gas  Ch’y. 
Kalama,  M.Ch. . . W.T. 

Keokuk,  S.  S Ia. 

Key  West,  T. Obs.,  Fla. 

Kingston Jamaica 

Kingston.  C.  H. . .C.  W. 

Knoxville Tenn. 

La  Crosse,  Ct.S. . . Wis. 

Lancaster Penn. 

Lavaca.  A.  S Tex. 

Leavenworth,  S.S..  Ks. 
Lexington Ky. 


Lima. . 


. Peru 


N. 

39  45 
. 35 

42  19  46 

39  10 

43  i3 
42  29  55 
46  48 

44  54  i5 
36  3 24 
36  1 7 58 

42  8 43 

40  48  11 
44  58  40 
30  40  18 

34  47  13 

35  47  35 

36  30  22 
42  12  10 
39  21  ] 

38  14 

39  24 

38  18  6 
46  3 
29  18  17 

32  22 

33  22  8 

42  36  46 

43  5 

44  39  4 

40  16 

41  45  59 
23  9 

25  5i  5 

41  27  13 

42  14 

34  36 

39  55 

28  32  28 
32  23 


W. 

o / 

io4  59  33 
93  37  16 

83  2 23 

75  30 
70  54 
9°  39  57 

92  8 
66  59  14 

76  36  3 

76  13  23 

80  412 
124  9 4 

93  10  30 

81  27  47 

87  41  40 
95  i5  10 

88  3 40 
104  47  43 

94  44 

84  40 

77  18 

77  27  38 
66  38  15 
94  47  26 
64  37  6 
79  16  49 
70  39  59 


NORTH  AND  SOUTH 
AMERICA. 

Lockport N.Y. 

Los  Angeles Cal. 

Louisville Ky. 

Lowell,  St.  Ann’s  Ch., 
Mass. 

Machias,  Th .Me. 

Macon,  Arsenal Ga. 

Madison,  Dome..  .Wis. 
Marblehead,  L ..Mass. 
Martinique,  S.P’t.W.I. 
Matagorda,  G.S. . .Tex. 
Matamoras. .... 

Matanzas Cuba 

Memphis,  S.S. . .Tenn. 

Mexico Mex. 

Milwaukee Mich. 

xMinneapolis,  U.C.,Min. 
Mississippi  City,  G.  S., 
Miss. 

Mobile,  E.  Ch Ala. 

Monterey,  Az.  S. . .Cal. 

Montevideo. . .Rat  Is’d 


N. 


42  38  46 
44  43 
’32  5°  25 

43  4 33 

42  3°  *4 

J4  27 
28  41  29 

25-52-50 
23  3 

35  7 

J9  25  45 

43  2 24 

44  58  38 

30  22  54 
30  41  26 

36  35 
S. 

34  53 
N. 


30  19  43 
19  30 

38  36 

40  43  28 
46  26 
40  23 
24  33  31 
17  58 
44  8 

35  59 
43  58  50 
40  2 36 
28  37  36 

39  29 
38  6 

S. 


Little  Rock Ark.  1 34  40 


32  22  45 
45  31 
37  4 47 
4i  23  24 

41  16  57 
36  9 33 
25  5 
3i  34 

4i  5 5 

41  38  10 
4.1  18  28 
41  21  16 
29  57  46 

40  42  44 

35  6 

41  30 

42  48  30 
39  39  36 
41  29  12 

36  50  47 
4i  2 50 
4i  33 
35  6 28 

44  45 

37 

47  3 
4i  15  43 

43  28  32 

45  23 

8 57  9 
39  16  2 
30  20  42 


18 


73  32  56 

89  12 
70  2 24 

70  5 57 
86  49  3 
77  21  2 
91  24  42 

101  21  24 

70  55  36 
72  55  45 
72 

90 

74  24 


5 29 
3 28 


Montgomery,  S.  H.,  Ala. 

Montreal C.  E. 

Mound  City 111. 

Nantucket,  L. . . . Mass. 

Nantucket,  S.  Tower, 

86  18  I Mass. 

63  35  Nashville,  U Tenn. 

76  50  Nassau,  L N.  P. 

72  40  45  Natchez Miss. 

82  21  23  Nebraska,  Junction  of 

Forks  of  Platte  Riv. 

77  10  6 New  Bedford,  B.  Ch. 

70  35  59  Mass! 

7346  New  Haven, Col., Conn. 

86  57  New  London,  P.Ch. 

86  5 New  Orleans,  Mint,  La. 

90  3 8 1 New  York,  C.  H. , N.  Y. 

Newbern,  E.  Sp. . .N.C. 

81  39  14  Newburg,  A.  Sp.,  N.Y. 

96  54  30  Newburyport,L.,Mass. 

92  8 New  Castle.  E.Ch., Del. 

74  224  Newport,  Sp R.  ' 

122  50  39  Norfolk,  C.  H Va. 

91  25  Norwalk Conn. 

81  48  31  Norwich 

76  46  Ocracoke,  L N.C. 

76  28  37  < )gdensburg,  L. . . N.  Y. 

83  54  Did  Point  Comfort,  Va. 

91  14  48  Olympia Wash.T. 

762033  Omaha,  P.  Ch. ..  .Neb. 

96  37  21  Oswego,  S.  S N.Y. 

9458  Ottawa Can. 

84  18  Panama,  Cath’l.  ..N.G. 

Parkersburg W.  Va. 

77  6 Pascagoula Miss. 

Pensacola,  Sq’re..Fla.  30  24  33 

92  12  I Petersburg,  C.  H. . .Va.|37  13  47  77  24  16 

G* 


W. 

o"  L ,J 

78  46 

[18  14  32 

85  30 


71  19  2 
67  27  21 
83  37  36 

89  24  3 
7°  5o  39 
60  55 

95  57  56 
97  27  50 
81  40 

90  7 

99  5 6 

87  54  4 
93  H 8 


1 57 

2 28 
121  52  59 

56  13 


77  5 

74  33 

70  52  28 

75  33  48 

71  18  49 

76  7 22 
73  25  35 

72  7 
75  58  51 

75  30 

76  18  6 
122  55 

95  56  14 
76  35  5 
75  42 
79  27  17 
34  12 
32  45 
87  12  53 


78 


LATITUDE  AND  LONGITUDE, 


ZLatitncLe  and  JLjon.gitu.cLe—  Continued. 

Latitude.  Longitude. 


Location. 

Latitude. 

NORTH  AND  SOUTH 

N. 

AMERICA. 

0 1 u 

Philadelphia,  S.  H.,  Pa. 

39  56  53 

Pike’s  Peak,  S.S.  .Col. 

38  48 

Pittsburg Penn. 

40  32 

Plattsburg,  Sp — N.Y. 

44  4i  57 

Plymouth,  Pier  . . .Ms. 

41  58  44 

Point  Hudson W.T. 

48  7 3 

Port  au  Prince. . . W.  I. 

18  33 

Port  Townshend,  A.S., 

Wash.  T. 

48  6 56 

Portland,  C.H Me. 

43  39  28 

Portland,  S.  S. ..... .0. 

45  30 

Porto  Bello N.  G. 

9 34 

Porto  Cabello,  Mara- 

caibo 

10  28 

Portsmouth,  L. . .N.  H. 

43  4 16 

Prairie  du  Chien..Wis. 

43  2 

Princeton,  S.Cap.,  N.  J. 

40  20  40 

Providence,  U.  Ch. , R.  I. 

41  49  26 

Provincetown,  Sp., 

Mass. 

42  3 

Puebla  de  los  Angelos, 

Mex. 

19  J5 

Quebec,  Citadel.  .Can'a 

46  49  12 

Queenstown....  “ 

43  9 

Raleigh,  Square.. N.C. 

35  46  50 

Richmond,  Cap Va. 

37  32  16 
3.. 

Rio  de  Janeiro,  S.  Loaf. 

22  56 

N. 

Rochester,  R.  H.  .N. Y. 
Rockland,  E.  Ch . . . Me. 
Sackett’s  Harbor,  N.  Y. 
Sacramento. ..... .Cal 

Salem,  So Mass. 

Salt  Lake  City,  Obs., 
Utah 

Saltillo Mex. 

San  Antonio Tex. 

San  Buenaventura, 

G.  S Cal. 

San  Diego,  A. S — “ 
San  Francisco,  C.  S. 

Station Cal. 

San  Jose,  Sp “ 

San  Luis  Obispo. . “ 

San  Pedro “ 

Sandusky,  L 0. 

Sandy  Hook,  L. . . N.  J. 
Santa  Barbara,  M.  Cli., 
Cal. 

Santa  Clara,  C.Ch.. 
Santa  Cruz,  F.  S. . 

Santa  Fe N.  Mex. 

Savannah,  Sp.. Ga. 

Schenectady N.Y. 

Sherman,  R. R. D.,Wy. 

Shreveport,  S.  S I .a. 

Smithville,  G.S. . .N.C. 

Springfield Mass. 

Springfield,  S.II 111. 

Springfield,  S.  S “ 

St.  Augustine. ... . .Fla. 


1 17 


Location. 


43 

44  6 
43  55 
38  34  41 

42  31 

40  46 
25  26  22 
29  25  22 

34  15  46 

32  43  68 

37  48 
37  19  50 

35  10  38 

33  43  20 

41  32  3° 

40  27  40 

34  26 
37  20  49 

36  57  31 

35  41  6 
32  4 52 

42  48 

41  7 50 

32  30 

33  54  58 

39  47  57 

42  6 


W. 

75  9 
104  59 
80  2 

73  26  54 
70  39  12 

122  44  33 
72  16  3 

122  44  58 
70  i5 
122  27  30 
79  4° 

68  7 

70  42  34 
91  8 35 

74  39  55 

71  24  19 

70  11  18 

98 

71  12  15 

79  8 
78  38 
77  26 

43  9 

77  5i 

69  6 52 

75  57 
121  27  44 

7°  53  58 


hi  53  47 
101  4 45 
98  29  15 


19  *5  56 

17  9 40 

122  23  19 

121  53  39 
120  43  31 

18  16 
82  v:  - 1 
74 

19  42  42 

[21  26  56 

122  I 29 
106 

81  5 26 
73  55 
105  23  33 
93  45 
78  1 

89  39  20 

72  36 


NORTH  AND  SOUTH 
AMERICA. 

St.  Augustine,  P.  Ch., 
Fla. 

St.  Bartholomew,  S. 

Point W.  I. 

St.  Christopher,  N.  Pt., 
W.  I. 

St.  Croix,  Obs “ 

St.  Domingo “ 

St.Eustatia,Town.  “ 

St.  Jago  de  Cuba,  En- 
trance   W.  I. 

St.  John N.  B. 

St.  Joseph .Mo. 

St.  Louis,  W.U.. . 

St.  Mark’s,  Fort.. Fla. 
St.  Martin’s,  Fort,W.  I. 
St.  Mary's,  M.  H. . .Ga. 

St.  Paul Minn. 

St.  Thomas,  Fort  Ch’n, 
W.  I. 

St.  Vincent’s,  S.  Point, 
W.  I. 

Staunton Va. 

Stockton,  S.S Tex. 

Stonington,  L. . .Conn. 
Sweetwater  River, 
Mouth  of.  ...Neb. T. 

Sydney,  S.  S N.  S. 

Syracuse . .N.Y. 

Tallahassee Fla. 

Tampa  Bay,  E. Key  “ 

Tampico,  Bar Mex. 

Taunton, T.  C.Ch.,  Mass. 
Tobago,  N.  E.  P’r.W  I. 

Toronto Can. 

Trenton,  P.Ch  . . .N.  J. 
Trinidad,  Fort...W.  I. 

Troy,  D. Ch N.Y. 

Tuscaloosa Ala. 

Utica,  Dut.Ch N.Y. 


29  53  J 
17  53  3° 

7 24 

17  44  30 

18  29 

17  29 

19  58 
45  i4 

23  3 i3 
38  8 3 

30  9 

18  5 

30  43  12 
44  52  46 


29  48  3c  81  35 


Valparaiso,  Fort.  .Chili 

Vanda  lia 111. 

Vera  Cruz Mex. 

Vicksburg,  S.  S..  .Miss. 

Victoria Tex. 

Vincennes Ind. 

Virginia  City, S.S.,M.T 

Washington.  . .Capitol 

Watertown,  Ars’l.  .Ms. 

West  Point N.Y 

Wheeling Va. 

Wilmington,  E.  Ch.. 

N.  C. 

Wilmington, T.H.  .Del. 
Worcester,  Ant.  H. . Ms. 

Yankton,  S.S Dak. 

Yazoo Miss. 

York Penn 

| Yorktown Va. 


13  9 
38  8 51 
30  50 

41  19  36 

42  27  18 
46  12 

43  3 
30  28 
27.  36 
22  15  30 

54 
11  20 
43  39  35 
40  13  10 
10  39 

42  43 

33  12 

43  6 49 

38  50 
19  11  52 

32  23 

28  46  57 
S8  43 
45  20 

38  53  20 
42  21  41 

23  26 
40  7 

34  J4 

39  44  27 
42  16 
42  45 

33  5 
39  58 
37  13 


W. 

/ n 
81  18  41 
62  56  54 

62  50 
64  40  42 
69  52 

63 

75  52 

66  3 30 
109  40  44 
90  12  4 
84  12  30 

63  3 

81  32  53 
95  4 54 

64  55  18 

61  14 
79  4 i5 
102  50 
71  54 

107  45  27 
60  12 

76  9 16 

84  36 

82  45  15 
97  5i  5i 
71  5 55 

60  27 
79  23  21 

74  45  5° 

61  32 

73  2 16 

87  42 

75  13 


77 


71  41 

89  2 

96  8 36 
9°  54 

97  1 

87  25 

112  3 

36 

71  9 45 
73  57  1 
80  42 

77  56  38 

75  33  3 

71  48  13 

97  3° 

90  20 

76  40 
76  34 


LATITUDE  AND  LONGITUDE. 


79 


Latitude  and  Longitude— Continued. 


Location. 


EUROPE,  ASIA,  AFRICA, 
AND  THE  OCEANS. 

Aleppo  

Alexandria,  L 

Algiers,  L 

Amsterdam 

Antwerp 

Archangel 

Athens 

Barcelona. 


Latitude. 


Batavia,  Obs 

Beucoolen,  Fort,  Su’a. 


Berlin,  Obs. . . 
Bombay,  F.  S. 


Botany  Bay,  C.  Roads , 

Bremen 

Bristol 


Brussels,  Obs. 
Bussorah 


Cadiz. 


Cairo 

Calais  . . . 
Calcutta  . 
Candia. . . 
Canton  . . 


Cape  Clear. 

Cape  of  G.  Hope.  Obs. 
Cape  St.  Mary,  Mad'r. 

Ceylon,  Port  Pedro  . . 
Christiana 


Congo  River 

Constantinople,  St.  S. . 

Copenhagen 

Corinth 

Cronstadt 

Dover 


Dublin 

Edinburgh 

Falkland  Islands, 
Helena,  Obs. 


St. 


N. 

36  11 
31  12 

36  47 
52  22 

51  13 

64  32 

37  58 
4i  23 

S. 

6 8 
3 48 
N. 

52  30  16 
18  56 

S. 

34  2 
N. 

53  5 

5i  27 

50  51  3 
30  3° 

36  32 

30  3 

50  58 

22  34 

35  31 

23  7 

51  26 
S. 

33  56 

25  f 

9 49 
59  55 
S. 

6 8 
N. 

41  1 
55  41 

37  54 
59  59 

8 


Longitude. 


E. 
o / 
37  10 
29  53 

3 4 

4 53 
4 24 

40  33 
23  44 


106  50 
102  19 

13  23  45 
72  54 

151  13 

8 49 
W. 

2 35 
E. 

4 22 
48 

XV. 

6 18 
E. 

31  18 
1 5i 
8 20 

25  8 
13  14 
W. 


28  45 
45  7 

80  23 
10  43 


Location. 


EUROPE,  ASIA,  AFRICA, 
AND  THE  OCEANS. 

Genoa 


Gibraltar. . . . 
Glasgow  . . . . 

Greenwich  . 


Hamburg . 
Havre 


Hawaii  or  Owyhee.. . 


Hongkong . 
Honolulu  . . 


Hood  Isl’d,Gallapagos. 
Hood’s  Island,  Mar 
quesas 


Jeddo  or  Tokio. 

Jerusalem 

Leghorn,  L 

Leipsic 

Leyden 


Lisbon 

Liverpool,  Obs. . 

Madras 


Madrid  . 


Fayal.  S.  E.  Point 

Feejee  Group,  Ovolai 
Obs 


Florence 

Funchal,  Madeira. . 
Geneva 


53  23  12 
55  57 
S. 
i5  55 
N. 

38  30 
S. 

1 7 4i 
N. 


28  59 
!2  34 
22  52 

29  47 


43  46 
32  38 

46  11  59 


6 20  30 
3 12 

5 45 


28  42 
E. 

*78  53 

11  16 
W. 

16  55 
E. 

6 9 15 


Majorca,  Castle. 
Malaga 


Malta,  Valetta. 

Manila 

Marseilles 

Messina,  L. 

Mocha 

Moscow 

Muscat 

Naples,  L 


N. 

44  24 

36  7 
55  52 
51  28  38 

53  33 


Longitude. 


22  16  30 
21  18  12 

1 23 
S 

9 26 
N. 

35  40 
3i  48 
43  32 

51  20  20 

52  9 28 

38  42 

53  24 

*4  4 9 
40  25 

39  34 

36  43 


5 22 
4 16 


New  Castle 

New,  t. Hebrides,  Table 

Island 

Niphon,  Cape  Idron, 

Japan ' 

Odessa 

Palermo,  L 

Paris,  Obs 

Pekin 


Plymouth 

Port  Jackson.  .N.S.W. 
Porto  Praya,  Cape  Verd 
Islands 


Prince  of  Wales  Island. 


35  54 
J4  36 
43  18 
38  12 
3 20 
55  40 
23  37 

49  50 

54  58 
S. 

5 28 
N. 

34  36  3 
46  28 

38  8 
48  50  13 

39  54 

50  21 
S. 

35  5i  32 
N. 

14  54 
S. 

10  46 


E. 

9 58 
6 
W. 
i55  54 
E. 

114  14  45 
157  30  36 
W. 

89  46 
E. 

r38  57 

i39  40 
37  20 
10  18 
12  22 
4 29  15 
W. 

9 9 

3e. 

80  15  45 
W. 

3 42 
E. 

2 23 
W. 

4 26 
E. 

*4  30 
121  2 

5 22 

15  35 
43  12 
35  33 
58  35 
14  16 

W. 

*37 

E. 

1 67  7 

38  50  35 
30  44 

13  22 

2 20 

16  28 

W. 

V 

51  18 

w. 

23  3 
E. 

142  12 


8o 


LATITUDE  AND  LONGITUDE. 


Latitude  and.  3L.ongitri.de— Continued. 


Location. 


JI 


Longitude. 


EUROPE,  ASIA,  AFRICA, 
AND  THE  OCEANS. 
Queenstown 


Rome,  St.  Peter’s. 
Rotterdam 


Santa  Cruz Ten’fe 

Scilly,  St.  Agnes,  L 

Senegal,  Fort 


Sevastopol . 

Seville 

Siam 


Sierra  Leone. 


Singapore. 
Smyrna — 


N. 

5i  47  ' 

41  54 
5 54 

28  28 
49  54 
16  1 

44  37 
36  59 
H 55 
S. 

8 30 
N. 

1 17 
38  26 


Southampton  . 


S. 

St.  Helena ••••II5  55 


W. 

°8 /g  " 

12  27 

16  16 
6 21 
16  32 
E. 

33  30 
5 58  ' 
100 

W. 

13  18 
E. 

103  50 
27  7 
W. 

1 30 

5 45 


Location. 


EUROPE,  ASIA,  AFRICA, 
AND  THE  OCEANS. 

St.  Petersburg 

Suez v*'**  ■ 

Surat,  Castle 


Sydney N.S.W. 

Tahiti  or  Otaheite 


Tangier 

Toulon 

Tripoli 

Tunis,  City. . . 

Venice 

Vienna 

Warsaw,  Obs. 


Wellington... New  Z’d 

Yokohama 

Zanzibar  Island,  Sp. . 


N. 

0 'a  ‘ 

59  56 
29  59 

11 
S. 

33  33  42 

7 45 
N. 

35  47 
43  7 

34  54 

36  47 

40  50 

48  13 
52  13 
S. 

41  14 
N. 

35  26 
S. 

6 28 


Longitude. 


E. 

O / // 

30  19 
32  34 
72  47 

151  23 

W. 

149  30 
E. 

5 54 
5 22 

13  11 
10  6 

14  26 
16  23 
2129 

i74  44 

139  39 

39  33 


0"bs© i*'V’atoi*ios. — 2Vo£  included  in  pvevious  Tuble. 
Longitude  given  in  Time. 


Albany,  Dudley  . . 
Alleghanv,  Penn. . 
Birr  Castle,  Earl 
of  Rosse 


Cambridge,  U.  S. . . 

Cambridge,  Eng... 

Cape  of  G.  Hope.. 

Copenhagen,  Un’y. 
Crescent  City,  A. 

S. , Cal 

Dublin 

Edinburgh 


Florence. 
Geneva. . 


Longitude. 


Georgetown,  U.S. . 
Gibbes’s,  Charles- 
ton, U.  S 


Greenwich  . 


Hamburg . 
Leipsic . . . 
Leyden . . . 


N. 

i ‘ //  ■ iti 

42  39  49-55 

40  27  36 

53  5 47 

42  22  52 

52  12  51.6 
S. 

33  56  3 
N. 

55  40  53 

41  44  43 

53  23  13 
55  57  23-2 

43  46  4*-4 
46  11  59.4 

38  53  39 

32  47  7 

51  28  38 

53  33  5 

51  20  20.1 

52  9 28.2 

53  24  47-8 


Liverpool . . . 

L.  M.  Rutherfurd, 

New  York 14°  43  49 


W. 

m.  s 

4 54  59-  52 

5 20  2.9 

31  40.9 

4 44  30-9 

E. 

22.75 

1 13  55 

50  19.8 
W. 

8 16  49.1 
25  22 
12  43.6 
E. 

45  3-6 
24  37-7 
W. 

5 8 12.5 
5 19  44-7 

E. 

39  54- 1 
49  28.5 
17  57-5 
W. 

12  o.  11 
4 55  57 


Madras 

Marseilles. . 


Mitchell’s,  Cin.,0. 


Moscow 

Munich,  Bogenli’n 
Palermo 


Portsmouth . 
Quebec 


Rome,  College. . . 
Salt  Lake  City, 

Utah 
San  Francisco,  Sq., 
Cal 


Latitude.  | Longitude. 


• N. 

1 u m 
13  4 8.1 

43  *7  5° 


39 


6 26 


Santiago  de  Chili. 

St.  Croix,  W.  I 

St.  Petersburg,  A. . 
Stockholm 


Sydney 

TiflVs,  Key  West. 

Fla 

Unkrechtsberg,  01- 
mutz 


Washington. . . 
West  Point,  N.Y. . 


55  45  19  8 
48  8 45 
38  6 44 

5°  48  3 
46  48  30 

41  53.52-2 

40  46  4 

37  47  55 

S. 

33  26  24.8 

N. 

17  44  30 
59  56  29.7 

59  20  31 
S. 

33  5n41' 
24  33  31 

49  35  4° 

38  53  39 

41  23  26 


E. 

h.  m.  s. 

5 20  57.3 
21  29 
W. 

5 37  59 
E. 

2 30  16.96 
46  26.5 
53  24-17 
W. 

4 23  9 
4 44  49.02 
E. 

49  54-7 
W. 

7 27  35- 1 

8 9 38.1 
4 42  18.9 

4 18  42.8 

E. 

2 1 13-5 

1 12  24.8 

10  4 59.86 
W. 

5 27  i4-i 

E. 


5 8 12.03 

4 55  48 


DIFFERENCE  IN  TIME, 


8l 


DIFFERENCE  IN  TIME. 

Difference  in  Time  at  Following  Locations. 

Longitude  computed  both  from  New  York  and  Greenwich . 

Exact  Difference  of  Time  between  New  York  and  Greenwich  is  4 h.  56  m. 
1.6  sec.,  but  in  following  table  2 seconds  are  given  when  the  decimal  in  any 
reduction  exceeds  .5  seconds. 


Location. 


Acapulco 

Albany 

Alexandria. . Egypt 

Algiers 

Amsterdam . 

Antwerp 

Apalachicola 

Astoria 

Atlanta 

Auburn 

Augusta Ga. 

Augusta Me. 

Austin 

Baltimore 

Bangor 

Barbadoes,  S.  Ft. 

Barnegat,  L 

Bath 

Baton  Rouge. . . 

Beaufort N.C. 

Beaufort S.  C. 

Belfast 

Benicia 

Berlin 

Bismarck 

Bombay,  F.S, 
Boston,  S.H. , 

Bremen 

Bridgeport  . . . 
Brooklyn,  N.  Yard. 

Brunswick Me. 

Brunswick Ga. 

Brussels 

Buenos  Ayres 

Buffalo,  L 

Burlington Ia. 

Burlington N.  J. 

Burl  ngton Vt. 

Bushnell Neb. 

Cadiz 

Cairo 

Cairo 111. 

Calais Me. 

Calcutta 

Callao 

Cambridge  . . Mass. 

Canton 

Cape  Girardeau 

Cape  of  Good  Hope. 

Cape  Horn 

Cape  May 

Cape  Race  

C.irthageoa 

Castinc 


F representing  Fast , and  S Slow. 
New  York,  j Greenwich.  Location. 


h.  m. 

1 43  15  S. 

1 F. 
6 55  34 
5 8 18 
5i6,S 

5 1338 
43  54  S. 

3 19  *7 
4i  32 
9 50 
31  34 
16  55  F 
1 34  55  S. 
10  26 
20  54  F. 
57  34 
22  S. 
16  46  F. 

I 9 10S. 

10  38 
26  39 
20  1 F. 

3 12  36  S, 

5 49  37  F. 

1 46  30  S. 
9 47  38  F 

11  47.6 
5 3i  18 

3 17 
4 

1 6 12 
29  56  S. 

5 13  30 F. 

2 34 
19  54  S. 

8 24 

3 29 
16  38 
59  30- 

4 30  5o  F. 

1 14 
43  S. 

2 6 57  F. 
49  22 
12  50  S. 

II  30 F. 

2 28  58 

2 10  S. 

9 57  F. 

26  58 
2 56  S. 

23  46  F. 

6 30  S. 

21  2 F. 


h.  m. 

6 39  i7  S. 

4 55 
1 59  32  F. 

12  16 
19  32 
17  36 

5 39  56  S. 
8 15  19 

5 37  33 
5 5 52 

5 27  36 

4 39  6 

6 30  27 

5 6 28 

4 35  8 

3 58  28 

4 56  24 

4 39  16 

6 5 12 

5 6 39 

5 22  40 
4 36  1 
8 8 38 

53  35  F. 

6 42  32  S. 

4 51  36 F. 
4 44  14  S. 

35  16  F. 

4 52  44  S. 

4 55  58 

4 39  5o 

5 25  58 
17  28  F 

3 53  28  S. 

5 15  56 

4 26 

4 59  30 

5 12  40 

6 55  32 
25  12 

5 12  F. 

5 56  45  S. 

4 29  4 

5 53  20  F 

8 52  S. 

4 44  3i 
32  56 F. 
58  12S. 

13  55  F. 
29  4S. 

58  58 

3 32  16 

5 2 32 

4 35 


New  York. 


Cedar  Keys 

Chagres 

Charleston 

Charlestown 

Cheboygan 

Chicago 

Chickasaw 

Cincinnati 

Cleveland 

Colorado  Springs. . 

Columbia 

Columbus 

Concord 

Constantinople 

Copenhagen 

Corpus  Christi 

Council  Bluffs 

Crescent  City 

Darien 

Davenport 

Dayton 

Deadwood 

Denver 
Detroit 

Dover Del. 

Dover. N.  H. 

Dublin 

Dubuque 

Duluth 

Eastport 

Edenton 

Edinburgh 

E 1 i zabe  th  C i ty , N.  C. 

Erie *. 

Eureka 

Falls  St.  Anthony. . 

Fernandina 

Fire  Island,  L 

Florence Ala. 

Fort  Gibson. 

Fort  Henry.  .Tenn. 

Fort  Laramie 

Fort  Leavenworth. 
Frederick 
Fredericksb’g. . Va. 
Fredericton. . .N.B. 

Funchal 

Galveston 

Geneva 

Geneva N.Y. 

Genoa 

Georgetown. . .Ber. 
Georgetown. . .S.  C. 
Gibraltar, 


A. 

36  9S. 
24  3 
23  4i 

11  48 F. 
4i  37  S. 
54  30 
56  24 
4i  57 
30  40 

2 3 15 

28  7 

35  57 
10  6 F. 

6 51  58 
5 4<5  18 
1 33  47  S. 

1 27  10 

3 20  44 

29  4i 
1 12  30 

40  42 

1 58  30 

2 3 57 

36  76 
35  58 
12  26  F. 

4 30  4° 

1 6 38  S. 

1 12  10 

28  6 F. 
10  24  S. 

4 43  14  F. 

8 52  S. 

24  15 

3 20  37 
1 16  40 

29  50 

3 10F, 
54  45  S. 

1 24  59 
56  13 

2 3 9 
1 22  54 

13  10 
*3  49 
29  29  F. 

3 48  22 

1 23  8 S. 

5 20  39  F. 
12  14  S. 

5 3i  34  F. 
37  33 
21  6 S. 

4 34  34  F. 


5 32  11  S. 
5 20  5 
5 19  43 

4 44  13 

5 37  38 
5 50  31 
5 52  26 
5 37  59 

5 26  42 

6 59  17 
5 24  8 

5 3i  59 

4 45  56 

1 55  56  F 
50  16 

6 29  48  S. 
6 23  12 

8 16  45 

5 25  43 

6 2 32 

5 36  44 

6 54  32 
6 59  58 
5 32  10 

5 2 

4 43  36 
25  22 

6 2 40 
6 8 32 

4 27  56 

5 6 26 
12  48 

5 4 54 

5 20  17 
8 16  39 

6 12  42 
5 25  51 

4 52  51 

5 5°  47 

6 21  1 

5 52  15 

6 59  n 
6 18  56 
5 9 12 

5 9 5i 

4 26  33 

1 7 40 

6 19  10 

24  37  F. 

5 8 16  S. 
35  32  F. 

4 18  28  S. 
5i  7 7 
21  28 


82 


DIFFERENCE  IN  TIME. 


Difference 


Location. 

New  York. 

Greenwich. 

t 

i.  m.  8.  1 

i.  m.  8. 

Glasgow 

4 38  58  F. 

17  4S.  : 

Gloucester 

13  22^ 

4 42  40 

Grafton 

24  5 S. 

5 20  7 

Grand  Haven 

49  10 

5 45 12 

Greenwich 

4 56  1 .6 

— 

Halifax 

41  42  F. 

4144° 

Hamburg 

5 35  54 

39  52  F. 

Harrisburg 

11  18  S. 

5 7 20  S. 

Hartford 

5 i9  F. 

4 50  43 

Havana,  Morro. . . 

33  24  S. 

5 29  26 

Havre 

4 56  26  F. 

24  F. 

Hawaii  or  Owyhee 

5 27  34  S.  : 

10  23  36  S. 

Hongkong : 

i2  27  I F. 

7 36  59  F- 

Honolulu : 

15  27  30 

10  31  28 

Hudson 

1 12 

4 54  40  S. 

Huntsville 

51  46  S. 

5 47  48 

Indianapolis 

48  18 

5 44  20 

Indianola 

1 30  2 

6 26  4 

Jackson L 

1 4 30 

6 0 32 

Jacksonville.. .... 

30  35 

5 26  37 

Jalapa 

1 31  36 

6 27  38 

Jeddo  or  Tokio . . . 

14  16  2 F. 

9 20  F. 

Jefferson  City. . . ; 

1 12  30  S. 

6 8 32  S. 

Jersey  City 

8 

4 56  i° 

Jerusalem 

7 25  22  F. 

2 29  20  F. 

Kalama 

3 i5  21  S. 

8 11  23  S. 

Keokuk 

1 9 38 

6 5 40 

Key  West 

31  *3 

5 27  14 

Kingston Can. 

9 53 

5 5 54 

Kingston Jam. 

11  2 

5 7 t 

Knoxville 

39  34 

5 35  36 

La  Crosse 

1 8 58 

6 4 59 

La  Guayra 

27  54  f • 

4 28  8 

Lancaster 

19  21  s. 

5 *5  22 

Lavaca 

1 30  27 

6 26  29 

Leavenworth 

1 23  40 

6 19  52 

Leghorn 

5 37  i4  F- 

41  12  F. 

Lexington 

41  10  S. 

5 37  12  S. 

Lima 

12  22 

5 8 24 

Lisbon 

4 19  26  F. 

36  36  F. 

Little  Rock 

1 12  46  S. 

6 8 48  Sj 

Liverpool 

4 44  2 F. 

12 

Lockport 

19  2 b. 

5*5  4 

Los  Angeles 

2 56  10 

7 52  18 

Louisville 

45  58  T7 

5 42  ^ 

Lowell. 

IO  45  F. 

4 45  16 

Machias  Bay 

26  12 

4 29  49 

Macon. 

38  28  s. 
1 1 35 
4 41  14  F 
4 38  18 

5 34  3° 

Madison . . . 

5 57  36 

Madrid.. 

Malaga 

14  48 
17  44 

M 

• 5 54  2 

• 13  10 

58  I 

Manila 

848 

Maracaibo 

9 2 

4 47  S 

Marblehead,  L. . . 

12  41 

4 43 

Marseilles 

. 5 17  20 

21  28  F 

Martinique 

Matagorda 

Matamoras. 

Matanzas 

. 52  22 

. 1 27  50  S 

• 1 33  50 

. 1 3°  38 

4 3 4oS 

. 6 23  52 

6 29  51 

5 26  40 

Memphis 

. 1 4 26 

6 0 28 

Mexico 

. 1 40  19 

6 36  20 

in  Time — Continued. 

Location. 


Monterey. 


Montreal. . . 

Montserrat 

Moscow 

Mound  City 

Nantucket 

Naples 

Nashville 

Nassau 

Natchez 

Nebraska 

New  Bedford 

New  Haven 

New  London 

New  Orleans 

New  York.  

Newbern 

Newburg 

Newbury  port 

New  Castle 

New  Castle. . . Del, 

Newport 

Norfolk 

Norwalk 

Norwich 

Ocracoke. ....... 

Odessa 

Ogdensburg. .. . . 
Old  Point  Comfort 

Olympia 

Omaha.. 

Oswego . . — 

Ottawa 

Paducah 

Palermo 

Panama 

Paris 

Parkersburg. 

Pekin  ...... 

Pensacola. . . 
Petersburg. . 
Philadelphia 
Pike’s  Peak. 
Pittsburg . . . 
Plattsburg . . 

Plymouth 

Plymouth..  .Mass. 
Port  Au  Prince, 

St.  Domingo 
Port  Townshend. . 

Portland 

Portland 

Porto  Praya. 

Porto  Rico. . 
Portsmouth. 


few  York. 

Greenwich. 

. m.  8.  1 

1.  m.  8. 

55  35  S. 

5 5i  36  S- 

1 16  55 

6 12  57 

1 6 

5 56  8 

56  7 

5 52  9 

18  22  F. 

4 37  40 

3 n 3oS- 

8 7 32 

1 11  10 F. 

3 44  52 

49  10  S. 

5 45  12 

1 50  F. 

4 54  12 

47  i4 

4 8 48 

7 18  14 

2 22  12  f. 

1 26  S. 

5 56  28  S. 

15  38  F. 

4 40  24 

5 53  6 

57  4F. 

51 15  s. 

5 47  I6  b- 

13  23 

5 9 24 

1 9 37 

6 5 39 

1 49  24 

6 45  26 

12  19  F. 

4 43  42 

4 19 

4 5i  43 

7 40 

4 48  22 

1 4 12  S. 

6 14 

— 

4 56  1.6 

12  18 

5 8 20 

1 

4 56  2 

12  32  F. 

4 43  30 

4 49  34 

6 28 

6 13  S. 

5 2 15 

10  46  F. 

4 45  15 

9 8S. 

5 5 9 

2 19  F. 

4 53  42 

7 34 

4 48  28 

7 54  S. 

5 3 55 

6 58  58  F. 

2 2 56  I 

5 58  S. 

5 2 £ 

9 11 

5 5 12 

3 i5  38 

8 11  40 

1 27  43 

6 23  45 

.Or. 


io  19 
6 46 
58  22 
5 49  30  F. 

1 21  48  S. 

5 5 22  F. 

30  15  S. 
12  41  54  F. 

52  50- s. 
13  35. 

4 34  6 

2 3 50 
24  6 

2 14  F, 
4 39  26 
13  25 

16  34 

3 14  58  S. 
15  2 F. 

3 13  48  S. 
3 23  50  F. 
33  26 
13  11 


5 6 20 
5 2 48 

5 54  24 
ca  28  F. 
5 17  49  s. 

9 20  F. 
5 26  37  S. 

7 45  52  F. 
5 48  52  S. 
5 9 37 

5 36*2 

6 59  52 

5 20  8 

$ 

4 42  37 

4 39  28 

8 11 
4 4i 

8 9 50 
1 32  12 
4 22  36 
4 42  5i 


DIFFERENCE  IN  TIME, 


83 


Difference  in  Time-  Continued. 

Location.  New  York.  Greenwich.  Location.  New  York. 


Prairie  du  Chien 
Princeton. . 
Providence 
Provincetown. . . 

Quebec 

Queenstown,  L. 

Raleigh 

Richmond 

Rio  de  Janeiro. . 
Rochester. . 
Rockland . . 

Rome 

Rotterdam . 
Sackett’s  Harbor, 
Sacramento 

Salem 

Salt  Lake  City. . 

Saltillo 

San  Antonio 
San  Buenaventura 

San  Diego 

San  Francisco, 

C.  S.  S. 

• San  Francisco,  P 

San  Jose 

Sandusky. . . 
Sandy  Hook. 

Santa  Barbara. . . . 
Santa  Clara 

Santa  Cruz. 

Santa  Cruz. Ten’ fe 

Santa  Fe 

Savannah 

Schenectady 

Seville 

Sherman 

Shreveport 

Siam 

Sierra  Leone 

Singapore 

Smithville 

Smyrna 

Southampton 

Springfield 111. 

Springfield.  .Mass. 

St.  Augustine 

St.  Croix,  Obs 

St.  Helena 

St.  .Jago  de  Cuba. . 

St.  John 

St.  Joseph Mo. 


833S. 
2 38 

10  24  F. 

15  16 

11  13 

4 42  46 

18  31  S. 
13  43 

2 3 26  F. 
15  22  S. 

19  34  F. 

5 45  50 
5 13  58 

7 46  S. 

3 9 49 

12  26  F. 
2 31  34  S. 

1 48  17 

1 37  55 
312 

2 52  37 

3 13  32 
3 13  5o 
3 n 33 

34  47 
iF. 

3 2 49  S. 

3 9 46 
3 12  4 

3 5o  58  F. 

8 4S. 

28 

22  F. 

4 3^  io 
5 33  S. 

1 18  58 
1 36  2 F. 

4 2 50S. 

1 51  22  F. 

16  3 S. 

6 44  30  F. 

4 5o  2 
2 36  S. 

5 39  F- 

29  13  s. 

37  19  F- 
4 33  2 
7 26  S. 

31  48  F.  | 

2 22  41  S.  | 


A. 

6 4 34  S, 
4 58  40 

4 45  37 
4 40  45 

4 44  49 
33  16 

5 14  32 
5 9 44 
2 52  36 
5 11  24 

4 36  27 
49  48  F. 
17  56 

5 3 48  S. 
8 5 51 

4 43  36 

7 27  35 

6 44  19 

6 33  57 

7 57  4 

7 47  39 

8 9 33 
8 9 51 
8 7 35 

5 30  49 
4 56  1 
7 58  50 

5 


8 8 


1 5 4 
7 4 5 

5 24  22 

4 55  40 

23  52 

7 1 34 

6 15 

6 40  F. 
53  12  S. 

6 55  20  F. 

5 12  5 S. 
1 48  28  F. 

6 S. 
5 58  37 

4 50  24 

5 25  15 

4 18  43 

23 

5 3 28 
4 24  14 

7 18  43 


St.  Louis 

St.  Mark’s 

St.  Mary’s 

St.  Paul 

St.  Petersburg. . . 
St.  Thomas,  Fort. 

Staunton 

Stockholm 

Stonington 

Suez. ; 

Sweetwater  River, 

Mouth  of. 

Sydney N.S. 

Sydney. . .N.S.W. 

Syracuse 

Tahiti  or  Otaheite. 

Tallahassee  

Tampa  Bay 

Tampico  Bar 

Taunton 

Toronto 

Toulon 

Trenton 

Tripoli 

Troy 

Tunis 

Turk’s  Island 

Tuscaloosa 

Utica 

Valparaiso 

Vandalia 

Venice 

Vera  Cruz 

Vicksburg 

Victoria Tex. 

Vienna 

Vincennes . . . 
Virginia  City. 

Warsaw 

Washington,  Obs. . 

West  Point. 

Wheeling 

Wilmington.  .Del. 
Wilmington.  .N.  C. 

Worcester 

Yankton 

Yazoo 

Yeddo 

Yokohama 

York 

York  town 


A.  m.  «. 

4 47  S. 
40  48 
30  10 
11  24  18 
6 57  18  F. 
36  20 
20  15  S. 

6 8 26  F. 
8 26 

7 6 18 


2 15 

8 2 

15  I34~ 

8 35  S. 

14  44  2 F. 
42  22  S. 
34  59 
1 35  25 
11  38  F. 
21  32  S. 
5 17  3°  F- 
3 2 S. 
5 48  36  F. 

3 54 
5 36  26 

11  22 

54  46  S. 

4 5o 

9 18 

1 6 

5 53  46  F. 
1 28  33  S. 

1 7 34 

1 32  2 

6 1 34  F. 
53  38  S. 

2 32  10 

6 20  11  F. 

12  10  S 
14  F. 

26  46  S. 

6 11 
15  45 
8 49  F. 

1 33  58  S. 

1 5 18 

14  14  42  F. 
14  14  43 
10  38  S. 
io  14 


j (Greenwich. 

A.  m.  s. 

6 48  S. 

5 36  50 

5 26  12 

6 20  20 

2 1 16  F. 

4 19  4i  S. 

5 16  17 

1 12  25  F. 
4 37  36  S. 

2 10  16  F. 

7 11  2 S. 

4 48 

10  5 32  F. 

5 4 37  S. 
9 48  F. 
5 38  24  S. 

5 3i  1 

6 31  27 

4 44  24 

5 1 7 33 
21  28  F. 

4 59  3 S. 

52  44  F. 
4 52  9 S. 
40  24  F. 

4 44  40  S. 

5 5o  48 

5 52 

4 46  44 

5 56  8 
57  44  F. 

6 24  34  S. 

6 3 36 

6 28  4 

1 5 32  F. 

5 49  40  S. 

7 28  12 

1 24  9 F. 

5 8 12  S. 

4 55  48 

5 22  48 
5 2 12 

5 11  47 
4 47  13 

6 30 

6 1 20 
9 18  40  F. 

9 18  41 
6 40  S. 

6 16 


fo  Compute  Difference  of  Time  between  ISTew  York  and 
Greenwich  and  any  Location  not  given  in  Table. 

Rule.  — Reduce  longitude  of  location  to  time,  and  if  it  is  W of  as- 
sumed meridian  it  is  Slow  ; if  E.,  it  is  Fast. 

If  difference  for  New  York  is  required,  and  it  exceeds  4 h.  z6  m. 
2»sec  subtract  this  time,  and  remainder  will  give  difference  of  time,  S.  • 
and  it  it  (4  h.  56  m 2 sec.)  does  not  exceed  it,  subtract  difference  from  it, 
and  remainder  will  give  difference  of  time,  F. 


84 


TIDES. 


TIDES. 

Tide-Table  for  Coast  of  United  States, 
Showing  Time  of  High-water  at  Full  and  New  Moon , termed  Establish- 
ment of  the  Fort , being  Mean  Interval  between  Time  of  Moon's  Transit 
and  Time  of  High-water.  (H.  S.  Coast  and  Geodetic  Survey.) 


Locations  and  Time. 

Spring. 

J Neap. 

1 

COAST  FROM  EASTPORT 

C 

TO  NEW  YORK. 

h.  m. 

Feet.  ] 

Feet. 

Eastport Me. 

11  30 

15 

( 

Campo  Bello*.... 

11 

25 

( 

Portland “ 

11  25 

9.9 

7.6  1 

Cape  Ann* 

11  30 

11 

Portsmouth N.H. 

11  23 

9.9 

7.2  1 

Newburyport. . .Mass. 

II  22 

6.6  ] 

Salem “ 

II  13 

10.0 

7.6  . 

Cape  Cod* “ 

11  3° 

6 

] 

Boston  Light  .. . “ 

II  12 

10.9 

8. 1 

Bostont “ 

II  27 

10.3 

8-5  ' 

Nantucket “ 

12  24 

3-6 

2.6 

Edgartovvn “ 

12  T6 

2-5 

1.6 

Holmes’s  Hole. . “ 

n 43 

1.8 

13  ; 

Tarpaulin  Cove  . “ 

8 4 

2.8 

1.8  < 

Wood’s  Hole,  n.  side. 

7 5o 

4-7 

3-i 

N.  Bedford  (Dump-) 

A 6 

2*8  1' 

ing  Rock)  J 

7 57 

New  York? N.Y. 

8 13 

5-4 

3-4  “ 

Albany* “ 

3 30 

1 

LONG  ISLAND  SOUND. 

Newport R.  I. 

7 45 

4.6 

3-i 

Point  Judith “ 

7 32 

3-7 

2.6 

Montauk  Point. . N.Y. 

8 20 

2.4 

1.8 

Watch  Hill R.I. 

9 

3-i 

2.4 

Providence* 

8 25 

5 

Stonington Ct. 

9 1 

3.2 

2.2 

Little  Gull  Isl’d.  N.Y. 

9 38 

2.9 

2-3 

New  London Ct. 

9 28 

3-i 

2.1 

New  Haven “ 

11  16 

6.2 

5-2 

Bridgeport “ 

11  11 

8 

4-7 

Oyster  Bay N.Y. 

11  7 

9.2 

5-4 

Sand’s  Point “ 

11  13 

8.9 

6.4 

New  Rochelle. . . u 

11  22 

8.6 

6.6 

Throg’s  Neck. . . “ 

11  20 

9.2 

6. 1 

Hell  Gate* u 

9 35 

6 

COAST  OF  NEW  JERSEY. 

Cold  Spring  Inlet,  N.  J. 

7 32 

5-4 

3-6 

Sandy  Hook N.J. 

7 29 

5-6 

4 

Amboy “ 

8 15 

5 

Cape  May  Landing  “ 

8 19 

6 

4-3 

Egg  Harbor*. ... . “ 

9 34 

5 

DELAWARE  BAY  AND 
RIVER. 

Delaware  Break watei 

r 8 

4-5 

3 

Higbee’s  (Cape  May). , 

• 8 33 

6.2 

3-9 

Egg  Isl’d  Light . .N.J 

• 9 4 

7 

5-i 

New  Castle Del 

• 11  53 

6.9 

6.6 

Philadelphia Penn 

• 13  44 

1 6.8 

5-i 

Locations  and  Time. 


CAROLINA,  GEOR-GIA, 
AND  FLORIDA. 


Beaufort  . 


Charlestonll  (C.  H.  ) 

Wharf S.C.  j 

Fort  Pulaski Ga. 

Savannah “ 

St.  Augustine Fla. 

Cape  Florida  ... . “ 

Key  West “ 

Tampa  Bay “ 

Cedar  Keys “ 


WESTERN  COAST. 


.Cal. 


San  Diego 

San  Pedro 

Cuyler’s  Harbor . 

San  Luis  Obispo.. 

Monterey 

South  Farallone  . 

San  Francisco . . . 

Mare  Island 

Benicia 

Ravenswood 

Bodega. 

Humboldt  Bay. . . 

Astoria Or. 

Nee-ali  Harbor,  Wash. 
Port  Townshend  “ 


13  i5 


t Refem  to  rise  and  fall  of  tide  alone. 


MISCELLANEOUS. 

Bay  of  Fundy*.  .N.S. 

Blue  Hill  Bay*. . 

St.  John’s* 

Kingston* Jam. 

Halifax* ...N.  S. 

Pensacola* Fla. 

Galveston* Tex. 

+ t § li  see  p.  85. 


Feet. 


1.6 


60 


30 


1.8 


1.6 


M.,n  i n ter vaV has  been  increased  12  A.  =6  min.  (hnlf  « menu  lunar  day)  for  aom.  nnrU  ir  Def 


Note.- 

aware 


TIDES. 


85 


Bench.  UVIarks  referred,  to  in  preceding  Table. 

t Boston.  — Top  of  wall  or  quay,  at  entrance  to  dry-dock  in  Charlestown  navy- 
yard,  14.76  feet  above  mean  low- water. 

t New  York.  —Lower  edge  of  a straight  line,  cut  in  a stone  wall,  at  head  of  wooden 
wharf  on  Governor’s  Island,  14.56  feet  above  mean  low- water. 

§ Old  Point  Comfort,  Va. — A line  cut  in  wall  of  light-house,  one  foot  from  ground, 
on  southwest  side,  n feet  above  mean  low- water. 

II  Charleston,  S.  C.  — Outer  and  lower  edge  of  embrasure  of  gun  No.  3,  at  Castle 
Pinckney,  10.13  feet  above  mean  low-water. 


Establishment  of  the  Port  for  several  Locations  in 
Europe,  etc. 

Port.  | Time.  ||  Port.  Time.  Port.  Time. 


Amsterdam 

Antwerp 

Beachy  Head . . Eng. 

Belfast 

Bordeaux 

Bremen 

Brest  Harbor 

Bristol. 

Bristol  Quay 

Cadiz 

Calais 

Calf  of  Man 

Cape  St.  Vincent. . . . 


h.  in. 

3 

4 25 
11  50 

10  43 

6 50 
6 

3 47 

7 21 
6 27 

1 40 

11  49 
n 5 

2 30 


Chatham 

Cherbourg 

Clear  Cape 

Cowes 

Dover  Pier 

Dublin  Bar 

Funchal 

Gravesend Eng. 

Greenock 

Holyhead 

Hull Eng. 

Land’s  End 

Lisbon 


h.  m. 
1 2 
7 49 
4 

10  46 

11  12 
11  12 
11  30 

1 i4 

8 

10  11 
6 29 
3 57 

2 30 


Liverpool 

London  Bridge 

Newcastle 

Portsmouth  D.-yard, 
Eng.' 

Quebec  

Ramsgate  Pier 

Rye  Bay Eng. 

Sheerness 

Sierra  Leone 

Southampton.  .Eng. 
Thames R.,mo’th  “ 
Woolwich “ 


h. 


n 41 
8 

10  27 

11  20 
57 

8 15 

11  40 

12 

2 15 


Rise  and  Fall  of  Tides  in  Grnlf  of  Mexico. 


Locations. 

C 

§ 

be 

.5 

m 

Neap. 

Locations. 

Mean. 

Spring. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

St.  George’s  Island.. . .Fla. 

1. 1 

1.8 

.6 

Isle  Derniere 

. . . . La. 

1.4 

1.2 

Fort  Morgan  ( Mobile ) 

Entrance  to  Lake 

Cal- 1 

Bav) Ala.  j 

1 

1-5 

•4 

PftS'ifM] 

La  j 

i-5 

i-4 

Cat  Island Miss. 

1.3 

1.9 

.6 

Aransas  Pass 

1. 1 

1.8 

Southwest  Pass La. 

1. 1 

1.4 

•5 

Brazos  Santiago. . . 

u 

•9 

1.2 

Tides  of  G-nlf  of  IVIexico. 


On  Coast  of  Florida,  from  Cape  Florida  to  St.  George’s  Island,  near  Cape  San  Bias, 
the  tides  are  of  the  ordinary  kind,  but  with  a large  daily  inequality.  From  St. 
George's  Island,  Apalachicola  entrance,  to  Derniere  Isle,  the  tides  are  usually  of  the 
single-day  class,  ebbing  and  flowing  but  once  in  24  (lunar)  hours.  At  Calcasieu  en- 
trance. double  tides  reappear,  and  except  for  some  days  about  the  period  of  Moon’s 
greatest  declination,  tides  are  double  at  Galveston,  Texas.  At  Aransas  and  Brazos 
Santiago  the  single-day  tides  are  as  perfectly  well  marked  as  at  St.  George’s,  Pensa- 
cola. Fort  Morgan,  Cat  Island,  and  the  mouths  of  the  Mississippi.  For  some  3 to  5 
toys,  however,  about  the  time  when  the  Moon’s  declination  is  nothing,  there  are 
generally  two  tides  at  all  these  places  in  24  hours,  the  rise  and  fall  being  quite  small. 

Highest  high  and  lowest  low  waters  occur  wjhen  greatest  declination  of  Moon 
happens  at  full  or  change.  Least  tides  when  Moon’s  declination  is  nothing  at  first 
or  last  quarter. 

Tides  of  Bacifxc  Coast. 

On  Pacific  coast  there  is,  as  a general  rule,  one  large  and  one  small  tide  during 
each  day,  heights  of  two  successive  high-waters  occurring,  one  A.M.,  and  other 
P.  M.  of  same  24  hours,  and  intervals  from  next  preceding  transit  of  Moon  are  very 
different.  These  inequalities  depend  upon  Moon’s  declination.  When  Moon’s  de- 
clination is  nothing,  they  disappear,  and  when  it  is  greatest,  either  North  or  South, 
they  are  greatest.  The  inequalities  for  low  wTater  are  not  same  as  for  high,  though 
they  disappear,  and  have  greatest  value  at  nearly  same  time. 

When  Moon’s  declination  is  North,  highest  of  twm  high  tides  of  the  24  hours  oc- 
curs at  San  Francisco,  about  11.5  hours  afte* Moon’s  southing  (transit);  and  wThen 
declination  is  South,  lowest  of  the  two  high  tides  occurs  about  this  interval. 

Lowest  of  two  low-waters  of  the  day  is  the  one  which  follows  next  highest  high- 
water. 


IT 


3 


86 


STEAMING  DISTANCES, 


STEAMING  DISTANCES. 

Distances  between  various  Forts  of* United  States 
and  Canada. 

By  Lake,  River,  and  Canal. 


Locations. 

Lake  1 
and  1 
River.  | 

Canal. 

Total. 

Miles. 

Miles. 

Miles. 

Duluth  to  Buffalo. . . 

1024 

1 

1025 

Chicago  to  Buffalo  . . 

925 

— 

925 

Duluth  to  Oswego. . . 

1133 

27 

1160 

Chicago  to  Oswego.. 

1034 

26 

1060 

Duluth  to  New  York, 
via  Buffalo 

1166 

353 

1519 

via  Oswego 

1294 

233 

1527 

Duluth  to  Montreal . 

1289 

72 

1361 

Chicago  to  New 
York,  via  Buffalo . 

1067 

352 

1419 

Locations. 


Chicago  to  NewYork, 

via  Oswego 

Chicago  to  Montreal. 
Buffalo  to  Colborne, 
via  Welland  Canal. 
Buffalo  to  New  York. 
Welland  Canal  to 

Montreal 

Montreal  to  Kingston 
Ottawa  to  Kingston 


Lake 

and 

River. 


ii95 

190 


142 


3°4-5 
126. 25 


232 

71 

26.77 

352 

7°-5 

120 

126.25 


Total. 


.427 

1261 

26.77 

494 

375 

246.25 

126.25 


Miles. 


Alexandria. . . 
Amsterdam . . 
Barbadoes . . . 

Batavia 

Bermudas  . . . 

Bombay 

Boston 

Bremen 

Bristol 

Buenos  Ayres 

Cadiz 

Calcutta . . 


iNT.Y, 

4893 

3291 

1855 

8972 

682 

8522 

356 

3428 

2979. 

6010 

3I25 

9350 


Miles. 


Lond. 

3 102 
262 
3.812 
1 1 492 
3 H2 
10703 

3030 

408 

’ 501 
6 280 

x ns 


Cape  Race. . 

Cowes 

Funchal . . . 
Galway .... 
Gibraltar  . . 
Glasgow  . . . 
Halifax .... 

Havana 

Hobart  Town. . 
Kingston,  Jam. 
Lima 


xi  531 1 Madras  . 


N.  Y. 

1 004 
3092 
2 760 

2 720 

3 260 
2913 

59° 
1 161 

9 i87 
1456 
10050 
8707 


2249 
200 
1 3°3 
72 

1 325 
765 

2 706 
4i97 
11  368 

4 305 
10  149 
10888 


Miles. 


New  Orleans. 

Norfolk 

Pensacola . . . 
Philadelphia. 

Quebec  

Queenstown . 
Rio  Janeiro. . 

St.  Johns 

Southampton 
Swan  River. . 
Tortugas  . . . 
Washington 


N.Y. 

1790 

308 

1623 

262 

1360 

2780 

4970 

1064 

3io3 

8480 

1151 

461 


Halifax  to 

Liverpool 

St.  Thomas 

St.  Johns,  N.  F.  . 
Quebec  to  Glasgow 
Liverpool  to 

Boston 

Quebec 

Philadelphia 

Callao 

Fastnet 

Cape  Race 

Aspinwall 

Port  Said 

Melbourne. . . . 

Rio  Janeiro 

San  Francisco.. 
via  Panama. . . 
via  Tehuantepec 


Miles. 


2563 
x 563 

520 

2563 

2 955 
2855 
3147 

11  379 
283 
1992 
4650 

3 290 
13290 

5125 

13  800 
7 378 
6400 


Poets. 


Liverpool  to 

Havana  

Portland 

Baltimore 

N.  Orleans  to  Havana 
Cape  Race  to 

Fastnet 

Halifax 

Boston 

St.  Johns,  N.  F.,  to 

Quebec 

Boston 

Greenock 

Bermudas  to  Nassau. 
Panama  to 

San  Juan  del  Sud  . 
Gulf  of  Fonseca. . . 
Acapulco. ......... 

Manzanilla 


4100 

2770 

3400 

57° 

1711 

457 

835 

891 

890 

1848 

804 

570 

739 

1416 

1724 


Poets. 


Panama  to 

San  Diego 

Monterey 

San  Francisco 

San  Francisco  to 
San  Juan  del  Sud. 

Acapulco 

Manzanilla 

San  Diego 

Monterey 

Humboldt 

Columbia  R.  Bar. . 

Vancouver. 

Portland 

Port  Townshend . . 

Victoria 

Yokohama 

Honolulu 

Honolulu  to  Callao. . 


Distances  between  various  Ports  and  New  York 
and  London. 

Not  included  in  preceding  Table. 


Miles. 

Lond. 

4 730 

3 447 

4 654 

3 404 
3080 

55i 
5200 
2 214 
211 
10  661 

4 182 
3612 


Distances  between  various  Forts  of  England, 
Canada,  United  States,  etc. 

Not  included  in  preceding  Table. 

Miles. 


2897 
3I98 
3240  i 

2685  j 
1841  i 
1543 
474 
105 
200 

53°  ‘ 
638 
650 
732 
7i5 
4750 
2080 
5H5 


> = J g g a : § 

: o 

«^.S  ^ :■© 

■ •Ialisg5^gl3fea-.|-  :-|l 
oSxSg=o«$a32  ^-jriS  3 

:ls!iill&-a2  l^!!”S 


STEAMING  DISTANCES. 


Puerto  Bello  | 1558  I 1410  I 972  I XI42 


fractions. 


89 


FRACTIONS. 

A Fraction,  or  broken  number,  is  one  or  more  parts  of  a Unit. 

Ti T USTRATION  -12  inches  are  1 foot.  Here,  1 foot  is  unit,  and  12  inches  its  parts ; 
inches therefore,  are  one  fourth  of  a foot,  for  3 is  fourth  or  quarter  of  12. 

A Vulvar  Fraction  is  a fraction  expressed  by  two  numbers  placed  one 
above  the  other,  with  a line  between  them  ; as,  50  cents  is  the  i of  a dollar. 

Upper  number  is  termed  Numerator , the  lower  Denominator.  Terms  of  a fi  ac- 
tion express  numerator  and  denominator;  as,  6 and  9 are  terms  of  -9. 

‘ A Proper  fraction  has  numerator  equal  to,  or  less  than  denominator;  as,  T,  etc. 

An  Improper  fraction  is  reverse  of  a proper  one;  as,  f,  etc. 

A Mixed  fraction  is  a compound  of  a whole  number  and  a fraction;  as,  5§-,  etc. 

A Compound  fraction  is  fraction  of  a fraction ; as,  i of  f,  etc. 

A Complex  fraction  is  one  that  has  a fraction  for  its  numerator  or  denominator, 

or  both;  as,  J.  or  JL,  or  *r  or  etc. 

’ 6 4 S 6 

Note.— A Fraction  denotes  division,  and  its  value  is  equal  to  quotient,  obtained  by 
dividing  numerator  by  denominator;  thus,  - is  equal  to  3,  and  5 is  equal  to  45. 


Fied. notion  of  Fractions. 

To  Compute  Common  Measure  or  greatest  Number 
tlx  at  will  divide  Two  or  more  IsT  umbers  without  a 
Remainder. 

Ruj.e. Divide  greater  number  by  less;  then  divide  divisor  by  remainder;  and  so 

on,  dividing  always  last  divisor  by  last  remainder,  until  there  is  no  remainder,  and 
last  divisor  is  greatest  common  measure  required. 

Example  i.— What  is  greatest  common  936)  1908  (2 
measure  of  1908  and  936  ? i872 

36)  936  (26 
72 

216.  Hence  36. 

2.— How  many  squares  can  there  be  obtained  in  an  area  of  90  by  160  feet? 

Here  10  is  greatest  common  measure. 

Hence,  A6/  = 16,  and  = 9;  therefore  16  X 9 = J44- 

To  Compute  least  Common.  ^Multiple  of*  Two  or  more 
ISTiimhers. 

Rule  —Divide  given  numbers  by  any  number  that  will  divide  the  greatest  num- 
ber of  them  without  a remainder,  and  set  quotients  with  undivided  numbers  in  a 

^JDiv^de Second  line  in  same  manner,  and  so  on,  until  there  are  no  two  numbers 
that  can  be  divided;  then  the  continued  product  of  divisors  and  last  quotients  will 
give  common  multiple  required. 

5)  40 . 50 . 25 
5)  8 . 10  ♦ 5 


Example.  — What  is  least 
common  multiple  of  40,  50, 
and  25  ? 


2)  8 . 


4 . 1 . 1.  Then  5X5X2X4X1X1  = 200. 

To  Reduce  a Fraction  to  its  Lowest  Term. 

RrLE Divide  terms  by  any  number  or  series  of  numbers  that  will  divide  them 

vithout  a remainder,  or  by  their  greatest  common  measure. 

Example.— Reduce  £§{j-  of  a foot  to  its  lowest  terms. 

It-S-  - 10  = it  - 8 = A - 3 = J. or  9 *«»• 


90 


FRACTIONS. 


To  Reduce  a Mixed  Fraction  to  its  Equivalent,  an  Im- 
proper Fraction. 

Rule.  — Multiply  whole  number  by  denominator  of  fraction  and  to  product  add 
numerator;  then  set  that  sum  above  denominator. 

Example  i. — Reduce  2^#  to  a fraction.  23  X f 2 = ~ . 

663 

2.— Reduce  inches  to  its  value  in  feet.  123  -4-  6 = 20|  = 1 foot  8^  ins. 


To  Reduce  a Complex  Fraction  to  a Simple  one. 

Rule. — Reduce  the  two  parts  both  to  a simple  fraction,  multiply  numerator  of  re- 
duced fraction  by  denominator  of  reduced  denominator,  and  denominator  of  numer- 
ator fraction  by  numerator  of  denominator  fraction. 


Example.— Simplify  complex  fraction  - 


2|=  f 
4*  = V 


8 X 5 =40 
3 X 24  = 72 


5_ 

9’ 


To  Reduce  a NVliole  Number  to  an  Equivalent  Fraction 
having  a given  Denominator. 

Rule.— Multiply  whole  number  by  given  denominator,  and  set  product  over  said 
denominator. 


Example. — Reduce  8 to  a fraction,  denominator  of  which  shall  be  9. 

8X9  = 72;  then  result  required. 


To  Reduce  a Compound  Fraction  to  an  Equivalent 
Simple  one. 

Rule. — Multiply  all  numerators  together  for  a numerator,  and  all  denominators 
together  for  a denominator. 

Note. — When  there  are  terms  that  are  common,  they  may  be  cancelled. 
Example.— Reduce  of  | of  § to  a simple  fraction. 

i X 4 X 3 = J4  = 4-  0r’  i x f x f — 4 » hy  cancellin9  2’s  and  3’s. 

To  Reduce  Fractions  of  different  Denominations  to 
Equivalents  liaving  a Commoix  Denominator. 


Rule.  — Multiply  each  numerator  by  all  denominators  except  its  own  for  new  nu- 
merators; and  multiply  all  denominators  together  for  a common  denominator. 

Note.  — In  this,  as  in  all  other  operations,  whole  numbers,  mixed  or  compound 
fractions,  must  first  be  reduced  to  form  of  simple  fractions. 

2.  When  many  of  denominators  are  same,  or  are  multiples  of  each  other,  ascertain 
their  least  common  multiple,  and  theu  multiply  the  terms  of  each  fraction  by  quo- 
tient of  least  common  multiple  divided  by  its  denominator. 


Example.  — Reduce  -|,  and  to  a 
common  denominator. 


1 X 3 X 4=  12) 
2X2X4  = 16 

3X2X3  = 18) 

2 X 3 X 4 = 24 


— 1 2 16  _ 18 

2 4 — 2 4 — T4> 

0r  3T»  A and  A* 


Addition. 


Rule. — If  fractions  have  a common  denominator,  add  all  numerators  together,  <*, 
and  place  sum  over  denominator. 


Note. — If  fractions  have  not  a common  denominator,  they  must  be  reduced  to 
one.  Also,  compound  and  complex  must  be  reduced  to  simple  fractions. 

Example  1.  — Add  ^ and  | together.  J + f = f = I* 


FRACTIONS. 


91 


Subtraction. 

Rule. —Prepare  fractions  same  as  for  other  operations,  when  necessary;  then 
subtract  one  numerator  from  the  other,  and  set  remainder  over  common  denom- 
inator. 

Example.— 
between  -J 


Multiplication. 

Rule.— Prepare  fractions  as  previously  required;  multiply  all  numerators  to- 
gether for  a new  numerator,  and  all  denominators  together  for  a new  denominator. 

Example  i. — What  is  product  of  and  ^ f X = g^g-  — 

2.  —What  is  product  of  6 and  | of  5 ? f X f of  5 = f X ^ = -6g°-  = 20. 


Division. 

Rule.— Pi  ipare  fractions  as  before;  then  divide  numerator  by  the  numerator, 
and  denominator  by  the  denominator,  if  they  will  exactly  divide;  but  if  not,  invert 
the  terms  of  divisor,  and  multiply  dividend  by  it,  as  in  multiplication. 

Example  i.—  Divide  2^5  by  2¥5  -4-  f = f = if. 

_ nivi/lo  5 2 5 2 — 5 V 15  . — 15  y 5 7 5 — 2 5 j 1 

2.— Divide  j ny  ^ . 15  — j X -y-  — <r  X j — i s “ 6 — 4^- 


Application  of  Deduction  of'  Fractions. 

To  Compute  Value  of  a Fraction,  in  Parts  of  a W hole 
Number. 

Rule. — Multiply  whole  number  by  numerator,  and  divide  by  denominator;  then, 
if  anything  remains,  multiply  it  by  the  parts  in  next  inferior  denomination,  and 
divide  by  denominator,  as  before,  and  so  on  as  far  as  necessary;  so  shall  the  quo- 
tients placed  in  order  be  value  of  fraction  required. 

Example  i. — What  is  value  of  f of  | of  9 ? 

1 ftf  2 _ 2 2V9_ 18  

2 01  3 — 6’  anQ  6 X • — ~6"  — 3- 

2.— Reduce  f of  a pound  to  an  avoirdupois  ounce.  4)  3 (o 

16  ounces  in  a lb. 

4)  48  (12  ounces. 

To  Reduce  a Fraction  from  one  Denomination  to 
another. 

Rule.— Multiply  number  of  required  denomination  contained  in  given  denomina- 
tion by  numerator  if  reduction  is  to  be  to  a less  name , but  by  denominator  if  to  a 
greater. 

Example  i. — Reduce  f of  a dollar  to  fraction  of  a cent. 

£ X IOO  = 

2. — Reduce  f of  an  avoirdupois  pound  to  fraction  of  an  ounce. 

1 v — 1 5 8 o 2 

0 X It)  — 6 — 3 — 2 g. 

3. — Reduce  f of  f of  a mile  to  the  fraction  of  a foot. 

t0fi=AX  5280  = JUj.680  = 2JM0. 

For  Rule  of  Three  in  Vulgar  Fractions,  see  Decimals,  page  94. 


What  is  difference 
if? 


6 X c 


8X9  = 72 


~ 7 2 — 12  — 8 6 — 12- 


92 


DECIMALS. 


DECIMALS. 

A Decimal  is  a fraction,  having  for  its  denominator  a unit  with 
as  many  ciphers  annexed  as  the  numerator  has  places ; it  is  usually  ex- 
pressed by  writing  the  numerator  only,  with  a point  at  the  left  of  it.  Thus, 
A is  .4;  T%%  is  .85 ; is  .0075 ; and  is  .00125.  When  there  is 

a deficiency  of  figures  in  the  numerator,  prefix  ciphers  to  make  up  as  many 
places  as  there  are  ciphers  in  denominator. 

Mixed  numbers  consist  of  a whole  number  and  a fraction;  as,  3.25,  which  is  the 
same  as  3 or 

Ciphers  on  right  hand  make  no  alteration  in  their  value;  for  .4,  .40,  .400  are  deci- 
mals of  same  value,  each  being  T^,  or  f . 


jAcLdltion. 

Rule.  — Set  numbers  under  each  other  according  to  value  of  their  places,  as  in 
whole  numbers,  in  which  position  the  decimal  points  will  stand  directly  under  each 
other;  then  begin  at  right  hand,  add  up  all  the  columns  of  numbers  as  in  integers, 
and  place  the  point  directly  below  all  the  other  points. 

Example.  —Add  together  25. 125  and  293. 7325.  25. 125 

2937325 

318.8575  sum. 


Subtraction. 

Rule. — Set  numbers  under  each  other  as  in  addition;  then  subtract  as  in  whole 
numbers,  and  point  off  decimals.as  in  last  rule. 

Example.— Subtract  15.15  from  89.1759.  89.1759 

74.0259  remainder. 


Multiplication. 

Rule  —Set  the  factors,  and  multiply  them  together  same  as  if  they  were  whole 
numbers:  then  point  off  in  product  just  as  many  places  of  decimals  as  there  are 
decimals ’in  both  factors.  But  if  there  are  not  so  many  figures  in  product,  supply 
deficiency  by  prefixing  ciphers. 

Example.— Multiply  1.56  by  .75.  1.56 

•75 

780 

1092 

1. 1 700  product 


By  Contraction. 

To  Contract  tLe  Operation  so  as  to  retain  only  as  many 
Decimal  places  in  Drod.nct  as  may  be  required. 

Rule  —Set  unit’s  place  of  multiplier  under  figure  of  multiplicand,  the  place  of 
which  is  same  as  is  to  be  retained  for  the  last  in  product,  and  dispose  of  the  rest  of 
fi wares  in  contrary  order  to  which  they  are  usually  placed.  14.  . 

°In  multiplying  reject  all  figures  that  are  more  to  right  hand  than  each  multiply- 
ing fi^ire  and  sit  down  the  products,  so  that  their  right-hand  figures  may  fall  in  a 

column  directly  below  each  other,  and  increase  first  figure  

in  every  line  with  what  would  have  arisen  from  figures 
omitted;  thus,  add  1 for  every  result  from  5 to  14  2 from 
15  to  24,  3 from  25  to  34,  4 from  35  to  44,  etc.  and  the  sum 
of  all  the  lines  will  be  the  product  as  required. 

Example.— Multiply  13-574  93  by  46-2051,  and  retain  only 
four  places  of  decimals  in  the  product. 


13-574  93 
1 502.64 

54  299  72  . „ 

8 144  96-}-  2 for  18 
271  50  -f-  2 11  18 
6 79  + 4 “ 35 
14  + 1 “ 5 


627.23  zi 

NoTK.-When  exact  result  is  required,  increase  last  figure  with  what  would  have  arisen  from  all  the 
figures  omitted. 


DECIMALS. 


93 


Division. 

Rule  —Divide  as  in  whole  numbers,  and  point  off  in  quotient  as  many  places  for 
decimals  as  decimal  places  in  dividend  exceed  those  in  divisor;  but  if  there  are  not 
so  many  places,  supply  deficiency  by  prefixing  ciphers. 

Example.  Divide  53  by  6.75.  6.75)  53.00000  (=7.851-]-. 

Here  5 ciphers  are  annexed  to  dividend  to  extend  division. 


By  Contraction. 

Rule.— Take  only  as  many  figures  of  divisor  as  will  be  equal  to  number  of  figures, 
both  integers  and  decimals,  to  be  in  quotient,  and  ascertain  how  many  times  they 
may  be  contained  in  first  figures  of  dividend,  as  usual. 

Let  each  remainder  be  a new  dividend;  and  for  every  such  dividend  leave  out 
one  figure  more  on  right-hand  side  of  divisor,  carrying  for  figures  cut  off  as  in  Con- 
traction of  Multiplication. 

Note.— When  there  are  not  so  many  figures  in  divisor  as  there  are  required  to  be  in  quotient,  con- 
tinue first  operation  until  number  of  figures  in  divisor  are  equal  to  those  remaining  to  be  found  in  quo- 
tient, after  which  begin  the  contraction. 


Example. — Divide  2508.92806 

92.4103I5)  2508.928I06  (27.1498 

13.849 

912 

by  92.41035,  so  as  to  have  only 

1848  207  -f-  1 

9 24x 

S32  + 4 

four  places  of  decimals  in  quo- 

660 721 

4 608 

80 

tient. 

646  872  + 2 

3 696 

74  + 2 

13S49 

912 

6 

RedvLctiorL  of  Decimals. 

To  Reduce  a Vulgar  Fraction  to  its  Equivalent  Decimal. 

Rule.— Divide  numerator  by  denominator,  annexing  ciphers  to  numerator  to  ex- 
tent that  may  be  necessary. 

Example. — Reduce to  a decimal.  5)  4.0 

.8 

To  Compute  Value  of  a Decimal  in  Terras  of  an  Inferior 
Denomination. 

Rule. — Multiply  decimal  by  number  of  parts  in  next  lower  denomination,  and 
cut  off  as  many  places  for  a remainder,  to  right  hand,  as  there  are  places  in  given 
decimal. 

Multiply  that  remainder  by  the  parts  in  next  lower  denomination,  again  cutting 
off  for  a remainder,  and  so  on  through  all  the  parts  of  integer. 

Example  i.— What  is  value  of  .875  dollars?  .875 

100 

Cents , 87.500 
10 

Mills,  5.000  = 87  cents  5 mills. 

2.  — What  is  volume  of . 140  cube  feet  in  inches  ? 

.140 

1728  cube  inches  in  a cube  foot. 

241 920  cube  ins. 

3. — What  is  value  of  .00129  of  a foot?  -01548  ins. 

To  Reduce  a,  Decimal  to  an  Equivalent  Decimal  of  a 
Higher  Denomination. 

Rule.— Divide  by  number  of  parts  in  next  higher  denomination,  continuing  op- 
eration as  far  as  required. 

Example  1.— Reduce  1 inch  to  decimal  of  a foot.  12 |i. 000 00 

I -o83  33+  foot- 

2.— Reduce  14"  12"'  to  decimal  of  a minute.  14"  12"' 

60 

60  852/" 

60  14.2" 

. 236  66'+  minute. 


DECIMALS. — DUODECIMALS. — MEAN  PROPORTION. 


94 


When  there  are  several  numbers , to  he  reduced  all  to  decimal  of  highest. 
Rule.— Reduce  them  all  to  lowest  denomination,  and  proceed  as  for  one  denomi- 


Rule. Prepare  the  terms  by  reducing  vulgar  fractions  to  decimals,  compound 

numbers  to  decimals  of  the  highest  denomination,  first  and  third  terms  to  same 
denomination ; then  proceed  as  in  whole  numbers. 


In  Duodecimals,  or  Cross  Multiplication,  the  dimensions  are  taken  in  feet, 
inches,  and  twelfths  of  an  inch. 

Rule.— Set  dimensions  to  be  multiplied  together  one  under  the  other,  feet  under 
feet,  inches  under  inches,  etc.  . , . ...  ..  „ . 

Multiply  each  term  of  multiplicand,  beginning  at  lowest,  by  feet  in  multiplier,  and 
set  result  of  each  immediately  under  its  corresponding  term,  carrying  i for  every 
I2  from  one  term  to  the  other.  In  like  manner,  multiply  all  multiplicand  by  inches 
of  multiplier,  and  then  by  twelfth  parts,  setting  result  of  each  term  one  place  farther 
to  right  hand  for  every  multiplier.  And  sum  of  products  will  give  result. 

Example.— How  many  square  inches  are  Feet.  ins.  Twelfths, 
there  iu  a board  35  feet  4.5  inches  long  and  12  35  4 0 

feet  3!  inches  wide  ? — \ — 

A2A  6 o 


Value  of  Duodecimals  in  Square  IPeet  and  Inches. 


Illustration. —What  number  of  square  inches  are  there  in  a floor  100  feet 
6 inches  long  and  25  feet  6 inches  and  6 twelfths  broad? 


Mean  Proportion  is  proportion  to  two  given  numbers  or  terms. 
Rule.— Multiply  two  numbers  or  terms  together,  and  extract  square  root  of  their 
product 

Example.— What  is  mean  proportionate  velocity  to  16  and  81? 

16  X 81  = 1296,  and  V1296  = 36  mean  velocity . 


Example.  — Reduce  5 feet  10  inches  and  3 
barleycorns  to  decimal  of  a yard. 


nation. 


Feet.  Ins.  Be. 

5 10  3 


12 

70 

3 


3 21 3- 

12  7r. 


3 5.9166 


1.9722-J-  yards. 


IRnle  of  Three. 


what  will  .625  of  a ton  cost? 


Example.— If  .5  of  a ton  of  iron  cost  .75  of  a dollar, 


.5 : .75 :: .625 
.625 

.5)  -46875 

• 9375)  dollar. 


DUODECIMALS. 


11  960 


434  3 0 0 


Sq.  Ft. 


Sq.  Ins. 


Sq.  Ft.  Sq.  Ins. 


1 Foot. . . 
1 Inch. . . 
I Twelfth 


or  144. 

11  T 2. 


12. 

I. 


of  i twelfth  = or  .083  333,  etc. 

Tf  of  TZ  of  “ = 20  7 iTS  “ -006 944)  etc- 


2566  feet  11  ins.  3 twelfths  = 2566  feet  135  ins. 


MEAN  PROPORTION. 


RULE  OF  THREE. — COMPOUND  PROPORTION 


RULE  OF  THREE. 

Rule  of  Three.  — It  is  so  termed  because  three  terms  or  numbers  are 
given  to  ascertain  a fourth. 

It  is  either  Direct  or  Inverse.  . , ..  . 

It  is  Direct  when  more  requires  more,  or  less  requires  less ; thus,  it  3 bar- 
rels of  flour  cost  $18,  what  will  10  barrels  cost? 

In  this  case  Proportion  is  Direct , and  stating  must  be, 

As  3 : 10  ::  18  • 60. 

it  Inverse  when  more  requires  less,  or  less  requires  more;  thus,  if  6 men  build 
a certain  quantity  of  wall  in  10  days,  in  how  many  days  will  8 men  build  like  quan- 
tity ? Or,  if  3 min  dig  too  feet  of  trench  in  7 days,  in  how  many  days  will  a men 
perform  same  work? 

Here  the  Proportion  is  Inverse , and  stating  must  be, 

As  8 : 6 ::  io  : 7.5,  and  2 : 3 ::  7 : 10.5. 

The  fourth  term  is  always  ascertained  by  multiplying  2d  and  3d  terms  together, 

"idrh?e“  for  the  stating,  two  of  them  contain  the 

supposition,  and  the  third  a demand. 

Rule. State  question  by  setting  down  in  a straight  line  the  three  necessary 

11  TetThi  id  term  be  °that  of  Supposition,  of  same  denomination  as  the  result  or  4 th 
term  is  to  be,  making  demanding  number  2d  term,  and  the  other  number  1st  terr 
when  question  is  in  Direct  Proportion , but  contrariwise  if  in  Inverse  Proportion , 
that  is.  let  demanding  number  be  ist  term.  » . , ... 

Multiply  2d  and  3d  terms  together,  and  divide  by  ist,  and  product  will  give  re- 
sult, or  4th  term  sought,  of  same  denomination  as  2d  term. 

Note. — If  first  and  third  terms  are  of  different  denominations,  reducethemto  same.  If,  J^divij 
ion,  there  is  any  remainder,  reduce  it  to  next  lower  denomination,  divide  by  dmeor  as  betore,  ana 
quotient  will  be  of  this  last  denomination. 

Sometimes  two  or  more  statings  are  necessary,  which  may  always  be  known  by 
nature  of  question. 

Tons.  Tons.  Dolls. 


Example  i. — If  20  tons  of  iron  cost  $225,  what  will 
500  tons  cost  ? 


20  : 500 


'.  225 
5°° 


2 |o)  11  250I0 


5625  dollars. 

2. . — A wall  that  is  to  be  built  to  height  of  36  feet,  was  raised  9 feet  by  16  men  in 
6 days;  how  many  men  could  finish  it  in  4 days  at  same  rate  of  w’orking ? 

Days.  Days.  Men.  Men. 

4 : 6 ::  16  : 24 

Then,  if  9 feet  requires  24  men,  what  will  27  men  require? 

9 : 27  24  : 72  men. 


COMPOUND  PROPORTION. 

Compound  Proportion  is  rule  by  means  of  which  such  questions  as 
would  require  two  or  more  statings  in  simple  proportion  (Rule  of  Three) 
can  be  resolved  in  one. 

As  rule,  however,  is  but  little  used,  and  not  easily  acquired,  it  is  deemed  prefer- 
able to  omit  it  here,  and  to  show  the  operation  by  two  or  more  statings  in  Simple 
Proportion. 

Illustration  i. — How  many  men  can  dig  a trench  135  feet  long  in  8 days,  when 
16  men  can  dig  54  feet  in  6 days? 


Feet.  Feet.  Men.  Men. 

First .As  54  : 135  16  : 40 

Days.  Days.  Men.  Men. 
Second As  8 : 6 ::  40  : 30 


g6  COMPOUND  PROPORTION. INVOLUTION. — EVOLUTION. 


2.  — If  a man  travel  130  miles  in  3 days  of  12  hours  each,  how  many  days  of  10 
hours  each  would  he  require  to  travel  360  miles? 

Miles.  Miles.  Days.  Days. 

First As  130  : 360  : : , 3 : 8.307-f- 

Hours.  Hours.  Days.  Days. 

Second As  10  : 12  8.307  : 9.9684 

3. — If  12  men  in  15  days  of  12  hours  build  a wall  30  feet  long,  6 wide,  and  3 deep, 

in  how  many  days  of  8 hours  will  60  men  build  a wall  300  feet  long,  8 wide,  and 
6 deep?  120  days. 

By-  Cancellation. 

Rule.— On  right  of  a vertical  line  put  the  number  of  same  denomination  as  that 
of  required  answer. 

Examine  each  simple  proportion  separately,  and  if  its  terms  demand  a greater 
result  than  3 d term , put  larger  number  on  right  and  lesser  on  left  of  line  ; but  if  its 
terms  demand  a less  result  than  3 d term , put  smaller  number  on  right  and  larger 

on  left  of  line. 

Then  Cancel  the  numbers  divisible  by  a common  divisor,  and  evolve  the  4th  term 
or  result  required. 

Take  Illustration  1.  page  95 : 3d  term,  or  term  of  supposition  of  same  denomination 
as  required  result,  16  men. 


Statement. 
16 
135 
6 


54 


135  feet  require  more  men  than  54  feet , 
and'  8 days  less  men  than  6 days. 


2 X 5 X 3 = 30  men. 
Illustration  3. — 3d  term,  15  days. 

Statement.  60  men  require  less  days  than  12  men, 

15  8 hours  more  days  than  12  hours,  300  feet 

more  days  than  30  feet,  8 feet  more  days 
than  6 feet,  and  6 feet  more  days  than 
3 feet. 


Result  by  Cancellation. 

W 2 
m 5 
0 3 


60 

8 

30 

6 

3 


12 

12 

300 

8 

6 


Result  by  Cancellation. 

n 
n 


3 X 4 X 10  = 120  days. 


30 

0 

3 


300  10 


INVOLUTION. 

Involution  is  multiplying  any  number  into  itself  a certain  number  of 
times.  Products  obtained  are  termed  Powers . The  number  is  termed  the 
Root , or  first  power. 

When  a number  is  multiplied  by  itself  once,  product  is  square  of  that 
number ; twice,  cube ; three  times,  biquadrate ; etc.  Thus,  of  the  number  5. 

5 is  the  Root , or  1st  power. 

5X5=  25  “ Square,  or  2d  power,  and  is  expressed  52. 

5X5X5  = 125  “ Cube , or  3d  power , and  is  expressed  53. 

5X5X5X5  = 625  “ Biquadrate , or  4th  power , and  is  expressed  54. 

The  lesser  figure  set  superior  to  number  denotes  the  power,  and  is  termed  the 
Index  or  Exponent. 


Illustration  i. — What  is  cube  of  9? 

2. — What  is  cube  of  J ? 

3. — What  is  4th  power  of  1.5  ? 


EVOLUTION. 

Evolution  is  ascertaining  Root  of  any  number. 

Sign  y/  placed  before  any  number  indicates  that  square  root  of  that  number  is  re- 
quired or  shown. 

Same  character  expresses  any  other  root  by  placing. the  index  above  it. 

Thus,  V25  = 5 ; 4 + 2 = V36* 

^27  = 3,  and  ^64=  4. 

Roots  which  only  approximate  are  termed  Surd  Roots. 


729. 

27 

5.0625. 


EVOLUTION. 


97 


To  Extract  Square  Root. 

Rule.— Point  off  given  number  from  units’  place,  into  periods  of  two  figures  each. 

Ascertain  greatest  square  in  left-hand  period,  and  place  its  root  in  quotient;  sub- 
tract square  number  from  this  period,  and  to  remainder  bring  down  next  period 
for  a dividend. 

Double  this  root  for  a divisor;  ascertain  how  many  times  it  is  contained  in  divi- 
dend. exclusive  of  right-hand  figure,  which,  when  multiplied  by  number  to  be  put 
to  right  hand  of  this  divisor,  product  will  be  equal  to,  or  next  less  than  dividend; 
place° result  in  quotient,  and  also  at  right  hand  of  divisor. 

Multiply  divisor  by  last  quotient  figure,  and  subtract  product  from  dividend; 
bring  down  next  period,  and  proceed  as  before. 

Note. — Mixed  decimals  must  be  pointed  off  both  ways  from  units. 

Example  i. — What  is  square  root  of  2? 

ii  2.000 000  (..41+.  *• what  is  s9uare  root  of  '44? 

Jl  1 J|  *44  (12 


24IIOO 

1 1 1 

41  96 

22|c>44 

281  400 

1 44 

1 281 

282 1 1 19  00 


Square  Roots  of  Fractions. 

Rule.— Reduce  fractions  to  their  lowest  terms,  and  that  fraction  to  a decimal, 
and  proceed  as  in  whole  numbers  and  decimals. 

Note.— When  terms  of  fractions  are  squares,  take  root  of  each  and  set  one  above  the  other ; as 
is  square  root  of 

Example.— What  is  square  root  of  T9^  ? .866  025  4. 

To  Compute  4t,li  or  8tli  Root  of  a IN'  umber,  etc. 
Rule. —For  the  4th  root  extract  square  root  twice,  and  for  8th  root  thrice,  etc. 


To  Extract  Cube  Root. 

Rule. — From  table  of  roots  (page  272)  take  nearest  cube  to  given  number,  and 
term  it  the  assumed  cube. 

Then,  as  given  number  added  to  twice  assumed  cube,  is  to  assumed  cube  added 
to  twice  given  number,  so  is  root  of  assumed  cube  to  required  root,  nearly  ; and  by 
using  in  like  manner  the  root  thus  found  as  an  assumed  cube,  and  proceeding  in 
like  manner,  another  root  will  be  found  still  nearer;  and  in  like  manner  as  far  as 
may  be  deemed  necessary. 

Example. — What  is  cube  root  of  10  517.9? 

Nearest  cube,  page  272;  10648,  root  22.  10648.  10  517.9 

2 2 

21296  21035.8 

10  517.9  10648. 

31 813.9  : 31 683.8  ::  22  : 21.9 -f-. 

To  Ascertain  or  to  Compute  the  Square  or  Cube  Roots  of 
Roots,  Whole  Numbers,  and  of  Integers  and  Decimals, 
see  Table  of  Squares  and  Cubes,  and  Rules,  pp.  272,  300. 

To  Extract  any  Root  whatever. 

Let  P represent  number.  I Let  A represent  assumed  power,  r its  root. 

n “ index  of  the  power.  | R li  required  root  of  P. 

Then,  as  sum  of  n -{-  1 x A and  n — 1 X P is  to  sum  of  n -}- 1 X P and  n — 1 X A. 
so  is  assumed  root  r to  required  root  R. 

Illustration. — What  is  cube  root  of  1500? 

Nearest  cube,  page  272,  is  1331,  root  n. 

P = 1500,  n = 3,  A = 1331,  r = n ; 
then,  n 1 XA  = 5324,  n- J- 1 X P = 6000 
n — 1 X P — 3000,  n — 1 X A = 2662 

8324  8662 ::  11 

I 


: 11.446-}-. 


98  EVOLUTION. PROPERTIES  OF  NUMBERS. POSITION. 

To  Compute  tlie  Root  of  an  Even  Rower  greater  tlian 
any  given  in  Table  of  Square  and.  Cube  Roots. 
Rule. — Extract  square  or  cube  root  of  it,  which  will  reduce  it  to  half  the  given 
power;  then  square  or  cube  root  of  that  power;  and  so  on  until  required  root  is  ob- 
tained. 

Example  i. — Suppose  a 12th  power  is  given ; the  square  root  of  that  reduces  it  to 
a 6th  power,  and  the  square  root  of  6th  power  to  a cube. 

2. — What  is  biquadrate,  or  4th  root,  of  2 560000? 

y/2  560  000  = 1600,  and  1600  = 40. 

Note. — For  other  rules  for  extraction  of  roots  see  pp.  301-4. 


PROPERTIES  OF  NUMBERS. 

1.  A Prime  Number  is  that  which  can  only  be  measured  (divided  without  a re- 
mainder) by  1 or  unity. 

2.  A Composite  Number  is  that  which  can  be  measured  by  some  number  greater 
than  unity. 

3.  A Perfect  Number  is  that  which  is  equal  to  the  sum  of  all  its  divisors  or  ali- 
quot parts ; as  6 |,  -|,  •§. 

4.  If  sum  of  the  digits  constituting  any  number  be  divisible  by  3 or  9,  the  whole 
is  divisible  by  them. 

5.  A square  number  cannot  terminate  with  an  odd  number  of  ciphers. 

6.  No  square  number  can  terminate  with  two  equal  digits,  except  two  ciphers  or 
two  fours. 

7.  No  number,  the  last  digit  of  which  is  2,  3,  7,  or  8,  is  a square  number. 


Rowers  of  tlie  first  UNTine  NTnixTbers. 


1st. 

2d. 

3d. 

4 th. 

5th. 

6th. 

7th. 

8th.  I 

9 th. 

I 

I 

1 

1 

1 

1 

1 

1 

1 

2 

4 

8 

16 

32 

64 

128 

256 

512 

3 

9 

27 

81 

243 

729 

2 187 

6561 

19683 

4 

16 

64 

256 

1 024 

4096 

16384 

65  536 

262  144 

5 

25 

125 

625 

3125 

15625 

78  125 

39°  625 

1953125 

~6~ 

36 

216 

1296 

7776 

46656 

279936 

1 679616 

10  077  696 

7 

49 

343 

2401 

16  807 

117649 

823  543 

5 764  801 

40  353  607 

8 

64 

512 

4096 

32768 

262  144 

2 097  152 

16777  216 

134  217  728 

9 

81 

729 

6561 

59  °49 

1 53i  44i 

4 782  969 

1 43046721 

387  420  489 

POSITION. 

Position  is  of  two  kinds,  Single  and  Double,  and  it  is  determined  by 
number  of  Suppositions. 

Single  ^Position. 

Rule. — Take  any  number,  and  proceed  with  it  as  if  it  were  the  correct  one;  then, 
as  result  is  to  given  sum,  so  is  supposed  number  to  number  required. 

Example  i. — A commander  of  a vessel,  after  sending  away  in  boats  J,  J,  and  ^ 
of  his  crew,  had  left  300;  what  number  had  he  in  command? 

Suppose  he  had 600. 

A of  600  is  200 
A of  600  is  100 
-Jof6ooisi5o  450 

150  : 300  *.  *.  600  : 1200  men. 


POSITION. FELLOWSHIP. 


99 


2.  — A person  asked  his  age,  replied,  if  £ of  my  age  be  multiplied  by  2,  and  that 
product  added  to  half  the  years  I have  lived,  the  sum  will  be  75.  How  old  was  he  ? 

37.5  years. 

Double  ^Position. 

Rule.— Assume  any  two  numbers,  and  proceed  with  each  according  to  conditions 
of  question;  multiply  results  or  errors  by  contrary  supposition;  that  is,  first  posi- 
tion by  last  error,  and  last  position  by  first  error. 

If  errors  are  too  great,  mark  them  and  if  too  little,  — . 

Then,  if  errors  are  alike , divide  difference  of  products  by  difference  of  errors;  but 
if  they  are  unlike , divide  sum  of  the  products  by  sum  of  errors. 

Example  i. — A asked  B how  much  his  boat  cost;  he  replied,  that  if  it  cost  him  6 
times  as  much  as  it  did,  and  $30  more,  it  would  have  cost  him  $300.  What  was 
price  of  the  boat? 

Suppose  it  cost  him. . 60 30 

6 times.  6 times. 


360 

and  30  more 


180 

and  30  more 


90 

90 


39° 

3°° 

30  2d  position. 
2700 
5400 


90— 

60  1st  position. 
5400 


180)  8100  (45  dollars. 

2.— What  is  length  of  a fish  when  the  head  is  9 inches  long,  tail  as  long  as  its  head 
and  half  its  body,  and  body  as  long  as  both  head  and  tail  ? 6 feet. 


FELLOWSHIP. 

Fellowship  is  a method  of  ascertaining  gains  or  losses  of  individuals 
engaged  in  joint  operations. 

Single  PCellowsLip. 

Rule. — As  the  whole  stock  is  to  the  whole  gain  or  loss,  so  is  each  share  to  the 
gain  or  loss  on  that  share. 

Example. — Two  men  drew  a prize  in  a lottery  of  $9500.  A paid  $3,  and  B $2  for 
the  ticket;  how  much  is  each  share? 

5 : 9500  3 : 5700,  A’s  share. 

5 : 9500  : : 2 : 3800,  B’s  share. 

Double  IPellowslxip, 

Or  Fellowship  with  Time. 

Rule.— Multiply  each  share  by  time  of  its  interest;  then,  as  sum  of  products  is  to 
product  of  each  interest,  so  is  whole  gain  or  loss  to  each  share  of  gain  or  loss. 

Example.— A cutter’s  company  take  a prize  of  $10000,  which  is  to  be  divided  ac- 
cording to  their  rate  of  pay  and  time  of  service  on  board.  The  officers  have  been 
on  board  6 months,  and  the  crew  3 months;  pay  of  lieutenants  is  $100,  ensigns  $50, 
nnd  crew  $10  per  month;  and  there  are  2 lieutenants,  4 ensigns,  and  50  men;  what 


is  each  one’s  share  ? 

2 lieutenants $100  = 200  x 6 = 1200 

4 ensigns 50  = 200X6  = 1200 

50  men 10  = 500  X 3 = 1500 

3900 

Lieutenants 3900  : 1200  10000  : 3076. 92-S-  2 = 1538.46  dolls. 

Ensigns 3900  : 1200  ” 10000  : 3076. 92  -4-  4=  769.23  “ 

Men 3900:1500:110000:3846.16-7-50=  76.92  u 


3 


ioo 


PERMUTATION. 


ti 


PERMUTATION. 

Permutation  is  a rule  for  ascertaining  how  many  different  ways  any 
given  number  of  numbers  of  things  may  be  varied  in  their  position. 

Permutation  of  the  three  letters  abc , taken  all  together , are  6 ; taken  two 
and  two,  are  6;  and  taken  singly,  are  3. 

Rule.— Multiply  all  the  terms  continually  together,  and  last  product  will  give 
result. 

Example  i. — How  many  variations  will  the  nine  digits  admit  of  ? 

1X2X3X4X5X6X7X8X9  = 362  880. 

2. How  many  years  would  there  be  required  to  elapse  before  10  persons  could 

be  seated  in  a varied  position  collectively,  each  day  at  dinner,  including  one  day  in 
every  4 years  for  a leap  year  ? 9935  years,  42  days. 


When  only  part  of  the  Numbers  or  Elements  are.  taken  at  once.  Rule.— 
Take  a series  of  numbers,  beginning  with  number  of  things  given,  decreasing  by  1, 
until  number  of  terms  equals  number  of  things  or  quantities  to  be  taken  at  a time, 
and  product  of  all  the  terms  will  give  sum  required. 

Example  i.— How  many  changes  can  be  made  with  2 events  in  5? 

5 — 1 ==  4,  and  4X5  = 2 terms.  Hence,  5X4  = 20  changes. 

2.— How  many  changes  of  2 will  3 playing  cards  admit  of? 

3 — 1 = 2,  and  2X3  = 2 terms.  Hence,  2X3  = 6 changes. 

2. How  many  changes  can  be  rung  with  4 bells  (taken  4 and  4 together)  out  of  6 ? 

4 _ 1 = 3,  and  3X4X5X61=4  terms  or  changes. 

Hence,  3X4X5X6  = 360  changes. 

When  several  of  the  Elements  are  alike.  Rule.— Ascertain  the  permutations 
of  all  the  numbers  or  things,  and  of  all  that  can  be  made  of  each  separate  kind  or 
division;  divide  number  of  permutations  of  whole  by  product  of  the  several  partial 
permutations,  and  quotient  will  give  number  of  permutations. 

Example. —How  many  permutations  can  be  made  out  of  the  letters  of  the  word 
persevere  (9  letters,  having  4 e’s  and  2 r’s)? 

1X2X3X4X5X6X7X8X9  = 362880; 

1  X 2 X ^ X 4 = 24  for  the  e’s;  1 x 2 = 2 for  the  r s,  and  24X2  = 48. 

Hence,  362880  = 48  = 7560. 

Or,  Add  logarithms  of  all  the  terms  together,  and  number  for  the  sum  will  give 
result. 

Example  1.— How  many  permutations  can  be  made  with  three  letters  or  figures? 

Log.  1 =*00 

2 = . 301 03 

3 = .4.77  121  3 

.778151  3 = log.  of  number  6. 

2. How  many  variations  will  15  numbers  in  16  places  admit  of? 

Add  logarithms  of  numbers  1 to  16  and  take  logarithm  of  their  sum— 
viz. , 13. 320  661  97  = 20  922  789  888  000. 

Number  of  positions  of  the  blocks  in  the  “ 15  puzzle  ” is  as  above  for  their  16  permutations. 

IPerrriuitatioiTS, 

Whereby  any  questions  of  Permutation  may  be  solved  by  Inspection , number  of 
terms  not  exceeding  20. 


I 

1 

! 5 

120 

9 

362  880 

13 

6 227  020  800 

2 

2 

6 

720 

10 

3 628  800 

M 

87  178  291  200 

3 

6 

7 

5040 

11 

39  916  800 

i5 

1 307  674  368  000 

4 

24 

8 

40320 

12 

479001  600 

16 

20  922  789  888  000 

355  687  428  096  000 
6 402  373  705  728  000 
121  645  100  408  832  000 
2 432  902003  176  640000 


ARITHMETICAL  PROGRESSION. 


IOI 


ARITHMETICAL  PROGRESSION. 


Arithmetical  Progression  is  a series  of  numbers  increasing  or  de- 
creasing by  a constant  number  or  difference ; as,  i,  3,  5,  7,  12,  9,  6,  3.  The 
numbers  which  form  the  series  are  designated  Terms;  the  first  and  last 
are  termed  Extremes , and  the  others  Means . 

When  any  three  of  following  elements  are  given , the  remaining  two  can  he  ascer- 
tained—x iz. , First  term,  Last  term,  Number  of  terms,  Common  Difference , and  Sum 
of  all  the  terms. 


When  Last  term , Number  of  terms,  and  Sum  of  series  are  given.  Rule.  — From 
quotient  of  twice  sum  of  series,  divided  by  number  of  terms,  subtract  last  term. 


ing  1st , l last , n number  of  and  S sum  of  all  terms , and  d common  difference. 

Illustration. — A man  travelled  390  miles  in  12  days,  travelling  60  miles  last  day. 
How  far  did  he  travel  first  day  ? 


When  First  term , Common  Difference , and  Number  of  terms  are  given.  Rule.— 
Multiply  the  number  of  terms  less  1,  by  common  difference,  and  to  product  add  first 
term. 

Example.— A man  travelled  for  12  days,  at  the  rate  of  5 miles  first  day,  10  second, 
and  so  on ; how  far  did  he  travel  the  last  day  ? 


When  First  term , Number  of  terms,  and  Sum  of  series  are  given.  Rule.  — Divide 
twice  sum  of  series  by  number  of  terms,  and  from  quotient  subtract  first  term. 


Illustration. — A man  travelled  360  miles  in  12  days,  commencing  with  5 miles 
first  day ; how  far  did  he  travel  last  day  ? 


When  Common  Difference  and  Extremes , or  First  and  Last  term,  are  qiven 
Rule.— Divide  difference  of  extremes  by  common  difference,  and  add  1 to  quotient] 

Example  —A  man  travelled  3 miles  first  day,  5 second,  7 third,  and  so  on  till  he 
went  57  miles  in  one  day;  how  many  days  had  he  travelled  at  close  of  last  day? 


When  Sum  of  series  and  Extremes  are  given.  Rule.— Divide  twice  sum  of  series 
by  sum  of  first  and  last  terms. 


To  Compute  First  Term, 


^ — = 65,  and  65  — 60  = 5 first  term. 
To  Compute  Last  Term. 


12-1X5  = 55?  and  55  + 5 = 60  miles. 


, S , din  — 1) 

and _ — l. 

n 2 


360  X 2 

— — — = 65,  and  65  — 5 — 60  miles. 

To  Compute  Number  of  Terms. 


57  ~ 3 2 — 27,  and  27  -f- 1 = 28  days. 


day;  how  many  days  was  he  travelling? 


Illustration. -A  man  travelled  840  miles,  walking  3 miles  first  day  and  <7  last' 
iv:  how  manv  davs  was  travail, n«9  00  J 0 


840  X 2 1680 

3 + 57  60 


-7 — = 28  days. 


102 


ARITHMETICAL  PROGRESSION. 


Or, 


To  Compute  Common  Difference. 

Extremes  are 
of  terms. 

I -|-  a X l — ® 


When  Number  of  terms  and  Extremes  are  given.  Rule.— Divide  difference  of 
extremes  by  i less  than  number  of  terms. 


2 S — 2 an 
n (n  — i)  ’ 


2 S — l — a * 


and 


i nl- 


n (n  — i) 


= & 


ILLCSTRATION.-Extremes  are  3 and  .5,  and  number  of  terms  7 ; what  is  common 
difference  ? 

15  - 3 (7 -»)  = and  X = ' 2 com' d if- 

To  Compute  Sum  of  the  Series  or  of  all  Terms. 
When  Extremes  and  Number  of  terms  are  given.  Rule.— Multiply  number  of 
terms  by  half  sum  of  extremes. 

l-\-aX{l  — a)  , l-\~a . 

Or,  2 a-\- d (n  — i)  X -5  n\  ^ ' 2 ’ 

and  2 i-(dXn-i)  X -5  n = s* 

Illustration.— How  many  times  does  hammer  of  a clock  strike  in  12  hours? 

12  X 12  + 1 = 156,  and  156-7-2  = 78  times. 

To  Compute  any  Number  of  Arithmetical  Means  or 
Terms  'between  two  Extremes. 

p _ Subtract  less  extreme  from  greater,  and  divide  difference  by  1 more 

than  number  of  means  or* terms  required  to  be  ascertained,  and  then  proceed  as 
in  rule. 

To  Compute  Two  Arithmetical  Means  or  Terms  between 
two  given  Extremes. 

er,  will  give  means. 

Example  1.— Compute  two  arithmetical  means  between  4 and  16. 

j6  — 4-7-3  = 4 com.dif \ 

4 + 4 = 8 one  mean. 

_ 4 = 12  second  mean. 

2.— Compute  four  arithmetical  means  between  5 and  30. 


30  — 5 = 25,  and  25  -T-  4 + 1 = 5 > = c ™l-  dV} 


c -4-  5 = 10  = 1st  mean, 
10+5  = 15  = ^ “ 


*5  + 5 — 20  = 3d  mean. 
20+5  = 25  = 4^  “ 


^Miscellaneous  Illustrations. 

* Earner  bavins  been  purchased  upon  following  terms— viz.:  $5000  upon 

1st.  How  many  months  must  elapse  before  final  payment? 

■2d  What  was  amount  of  purchase-money,  or  sum  of  series  ? 

Here  aro  first  and  last  terms— viz.,  500  and  5000,  and  common  difference , 500. 
Hence , To  compute  number  of  terms  and  amount  of  purchase, 

5000  - 500  = 500  = 9,  and  9 + 1 = 10  = number  of  terms  or  months , and  10  X 

5000+500  _ IO  x 2750  = $ 27  500,  amount  of  purchase. 

from  first  stone?  , . r 

First  term  2,  last  term  200,  and  number  of  terms  100. 

Hence,  joo  X = 10 100  yards. 


GEOMETRICAL  PROGRESSION. 


103 


_ If  in  the  sinking  of  curb  of  a well,  $3  is  to  bo  given  for  first  foot  in  depth,  $5 
for  second,  $7  for  third,  and  increasing  in  like  manner  to  a depth  of  20  feet,  what 
would  it  cost? 

First  tei'm  3,  common  difference  2,  and  number  of  terms  20. 

Hence,  20  — 1 X 2 + 3=41,  last  term. 


4 If  a contractor  engaged  to  sink  a curb  to  depth  of  20  feet  for  $400,  and  the 
contract  was  annulled  when  he  had  reached  a depth  of  8 feet;  how  much  had  he 
earned  ? . 

400  — 20 — number  of  terms.  But  inasmuch  as  400  may  be  divided  into  20  terms 
in  arithmetical  proportion  in  many  different  ways,  according  to  value  of  1st  term, 
it  becomes  necessary  to  assume  the  value  of  the  first  foot  as  value  of  1st  term. 

Assuming  it  at  $5,  the  required  proportion  will  be,  1st  term  5,  number  of  terms  20, 
sum  of  ser  ies  400. 


Then,  5 -f-  x 7 = i6yg-  = 1 si  term  -{-product  of  common  difference  and  8th 
term  less  1,  which  added  to  5 — 21N-,  and  X 4 = half  number  of  terms  for  which 
cost  is  sought  = 84^  dollars , sum  earned. 


Geometrical  Progression  is  any  series  of  numbers  continually  in- 
creasing by  a constant  multiplier,  or  decreasing  by  a constant  divisor,  as 
1,  2,  4,  8,  16,  etc.,  and  15,  7.5,  3.75,  etc. 

The  constant  multiplier  or  divisor  is  the  Ratio. 

When  any  three  of  following  elements  are  given , remaining  two  can  be  computed , 
viz. : First  term,  Last  term,  Number  of  Terms,  Ratio , and  Sum  of  all  Terms. 


When  Ratio,  Last  Term , and  Number  of  Terms  are  given.  Rule.  — Divide  last 
term  by  ratio  raised  to  a power  denoted  by  number  of  terms  less  1. 


S sum  of  all  terms,  and  r ratio. 

Illustration.  — Last  term  is  4374,  number  of  terms  8,  and  ratio  3;  what  is  first 
term  ? 


When  First  Term  and  Ratio  are  Equal.  Rule.— Write  a few  of  leading  terms 
of  series  and  place  their  indices  over  them,  beginning  with  a unit.  Add  together 
the  most  convenient  and  least  number  of  indices  to  make  the  index  to  term  required. 

Multiply  terms  of  the  series  of  these  indices  together,  and  product  will  give  term 
required. 

Or,  Multiply  first  term  by  ratio  raised  to  a power,  denoted  by  number  of  terms 
less  1. 

Example  1.— First  term  is  2,  ratio  2,  and  number  of  terms  13;  what  is  last  term? 


Then,  5-1-5  + 3 = 13  = sum  of  indices,  and  32  X 32  X 8 = 8192  ==  last  term. 

Or,  2X2  1 = 8192.  Also  by  inspection  of  table,  page  105, 13th  term  = 8192. 


Then,  3 + 41  X — = 440,  sum  of  all  terms,  or  cost  of  curb. 


Hence, 


400  — 5 X 20  X 2 600 

20  X (20  — 1)  380 


--  — 1 11  common  difference. 
o 1 y 


GEOMETRICAL  PROGRESSION. 


To  Compute  First  Term 


Or,  — — and  rl  — S (r  — 1)  = a.  a representing  1st  term,  l last,  n number  of, 


To  Compute  Last  Term. 


Indices,  1234  5 

Terms,  2,  4,  8,  16,  32. 


2> The  price  of  12  horses  being  4 cents  for  first,  16  for  second,  and  64  for  third, 

and  so  on;  what  is  price  of  last  horse? 

Indices,  1234 
Terms,  4,  16,  64,  256. 

Then,  4 + 4 + 4 = 12  — sum  of  indices , and  256  X 256  X 256  = 256  3 — $ 167  772 10. 

When  First  Term  and  Ratio  are  Different.  Rule.— Write  a few  of  leading  terms 
of  series  and  place  their  indices  over  them,  beginning  with  a cipher.  Add  together 
the  most  convenient  indices  to  make  an  index  less  by  1 than  term  sought. 

Multiply  terms  of  these  series  belonging  to  these  indices  together,  and  take 
product  for  a dividend. 

Or  Raise  first  term  to  a power,  index  of  which  is  1 less  than  number  of  terms 
multiplied;  take  result  for  a divisor;  proceed  with  their  division,  and  quotient  will 
give  term  required. 

Example  1.— First  term  is  1,  ratio  2,  and  number  of  terms  23;  what  is  the  last 
term? 

Indices,  01234  5 
Terms,  1,  2,  4,  8,  16,  32. 

Then  5 + 5 + 5 + 5 + 2 = 22  = sum  of  indices , and  32  X 32  X 32  X 32  X 4 = 
4 194  304,  and  4 194  304  + the  5th  power  (6  — 1)  of  1 = 1 = 4 *94  3Q4- 

Or,  1 X 2 23 — 1 = 4 194  304.  By  inspection  of  table,  page  105,  23d  term  = 4 194  3°4- 

2 —If  1 cent  had  been  put  out  at  interest  in  1630,  what  would  it  have  amounted 
to  in  1834,  if  it  had  doubled  its  value  every  12  years? 

1834  — 1630  = 204,  which  -4- 12  = 17,  and  17  + 1 — 18  = number  of  terms. 
Indices,  01234  7 

Terms,  1,  2,  4,  8,  16,  128. 

Then,  7 + 4 + 3 + 2 + 1 = 17,  and  128  X 16  X 8 X 4 X 2 X 1 = 131 072,  andi3io72 

-4-  1 , the  4th  power  (5  — 1)  of  1 = $ 1 310  72* 

When  First  Term , Ratio , and  Sum  of  the  series  are  given.  Rule. — From  sum  of 
series  subtract  quotient  of  first  term  subtracted  from  sum  of  series,  divided  by 
ratio.  Or,flXV*-i=t 

Example.— First  term  is  2,  ratio  3,  and  sum  of  series  2186;  what  is  last  term? 

2186 — — - = 2186 — 728  = 1458,  last  term. 


To  Compute  TsTnm'ber  of*  Terms. 

When  Ratio  First , and  Last  Terms  are  given.  Rule.— Divide  logarithm  of  quo- 
tient of  product  of  ratio  and  last  term,  divided  by  first  term,  by  logarithm  of  ratio. 


Or, 


log.  (a  + S r — 1)  — log-  a . 


log.  I — log,  a 


log.  r 


and 


log.  I — log,  (r  l — - r 


log.  (S  - 
_ S) 


-a)  — log.  (S  — l) 
+ 1 = n. 


+ > 


Example.  — Ratio  is  2,  and  first  and  last  terms  are  1 and  131072;  what  is  num- 
ber of  terms  ? 

log.  2 X 131  °g  = log.  262 144  = 5.418  54,  and  5.418  54  log.  of  2 = = lS- 


To  Compute  Sum  of  Series. 


When  First  Term,  Ratio , and  Number  of  Terms  are  given.  Rule.— Raise  ratio  to 
a power  index  of  which  is  equal  to  number  of  terms,  from  which  subtract  i ; then 
divide  remainder  by  ratio  less  i,  and  multiply  quotient  by  first  term. 


GEOMETRICAL  PROGRESSION". 


I05 


Or, 


Illustration  i.— First  term  is  2,  ratio  2,  and  number  of  terms  13;  what  is  sum 
of  series  ? 

% 

2 13  1 = 8192  — 1 = 8191,  and  8191  -4-  (2  — 1)  = 8191,  and  8191  x 2 = 16  382. 

2.  — If  a man  were  to  buy  12  horses,  giving  2 cents  for  first  horse,  6 cents  for 
second,  and  so  on,  what  would  they  cost  him  ? $5314.40. 

To  Compote  Ttatio. 

When  First  Term , Last  Term , and  Numbers  of  Terms  are  given.  Rule.— Divide 
last  term  by  first,  and  quotient  will  be  equal  to  ratio  raised  to  power  denoted  by  1 
less  than  number  of  terms;  then  extract  root  of  this  quotient. 

S^  — a 

Illustration.— First  term  is  2,  last  term  4374,  and  number  of  terms  8;  what  is 
ratio  ? 

4374  8—  1. 

— £=  2187,  and  2187  — 3,  ratio. 

Miscellaneous  Illustrations. 

1.  What  is  9th  term  in  geometrical  progression  3,  9,  27,  81,  etc.?  and  what  is 
sum  of  terms? 

1st  term  = 3,  number  of  terms  9,  and  ratio  3. 

Hence,  by  rule  to  compute  last  term,  1st  term  and  ratio  being  equal — 

Indices,  1234 
Terms,  3,  9,  27,  81. 

Then,  2 -f-  3 -I-4  = 9 ==  sum  of  indices,  and  9 X 27  X 81  = 19  683  = last  term. 

By  rule  to  compute  sum  of  terms— 

39 — 1 19682 

— _ ^ X 3 — — - — = 9841  x 3 = 29  523,  sum  of  terms. 

2.  First  term  is  1,  ratio  2,  and  last  term  131 072 ; what  is  sum  of  series? 

131 072  X 2 — 1 = 262  143,  and  262  143  -4-  2 — 1 = 262  143. 

3.  What  are  the  proportional  terms  between  2 and  2048? 


4 -f-  2 = 6,  and  6 — 1 = 5,  and 


5 /2048 

V— =4- 


Hence,  2 : 8 : 32  : 128  : 512  : 2048. 

4.  Sum  of  series  is  6560,  ratio  3,  and  number  of  terms  8;  what  is  first  term  ? 

3 — 1 2 

- = 6560  X — 2,  first  term. 


6560  X - 


3 8 — 1 “ ' ' 6560 

Geometrical  Progressions, 

Whereby  any  questions  of  Geometrical  Progression  and  of  Double  Ratio  may  be 
solved  by  Inspection , number  of  terms  not  exceeding  56. 

16384  29  268435456  43  4398046511104 

32768  30  536870912  44  8796093022208 

65536  31  1073741824  45  17592186044416 

131072  32  2147483648  46  35184372088832 

262144  33  4294967296  47  70368744177664 

524288  34  8589934592  48  14°  737  488  355  32B 

1048576  35  17179869184  49  281474976710656 

2097152  36  34  359  738  368  50  562949953421312 

4I94  3°4  37  68719476736  51  1125899906842624 

8388608  38  137  438  953  472  52  2251799813685248 

16777216  39  274877906944  53  4503599627370496 

33  554  432  40  549755813888  54  9007199254740992 

67108864  41  1099511627776  55  18014398509481984 

134217728  42  2199023255552  56  36028797018963968 

Illustrations. — 12th  power  of  2 = 4096,  and  7th  root  of  128  — 2. 


I 

15 

2 

16 

4 

17 

8 

18 

1 6 

*9 

32 

20 

64 

21 

128 

22 

256 

23 

512 

24 

1024 

25 

2048 

26 

4096 

27 

8192 

28 

io6 


ALLIGATION. 


ALLIGATION. 

Alligation  is  a method  of  finding  mean  rate  or  quality  of  different  ma- 
terials  when  mixed  together. 

To  Compute  Mean.  IPrice  of*  a Mixture. 

When  Prices  and  Quantities  are  known.  Rule.  - Multiply  each  quantity  by  its 
rat<T  divide  sum  of  products  by  sum  of  quantities,  and  quotient  will  give  rate  ot  the 
composition. 

Example.— If  io  lbs.  of  copper  at  20  cents  per  lb.,  1 lb.  of  tin  at  5 cents,  and  1 lb. 
of  lead  at  4 cents,  be  mixed  together,  what  is  value  of  composition . 

10  X 20  = 200 
1X5=  5 

j X 4 = 4^ 

12  ) 209  (17.416  cents. 

To  Compute  Qnantity  of*  each.  Article. 

When  Prices  and  Mean  Price  are  given.  Rule. — Write  prices  of  ingredients,  one 
under  the  other  in  order  of  their  values,  beginning  with  least,  and  set  mean  price 
at  left.  Connect  with  a line  each  price  that  is  less  than  mean  rate  with  one  or  more 

l^Wrfte  difference  between  mixture  rate  and  that  of  each  of  simples  opposite  price 
wK“=ed;  then  sum  of  differences  against  any  price  will  express 
quantity  to  be  taken  of  that  price. 

Example.— How  much  gunpowder,  at  72,  54,  and  48  cents  per  pound,  will  compose 
a mixture  worth  60  cents  a pound? 

(48  \ 12,  at  48  cents. 

60  54V  12,  at  54  cents. 

( 72 / 12  6 — 18,  at  72  cents. 

Here,  72  — 60  = 12  at  48,  72 — 60  = 12  at  54,  60—48  = 12,  and  60  — 54  = 6 = 

12  d-  6 — 18  at  72.  — — — — , 

Then  12  X 48+ 12  X 54  + *8  X 7* = ^ ° , 2520-r-  12  + 12  + 12  + 6 = 60  cents. 

Thus,  had  it  been  required  to  mix  18  pounds  at  48  cents,  result  would  be  18  at  48, 
18  at  54,  and  27  at  72  cents  per  pound. 

When  the  whole  Composition  is  limited.  Rule. -As  sum  of  relative  quantities, 
as  ascertained  by  above  rule,  is  to  whole  quantity  required,  so  is  each  quant, ty  so 
ascertained  to  required  quantity  of  each. 

Example.— Required  ioo  pounds  of  above  mixture 

Then,  12  + 12  + 18  = 4a-  Then,  42  : 100  : : 12  : 28. 571 

’ 42  : 100  ..  12  . 28.571  pounas. 

42  : 100  ::  18  : 42.857  pounds. 

When  Price  of  Several  Articles  and  Quantity  of  one  of  them  is  given.  Rule.-As- 
certain  proportionate  quantities  of  ingredients  by  previous  rule. 

Then  as  number  opposite  ingredients,  quantity  of  which  is  given,  .8 .to  given 
quant”tyfs?“s  number  opposite  to  each  ingredient  to  quantity  required  of  that  ,n 
gredient. 

Example  -Having  « lbs.  of  tobacco,  worth  60  cents  per  pound,  how  much  of 

other  qualities,  worth  65,  70,  and  75  cents  per  pound,  must  ho  mixed  with  it,  so  as  to 

sell  mixture  at  68  cents  per  pound? 

B a?75Pc^ 

kinds  must  be  increased  in  like  manner. 

Henco,  7:35-  : 2 : 10  = 10  at  65  cents. 

7 : 35  *•  : 3 • I5  = IS  “ 7°  cents- 
7 : 35  : : 8 : 40  = 40  “ 75  cents. 


SIMPLE  INTEREST. 


IQ/ 


SIMPLE  INTEREST. 

To  Compute  Interest  on  any  Griven  Sum  for  a Period 
of*  One  or  more  Years. 

Rule. — Multiply  given  sum  or  principal  by  rate  per  cent,  and  number  of  years- 
point  off  two  figures  to  right  of  product,  and  result  will  give  interest  in  dollars  and 
cents  for  1 year. 

Example.— What  is  interest  upon  $ 1050  for  5 years  at  7 per  cent.  ? 

1050  X 7 X 5 = 36  75°,  and  367. 50  = $ 367.50. 

When  Time  is  less  than  One  Year.  Rule.— Proceed  as  before,  multiplying  bv 
number  of  months  or  days,  and  dividing  by  following  units— viz.  12  for  months 
and  365  or  366,  as  the  case  may  be,  for  days. 

Example.— What  is  interest  upon  $1050  for  5 months  and  30  days  at  7 per  cent,? 

5 months  and  30  days  = 183  days.  l83  = 3685,  and  36.85  = $ 36.83. 

The  operation  of  computing  interest  may  he  performed  thus : 

Assuming  interest  upon  any  sum  at  6 per  eent.=-i  per  cent,  for  2 months 
Interest  at  5 per  cent,  is  Jth  less  than  at  6 per  cent. 

Interest  at  7 per  cent,  is  Jth  greater  than  at  6 per  cent. 

Taking  preceding  example— 2 months  = 1 per  cent.=  10.50 
2 “ = 1 “ 10.50 

1 ..  “ =i  “ 5-25 

30  days  = 1 month  = 5. 25 

3i-5o 

Add  f for  7 per  cent.  = 5.25 

$36-75  ' 

precedlns  arise8  from  183  d“ys  ^ talen  in 

In  every  computation  of  interest  there  are  four  elements— viz.,  Principal  Time 
be  Mcerteined1*6811  °F  Amount’  any  three  of  which  beiuS  Siven»  remaining  one  can 

To  Compute  Principal. 

When  Time,  Rate  per  Cent , and  Interest  are  given.  Rule.  —Divide  given  interest 
by  interest  of  $1,  etc.,  for  given  rate  and  time.  S mierest 

Example.— What  sum  of  money  at  6 per  cent,  will  in  14  months  produce  $ 14? 

14  -4-  .07  = 200  dollars. 

To  Compute  Yl ate  per  Cent. 

When  Principal,  Interest,  and  Time  are  given.  Rule.— Divide  given  interest  bv 
interest  of  given  sum,  for  time,  at  1 per  cent.  h merest  by 

wa^ha^erTenl^32'66  diSCOUUted  from  a note  of  *4°°  for  14  months,  what 
Interest  on  400  for  14  months  at  1 per  cent.  = 4. 66. 

Then  32.66-4-4.66  = 7 per  cent. 

To  Compute  Time. 


# J TJlT  Pr\nciPfi  per  Cent. , and  Interest  are  given. 

terest  by  interest  of  sum,  at  rate  per  cent,  for  one  year. 


Rule.— Divide  given  In- 


Example-Id  what  time  will  $ 108  produce  $ n.34,  at  7 per  cent.  ? 
Interest  on  108  for  one  year  is  7.56. 


ri. 34 -f-  7. 56  = 1. 5 years. 

rate  onmeresfhavh/g  been^^er^ent  ^whatw^  principal  invested  ?°^  13  $*2006?*^ 
If  fiooo  in  18  months  will  produce  $ ,090,  what  is  rate?  6 per  cmC 


is  rate? 


io8 


COMPOUND  INTEREST. 


COMPOUND  INTEREST. 

If  any  Principal  be  multiplied  by  number  (in  following  table)  opposite 
years,  and  underP rate  per  cent.,  sum  will  be  amount  of  that  principal  at  com- 
pound  interest  for  time  and  rate  taken. 

Example.— What  is  amount  of  $500  for  10  years  at  6 per  cent.  ? 

Tabular  number. . . . i-79o84,  and  *-79°  84  X 500  = 895.42  dollars 


3 

Per  Cent. 


1.03 
1.0609 
1.092  73 

1.12551 
i- 15927 
I-I94  05 
1. 229  87 
1. 266  77 
1.30477 
i- 343  92 
1.384  24 


Per  Cent. 

Per  Cent. 

1.04 

1.05 

1. 081  6 

1. 102  5 

1. 124  86 

1. 157  62 

1. 169  86 

1-2155 

1.216  68 

1.27698 

1.265  32 

i-34 

i-3I593 

1.407  1 

1.368  57 

1-477  45 

1.423  31 

x-  55X32 

1.480  24 

1.628  89 

1-539  45 

1 - 7 xo  33 

j 1. 601  03 

1 I-795  85 

4 I 5 

Per  Cent.  Per  Cent. 


[.66507 
I-731 67 
1.80095 
1.872  98 
1.947  99 
2.025  81 
2. 106  84 
2.191 13 
2.278  76 
2.369  92 
.464  21 


2.032  79  I 2.5633 


.88564 
1.979  93 
2.078  92 
2.182  87 
2. 292  01 
2.40661 
2.52695 
2.65329 
2. 785  96 
2.92526 
3-°  7*52 
3.22509  I 


6 

Per  Cent. 


1.132  92 
:.  260  9 
s-396  55 
-•  54°  35 
>.692  77 

2- 85433 
$•02559 
3.207  13 

3- 399  56 

3-60353 

81974 

04873 


Vnr  nnv  other  Kale  or  jreriuu. — 
number  tor  logarithm  will  give  tabular  amount  as  above 


Rate.  i 

Time.  || 

Per  cent. 
1 

69.68 

2 

35 

3 

23-44  II 

Rate. 

Per  cent. 

4 

5 

6 


Rate. 


17.67 

14.21 


Per  cent. 

7 

8 

9 


10.34 

9.01 

8.04 


30 


7.27 

3- 8 
2.64 


1 - 4 

Years.  per  Cent.  Per  Cent. 


•5 

1 

1- 5 

2 

2- 5 

3 

3- 5 

4 

4- 5 

5 

5- 5 

6 


1. 015 
1.0302 

1-0457 

1.0614 

1.0773 

1.0934 
1. 1098 
1.1265 
1. 1434 
1.1604 
1. 178 

1.1956 


1.02 
1.0404 
1.0612 
1.0824 
1. 1041 
1. 1262 
1.1487 
1. 1717 
1. 1951 
1. 219 
1.2434 
1.2689 


5 

Per  Cent. 


of*  13  Years. 

3 I 4 ! 5 1 6 

Years.  per  Cent.  Per  Cent,  j Per  Cent,  j Per  Cent. 


1.025 

1.0506 

1.0769 

1.1038 

1.1314 
I- 1597 
1.1887 
1.2184 
1.2489 
1.2801 
1. 3121 
3449 


6 | 

Per  Cent. I 


1.03 

1.0609 

1.0927 

i-i255 

i-i593 

1. 1941 

1.2299 

1.2668 

1.3048 

1-3439 

1.3842 

[.4258 


6.5 

7 

7-5 

8 

8.5 
9 

9-5 

10 

10.5 

11 

n-5 

12  | 


1. 2134 
1. 2317 
1.2502 
1.269 
1. 288 
1.3073 
1.3269 
1.3469 
1.3671 
1.3876 
1.4084 
4295 


1.2936 

i.3i95 

1-3459 

1.3728 

1.4002 

1.4282 

1.4568 

1,486 

I-5I57 
1 546 

1-5769 

.6084 


1. 3785 

i-4i3 

1.4483 

1.4845 

1.5216 

1.5597 

i-5987 

1.6386 

1 6796 

1.7216 

1.7606 

1.8087 


1.4684 

1 5102 

i-558 

1.6047 

1.6528 

1.7024 

1-7535 

1. 8061 

1.8603 

1.9161 

x-9736 

2.0356 


) 1. 1950  - , 

Illustration.— What  is  amount  of  $500  at  semi-annual  interest  o 5 per 

compounded  for  10  years?  e 

Tabular  number  1.6386.  Then,  500  X 1.628  89  - $ 8x4-44- 5- 

To  Compute  Interest  on  any  Given  Sum 

A - */A 

For  a Period  of  Years.  P (1  -f  r) n = A ; 


= p; 


— W (l  + ’*)n  V’  v p 

and  log-  A-i°g-p  = n P representing  principal,  r rate  per  cent,  per  annum , n 

numberlf  years]  and  A amount  of  principal  and  interest. 


DISCOUNT  OR  REBATE. EQUATION  OF  PAYMENTS.  IO9 


Illustration.— Assume  as  preceding,  $500  at  5 per  cent,  for  10  years. 


500  X 1.0510  = 500  X 1.628  89  = 1814.44.5,  amount,  — 500,  principal. 


500  X 1.05 


V 


For  any  Period.  — Assume  elements  of  preceding  case,  interest  payable  semi- 
J .05  , 

annually.  10  X 2 = 20,  number  of  payments ; ——  — 025,  rate. 

Then,  500  X i.o252°=  500  X 1-638  62  = $ 819.31. 

When  term  of  payments  and  rate  are  not  given  in  table. 


DISCOUNT  OR  REBATE. 

Discount  or  Rebate  is  a deduction  upon  money  paid  before  it  is  due. 

To  Compute  lie  To  ate  upon  any  Su.m. 

Rule.— Multiply  amount  by  rate  per  cent,  and  by  time,  and  divide  product  by 
sum  of  product  of  rate  per  cent,  and  time,  added  to  100. 

Example  1.— What  is  discount  upon  $ 12075  for  3 years,  5 months,  and  15  days, 
at  6 per  cent.  ? 


2. — What  is  present  value  of  a note  for  $963.75,  payable  in  7 months,  at  6 per 
cent.  ? 

6 rate.  7 months  = T7^-  of  1 year  = 6 X 7-4-12  = 3.5,  and  3.54-100  = 103.5  = 1.035. 
963-75 I.°35=  $93i*i6. 

To  Compute  tlie  Sum  for  a given  Time  and.  Rate,  to  yield 
a Certain  Sum. 

Rule.— Divide  given  sum  by  proceeds  of  $ 1 for  given  time  and  rate. 

Example.— For  what  sum  should  a note  be  drawn  at  90  days,  that  when  dis- 
counted at  6 per  cent,  it  will  net  $ 200? 

Discount  on  $ 1 for  904-  3 days  at  6 per  cent.=  $ .0155. 

Hence,  $1 — .0155  = .9845, proceeds,  and  $200 -4-. 9845  = $203. 14. 9. 


EQUATION  OF  PAYMENTS. 

Rule. — Multiply  each  sum  by  its  time  of  payment  in  days,  and  divide  sum  of 
products  by  sum  of  payments. 

Example.— A owes  B $300  in  15  days,  $60  in  12  days,  and  $350  in  20  days;  when 
is  the  whole  due  ? 


Illustration. — Assume  $1000  for  30  years,  at  7 per  cent,  half-yearly. 


log.  4-  1 = .014  940  3,  and  log.  .014  940  3 X 3°  X 1000  = $ 2806.78. 
2 


3 years  5 months  and  15  days  = 3. 4574  years. 


12  075  X 6 X 3-  4574  _ 2 50  48S.  63  _ 2Q74  ^ $ 


100  4- (6  X 3-4574)  120.7444 


= 2074.53  = $2074.53. 


300  x 15  = 4500 

60  X 12  = 720 
350  X 20  = 7 000 


710  ) 12220  (17  4~  days. 

K 


IIO 


ANNUITIES. 


ANNUITIES. 

To  Compute  Amount  of  Annuity. 

When  Time  and  Ratio  of  Interest  are  Given.  Rule. — Raise  the  ratio  to  a power 
denoted  by  time,  from  which  subtract  i ; divide  remainder  by  ratio  less  i,  and  quo- 
tient, multiplied  by  annuity,  will  give  amount. 

Note.— $ i added  to  given  rate  per  cent,  is  ratio,  and  preceding  table  in  Compound  Interest  is  a 
table  of  ratios. 

Example.— What  is  amount  of  an  annual  pension  of  $ioo,  interest  5 per  cent., 
which  has  remained  unpaid  for  four  years? 

1.05  ratio;  then  1.054— 1 = 1.21550625  — i = . 215 506 25,  and  .215  506  25  = (1.05 
— 1). 05  = 4.310 125,  which  x 100  = 1431.01.25. 


To  Compute  Present  'VVprtli  of  an.  Annuity. 

When  Time  and  Rate  of  Interest  are  Given.  Rule.— Ascertain  amount  of  it  for 
whole  time;  divide  by  ratio,  involved  to  time,  and  result  will  give  worth. 

Example.— What  is  present  worth  of  a pension  or  salary  of  $500,  to  continue  10 
years  at  6 per  cent,  compound  interest? 

$ 500,  by  last  rule,  is  worth  $6590.3975,  which,  divided  by  1.06 10  (by  table,  page 
108,  is  1. 790  84)  = $ 3680.05. 

Or,  Multiply  tabular  amount  in  following  table  by  given  annuity,  and  product 
will  give  present  worth. 


Illustration  l—  As  above;  10  years  at  6 per  cent.  = 7. 360 08,  and  7.36008  X 50° 
= 3.68.004  dollars. 

2.  What  is  present  worth  of  $150  due  in  one  year  at  6 per  cent,  interest  per  annum  ? 
•943  39  X 150  = 1141.50.85. 


Present  ‘Worth.  of  an  Annuity  of  $1,  at  A,  £>,  and  6 
Per  Cent.  Compound  Interest  for  Periods  under  So 
Years. 


Years. 

4 Per  Cent. 

5 Per  Cent. 

6 Per  Cent. 

Years. 

4 Per  Cent. 

5 Per  Cent. 

6 Per  Cent. 

1 

•961  54 

.95238 

•943  39 

*3  • 

9.98562 

9-393  57 

8.85268 

2 

1.886  09 

1.85941 

I-833  39 

14 

10. 563  07 

9.898  64 

9. 29498 

3 

2-775  1 

2,72325 

2.673  01 

15 

11.11843 

10.37966 

9.71225 

4 

3.6299 

3-545  95 

3-465  1 

16 

11.651  28 

10.837  78 

10.105  89 

5 

4-452  03 

4.32948 

4.212  36 

17 

12. 166  26 

11.27407 

10.477  26 

6 

5.242  15 

5-07569 

4.9x732 

18 

12.65926 

11.689  58 

10.827  6 

7 

6.002  03 

5-78637 

5-  582  38 

J9 

I3-I33  88 

12.085  32 

11.158  11 

8 

6.731  76 

6.463  21 

6.209  79 

20 

13.59029 

12.462  21 

11.469  92 

9 

7- 4364 

7. 107  82 

6. 801  69 

21 

14.029  12 

12.821 15 

11.76407 

10 

8.11085 

7.72173 

7.36008 

22 

14.451 12 

13-163 

12.041  58 

11 

8.76044 

8.306  41 

7.886  87 

23 

14.856  82 

13.48807 

12.303  38 

12 

9-38505 

8.86325 

8. 383  84 

24 

15.24695 

13.798  64 

12.55035 

For  a Rate  of  Interest  and  Term  of  Years  not  given  in  either  Table . 

'==  A.  Notation  as  preceding. 

(i  + r)»J 

Illustration. — Take  $ 1 at  4 per  cent,  for  24  years. 

Log.  1.04  = .017033,  which  X -24  = .408  799.  log.  .408  799  = 2.5633  = ratio  raised 
to  power  of  24. 

Then,  — X (1 ~r~)  = 25X1—  39° 122  = $ I5-  24-  695- 

.04  V 2.5633/ 

To  Compute  Yearly  Amount  tliat  will  Tiiqnidate  a Delot 
in  a Given  Number  of  Years  at  Compound  Interest. 


P 

r 


a.  Illustration.  — What  is  amount  of  an  annual  payment  that 

( 1 -j-  r) n — 1 

will  liquidate  a debt  of  $100  in  6 years  at  5 per  cent,  compound  interest? 


ANNUITIES. 


I I I 


(i -f. 05) 6 per  table,  page  i°8,  100  X .05  (f-^t  -ps)6  __  5 X T-34  ^7  __  ^ 

= i-34-  (1  + .05)6  — 1 I,34  1 *34 

When  Annuities  do  not  commence  till  a certain  period  of  time , they  are  said  to  be 
in  Reversion. 

To  Compute  Present  Worth  of  an.  Annuity  in  Reversion. 

rcle Take  two  amounts  under  rate  in  above  table — viz.,  that  opposite  sum  of 

two  given  times  and  that  of  time  of  reversion;  multiply  their  difference  by  an- 
nuity, and  product  will  give  present  worth. 

Example.— What  is  present  worth  of  the  reversion  of  a lease  of  $40  per  annum, 
to  continue  for  6 years,  but  not  to  commence  until  end  of  2 years,  at  rate  of  6 per 
cent.  ? 

6 -f-  2 = 8 years 6. 209  79 

2 “ 1.833  39 

4.37640  X 40  = 1175.05.6. 

Amount  of  Annuity  of  Si,  etc..  Compound  Interest, 
from  1 to  SO  Years. 


4 

Per  Cent. 


2.04 
3.121  6 
4. 246  46 
5.41632 
6.632  97 
7.898  29 
9.2x423 
10. 582  79 
12.006  11 


5 

Per  Cent. 


2.05 
3-I52  5 
4.310  12 
5-525  63 
6.801  91 
8.14201 
9. 549  1 1 
11.026  56 
12.577  " 


6 

Per  Cent. 


2.06 
3-1836 
4.374  62 
5-637  09 
6.97532 
8-393  84 
9.897  47 
1 1- 49*32 
13.18079 


7 

Per  Cent. 


2.07 

3.2149 

4- 439  94 

5- 750  74 
7-*53  29 
8.654  02 

10.259  8 
11.97799 
13.816  45 


4 

Per  Cent. 


13-486  35 
15.025  8 
16.  626  84 
18.291 91 
2Q.  023  59 
21.82453 
23.69751 
25.64541 
27.671 23 
29.77808 


5 

Per  Cent. 


14.206  79 
15-9*7*3 
17.712  98 
19.59863 
21.57856 

23-657  49 
25.84037 
28.132  38 
30.539 
33-o65  95 


6 

Per  Cent. 


14.97164 
16.869  94 
18.882  14 
21.01507 
23.275  97 
25-672  53 
28.212  88 
30.90565 
33-75999 
36.785  59 


7 

Per  Cent. 


15.7836 
17.88845 
20. 140  64 
22.55049 
25.129  02 
27.88805 
30. 840  22 
33.99903 
37-378  96 
40.995  49 


Illustration. — What  is  amount  of  $ 1000  for  20  years  at  5 per  cent.? 

5 per  cent,  for  20  years  = 33.065  95 ; hence,  1000  X 33.06595  = $33.06.595. 

To  Compute  Amount  of  an  Annuity  for  any  Reriod 
and  Rate. 

Rule. — From  table  for  Compound  Interest,  page  108,  take  value  for  rate  per  cent, 
for  1 year,  and  raise  it  to  a power  determined  by  time  in  years,  from  which  subtract 
1,  divide  remainder  by  rate,  and  quotient  multiplied  by  annuity  will  give  amount 
required. 

Example.— What  will  an  annuity  of  $ 50,  payable  yearly,  amount  to  in  4 years,  at 
5 per  cent.  ? 

By  table,  page  108, 1.054  = 1.2155. 

1. 2155  — 1 -4- (1.05  — i)  = 4.3i>  and  4.31  X 50  = 1215.50. 


For  Half-yearly  and  Quarterly  Payments. 
Multiply  annuity  for  given  time  by  amount  in  following  table: 


Rate  per  cent. 

Half-yearly. 

Quarterly. 

Rate  percent. 

Half-yearly. 

Quarterly. 

3 

1.007445 

1.011 181 

5-5 

1.013  567 

1.020  395 

3-5 

1.008  675 

1. 013  031 

6 

1.014781 

1.022  227 

4 

1.009  902 

1. 014  877 

6-5 

i-oi5  993 

1.024  055 

4-5 

1. 011  126 

1. 016  729 

7 

1. 017  204 

1.025  88 

5 

1. 012  348 

1. 018  559 

7-5 

1. 018  414 

1.027  704 

Illustration  i. — Annuity  as  determined  in  previous  case  = $215. 50. 

Hence,  215.50  X 1.012348  from  above  table  = $218. 16  for  half  yearly  payments. 

2.  A person  30  years  of  age  has  an  annuity  for  10  years,  present  worth  of  it  being 
$1000,  provided  he  may  live  for  10  years.  What  is  annuity  worth,  assuming  that 
60  persons  out  of  every  3550,  between  the  ages  of  30  and  40,  die  annually? 

3550  — 600  (-60  X xo)  = 2950  would  therefore  be  living. 

And,  3550  : 2950  ::  1000  = 1830.98. 


1 1 2 


PERPETUITIES. — COMBINATION. 


PERPETUITIES. 

Perpetuities  are  such  Annuities  as  continue  forever. 

To  Compute  Value  of  a Perpetual  Annuity. 
Rule.— Divide  annuity  by  rate  per  cent.,  and  multiply  quotient  by  unit  in  pre- 
ceding table. 

Example.— What  is  present  worth  of  an  annuity  for  $ ioo,  payable  semi-annually, 
at  5 per  cent.  ? 

100-^.05  = 2,  and  2 X 1.012348,  from  preceding  table  = 2.024.70. 

To  Compute  Value  of  a ^Perpetuity-  in  Reversion. 
Rule.— Subtract  present  worth  of  annuity  for  time  of  reversion  from  worth  of 
annuity,  to  commence  immediately. 

Example.— What  is  present  worth  of  an  estate  of  $50  per  annum,  at  5 per  cent., 
to  commence  in  4 years  ? 

50 -4-  .05 — 1000 

$50,  for  4 years,  at  5 per  cent.  = 3. 545  95  (from  table,  page  no)  X 50=  177.2975 

822.7025 

which  in  4 years,  at  5 per  cent,  compound  interest,  would  produce  $1000. 


COMBINATION. 

Combination  is  a rule  for  ascertaining  how  often  a less  number  of  num- 
bers or  things  can  be  chosen  varied  from  a greater,  or  how  many  different 
collections  may  be  formed  without  regard  to  order  of  each  collection. 

Combinations  of  any  number  of  things  signify  the  different  collections 
which  may  be  formed  of  their  quantities,  without  regard  to  the  order  of  their 
arrangement. 

Thus,  3 letters,  a,  b,  c,  taken  all  together,  form  but  one  combination,  abc . 
Taken  two  and  tzoo,  they  form  3 combinations,  as  ab , ac , be. 

Note.— Class  of  the  combination  is  determined  by  number  of  elements  or  things  to  be  taken  ; if  two 
are  taken,  the  combination  is  of  2d  class,  and  so  on. 

Rule. — Multiply  together  natural  series  1,  2,  3,  etc.,  up  to  the  number  to  be  taken 
at  a time.  Take  a series  of  as  many  terms,  decreasing  by  1,  from  number  out  of 
which  combination  is  to  be  made,  ascertain  their  continued  product,  and  divide 
this  last  product  by  former. 

Example  i. — How  many  single  combinations,  as  ab,  ac,  may  be  made  of  2 letters 
out  of  3?  lX2=  2 = 6 = 

3X2  6 2 3' 

2.  — How  many  combinations  may  be  made  of  7 letters  out  of  12? 

ix  2 x 3X4X5X6X7 _ 5040  and  3 99i68o _ 

12  x II  X IO  x 9 X 8 x 7 x 6 3991  680’  5040 

3. — How  many  different  hands  of  cards  may  be  held,  as  at  whist,  combinations 

13  out  of  52  ? 635  013  559  600. 

When  two  Numbers  or  Tilings  are  Combined. 

Rule.— Multiply  together  natural  series  1,  2,  3,  etc.,  to  one  less  term  than  number 
of  combinations;  ascertain  their  continued  product,  and  proceed  as  before. 

Example. — There  are  3 cards  in  a box,  out  of  which  two  are  to  be  drawn  in  a re- 
quired order.  How  many  combinations  are  there? 

Here  there  are  2 terms ; hence,  2 — 1 = 1,  and  — - — = — = 6 -4- 1 = 6. 

3X2  6 

To  Compute  1ST  umber  of*  "Ways  in  whioh  any  Number  of 

Distinct  Objects  can  be  Divided  among  any  Number. 

Rule.— Multiply  together  numbers  equal  to  number  given,  as  often  as  objects 
are  to  be  divided  among  them. 

Example.— In  how  many  different  ways  can  10  different  cards  be  divided  among 
3 persons?  3X3X3X3X3X3X3X3X3X30*  3IO  = 59049. 


COMBINATION. — CIRCULAR  MEASURE. 


“3 


Combinations  with.  Repetitions. 

In  this  case  the  repetition  of  a term  is  considered  a new  combination.  Thus, 
1 2,  admits  of  but  one  combination,  if  not  repeated;  if  repeated,  however,  it  admits 
of  three  combinations,  as  1, 1 ; 1,  2;  2,  2. 

rule> To  number  of  terms  of  series  add  number  of  class  of  combination,  less  1 ; 

multiply  sum  by  successive  decreasing  terms  of  series,  down  to  last  term  of  series; 
then  divide  this  product  by  number  of  permutations  of  the  terms,  denoted  by  class 
of  combination. 

Example.— How  many  different  combinations  of  numbers  of  6 figures  can  be 
made  out  of  11? 

j!  j_  (6  _ j)  — — sum  of  number  of  terms,  and  number  of  class,  less  1. 

16  X 15  X 14  X 13  X 12  X u = 5 765  760= product  of  sum,  and  successive  terms  to 
last  term.  , . 

iX2XsX4X5X6  = 720  permutations  of  class  of  combination. 

^,”£5760= 8008. 

720 

‘V'ariations  -with  Repetitions. 

Every  different  arrangement  of  individual  number  or  things,  including  repeti- 
tions, is  termed  a Variation. 

Class  of  Variation  is  denoted  by  number  of  individual  things  taken  at  a time. 

Rule.— Raise  number  denoting  the  individual  things  to  a power,  the  exponent 
of  which  is  number  expressing  class  of  variation. 

Example  i. — How  many  variations  with  4 repetitions  can  be  made  out  of  5 fig- 
ures? 54  = 625. 

2. — How  many  different  combinations  of  4 places  of  figures  can  be  made  out  of 
the  9 digits  ? 

. , , 12  x II  X 10  X 9 II 880 

9 + (4  - X)  = 12,  and  JX2X3X4  = = 495- 

Coin  "bin  eition.  without  Repetitions. 

Rule.— From  number  of  terms  of  series  subtract  number  of  class  of  combination, 
less  1 ; multiply  this  remainder  by  successive  increasing  terms  of  series,  up  to  last 
term  of  series;  then  divide  this  product  by  number  of  permutations  of  the  terms, 
denoted  by  class  of  combination. 

Example  1 — How  many  combinations  can  be  made  of  4 letters  out  of  10,  exclud- 
ing any  repetition  of  them  in  any  second  combination  ? 

10  — (4  — 1)  =:  7 = number  of  terms  — number  of  class,  less  1. 

7X8X9X10  = 5040  —prod,  of  remainder  7,  and  successive  terms  up  to  last  term. 

iX2X3X4  = 24  = permutations  of  class  of  combination. 

Then,  - 2io. 

24 

2- — How  many  combinations  of  the  5th  class,  without  repetitions,  can  be  made 
of  12  different  articles? 


12  — (5  — 1)  = 8,  and 


l X 9 X 10  X 11  X 12  _ 85  040 

1X2X3  X 4 x 5 


• = 792- 


CIRCULAR  MEASURE. 

Unit  of  Circular  Measure  is  an  angle  which  is  subtended  at  centre  of  a circle 
by  an  arc  equal  to  radius  of  that  circle,  being  equal  to 
l8o° 

371416  — 57.296°. 

Circular  measure  of  an  angle  is  equal  to  a fraction  which  has  for  its  numerator 
the  arc  subtended  by  that  angle  at  centre  of  any  circle,  and  for  its  denominator  the 


radius  of  that  circle. 


CIRCULAR  MEASURE. — PROBABILITY. 


1 14 

To  Compute  Circular  Measure  of  an  -A-iigle. 

Rule. — Multiply  measure  of  angle  in  degrees  by  3.1416,  and  divide  by  180. 
Example. — What  is  circular  measure  of  240  10'  8"? 

24°  10'  8"  X 3- _ 87008  x 3.1416  _ g 
180  180X60X60  -°421 

To  Compute  Measure  of  an  -A_iigle,  its  Circular  Measure 
being  Given. 

Rule.— Multiply  circular  measure  of  angle  by  180,  and  divide  by  3.1416. 


PROBABILITY. 

Probability  of  any  event  is  the  ratio  of  the  favorable  cases,  to  all  the 
cases  which  are  similarly  circumstanced  with  regard  to  the  occurrence.  If 
an  event  have  3 chances  for  occurring  and  2 for  failing,  sum  of  chances 
being  5,  the  fraction  f will  represent  probability  of  its  occurring  and  is  taken 
as  measure  of  it.  Thus,  from  a receptacle  containing  1 white  and  2 black 
balls,  the  probability  of  drawing  a white  ball,  by  abstraction  of  1,  is  i;  prob- 
ability of  throwing  ace  with  a die  is  : in  other  words,  the  odds  are  2 to  1 
against  first,  and  5 to  1 against  second. 

If  m -f  n — whole  number  of  chances,  m representing  number  which  are  favorable, 

andn  unfavorable . Therefore  — ^ — =z  probability  of  event 
m-\-n 

Probabilities  of  two  or  more  single  events  being  known,  probability  of  their  oc- 
curring in  succession  may  be  determined  by  multiplying  together  the  probabilities 
of  their  events,  considered  singly. 

Thus,  probability  of  one  event  in  two  is  expressed  by  -J;  of  its  occurring  twice  in 
succession,  X or  ; of  thrice  in  succession,  ^ X ^ X or  -g,  etc. 

Illustration  i.—  If  a cent  is  thrown  twice  into  the  air,  the  probability  of  its  fall- 
ing with  its  head  up,  twice  in  succession,  is  as  1 to  4.  Thus,  it  may  fall: 

1.  Head  up  twice  in  succession.  \ 

2.  Head  up  1st  time  and  wreath  2d  time,  f „ 1 _ _ J_  _ times 

3.  Wreath  up  1st  time  and  head  2d  time.  » ’ i-f-3  ’ ^ .25  * 

4.  Wreath  up  twice  in  succession.  ) 

These  are  the  only  results  possible,  and  being  all  similarly  circumstanced  as  to 
probability,  the  probability  of  each  case  is  as  1 to  4.  or  odds  are  as  3 to  1. 

* Probability  of  either  head  or  wreath  being  up  twice  in  succession  is  as  1 to  1,  or 
chances  are  even,  because  1st  and  4th  cases  favor  such  a result;  probability  of  head 
once  and  wreath  once  in  any  order  is  as  1 to  2,  because  2d  and  3d  cases  favor  such  a 
result;  and  probability  of  head  or  wreath  once  is  as  3 to  4,  or  odds  are  as  3 to  1,  be- 
cause 1st,  2d,  and  3d,  or  2d,  3d,  and  4th  cases  favor  such  a result. 

Note. — 1 to  2 is  an  equal  chance,  for  i out  of  2 chances  = 1 to  1,  being  an  equal  chance  ; again,  1 to 
5 is  4 to  1,  for  1 out  of  5 chances  is  1 to  4. 

2.— If  there  are  4 white  balls  and  6 black  in  a bag,  what  is  the  chance  of  a person 
drawing  out  2 black  at  two  successive  trials? 

This  is  a combination  without  repetition.  Hence,  6 — (2  — 1)  = 5, 


and 


5 X 6 _ 


: = 11  which  x 2 for  successive  trials  = — or  — . 


1 X 2 2 1 7 ' 2 15 

3. — Suppose  with  two  bags,  one  containing  5 white  balls  and  2 black,  and  the  other 
7 white  and  3 black. 

Number  of  cases  possible  in  one  drawing  from  each  bag  is  (5  + 2)  X (7  + 3)  =7 
X 10  = 70,  because  every  ball  in  one  bag  may  be  drawn  alike  to  one  in  the  other. 


PROBABILITY. 


115 

Number  of  cases  which  favor  drawing  of  a white  ball  from  both  bags  is  5 X 7 = 35i 
for  every  one  of  the  5 white  balls  in  one  bag  may  be  drawn  in  combination  with  every 
one  of  the  7 in  the  other.  For  a like  cause,  number  of  cases  which  favor  drawing  of 
a white  ball  from  1st  bag  and  a black  one  from  2d  is  5 X 3 = !5 ; a black  ball  from  1st 
bag  and  a white  ball  from  2d  is  7 X 2 = 14;  and  a black  ball  from  both  is  3 X 2 = 6. 
Probability,  therefore,  of  drawing  is  as 

;L*LZ.  — — — — i to  1,  a white  ball  from  both  bags.  - ^ - = — = — = 3 to  u, 

7o  70  2 7°  7°  I4 

a white  ball  from  1st , and  a black  from  2 d.  ^ = y = 1 to  4,  a black 

ball  from  1st,  and  a lohite  from  2 d.  “ = 3 tov  32,  a black  ball  from 

e V 1-4-5  Y 7 20 

both. 


L 5X3  + 2X7_.29_^  to  4Ij  a white  ball  from  one , arid  a black  from  other , 


70 


70 


- = — = 10  to  30,  a white  ball 
49  49 


, 1 , 3 29  5X7  + 5X3+2X7 

for  both  2d  and  3d  cases  favor  this  result ; hence,  — + — = — 

= — = — = 32  to  3,  at  least  one  white  ball , for  the  1st,  2d,  and  3d  cases  favor  this 
70  35 

result ; hence,  + + — + — = —• 

’ 2 14  5 35 

Again,  if  number  of  white  and  black  balls  in  each  bag  are  same,  say  5 white  and 
2 black,  5 + 2 X 5 + 2 = 49,  then  probability  of  drawing  is  as 

5X5  — ?*>  = 25  to  24,  a white  ball  from  both.  5 X - - 
49  49 

from  1 st  and  a black  from  2d.  — — = 10  to  39,  a black  ball  from  1 st,  and  a 

J 49  49 

white  from  2d.  - + = 4 to  45,  a black  ball  from  both. 

49  49 

4 when  two  dice  are  thrown,  probability  that  sum  of  numbers  on  upper  sides 

is  any  given  number,  say  7,  is  as  follows : 

As  every  one  of  the  six  numbers  on  one  die  may  come  up  alike  to,  or  in  combi- 
nation with  the  other,  number  of  throws  is  6 X 6 = 36. 

!i  and  6 1 

2 “ 5>  ; and  as  these  numbers  may  be 

upon  either  die,  there  are  3 x 2 = 6 throws  in  favor  of  the  combination  of  7;  hence 
. 6 1 

probability  of  throwing  7 is  — = -g-,  or  as  1 to  5. 

5.  —Probability  of  a player’s  partner  at  Whist  holding  a given  card  is  as  follows: 
Number  of  cards  held  by  the  other  3 players  is  3 X 13  = 39  i probability,  there- 
fore, that  it  is  held  by  partner  is  +,  but  it  may  be  one  of  the  13  cards  which  he 

holds;  hence  probability  is  — X 13  = — = — , or  as  1 to  2- 
’ 39  39  3 

6.  —Probability  of  a player’s  partner  at  Whist  holding  two  given  cards  is  as  follows: 

30  X 38 

Number  of  combinations  of  39  things,  taken  2 and  2 together,  is  — = 741 ; 


therefore,  probability  that  these  2 cards  are  in  partner’s  hand  is  39  x 38  — 


1 X 2 


39  X 19 


= _i_  = x to  740 ; but  they  may  be  any  2 cards  in  partner’s  hand;  therefore,  since 
741  ^ ^ 

number  of  combinations  of  13  cards,  taken  2 and  2 together,  is  = -—  ==  78, 


78  2 

probability  required  is  ~ = — , or  as  2 to  17. 


1X2 


Similarly,  probability  that  he  holds  any  3 given  cards  is  as  — , or  as  22  to  681. 


Probabilities  at  a game  of  Whist  upon  following  points  are: 

9 to  '7,  that  one  hand  has  tiuo  honors , and  two  hands  one  ; 

9 to  55,  that  two  hands  have  each  two  honors  ; 

3 to  29,  that  each  hand  holds  an  honor  ; 

3 to  13,  that  one  hand  has  three  honors , and  one  hand  one  ; 

1 to  63,  that  four  honors  are  held  by  one  hand. 

7. if  3 half-dollars  are  thrown  into  the  air,  probability  of  any  of  the  possible  com* 

binations  of  their  falling  is  determined  as  follows : 

Hence,  ^=  -125  = 1 to  7 in  favor  of  3 heads. 


“ 2 heads  and  1 tail. 

“ 1 head  and  2 tails. 

“ 3 tails. 


And  in  like  manner,  if  5 were  thrown  up,  probability  of  any  of  their  possible 
combinations  would  be  determined  as  follows : 

/ I , i\5  /i\S,  5 /iN5,  5 X 4 / 1 \5  , 5X4X3/i\5.  5X4X3X2/i\S 

17  + T/  = (+  + Tw  + U j + o<7^3  W + *x  2 x 3 x 4 W 
, 5X4X3X2X1  /_^\5 
tlX2X3X4XS  \2/ 


Hence,  ^5=  .03125  = 1 to  31  in  favor  of  5 heads; 

Y (“)5=  -15625  = 5 to  27  “ 

“ 4 heads  and  1 tail ; 

f$i(v)  = -3125  = 10 1022  “ 

“ 3 heads  and  2 tails 

5 X 4 X 3 / 1 \5  u 

I — )=.3i2  5 = 10  to  22  “ 

1X2X3V2/  J 3 

“ 2 heads  and  3 tails  , 

SX4X3X2/i\5  . 

xX2X3X4(-)=-15  S^'  7 

“ 1 head  and  4 tails  ; 

5X4X3X2Xi/iy=  i. 

IX2X3X4X5W  J 5 0 

“ 5 tails. 

All  Wagers  are  founded  upon  the  principle  of  product  of  the  event, 
and  contingent  gain,  being  equal  to  amount  at  stake. 

Illustration  i. — Suppose  3 horses,  A,  B,  and  C,  are  entered  for  a race,  and  X 
wagers  12  to  5 against  A,  n to  6 against  B,  and  10  to  7 against  C. 

If  A wins,  X wins  6 + 7 — 12  = 1. 

“ B “ X “ 5 + 7 — 11  = 1. 

“C  “ X “ 5 + 6 — 10=1. 

Hence,  X wins  1,  whichever  horse  wins,  from  having  taken  field  against  each 
horse  at  odds  named. 

~ . . . . ( A are  5 to  12  ) ( TV  in  favor  of  A> 

Odds  given  in  fa-  1 _ ..  *;  LL  r ; corresponding  probabil-  1 V u 

vorof  *JB  “ 6 “ u l ityte  B, 


7 10  / l IT 

and  — + — + — = — = 1.06  = 1.06  to  1 in  favor  of  taker  of  odds. 

17  17  17  17 


PROBABILITY. 


II  7 


2. —Odds  given  upon  first  seven  favorite  horses  for  Oaks  Stakes  of  1828  were  so 
great,  that  probability  in  favor  of  taker  of  the  odds  when  reduced  was  as  follows : 
1st,  5 to  2 : 2d,  5 to  2 ; 3d,  4 to  1 ; 4th,  7 to  1 ; 5th,  14  to  1 ; 6th,  14  to  1 ; 7th,  15  to  1 

( 4 X 3 X 16  = 192 

5.  , 3 

16 


73  ’ 


I X 7 X 16  = 112 
3 X 7 X 3—63 
7 x 3 x 16  336 


— 267  -r-  336  = 1.092  = 1.092  to  1,  in  favor  of. taker  of  odds,  yet  neither  of  the  horses 
upon  which  these  odds  were  given  won. 

3._If  odds  are  3 to  1 against  a horse  in  a race,  and  6 to  1 against  another  horse 
in  a second  race,  probability  of  1st  horse  winning  is  J,  and  of  other  i Therefore 
probability  of  both  races  being  won  is  ^g-,  and  odds  against  it  27  to  i,or  1000  to  37.037. 
Odds  upon  such  an  event  were  given  in  1828  at  1000  to  60,  or  16.67  to  1. 

4 —Two  persons  play  for  a certain  stake,  to  be  won  by  winner  of  three  games  or 
results.  One  having  won  one  and  the  other  two,  they  decide  to  divide  the  sum, 
proportionate  to  their  interest.  How  much  of  it  should  each  one  receive? 

Operation.— If  winner  of  two  games  should  win  game  to  be  played,  he  would  be 
entitled  to  the  whole  sum ; if  he  lost,  he  would  be  entitled  to  half  of  it.  Now  as 


- = — , half  of  which  : 


, or  share 


one  event  is  as  probable  as  the  other,  --  -f-  - 
of  winner  of  two  games. 

When  events  are  wholly  independent,  so  that  occurrence  of  one  does  not 
affect  that  of  the  other,  probability  that  both  will  occur  is  product  of  proba- 
bilities that  each  will  occur. 

Note.— It  is  indifferent  whether  events  are  to  occur  together  or  consecutively. 

Illustration  i. — Assume  three  boxes,  each  containing  white  and  black  balls  as 

6 white,  5 black;  7 white,  2 black;  8 white,  10  black.  What  is  chance  of  drawing 
from  them  a white,  black,  and  a white  ball? 


Probabilities  are  — , — , and  product  of  which  = 6~^2~^~8 
11  9 18  297 


: 17.625  tO  I. 


2. — A gives  an  answer  correctly  3 times  out  of  4,  B 4 times  out  of  5,  and  C 6 out 
of  7.  What  is  probability  of  an  event  which  A and  B declare  correct  and  C denies? 
Operation. — Compound  probability  that  A and  B answer  correctly  and  C denies 


(all  3 of  which  are  in  favor  of  event)  is 


X — X — = — = 

4 5 7 I4°  35 

Compound  probability  that  A and  B deny  and  C is  correct  (all  3 of  which  are 
6 _ 3 
140  70’ 

Then  correct,  divided  by  sum  3 / 3 , 3\  _ .8714  _ <Q  2 

of  correct  and  incorrect,  • V35  + 70)  ~ .857  14 + .428  57  ~ 3 ’ 


against  event)  is  — X — X — 
4 5 7 


Odds  between  Results  or  Chances,  and  between  any 
Number  and  Whole  NT n mher,  at  various  Odds  against 
each.,  also  Value  of  each  Chance  in  parts  of  IOO. 


Odds  against 
each. 

Value  of 
Chance. 

Odds  against 
each. 

Value  of 
Chance. 

Odds  against 
each. 

Value  of 
Chance. 

Odds  against 
each. 

Value  of 
Chance. 

Even 

50 

2 to 

1 

33-33 

6. 5 to  1 

13-33 

15  to  1 

6.25 

11  to  10 

47.62 

2-5  “ 

1 

28.57 

7 “ 1 

12.5 

18  “ I 

5.26 

6 “ 5 

45-45 

• 3 “ 

1 

25 

7- 5 1 

11.76 

20  u I 

4.76 

5 “ 4 

44.44 

3-5  “ 

1 

22.22 

8 “ 1 

11. 11 

25  “ 1 

3-84 

5-5  “ 4 

42.1 

4 “ 

1 

20 

8.5  “ 1 

10.52 

3°  “ 1 

3.22 

6 “ 4 

40 

4-5  “ 

1 

18.18 

9 “ 1 

10 

4°  1 

2.44 

6.5  “ 4 

38.1 

5 “ 

1 

16.66 

9-5  “ 1 

9-52 

50  u 1 

1.96 

7 “ 4 

36.36 

5-5  “ 

1 

I5-38 

10  “ 1 

9.09 

60  “ 1 

1.64 

7-5  “ 4 

34-78 

6 “ 

1 

14.28 

12  “ 1 

7-7 

IOO  “ 1 

•99 

Operation.  — Divide  100,  or  unit,  as  case  may  be,  by  sum  of  odds,  and  multiply 
quotient  by  lesser  chance  or  odds. 

Illustration. — 6 to  4.  6 -{-  4 = 10,  and  100  -4- 10  X 4 = 40,  value  of  chance. 


1 1 8 WEIGHTS  OF  IRON,  STEEL,  COPPER,  ETC, 


WEIGHTS  OF  IRON,  STEEL,  COPPER,  ETC. 


NVronglit  Iron,  Steel,  Copper,  and.  Brass  I?lates6 
soft  rolled.  ( American  Gauged) 


No.  of 
Gauge. 

Thickness. 

Iron. 

Per  Sqi 
Steel. 

fare  Foot. 
Copper. 

Brass. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

oooo 

.46  or  T7g  full 

iS-4575 

18.7036 

20.838 

19.688 

ooo 

.409  64 

I6.4368 

16.6559 

18.556  7 

17.5326 

oo 

.364  8 or  | light 

i4-6376 

14.8328 

16.525  4 

I5-6I3  4 

o 

.324  86  or  .1  “ 

I3-035I 

13.2088 

14.716  2 

13.904 

i 

.2893 

11.6082 

11.7629 

13-105  3 

12.382 

2 

.257  63  or  1 full 

io.3374 

10.4752 

11.670  6 

11.026  6 

3 

.229  42 

9-2055 

9-3283 

10.392  7 

9.819  2 

4 

.204  31  or  full 

8.1979 

8.3073 

9-255  2 

8.744  5 

5 

.181  94  or  t3b-  light 

7.3004 

7-3977 

8.241  9 

7.787 

6 

.162  02 

6.5011 

6.5878 

7-339  5 

6.934  5 

7 

.144  28 

5-7892 

5.8664 

6.5359 

6.175  2 

8 

.128  49  or  J full 

5-I557 

5.2244 

5.8206 

5-4994 

9 

•ii4  43 

4-59I5 

4.6527 

5-183  7 

4.897  6 

IO 

.101  89  or  -j1^  full 

4.0884 

4.1428 

4.615  6 

4.3609 

ii 

.090  742 

3.641 

3.6896 

4.110  6 

3.8838 

12 

.080  808 

3.2424 

3.2856 

3.660  6 

3-458  6 

13 

.071  961 

2.8874 

2.9259 

3-259  8 

3-079  9 

14 

.064  084 

2.5714 

2.6057 

2.903 

2.742  8 

15 

.057  068 

2.2899 

2.3204 

2.585  2 

2.4425 

16 

.050  82  or  full 

2.0392 

2.0664 

2.302  1 

2.175  1 

17 

•045  257 

1.8159 

1.8402 

2.050  1 

1-937 

18 

.040  303 

1.6172 

1.6387 

1.825  7 

1-725 

19 

•035  89 

1.44 

1-4593 

1.625  8 

i-536i 

20 

.031  961 

1.2824 

1.2995 

1.4478 

1.3679 

21 

.028  462 

1. 142 

I-I573 

1.2893 

1.218  2 

22 

•025  347 

1. 01 7 

1.0306 

1.148  2 

1.0849 

23 

.022  571 

.9057 

•9*77 

1.022  5 

.96604 

24 

.021  1 

.8065 

•8i73 

•910  53 

.860  28 

25 

.0179 

.7182 

.7278 

.810  87 

.766  12 

26 

.015  94 

.6396 

.6481 

.722  08 

.682  23 

27 

.014  195 

.5696 

•5772 

•643  03 

•607  55 

28 

.012  641 

.5072 

•5i4 

.572  64 

•541  03 

29 

.011  257 

•4517 

•4577 

•509  94 

.481  8 

30 

.010025 

.4023 

.4076 

•454  13 

.429  07 

31 

.008  928 

•3582 

•363 

•404  44 

.382  12 

32 

.007  95 

•3*9 

.3232 

.360  14 

.340  26 

33 

.007  08 

.2841 

.2879 

.320  72 

•303  02 

34 

.006  304 

.2529 

•2563 

•285  57 

.269  81 

35 

.005  614 

•2253 

.2283 

•254  31 

.240  28 

36 

.005 

.2006 

•2033 

( .2265 

.214 

37 

.004  453 

.1787 

.181 

.201  72 

.190  59 

38 

.003965 

•I59i 

.1612 

.17961 

.1697 

39 

•003  531 

.1417 

.1436 

•159  95 

•151  13 

40 

.003  144 

.1261 

.1278 

.142  42 

•134  56 

Specific  Gravities 

7.704 

7.806 

8.698 

8.218 

Weights  of  a Cube  Foot . . 

481-75 

487-75 

543-6 

513-6 

u 

“ Inch . . 

•2787 

.2823 

.3146 

.297  2 

WEIGHTS  OF  IRON,  STEEL,  COPPER,  ETC.  II9 
NWronglit  Iron,  Steel,  Copper,  and.  Brass  IPlates. 


( Birmingham  Gauge. ) 


No.  of 
Gauge. 

Thickness. 

Iron. 

Per  Squai 
Steel. 

he  Foot. 
Copper. 

Brass. 

0000 

Inch. 

•454  or  re  fuU 

Lbs. 

18.2167 

Lbs. 

18.4596 

Lbs. 

20.5662 

Lbs. 

19.4312 

000 

.425 

I7-053I 

17.2805 

19.2525 

18.19 

00 

.38  or  f full 

15-2475 

15.4508 

17.214 

16.264 

0 

•34  ori  “ 

I3.6425 

13.8244 

15.402 

14-552 

1 

•3 

12.0375 

12.198 

13.59 

12.84 

2 

.284 

H-3955 

11.5474 

12.8652 

12.1552 

3 

.259  or  i full 

IO.3924 

IO.5309 

11.7327 

11.0852 

4 

.238 

9-5497 

9.6771 

10.7814 

10.1864 

5 

.22 

.203  or  ^ full 

8.8275 

8.9452 

9.966 

9.416 

6 

8.1454 

8.254 

9- *959 

8.6884 

7 

.18  or  ^ light 

7.2225 

7.3188 

8.154 

7.704 

8 

.165  or  I “ 

6.6206 

6.7089 

7-4745 

7.062 

9 

.148  or  i full 

5-9385 

6.OI77 

6.7044 

6-3344 

10 

•i34 

5.3767 

5.4484 

6.0702 

5.7352 

11 

.12  or  J light 

4.815 

4.8792 

5.436 

5-I36 

12 

.109 

4.3736 

4.4319 

4-9377 

4.6652 

13 

•°95  or  light 

3.8119 

3.8627 

4.3035 

4.066 

14 

.083 

3.3304 

3.3748 

3-7599 

3-5524 

15 

.072 

2.889 

2.9275 

3.2616 

3.0816 

16 

.065 

2.6o8l 

2.6429 

2.9445 

2.782 

17 

.058 

2.3272 

2.3583 

2.6274 

2.4824 

18 

.049  or  ^ light 

I.9661 

1.9923 

2.2197 

2.0972 

19 

.042 

I.6852 

I.7077 

1.9026 

1.7976 

20 

•035 

I.4044 

I.423I 

1-5855 

1.498 

21 

.032 

I.284 

1. 301 1 

1.4496 

1.3696 

22 

.028 

I*i235 

1.1385 

1.2684 

1.1984 

23 

•025  or  ^ 

1. 003 1 

I.O165 

1-1325 

1.07 

24 

.022 

.8827 

.8945 

.9966 

.9416 

25 

.02  or^ 

.8025 

.8132 

.906 

.856 

26 

.Ol8 

.7222 

•7319 

•8154 

.7704 

27 

.Ol6 

.642 

.6506 

.7248 

.6848 

28 

.014 

.5617 

.5692 

•6342 

•5992 

29 

.013 

•5216 

.5286 

.5889 

.5564 

30 

.012 

.4815 

•4879 

.5436 

•5136 

31 

•OI  or  thr 

.4012 

.4066 

•453 

.428 

32 

.009 

.3611 

•3659 

.4077 

.3852 

33 

.008 

.321 

.3253 

.3624 

.3424 

34 

.007 

.2809 

.2846 

.3171 

.2996 

35 

.005  or  gw 

.2006 

.2033 

.2265 

.214 

36 

.004  or 

.1605 

.1626 

.1812 

.1712 

Thickness  of  Sheet  Silver,  Grold,  etc. 
By  Birmingham  Gauge  for  these  Metals. 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

1 

.004 

7 

.015 

13 

.036 

19 

.064 

25 

•095 

31 

*I33 

2 

.005 

8 

.016 

14 

.041 

20 

.067 

26 

.103 

32 

.143 

3 

.008 

9 

.019 

15 

•047 

21 

.072 

27 

.113 

33 

•-I45 

4 

.01 

10 

.024 

16 

.051 

22 

.074 

28 

.12 

34 

.148 

5 

.013 

11 

.029 

17 

•057 

23 

-077 

29 

.124 

35 

.158 

6 

.013 

12 

•034 

18 

.061 

24 

.082 

30 

.126 

36 

.167 

120 


WEIGHTS  OF  IRON,  STEEL,  COPPER,  ETC. 


Wronglit  Iron,  Steel,  Copper,  and  Brass  NWire. 

American  Gauge,  f.  full,  1.  light. 


No.  of  1 
Gauge. 


Diameter. 


oooo 

OOO 


13 

14 

15 

16 

17 

18 

19 

20 


Iron. 


23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 


Inch. 

.46  or  f. 
.4O964 
.364  8 or  § 1. 
.324  86  or  T\  f. 
.2893 

.257  63' or  J 
.229  42 

.204  31  or  | f. 
.181  94  or  1. 
.162  02 
,144  28 

.128  49  or  J f. 
•H4  43 

.IOI  89  Or  f. 
.090  742 
.080  808 
.071  961 
.064  084 
.057  068 
.050  82  or  ^ f . 

•045  257 

.040  303 

•035  89 

.031  961 
.028  462 

•025  347 
.022  571 
.020  1 or  dy  f. 
.0179 
.015  94 
.014  195 
.012  641 
.011  257 
.010  025  or 
.008  928 
007  95 
007  08 
.006  304 
.005  614 
•°°5  or 
.004  453 
.OO3965 
•003  531 
.003  144 


Lbs. 

.560  74 

.444683 

.352  659 
.279  665 
.221  789 
.175  888 
.139  48 
.110  616 
.087  72 
.069  565 
.055  165 

.043  75i 

,034699 
.027  512 
.021  82 
.017  304 
.013  722 
.010  886 
.008  631 
.006  845 
.005  427 
.004  304 
.003  413 
.002  708 
.002  147 
.001  703 
.001  35 
.001  071 
.000  849  1 
.000  673  4 
.000  534 
.000  423  5 
.000  335  8 
.000  266  3 
.000  211  3 
.000  167  5 
.000  132  8 
.000  105  3 
.000083  66 
.000066  25 
.000  052  55 
.000  041  66 
.000  033  05 
.000  026  2 


Per  Lineal  Foot. 

Steel.  I Copper. 


Lbs. 

566  03 
448  879 
>355  986 
282  303 
,223  89I 
.177  548 
.140  796 
.111  66 
.088  548 
.070  221 
.055  685 
.044  164 
.035  026 
.027  772 
.022  026 
.017  468 
.013  851 

.OIO  989 
.008  712 
.006  909 
.005  478 
.OO4344 
.OO3445 
.002  734 
.002  167 

.OOI  719 
.OOI  363 
.OOI  o8l 
.OOO  857  I 
.OOO  679  7 
.OOO  539  I 
.OOO  427  5 
.OOO  338  9 

.000  268  8 
.000  213  2 
.000  169  1 
.000  134  1 
.000  106  3 
.000  084  45 
.000066  87 
,000  053  04 
.ooo  042  05 


Lbs. 

.640513 
.507  946 
.402  83 
•3*9  451 
.253  342 
.200911 
•159  323 
.126  353 
.100  2 
.079  462 
.063  013 
.049  976 
.039  636 
.031  426 
.024  924 
.019  766 

.015  674 

.012  435 
.009  859 
.007  819 
.006  199 
i .004916 
! .003  899 
! .003  094 
1 .002  452 
! .001  945 
! .OOI  542 
.001  223 
.000  969  9 
.000  769  2 
.000  609  9 
.000  483  7 
.000  383  5 
.000  304  2 
.000  241  3 
.000  191  3 
.000  151  7 
.000  120  4 
.000  095  6 
.000  075  7 
.00006003 
.000  047  58 


Brass. 


j .000  033  36  .000  037  75 
! .000  026  44  I .000  029  92 


Lbs. 

.605  176 
.479  908 
.380  666 
.301  816 
.239  353 
189  818 
150  522 
.119376 
.094  666 

.075  075 
.059  545 
.047  219 
.037  437 
.029  687 
.023  549 
.018  676 
.014  809 
.011  746 
.009  315 
.007  587 
.005  857 
.004  645 
.003  684 
.002  92 
.002  317 
.001  838 
.001  457 
.001  155 
.0009163 
.000  726  7 
.0005763 
.000457 
.000  362  4 
.000  287  4 
.000  228 
.000  180  8 
.000  143  4 
.000  113  7 
.000090  15 
.000  071  5 
.000056  71 
j .00004496 
.000  035  66 
I .000  028  27 


Specific  Gravities 7-774 

Weights  of  a Cube  Foot . . 485.87 
“ “ Inch . . .2812 


7.847 

490.45 

.2838 


8.88  | 8.386 

554.988  | 524-16 

.3212  | .3033 


Specific  Gravities  to  determine  the  computations  of  these  weights  were  made  by 
author  for  Messrs.  J.  R.  Browne  & Sharpe,  Providence,  R.  I. 


121 


WEIGHTS  OF  IKON,  STEEL,  COPPER,  ETC. 


'W' rought  Iron,  Steel,  Copper,  and  Brass  Wire. 
Birmingham  Wire  Gauge,  f.  full,  1.  light. 


No.  of 
Gauge 

Thickness. 

I Iron. 

Per  Lini 
| Steel. 

sal  Foot. 
Copper. 

Brass. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

oooo 

•454  or  xg-  £ 

.546  207 

•SSI  36 

.623  913 

.589  286 

ooo 

.425  „ 

.478  656 

.483 172 

•546  752 

.516407 

oo 

.38  orf-f. 

.382  66 

.386  27 

•437  099 

.412  84 

o 

•34  or  if. 

•30634 

.309  23 

•349  921 

•330  5 

i 

*3 

•2385 

.240  75 

.272  43 

•257  31 

2 

.284 

.213  738 

•215  755 

.244  146 

.230  596 

3 

.259  or  if. 

.177  765 

.179  442 

.203  054 

.191  785 

4 

.238 

.150  107 

•151  523 

.171  461 

.l6l  945 

5 

1 .22 

.128  26 

.129  47 

.146507 

•138  376 

6 

.203  or  J f. 

, .109  204 

.110  234 

.124  74 

.117  817 

7 

.18  or  T\  1. 

.085  86 

.086  667 

.098  075 

.092  632 

8 

.165  or  i 1. 

.072  146 

.072  827 

.082  41 

.077  836 

9 

.148  or  1 f. 

.058  046 

•058  593 

.066  303 

.062  624 

IO 

•i34 

.047  583 

.048  032 

•054  353 

.051  336 

ii 

! .12  or  1. 

.038  16 

.038  52 

•043  589 

.041  17 

12 

! .109 

.031  485 

.031  782 

.035  964 

.033  968 

13 

[ -095  or  ^ 1. 

.023  916 

.024  142 

.027  319 

.025  802 

. 14 

! .083 

.018  256 

.018  428 

.020  853 

.019  696 

15 

.072 

.013  728 

.013  867 

.015  692 

.014  821 

16 

i -065 

.011  196 

.01 1 ^302 

.012  789 

.012  O79 

17 ! 

.058 

.008  915 

.008  999 

.010  183 

.OO9  6l8 

18  ! 

•°49  or  ^ 1. 

.006363 

.006  423 

.007  268 

.006  864 

19 

.042 

.004  675 

.004  719 

•00534 

•005  O43 

20 

•035 

.003  246 

.003  277 

.003  708 

.003  502 

21  i 

.032 

.002  714 

.002  739 

.003  1 

.002  928 

22 

.028 

.002  078 

.002  097 

.002  373 

.002  24I 

23 

•025  or  ,L. 

.001  656 

.001  672 

.001  892 

.OOI  787 

24 

.022 

.001  283 

.001  295 

.001  465 

.OOI  384 

25 

.02  or  -6V 

.001  06 

.001  070 

.001  211 

.OOI  I44 

26 

.Ol8 

.000  858  6 

.000  866  7 

.000  980  7 

•OOO  926  3 

27 

• Ol6 

.000  678  4 

.000  684  8 

.000  774  9 

.OOO  731  9 

28 

.OI4 

.0005194 

.000  524  3 

•000  593  3 

.OOO  560  4 

29 

.013 

.000  447  9 

.000  452  1 

.0005116 

•OOO  483  2 

30 

.012 

.000  381  6 

.000  385  2 

•0004359  | 

.OOO4II  7 

31 

•OI  orTOT 

.000  265 

.000  267  5 

.000  302  7 

.OOO  285  9 

32 

.009 

.000  214  7 

.000  216  7 

.000  245  2 

•ooo  231  6 

33 

•008 

.000  169  6 

.000  171  2 

•000  193  7 

.OOO  183 

34 

.OO? 

.000  129  9 

.000  131  1 I 

. .000  148  3 

•ooo  140  I 

35 

•°°5  or  a-J  o 

.000  066  25  i 

.000  066  88 

.000  075  68 

•ooo  071  48 

36 

.004  or  ttIs 

.000  042  4 

.000  042  8 

.000  048  43 

.OOO  O45  74 

Thickness  of  Plates. 


No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

1 

•3125 

9 

•156  25 

17 

•056  25 

25 

.023  44 

2 

.281  25 

10 

.140625 

18 

•05 

26 

.021  875 

3 

.25 

11 

.125 

!9 

•043  75 

27 

.020312 

4 

•234  375 

12 

.112  5 

20 

•037  5 

28 

.018  75 

5 

a 

.218  75 

13 

.1 

21 

•034  375 

29 

.017  19 

0 

.203  125 

14 

.0875 

22 

•031  25 

30 

•015  625 

7 

8 

•!87  5 
.171  875 

15 

16 

•075 
.062  5 

23 

24 

r 

.028  125 
.025 

31 

32 

.014  06 
.012  5 

122 


WIRE  GAUGES, 


WIRE  GAUGES.  (English.) 

Warrington  ( Hylands  Brothers). 


No. 

Inch. 

No.  | 

Inch. 

No. 

Inch. 

No.  • 

Inch. 

| No. 

Inch. 

7/0 

6/0 

5/° 

4/0 

3/  ° 
2/0 

No. 

1 

% 

% 

11/ 

/as 

Inch. 

0 

1 

2 

3 

4 

5 

No. 

.326 

•3 

.274 

•25 

.229 

.205 

Sir  Jose 
Inch. 

6 

7 

8 

9 

10 

10.5 
ph  W) 

No. 

.191 

.174 

•159 

.146 

•133 

.125. 

litworth  * 
Inch. 

11 

12 

13 

14 

15 

16 

tfc  Co. 
No. 

.117 

.1 

.09 

.079 

.069 

.0625 

,’s. 

Inch. 

17 

18 

19 

20 

21 

22 

No.  | 

•053 

.047 

.041 

.036 

•0315 

.028 

Inch. 

1 

.001 

14 

.0.14 

34 

•034 

85 

.085 

240 

.24 

2 

.002 

15 

.015 

36 

.036 

90 

.09 

260 

.26 

3 

.003 

16 

.016 

38 

.038 

95 

.09 

280 

.28 

4 

.004 

17 

.017 

40 

.04 

100 

.1 

300 

•3 

5 

.005 

18 

.018 

45 

•045 

no 

.11 

325 

•325 

6 

.006 

19 

.019 

50 

•05 

120 

.12 

350 

•35 

7 

.007 

2Q 

.02 

55 

•055 

135 

•135 

375 

•375 

8 

.008 

22 

.022 

60 

.06 

150 

.15 

400 

•4 

9 

.009 

24 

.024 

65 

.065 

165 

.165 

425 

•425 

10 

.01 

26 

,026 

70 

.07 

180 

.18 

450 

•45 

11 

.011 

28 

.028 

*75 

•075 

200 

.2 

475 

•475 

12 

.012 

3<=> 

•03 

80 

.08 

220 

.22 

500 

•5 

13 

.013 

32 

.032 

1 

1 

Sir  Joseph  Whitworth,  in  1857,  introduced  a Standard  ire-uauge,  rang- 
ing from  half  an  inch  to  a thousandth,  and  comprising  62  measurements. 
It  commences  with  least  thickness,  and  increases  by  thousandths  or  an  inch 
up  to  half  an  inch.  Smallest  thickness,  ^ of  an  inclb  i%No‘1I  ’ ®0'  2 
is  2 and  so  on,  increasing  up  to  No.  20  by  intervals  of  two  ? 

No.  20  to  No.  40  by  yow*’  and  from  No*  to  No*.  100  by,T^o?'  +1ihe 
thicknesses  are  designated  or  marked  by  their  respective  numbers  m thou- 
sandths of  an  inch.  . 

This  gauge  is  entering  into  general  use  m England. 

of  Grreat  Britain, 


UNTew  Standard.  Wire  Gauge 
1884,. 


7/0 

6/0 

5/° 

4/0 

3/° 

2/0 


Inch.  J 

No. 

Inch.  |j 

No. 

Inch.  j 

No. 

• S 

8 

.160  I 

22 

.028 

36 

.464 

9 

.144 

23 

.024 

37 

•432 

10 

.128 

24 

.022 

38 

.4. 

11 

.Il6 

25 

.02 

39 

.372 

12 

• io4 

26 

,Ol8 

40 

•348 

13 

.092 

27 

.0164 

41 

•324  | 

14 

.08 

28 

.OI48 

42 

• 3 I 

15 

.072 

29 

.OI36 

43 

.276 

16 

.064  1 

30 

.0124 

44 

.252 

17 

.056 

31 

.OIl6 

45 

.232  ' 

l 18 

.048 

32  . 

.0108 

46 

.212  1 

19 

.04 

1 33 

.OI 

47 

.192 

20 

.036 

i 34 

.0092 

48 

.176 

1 21 

.032 

II  35 

.0084 

49 

No.  50. 

.001  inch. 

Inch. 


.OO76 

.0068 

.006 

.0052 

.OO48 

.OO44 

.OO4 

•OQ36 

.OO32 

.0028 

.0024 

.002 

.OOl6 

.0012 


WIRE  GAUGES. GAS  PIPES  AND  WIRE  COED.  1 23 


French.  ( Jauges  de  Fils  de  Fer). 

French  wire-gauges,  alike  to  the  English,  have  been  subjected  to  variation.- 
Following  table° contains  diameters  of  the  numbers  of  the  Limoges  gauge. 


Wire-Gauge  (Jauge  de  Limoges). 


Number.  Millimetre. 

Inch.  I 

O 

•39 

.OI54 

I 

•45 

.OI77 

2 

•56 

.0221 

3 

.67 

.0264 

4 

•79 

.0311 

5 

•9 

•0354 

6 

I.OI 

.0398 

7 

1. 12 

.O44I 

8 

1.24 

.O488 

Number. 

Millimetre. 

Inch. 

Number. ' 

Millimetre. 

Inch. 

9 

i-35 

•0532 

18 

3-4 

.134 

10 

1.46 

•0575 

19 

3-95 

.156 

11 

1.68 

.0661 

20 

4-5 

.177 

12 

1.8 

.0706 

21 

5-i 

.201 

13 

1.91 

.0752 

22 

5-65 

.222 

!4 

2.02 

•0795 

23 

6.2 

.244 

15 

2.14 

.0843 

24 

6.8 

.268 

16 

2.25 

.0886 

17 

2.84 

.112 

For  GJ-alvanizeid  Iron  Wire. 


Number. 

Millimetre. 

Inch.  I 

Number. 

Millimetre. 

Inch. 

Number. 

Millimetre. 

| Inch. 

I 

.6 

.0236 

9 

1,4 

.0551 

17 

3- 

.Il8 

2 

•7 

.0276 

10 

i-5 

.O59I 

18 

3-4 

•134 

3 

.8 

•0315 

11 

1.6 

.063 

19 

39 

•154 

4 

•9 

•0354 

12 

1.8 

.O709 

20 

4.4 

•173 

5 

1. 

•0394 

13 

2. 

.0787 

21 

4.9 

•193 

6 

1. 1 

•0433 

14 

2.2 

.0866 

22 

5-4 

.213 

7 

1.2 

•0473 

15 

2.4 

•0945 

23 

5-9 

.232 

8 

i-3 

.0512 

16 

2.7 

.106 

For  "Wire  and.  Bars. 


Mark. 

Millimetre. 

Mark.|  Millimetre. 

P 

5 

7 

12 

I 

6 

8 

13 

2 

7 

9 

14 

3 

8 

10 

i5 

4 

9 

11 

16 

5 

10 

12 

18 

6 

11 

[ Mark. 

Millimetre. 

Mark. 

Millimetre. 

Mark. 

Millimetre. 

13 

20 

19 

39 

25 

70 

14 

22 

20 

44 

26 

76 

15 

24 

21 

49 

27 

82 

l6 

27 

22 

54 

28 

88 

17 

30 

23 

59 

29 

94 

18 

34 

24 

64 

30 

100 

Thickness  of  Gras  Bipes. 


Diameter. 

Thickness.  II 

Diameter. 

Thickness. 

I Diameter. 

Thickness. 

1.5  t0  3 

•25 

8 to  10 

•5 

14  to  15 

•75 

4 “6 

•375  II 

12  “ 13 

.625 

1 16  “ 48 

.875 

Copper  Wrire  Cord. 

Circumference  and  Safe  Load. 


Inch.  Inch.  Inch.  Inch.  Inch.  Inch.  Iris.  Ins. 

Circumference 25  .375  .5  .625  .75  1 1.125  1.25 

Safe  load  in  Lbs 34  50  75  112  168  224  336  448 

Zinc— sheets. 

Thickness  and  "Weight  per  Square  IToot. 

Inch.  I Inch.  | Inch. 

.0311  = IO  OZ.  *0534  = 14  oz.  .0686  = 18  OZ. 

.O457  = 12  OZ.  I .o6lI  = l6  OZ.  I 10761  = 20  OZ. 


124  WEIGHT  AND  STRENGTH  OF  WIRE,  IRON,  ETC. 


WEIGHT  AND  STRENGTH  OF  WIRE,  IRON,  ETC. 
"Weight  and.  Strength,  of  "Warrington  Iron  Wire. 
Manufactured  by  Rylands  Brothers.  (England.) 

Weight  per  ioo  Lineal  Feet. 


No. 

Diame- 

ter. 

Weight 

Breaking 

An- 

nealed. 

Weight. 

Bright. 

No. 

Diameter. 

Weight. 

Breaking 

An- 

nealed. 

Weight. 

Bright. 

Gauge. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Gauge. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

7/0 

X 

64.46 

3490 

5233 

9 

.146 

5*5 

298 

447 

6/0 

% 

56.66 

3066 

4603 

10 

•133 

4*43 

247 

370 

5/0 

% 

49-36 

2673 

4OOO 

10.5 

.125 

4*03 

2l8 

327 

4/0 

% 

4253 

2303 

3457 

11 

.117 

3*53 

I9I 

288 

3/0 

% 

36.26 

1963 

2945 

12 

.1 

2.66 

145 

217 

2/0  . 

X 

3O.46 

1653 

2473 

13 

.09 

2.1 

113 

169 

O 

.326 

27.36 

i486 

2226 

14 

.079 

1.6 

87 

130 

I 

•3 

2 3-3 

1257 

1885 

i5 

.069 

1.23 

66 

99 

2 

.274 

19.36 

IO46 

1572 

16 

.0625 

.96 

53 

77 

3 

•25 

16.13 

873 

1309 

17 

•053 

•73 

39 

59 

4 

.229 

13*53 

732 

1098 

18 

.047 

•56 

3* 

46 

5 

.209 

11.26 

6lO 

9i3 

19 

.041 

•43 

23 

35 

6 

.191 

9.4 

509 

763 

20 

.036 

•33 

18 

27 

7 

•I74 

7.8 

422 

633 

21 

•031  25 

.26 

14 

21 

8 

•159 

6-53 

353 

5i9 

22 

.028 

.2 

11 

16 

To  Compute  Length  of  IOO  Pounds  of  Wire  of  a (Given 
Diameter. 

Rule. — Divide  following  numbers  by  square  of  diameter,  in  parts  of  an 
inch,  and  quotient  is  length  in  feet. 

37.68  for  wrought  iron.  I 33.42  for  copper.  I 28  for  silver. 

37.45  for  steel.  | 34-41  for  brass.  | 15.3  for  gold. 

13.64  for  platinum. 


Window  Glass. 

Thickness  and  "Weight  per  Square  Foot. 


No. 

Thickness. 

Weight. 

No. 

Thickness. 

Weight: 

No. 

Thickness. 

Inch. 

Oz. 

Inch. 

Oz. 

26 

Inch. 

12 

•059 

12 

17 

.083 

17 

.125 

G 

.063 

13 

.091 

19 

•154 

15 

.071 

15 

21 

. 1 

21 

36 

.167 

16 

.077 

16 

24 

.III 

24 

42 

.2 

W eight. 


Oz. 

26 
32 
36 

42 

Terne  IPlates. 

Terne  Plates — Are  of  iron  covered  with  an  amalgam  of  lead. 
Thickness  and  Weigh,  t of  Galvanized  Slieet  Iron. 
Sheet  2 Feet  in  Width  by  from  6 to  9 Feet  in  Length  (M.  Lejferts). 


£6 

No. 

29 

28 

27 


Weight 

per 

Sq.  Foot. 

Wire 

Gauge. 

Weight 

per 

Sq.  Foot. 

Wire 

Gauge. 

Weight 

per 

Sq  Foot 

If 

Weight 

per 

Sq.  Foot. 

Wire 

Gauge. 

Weight 

per 

Sq.  Foot. 

Wire 

Gauge. 

Weight 

per 

Sq.  Foot. 

Oz. 

No. 

Oz. 

No. 

Oz. 

No. 

Oz. 

No. 

Oz. 

No. 

Oz. 

12 

26 

15 

2 3 

20 

20 

27 

17 

36 

14 

53 

13 

25 

l6 

22 

22 

19 

30 

16 

42 

13 

6l 

14 

24 

l8 

21 

24 

18 

35 

15 

46 

12 

70 

WEIGHTS  OF  METALS. 


125 


Wrought  Iron. 

"Weight  of  Square  Rolled.  Iron, 
From  .125  Inch  to  10  Inches . one  foot  in  length. 


Side. 

Weight. 

Side. 

Weight. 

Side.  , | 

Weight. 

Side. 

Weight. 

Tn. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

•053 

2.125 

I5-263 

4*I25 

57-5W 

6.25 

I32.O4 

.25 

.211 

•25 

I7.II2 

.25 

61.055 

‘5 

142.816 

•375 

•475 

•375 

19.066 

•375 

64.7 

•75 

154.012 

•845 

.5 

21.12 

•5 

68.448 

7 

165.632 

.625 

1.32 

.625 

23.292 

.625 

72  305 

•25 

177.672 

•75 

1.901 

•75 

25-56 

•75 

76.264 

>5 

190.136 

.875 

2.588 

.875 

27-939 

.875 

80.333 

•75 

203.024 

1 

3-38 

3 

30.416 

5 

84.48 

8 

216.336 

.125 

4.278 

.125 

33-oi 

.125 

88.784 

•25 

230.068 

.25 

5.28 

•75 

35-704 

•25 

93.168 

•5 

244.22 

•375 

6-39 

•375 

38.503 

•375 

97-657 

•75 

258.8 

.5 

7.604 

•5 

41.408 

•5 

IO2.24 

9 

273.792 

.625 

8.926 

•625 

44.418 

.625 

106.953 

•25 

289.22 

•75 

10352 

•75 

47-534 

•75 

111.756 

•5 

305  056 

•875 

11.883 

.875 

50.756 

.875 

116.671 

-75 

321-33 

2 

13-52 

4 

54.084 

6 

121.664 

10. 

327.92 

Illustration.— What  is  weight  of  a bar  1.5  inches,  by  12  inches  in  length? 

In  column  1st.  find  1.5;  opposite  to  it  is  7.604  lbs.,  which  is  7 lbs.  and  .604  of  a lb. 
If  lesser  denomination  of  ounces  is  required,  result  is  obtained  as  follows: 

Multiply  remainder  by  16,  point  off  the  decimals,  and  the  figures  remain- 
ing on  left  of  the  point  will  give  number  of  ounces. 

Thus,  .604  of  a lb.  = .604  X 16  = .9.664  = 7 lbs.  9.664  ounces. 

To  Compute  Weight  for  less  than  a Foot  in  Length. 
Operation.— What  is  weight  of  a bar  6.25  inches  square  and  10.5  inches  long? 

In  column  7th,  opposite  to  6.25  is  132.04,  which  is  weight  for  a foot  in  length. 

6.25  X 12  inches  = 132.04  6 ins.=.5  =66.02 

3 “ =.25  =33-oi 

1.5“  = .125  = 16.505 

*i5-535 

"WeigHt  of  -A.ngle  Iron, 

From  1.25  to  4.5  Inches,  one  foot  in  length. 

Thickness  measured  in  Middle  of  each  Side. 


L Equj 

Sides. 

il  Sides 

Thick- 

ness. 

i. 

Weight 

L Unec 

Sides. 

iUAL  Sll 

Thick- 

ness. 

DES. 

Weight. 

L Unequ 

Sides. 

al  Side 

Thick- 

:s. 

Weight. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

I.25XI.25 

•1875 

i-5 

3 X2.5 

•375 

6.25 

6 X3-5 

.625 

18 

1-5  XI.5 

•1875 

2 

3-5x3 

•4375 

7-75 

6 X4-5 

.625 

20 

1-75X1.75 

•25 

3 

3-5X3 

•4375 

9.6 

y 

2 X 2 

•25 

3-5 

4 X3 

•5 

11 

2.25X2.25 

•3125 

4-5 

4 X3-5 

•5 

II-5 

2 X 2.375* 

•375 

5-5 

2.5  X2.5 

•3125 

5 

4 X3-5 

•5 

n-75 

2.5X2.875 

•375 

6-5 

3 X 3 

•375 

7 

4-5X3 

•5 

n-75 

3-5  X 3-5 

•4375 

10.5 

3-5  X3.5 

•4375 

9 

5 X3 

•5 

12.65 

4 X3.5  1 

•4375 

I3 

4 X 4 

•5 

12.5 

5 X3 

•5625 

13-7 

•75 

4-5  X4.5 

•5 

14 

5-5  X 3-5 

•5 

14-5 

4 X3.5 

•75 

I3-5 

4-5  X4-5 

•5625 

16 

5-5  X 3-5 

•5625 

15.6 

* Tbi»  column  gives  depth  of  web  added  to  the  thickness  of  base  or  flange. 

L* 


126 


WEIGHTS  OF  METALS. 


"Weight  of  Round  Rolled.  Iron, 

From  .125  Inch  to  12  Inches  in  Diameter . one  foot  in  length. 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter.  | 

Weight. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

.041 

2 

IO.616 

4-375 

50.815 

7-5 

149.328 

•25 

.165 

.125 

II.988 

•5 

53-76 

•75 

l'59-456 

.3125 

•259 

•25 

13-44 

.625 

56.788 

8 

169.856 

•375 

•373 

•375 

14-975 

•75 

59-9 

•25 

180.696 

•4375 

.508 

•5 

16.588 

5 

66.35 

•5 

191.808 

•5 

.663 

.625 

18.293 

.125 

69.731 

•75 

203.26 

•5625 

.84 

•75 

20.076 

•25 

73-I72 

9 

215.04 

.625 

1.043 

.875 

21.944 

•375 

76.7 

•25 

227.152 

.6875 

1.254 

3 

23.888 

•5 

80.304 

•5 

239.6 

•75 

1-493 

.125 

25.926 

.625 

84.001 

•75 

252.376 

.875 

2.032 

•25 

28.04 

•75 

87.776 

10 

265.4 

1 

2.654 

♦375 

30.24 

6 

95-552 

•25 

278.924 

.125 

3-359 

•5 

32-512 

•25 

103.704 

•5 

292.688 

•25 

4- *47 

.625 

34.886 

•375 

107.86 

•75 

306.8 

•375 

5.019 

•75 

37-332 

•5 

112.16 

11 

321.216 

•5 

5-972 

.875 

39.864 

.625 

116.484 

•25 

336.OO4 

.625 

7.01 

4 

42.464 

•75 

120.96 

•5 

351-104 

•75 

8.128 

.125 

45-174 

7 

130.048 

•75 

366.536 

.875 

9-333 

•25 

47-952 

•25 

139-544 

12 

382.208 

Weight  of  Iflat  Rolled  Iron, 

From  .5X.125  Inch  to  5.5 X 4-5  Inches*  one  foot  in  length. 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness, 

Weight. 

Thickness. 

Weight. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.5 

.875 

1.25 

1.5 

.125 

.211 

•75 

2.217 

•5 

2.112 

•75 

3.802 

•25 

.422 

.875 

2.583 

.625 

2.64 

.875 

4-435 

•375 

•634 

•75 

3.168 

1 

5.069 

•5 

•845 

1 

-875 

3.696 

1.125 

5-703 

.125 

.422 

1 

4.224 

1.25 

6.337 

.625 

•25 

.845 

1. 125 

4.752 

1-375 

6-97 

.125 

•25 

.264 

.528 

-375 

•5 

1.267 

1.69 

1.375 

1.625 

•375 

•792 

.625 

2.1 12 

.125 

.58 

.125 

.686 

•5 

1.056 

•75 

2-534 

•25 

1.161 

•25 

1.372 

.625 

1.32 

.875 

2.956 

•375 

1.742 

-375 

2.059 

.75 

1.125 

•5 

.625 

2.325 

2.904 

•5 

.625 

2.746 

3-432 

.125 

•3l6 

.125 

•475 

•75 

3-484 

•75 

4.H9 

•25 

•633 

•25 

•95 

.875 

4-065 

•875 

4.805 

•375 

•95 

•375 

1.425 

1 

4.646 

1 

5-492 

•5 

1.265 

•5 

1. 901 

1. 125 

5.227 

1. 125 

6.178 

.625 

1.584 

.625 

2-375 

1.25 

5.808 

i-5 

6.864 

•75 

1.9 

•75 

2.85 

i-375 

6.389 

1-375 

7-551 

.125 

.875 

•369 

.875 

1 

3-326 

3.802 

1.25 

.125 

1.5 

•633 

i-5 

8.237 

1.75 

•25 

•738 

•25 

1.266 

.125 

•739 

•375 

1. 108 

.125 

.528 

-375 

1.9 

•25 

1.479 

•5 

i*477 

•25 

1.056 

•5 

2-535 

•375 

2.218 

.625 

1.846 

•375 

1.584 

.625 

3-!68 

•5 

2-957 

* For  weights  of  square  bars  6ee  preceding  page. 


WEIGHTS  OF  METALS. 


127 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

1.75 

1.125 

2.5 

2.875 

.625 

3.696 

1-5 

10.772 

1.25 

10.56 

.125 

1.2 1 5 

•75 

4-435 

I.625 

11.67 

1-375 

11.616 

•25 

2.429 

.875 

5.178 

1-75 

12.567 

1-5 

12.672 

•375 

3.644 

1 

5-9i4 

1.875 

13465 

I.625 

13.728 

•5 

4.858 

1. 125 

6.653 

2 

14.362 

1-75 

14.784 

.625 

6.072 

1.25 

7-393 

O 05 

1.875 

15.84 

-75 

7.287 

1-375 

8.132 

tC.tio 

2 

16.896 

•875 

8.502 

i-5 

8.871 

.125 

•95 

2.125 

17-952 

1 

9.716 

1.625 

9.61 

-25 

1.9 

2.25 

19.008 

1.125 

10.931 

1.875 

•375 

2.851 

2-375 

20.064 

1.25 

12.145 

•5 

3.802 

9 f?95 

1.375 

13.36 

.125 

.792 

.625 

4.752 

i-5 

14-574 

•25 

1.584 

•75 

5.703 

.125 

1. 109 

1.625 

15.789 

-375 

2.376 

.875 

6.653 

.25. 

2.218 

1-75 

17.003 

•5 

3-i68 

1 

7.604 

•375 

3.327 

1.875 

18.218 

.625 

3.96 

1. 125 

8.554 

•5 

4.436 

2 

19.432 

•75 

4-752 

1.25 

9-505 

.625 

5.545 

2.125 

20.647 

.875 

5-544 

i-375 

10.455 

•75 

6.654 

2.25 

21.861 

1 

6.336 

i-5 

11.406 

.875 

7.763 

2.375 

23.076 

1. 125 

7.129 

1.625 

12.356 

I 

8.872 

2.5 

24.29 

1.25 

7.921 

i-75 

13-307 

1. 125 

9.981 

2.625 

25.505 

1-375 

8-713 

1-875 

14.257 

1.25 

11.09 

2.75 

26.719 

i-5 

9-505 

2 

15.208 

1-375 

12.199 

3 

1.625 

10.297 

2.125 

16.158 

1-5 

13.308 

1*75 

11.089 

9 875 

1.625 

14.417 

.125 

1.267 

0 

i-75 

15.526 

.25 

2-535 

& 

.125 

I.003 

1.875 

16.635 

-375 

3.802 

.125 

■845 

•25 

2.006 

2 

17.744 

•5 

5.069 

.25 

I.689 

•375 

3-009 

2.125 

18.853 

•625 

6.337 

•375 

2-534 

•5 

4.013 

2.25 

19.962 

•75 

7.604 

•5 

3-379 

.625 

59i6 

2.375 

21.071 

.875 

8.871 

.625 

4.224 

-75 

6.019 

2-5 

22.18 

1 

10.138 

•75 

5.069 

.875 

7.022 

1. 125 

11.406 

•875 

5.914 

1 

8.025 

A.  itJ 

1.25 

12.673 

1 

6.758 

1. 125 

9.028 

.125 

I.l62 

1-375 

13.94 

1-125 

7.604 

1.25 

10.032 

•25 

2.323 

i.5 

15.208 

1.25 

8.448 

i-375 

11-035 

•375 

3485 

1.625 

16.475 

1-375 

9.294 

i-5 

12.038 

•5 

4.647 

i-75 

17.742 

i-5 

IO.I38 

1.625 

13.042 

.625 

5.808 

1.875 

19.01 

1.625 

IO.983 

i-75 

14.045 

•75 

6.97 

2 

20.277 

i-75 

II.828 

1.875 

15.048 

.875 

8.132 

2.25 

22.811 

1-875 

12.673 

2 

16.051 

1 

9.294 

2-5 

25.346 

2.125 

2.125 

. 17.054 

1. 125 

10.455 

2.75 

27.881 

2.25 

18.057 

1.25 

II.617 

q nc 

.125 

.898 

2.5 

1-375 

12.779 

0.4.0 

•25 

1-795 

i-5 

13-94 

.125 

1-373 

•375 

2.693 

.125 

1.056 

1.625 

15.102 

.25 

2.746 

•5 

3-59i 

•25 

2.112 

i-75 

16.264 

•375 

4-II9 

.625 

4.488 

-375 

3.168 

1.875 

17425 

•5 

5.492 

•75 

5.386 

-5 

4.224 

2 

18.587 

.625 

6.865 

•875 

6.283 

.625 

5.28 

2.125 

19.749 

•75 

8.237 

1 

7.181 

-75 

6.336 

2.25 

20.91 

.875 

9.61 

1-125 

8.079 

.875 

7-392 

2.375 

22  O72 

1 

10.983 

1.25 

8.977 

1 

8.448 

2-5 

23.234 

1.125 

12.356 

I-375 

9.874 

1. 125 

9504 

2.625 

24-395  ; 

1.25 

13-73 

128 


WEIGHTS  OF  METALS. 


Thickness.  | 

Weight.  | 

Thickness. 

Weight. 

Thickness. 

Weight.  1 

Thickness. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

3.25 

3.75 

4.5 

5 

1-375 

15.102 

1-875 

23.762 

-75 

II.406 

3-25 

54-9*6 

i-5 

16.475 

2 

25-346 

1 

15.208 

3-5 

59-14 

1.625 

17.848 

2.25 

28.514 

1.25 

19.OI 

3-75 

63-365 

i-75 

19.221 

2-5 

31.682 

1.5 

22.8l2 

4 

67.589 

i-875 

20.594 

2.75 

34-851 

i-75 

26.614 

4-25 

71.813 

2 

21.967 

3 

38.019 

2 

30.415 

4-5 

76.038 

2.25 

24.712 

3-25 

41.187 

2.25 

34-217 

4-75 

80.262 

2.5 

27.458 

3-5 

44-355 

2-5 

38.OI9 

5.25 

2-75 

30.204 

a 

2-75 

41.82 

4 436 

3 

32.95 

3 

45.623 

.25 

.125 

1.69 

3-25 

49.425 

•5 

8.871 

0. 0 

•25 

338 

3-5 

53.226 

•75 

13-307 

.125 

1.479 

•5 

6-759 

3-75 

57.028 

1 

17.742 

•25 

2.957 

•75 

IO.I38 

4 

60.83 

1.25 

22.178 

•375 

4-436 

1 

I3-5I8 

4-25 

64.632 

i-5 

26.613 

•5 

5-9i4 

1.25 

I6.897 

4 75 

i-75 

3I-°49 

.625 

7-393 

i-5 

20.277 

3t.  1 

2 

35-484 

•75 

8.871 

i-75 

23.656 

•25 

4.013 

2.25 

39*92 

.875 

10.35 

2 

27.036 

•5 

8.026 

2-5 

44-355 

1 

11.828 

2.25 

30.415 

-75 

12.036 

2.75 

48.791 

1. 125 

13-307 

2-5 

33-795 

1 

16.052 

3 

53.226 

1.25 

14-785 

2-75 

37-I74 

1.25 

20.066 

3-25 

57.662 

1-375 

16.264 

3 

40-554 

i-5 

24.079 

3-5 

62.097 

i-5 

17.742 

3-25 

43-933 

I-75 

28.092 

3-75 

66-533 

1.625 

19.221 

3-5 

47-3I3 

2 

32.105 

4 

7O.968 

i-75 

20.699 

3-75 

50.692 

2.25 

36.118 

4-25 

75-404 

i-875 

2 

22.178 

23.656 

4.25 

2-5 

2-75 

40.131 

44.144 

4-5 

4-75 

79-839 

84.275 

2.25 

26.613 

.125 

1-795 

3 

48.157 

5 

88.71 

2-5 

29-57 

•25 

3-59i 

3-25 

52.17 

2-75 

32.527 

-5 

7.181 

3-5 

56.184 

0.0 

4.647 

3 

35.485 

-75 

10.772 

3-75 

60.197 

.25 

3-25 

38.441 

1 

14.364 

4 

64.21 

•5 

9.294 

0 

1.25 

17.953 

4-25 

68.223 

•75 

13-94 

2.75 

i-5 

21.544 

4-5 

72-235 

1 

18.587 

.125 

1.584 

i-75 

25.135 

a 

1.25 

23-234 

•25 

3.168 

2 

28.725 

O 

1-5 

27.881 

•375 

4.752 

225 

32.31:6 

•25 

4.224 

i-75 

32.527 

•5 

6.336 

2-5 

35-907 

-5 

8.449 

2 

37-174 

.625 

7.921 

2-75 

39-497 

•75 

12.673 

2.25 

41.821 

•75 

9-505 

3 

43.088 

1 

16.897 

2-5 

46.468 

.875 

n.089 

3-25 

46.679 

1-25 

21.122 

2-75 

51.114 

1 

12.673 

3-5 

50.269 

i-5 

» 25.346 

3 

55-76i 

1. 125 

14.257 

3-75 

53-86 

i-75 

29-57 

3-25 

60.408 

1.25 

15.841 

4 

57-45 

2 

33-795 

3-5 

65055 

1-375 

17-425 

4 K 

2.25 

38.019 

3-75 

69.701 

i-5 

19.009 

2-5 

42.243 

4 

74-348 

1.625 

20.594 

.25 

3.802 

2-75 

46.468 

4-25 

78-995 

i-75 

22.178 

1 -5 

7.604 

1 3 

50.692 

4-5 

83.642 

Illustration.— What  is  weight  of  a bar  of  iron  5.25  ins.  in  breadth  by  .75  inch 
in  thickness? 


In  column  7,  as  above,  find  5.25;  and  below  it,  in  column,  .75;  and  opposite  to 
that  is  13.307,  which  is  13  lbs.  and  .307  of  a pound. 

For  parts  of  a pound  and  of  a foot,  operate  according  to  rule  laid  down  for  table, 
page  125. 


WEIGHT  OF  SHEET  AND  HOOP  IRON. 


129 


"Weight  of*  SHeet  Iron.  {English.  D.  K.  Clark.) 
Per  Square  Foot  (at  480  lbs. per  Cube  Foot). 


Thickness.  Weight. 


As  by  Wire-gauge  used  in  South 

Square 


Staffordshire,  England. 


Inch. 

.0125 

.OI4I 

.OI56 

.0172 

.Ol88 

.0203 

.0219 

.0234 

.025 

.0281 

•0313 


Lbs. 

•5 

.562 

.625 

.688 

•75 

.813 

.875 

•938 

1 

I*I3 

1.25 


,qi 
F< 

in  1 ton. 


No. 

4480 

3986 

3584 

3256 

2987 

2755 

2560 

2388 

224O 

1982 

1792 


Thickness. 


Inch. 

•0344 

•0375 

.0438 

•05 

•0563 

.0625 

•075 

.0875 

.1 

.1125 

.125 


Weight. 


Lbs. 

1.38 

i-5 

1- 75 

2 

2.25 

2- 5 

3 

3- 5 

4 

45 

5 


Square 
Feet 
in  1 ton. 


No. 

1623 

1493 

1280 

1120 

996 

896 

747 

640 

560 

498 

448 


Thickness.  Weight. 


Inch. 

.1406 

.1563 

.1719 

.1875 

.2031 

.2188 

•2344 

•25 

.2813 

•3I25 


Lbs. 

5-63 

6.25 
6.88 
7-5 
8.13 
8.75 
9-38 

10 

11.25 
12.5 


Square 
Feet 
in  1 ton. 


No. 

398 

358 

326 

299 

276 

256 

239 

224 

199 

179 


Width. 


Ins. 

.625 

•75 

.875 


19 

18 


■Weight  of*  Hoop  Iron. 

Per  Lineal  Foot. 

Width. 


{English.) 


Weight. 


Lbs. 

.067 

.0875 

.I2l6 

.1636 


Ins. 
1. 125 
1.25 
i-375 
I*5 


W.  G. 

Weight. 

Width. 

W.  G. 

Weight. 

No. 

Lbs. 

Ins. 

No. 

Lbs. 

17 

.21 

i*75 

14 

.484 

16 

.27 

2 

13 

•634 

15 

•33 

2.25 

13 

.714 

15 

•36 

2-5 

12 

.91 

Weight  of  33 lack  and.  Galvanized  Sheet  Iron. 

(Morton's  Table,  founded  upon  Sir  Joseph  Whitworth  Sf  Co.  s Standard  Bir- 
mingham Wire-Gauge .)  (D.  K.  Clark.') 

Note.— Numbers  on  HoltzapffePs  wire-gauge  are  applied  to  thicknesses  on  Whit- 
worth gauge. 


Gauge  and  Weight  of 
Black  Sheets. 

Approximate  number 
of  Sq.  Ft.  in  i ton. 
Black.  (Galvanized. 

Gauge  and  Weight  of 
Black  Sheets. 

Approximate  number 
of  Sq.  Ft.  in  i ton. 
Black.  | Galvanized. 

No. 

Inch. 

Lbs. 

Sq.  Ft. 

Sq.  Ft. 

No. 

Inch. 

Lbs. 

Sq.Ft. 

Sq.Ft. 

I 

•3 

12 

187 

185 

17 

.06 

2.4 

933 

876 

2 

.28 

II. 2 

200 

197 

18 

•05 

2 

1120 

IO38 

3 

.26 

IO.4 

215 

212 

19 

.04 

1.6 

1400 

I274 

4 

.24 

9.6 

233 

229 

20 

.036 

1.4 

1556 

1403 

5 

.22 

8.8 

254 

250 

21 

.032 

1.28 

1750 

1558 

6 

.2 

8 

280 

275 

22 

.028 

1. 12 

2000 

1753 

7 

.18 

7.2 

311 

304 

23 

.024 

.96 

2333 

2004 

8 

.165 

6.6 

339 

331 

24 

.022 

.88 

2545 

2159 

9 

•15 

6 

373 

363 

25 

.02 

.8 

2800 

2339 

10 

•135 

5-4 

4i5 

403 

26 

.Ol8 

‘72 

3111 

2553 

11 

.12 

4.8 

467 

452 

27 

.Ol6 

.64 

35oo 

2808 

12 

.11 

4.4 

509 

49I 

28 

.014 

•56 

4000 

3122 

13 

•095 

3-8 

589 

566 

29 

•013 

•52 

4308 

3306 

14 

.085 

3-4 

659 

63O 

30 

.012 

.48 

4667 

35 13 

15 

.07 

2.8 

800 

757 

31 

.01 

•4 

5600 

4017 

16 

.065  . 

2.6 

862 

813 

32 

.009 

•36 

6222 

4327 

130 


WEIGHT  OF  ANGLE  AND  T IRON, 


"Weigh-t  of  Englisli  Single  and  T Iron.  {D.  K.  Clark.) 

ONE  FOOT  IN  LENGTH. 

Note. — When  base  or  web  tapers  in  section,  mean  thickness  is  to  be  measured. 

, , Sum  of  Width  and  Depth  in  Inches. 

Thick- 


□ess. 

i-5 

1 .625 

i-75 

1-875 

2 

2.125 

2.25 

2.375 

2.5 

2.625 

275 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.125 

•57 

.62 

.68 

•73 

.78 

•S3 

.88 

•94 

•99 

I.04 

I.09 

•1875 

.81 

.89 

.97 

1.05 

I*  13 

1. 21 

I.29 

1-37 

i.45 

1.52 

1.6 

•25 

I.04 

I*I5 

1.25 

1-36 

I.46 

1.56 

I.67 

I.77 

1.88 

I.98 

2.08 

•3125 

I.24 

i-37 

i-5 

1.63 

I.76 

I.89 

2.02 

2.15 

2.28 

2.41 

2-54 

2.875 

3 

3.125 

3.25 

3-375 

3-5 

3.625 

375 

3-875 

4 

4-25 

.125 

1. 14 

1.2 

1.25 

1-3 

1*45 

1. 41 

I.46 

I.5I 

1.56 

1.62 

I.72 

•i«7S 

1.68 

I.76 

1.84 

1.91 

1.99 

2.07 

2.15 

2.23 

2.3 

2.38 

2-54 

•25 

2.19 

2.29 

2.4 

2-5 

2.6 

2.71 

2.8l 

2.92 

3-02 

3.13 

3-33 

•312S 

2.67 

2.8 

2-93 

3.06 

3.19 

3-32 

345 

3.58 

3.7i 

3.84 

4.1 

•375 

3-i3 

3.28 

3-44 

3-59 

3-75 

3.9i 

4.06 

4.22 

4.38 

4-53 

4.84 

•4375 

3-57 

3-75 

3*93 

4.11 

4.29 

448 

4.66 

4.84 

5.02 

5-2 

5-56 

4-5 

4-75 

5 

5-25 

5-5 

575 

6 

6.25 

6.5 

6-75 

7 

1875 

2.7 

2.85 

3-QI 

3-j6 

3-32 

3.48 

3-63 

3-79 

3-95 

4.1 

4.26 

25 

3-54 

3-75 

3-96 

4.17 

438 

4-58 

4*79 

5 

5.21 

542 

5.63 

3125 

4-36 

4.62 

4.88 

5.14 

5-4 

5.66 

5 92 

6.18 

6-45 

6.71 

6.97 

375 

5.16 

5-47 

5.78 

6.09 

6.4I 

6.72 

703 

7-34 

7.66 

7-97 

8.28 

4375 

5-92 

6.29 

6.65 

7.02 

7-38 

7-75 

8.11 

8.48 

8.84 

9.21 

9-57 

5 

6.67 

7.08 

7-5 

7.92 

8-33 

8-75 

9.17 

9.58 

10 

10.42 

10.83 

5625 

7-38 

7-85 

8.32 

8.79 

9.26 

9-73 

10.2 

10.66 

11. 13 

12.6 

12.07 

7.25 

7-5 

7-75 

8 

8.25 

8.5 

8-75 

9 

925 

9-5 

975 

25 

5.83 

6.04 

6.25 

6.46 

6.67 

6.88 

7.08 

7.29 

7-5 

7.71 

7.92 

3125 

7-23 

7-49 

7-75 

8.01 

8.27 

8-53 

8.79 

9.05 

9.3i 

957 

9-83 

375 

8.59 

8.91 

9.22 

9-53 

9.84 

10.16 

10.47 

10.78 

11.09 

1 1. 41 

11.72 

4375 

9-93 

10.3 

10.66 

11.03 

n-39 

11.76 

12.12 

12.49 

12.85 

13.22 

I3-58 

5 

11.25 

11.67 

12.08 

12.5 

12.92 

I3-33 

13-75 

14.17 

14.58 

15 

15-42 

5625 

12.54 

13.01 

13.48 

13-94 

I4.4I 

14.88 

15-35 

15.82 

16.29 

16.76 

17.23 

625 

13.8 

14.32 

14.84 

15-36 

15.89 

16.41 

16.93 

1745 

17.97 

18.49 

19.01 

1 0 

10.5 

i i 

1 1.5 

1 2 

12.5 

13 

» 3-5 

'4 

'4-5 

•5 

375 

12.03 

12.66 

13.28 

13-91 

14-53 

18.31 

4375 

13-95 

14.67 

15.4 

16.13 

l6.86 

17-59 

19.04 

19.77 

20.5 

21.22 

5 

15-83 

16.67 

17-5 

18.33 

19.17 

20 

20.84 

21.67 

22.5 

23.34 

24-17 

5625 

17.7 

18.63 

19.57 

20.51 

21-44 

22.38 

23.31 

24.25 

25.19 

26.12 

'27.06 

625 

19-53 

20.57 

21.61 

22.66 

23-7 

24.74 

25.78 

26.83 

27.87 

2S.91 

2995 

75 

23-13 

24.38 

25.63 

26.88 

28.13 

2937 

30.63 

31.88 

33.13 

34  3s 

35  63 

1 2 

'2-5 

13 

13.5 

'4 

'5 

16 

»7 

18 

19 

20 

.625 

23-7 

24.74 

25.78 

26.83 

27.87 

29-95 

32.03 

34.12 

36.2 

38.28 

40.36 

•75 

28.13 

29-37 

30.63 

31.88 

33-13 

35.63 

38.13 

40.63 

41.13 

43.63 

46-13 

.875 

32.45 

33-91 

35.36 

36.82 

38.28 

I41.19 

44.12 

47.02 

49-95 

52.87 

55-78 

1 

36.67 

38.33 

40 

41.67 

43-33 

46.67 

50 

53-33 

56.67 

60 

[63-33 

Note.— American  rolled  is  slightly  heavier. 


WEIGHT  OF  HOOP  IRON. CAST  IRON. — METALS.  I 3 I 


"Weigh. t of  Hoop  Iron.  (D.  K.  Clark.) 

ONE  FOOT  IN  LENGTH. 

As  by  Wire-gauge  used  in  South  Staffordshire  (England). 


Width  in  Inches. 


Thickness. 

.625 

•75 

.8751  ' 

1.125 

! '-25  1 

I 1 -375  | 

'•5  1 

1.625  i 

'•75 

2 

No. 

Inch. 

Lb. 

Lb. 

Lb. 

Lb. 

Lb 

Lb. 

Lbs. 

Lbs.  | 

Lbs. 

Lbs. 

Lbs. 

21 

•0344 

.0716 

.0861 

.1 

.115 

.129 

•T44 

•158 

.197 

.201 

.229 

20 

.0375 

.0781 

.0938 

.IOQ 

.125 

.141 

•i56 

.172 

.188 

.203 

.219 

•25 

19 

.0438 

.0911 

.109 

.128 

.146 

.164 

.182 

.2 

.2I91 

.238 

•257 

.292 

iS 

.05 

.IO4 

.125 

a 46 

.167 

.188 

.208 

.229 

•25 

.271 

.292 

•333 

17 

.0563 

.117 

.141 

.164 

.188 

*211 

•234 

.258 

.281 

•305 

.328 

•375 

16 

.0625 

•13 

.156 

.182 

.208 

.234 

.26 

.286 

•313 

•339 

•365 

.417 

15 

•075 

.156 

.188 

.219 

•25 

.281 

•3i3 

•344 

•375 

•307 

•438 

•5 

14 

.0875 

.183 

.219 

.256 

•293 

•329 

.366 

.402 

•438 

•475 

.512 

.585 

13 

ii 

.208 

•25 

.292 

•333 

•375 

.416 

.458 

•5 

•543 

•584 

.667 

12 

.1125 

•234 

.281 

.328 

•375 

.422 

.469 

.516 

•563 

.609 

.656 

•75 

II 

.125 

.26 

•313 

•365 

.417 

.469 

.521 

•573 

.625 

.677 

.729 

•833 

IO 

.1406 

•293 

•352 

.41 

.469 

•527 

.586 

.645 

•703 

.762 

.82 

•938 

9 

•1563 

.326 

•391 

•456 

.522 

•587 

.652 

•7I7 

•783 

.848 

•9I3 

1.04 

8 

.1719 

•358 

•43 

.501 

•573 

.644 

.716 

.788 

•859 

•93i 

1 

I*I5 

7 

•1875 

•391 

.469 

•547 

.625 

•703 

.781 

•859 

•938 

1.02 

1.09 

1.25 

6 

.2031 

•423 

.508 

•593 

.677 

.762 

.836 

•931 

1.02 

1. 1 

1. 19 

i-35 

5 

.2188 

•456 

•547 

.638 

.729 

.82 

.912 

1 

1.09 

1. 19 

1.28 

1.46 

4 

.2344 

.488 

.586 

.683 

.781 

.879 

•977 

1.07 

I.!7 

1.27 

i-37 

1.56 

CAST  IRON. 

To  Compute  "W eight  of  a Cast  Iron  Bar  or  Rod. 

Ascertain  weight  of  a wrought  iron  bar  or  rod  of  same  dimensions  in 
preceding  tables,  or  by  computation,  and  from  weight  deduct  ^,-tli  part. 

Or,  As  .1000  : .9257  ::  weight  of  a wrought  bar  or  rod  : to  weight  re- 
quired. Thus,  what  is  weight  of  a piece  of  cast  iron  4 X 3.75  X 12  inches? 

In  table,  page  128,  weight  of  a piece  of  wrought  iron  of  these  dimensions 
is  50.692  lbs.  Then,  1000  : .9257  ::  50.692  : 46.93  lbs. 

Braziers’  and.  Sheathing  Copper. 

Braziers’  Sheets,  2X4  feet  from  5 to  25  lbs.,  2.5  X 5 feet  from  9 to  150  lbs.,  and 
3X5  feet  and  4X6  feet,  from  16  to  300  lbs.  per  sheet. 

Sheathing  Copper,  14  x 48  inches,  and  from  14  to  34  oz.  per  square  foot. 

Yellow  Metal,  14  x 48  inches,  and  from  16  to  34  oz.  per  square  foot. 


"W eight  of  Corrugated  Iron  Roof  Blates. 
per  square  FOOT.  (Birmingham  Gauge.) 


No. 

Black. 

Galvanized. 

| ' No. 

Black. 

Galvanized. 

No. 

Black. 

Galvanized. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

20 

26 

29 

23 

20 

22 

25 

l6 

18 

22 

22 

24 

1 24 

18 

20 

, 26 

14 

l6 

METALS. 

To  Compute  "Weight  of  Metals  of  a ny  Dimen- 
sions or  Form. 

# By  rules  in  Mensuration  of  Solids  (page  360 ),  ascertain  volume  of  the 
piece,  multiply  it  by  weight  of  a cube  inch,  and  product  will  give  weight 
in  pounds. 


i32 


WEIGHT  OF  CAST  IRON  PIPES. 


"Weiglit  of  Cast  Iron  IPipes  or  Cylinders. 
From  i to  70  Inches  in  Internal  Diameter . 


ONE  FOOT  IN  LENGTH. 


Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

Ins. 

I 

Inch. 

•25 

Lbs. 

3.°6 

Ins. 

4-75 

Inch. 

•375 

Lbs. 

18.84 

Ins. 

II 

Inch. 

.875 

Lbs. 

IOI.85 

•375 

5-05 

•5 

25.72 

n-5 

•5 

58.81 

1.25 

.25 

3.68 

.625 

32.93 

.625 

74.28 

•3I25 

4-79 

•75 

40.43 

•75 

90.06 

•375 

5.97 

5 

•375 

19.76 

.875 

106.13 

1.5 

•375 

6.89 

•5 

26.95 

12 

•5 

61.26 

•4375 

8.31 

.625 

34-46 

.625 

77-34 

•5 

9.8 

•75 

42.27 

•75 

93-73 

i-75 

•375 

7.81 

5-5 

•375 

21.59 

*875 

110.42 

•4375 

9-38 

•5 

29.4 

12.5 

•5 

63-71 

•5 

11.03 

.625 

37-52 

.625 

80.4 

2 

•375 

8-73 

•75 

45-95 

•75 

97-4 

•4375 

io-45 

6 

•375 

2343 

.875 

114.71 

.5 

12.25 

•5 

3i,86 

13 

•5 

66.16 

2.25 

•375 

965 

.625 

40-59 

.625 

8347 

•4375 

11.52 

•75 

49.62 

•75 

101.08 

•5 

13.48 

6.5 

•375 

25.27 

•875 

1 19 

2.5 

•375 

10.57 

•5 

34-31 

I3*5 

•5 

68.61 

•4375 

12.6 

.625 

4365 

.625 

86.53 

.5 

14.7 

•75 

53-3 

•75 

104.76 

2.75 

•375 

11.49 

7 

•5 

36.76 

•875 

123.29 

•4375 

14.67 

•5625 

41.7 

14 

•5 

71.06 

.5 

15*93 

.625 

46.71 

.625 

89.6 

3 

•375 

12.4 

•75 

56-97 

•75 

108.43 

•5 

I7-I5 

7-5 

•5 

39.21 

•875 

127.58 

.625 

22.2 

•5625 

44-45 

14-5 

•5 

73-51 

•75 

27-57 

•625 

49-77 

.625 

92.66 

3-25 

•375 

.5 

I3-32 

18.38 

8 

•75 

•5 

60.65 

41.66 

•75 

•875 

112.11 

131.87 

.625 

23-74 

•5625 

47.21 

15 

•5 

75-96 

•75 

29.4 

.625 

52.84 

•625 

95-72 

3-5 

•375 

.5 

14.24 

19.6 

9 

•75 

•5 

64.32 

46.56 

•75 

•875 

115.78 

136.16 

.625 

25.27 

5625 

52.72 

15-5 

•5 

78.47 

•75 

3I-24 

.625 

58.96 

•625 

98.78 

3-75 

•375 

15.16 

•75 

7i>67 

•75 

119.46 

.5 

20.83 

9-5 

•5 

49.01 

16 

•875 

140.44 

.625 

26.8 

•5625 

55-48 

•625 

101.85 

•75 

33-°8 

•625 

62.06 

•75 

123.14 

4 

•375 

.5 

16.08 

22.05 

10 

•75 

•5 

75-35 

5i-45 

16.5 

•875 

1 

144-73 

166.63 

.625 

•75 

28.33 

34-92 

•625 

•75 

65.09 

79-°3 

•625 

•75 

104.9 

126.75 

4*25 

•375 

17 

.875 

93-2; 

•875 

149.02 

.5 

23.28 

10.5 

•5 

53-91 

1 

I7I-53 

.625 

29.86 

•625 

68.15 

17 

.625 

107.97 

•75 

36.76 

•75 

82.7 

•75 

130.48 

4-5 

•375 

.5 

17.92 

23.88 

ji 

•875 

•5 

97-56 

56-36 

•875 

1 

153-3 

176-43 

.625 

3i-4 

.625 

71.21 

17-5 

.625 

111-03 

•75 

3859 

•75 

86.38 

•75 

134-  J6 

WEIGHT  OF  CAST  IRON  PIPES. 


133 


Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

Weight. 

Diameter. 

Thickn. 

| Weight. 

Ins. 

Inch. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

17-5 

.875 

157-59 

29 

•7/> 

218.7 

40 

.875 

350.56 

I 

181.33 

.875 

256.23 

I 

4OI.86 

18 

.625 

II4.I 

I 

294.05 

1. 125 

453.46 

•75 

I37-84 

30 

•75 

226.05 

1.25 

505-4I 

•875 

161.88 

.875 

264.8 

42 

.875 

367.69 

1 

186.23 

I 

303.86 

I 

421.45 

19 

.625 

120.23 

1.125 

343-22 

1. 125 

472.52 

•75 

145-19 

31 

•75 

233.4I 

I.25 

529.87 

.875 

170.46 

.875 

273.38 

44 

.875 

384.88 

1 

196.03 

1 

313.66 

I 

44I. I 

20 

.625 

126.35 

1. 125 

354-24 

1. 125 

497.58 

•75 

152.54 

32 

•75 

240.75 

I.25 

554.42 

.875 

179.03 

.875 

281.95 

46 

•875 

402.01 

1 

205.84 

1 

323.46 

I 

460.07 

21 

.625 

132.48 

1.125 

36S-27 

1. 125 

519.64 

•75 

159.89 

33 

•75 

248. 11 

I.25 

578.88 

•875 

187.61 

•875 

290.53 

48 

•875 

419.17 

1 

215.64 

1 

333.26 

I 

480.29 

22 

.625 

138.61 

1.125 

376.29 

1. 125 

541.69 

•75 

167.24 

34 

•75 

255  46 

I.25 

603.44 

.875 

196.19 

•875 

299.ll 

50 

.875 

436.43 

1 

225.44 

1 

343-06 

I 

499.89 

23 

.625 

144-73 

1. 125 

387.33 

1. 125 

563.75 

•75 

174-59 

35 

•75 

262.81 

I.25 

627.93 

.875 

204.76 

.875 

307.68 

52 

.875 

453-49 

1 

235-24 

1 

352.87 

I 

519.5 

24 

.625 

150.86 

1.125 

398.35 

1. 125 

585-81 

•75 

181.95 

36 

•75 

270.16 

I.25 

654.42 

.875 

213-34 

.875 

316.26 

55 

•875 

479-23 

1 

245.04 

1 

362.67 

I 

548.9 

25 

.625 

156.98 

1. 125 

409.28 

1. 125 

618.91 

•75 

189.3 

1.25 

456.37 

I.25 

689.21 

•875 

221.92 

37 

•75 

277.51 

58 

I 

578.29 

1 

254-85 

.875 

324.84 

1. 125 

651.96 

26 

.625 

163  11 

1 

372.47 

I.25 

725-93 

•75 

196.65 

1.125 

420.4 

1-375 

800.22 

•875 

230-5 

1.25 

468.65 

60 

1 

597-92 

1 

264.65 

38 

•75 

284.86 

1. 125 

674.01 

27 

.625 

169.23 

.875 

333.41 

1.25 

750.45 

•75 

204 

1 

382.27 

1-375 

827.17 

•875 

239.07 

1.125 

431.41 

65 

1 

646.93 

1 

274-45 

1.25 

480.89 

1. 125 

729.18 

28 

.625 

I75-36 

39 

•75 

292.21 

1.25 

811.73 

•75 

211-35 

•875 

341.97 

1-375 

894.6 

•875 

247.65 

1 

392.08 

70 

1 

69592 

1 

284.25 

1.125 

442.44 

1.25 

872.98 

29 

.625 

181.49 

1.25 

493.14 

i-5 

1051.25 

Equivalent  Length  of  Pipe  for  a SocTcet. 

7 + — = d representing  diameter  of  pipe  and  l length  in  inches. 

Additional  weight  of  two  flanges  for  any  diameter  is  computed  equal  to  a lineal 
foot  of  the  pipe. 

Note.— These  weights  do  not  include  any  allowance  for  spigot  and  socket  ends. 
2.— For  rule  to  compute  thicknesses  of  pipes,  flanges,  etc.,  see  page  560. 

M 


134  WEIGHT  OF  FLAT  ROLLED  BAR  AND  SQUARE  STEEL. 

NVeiglit  oF  Flat  Foiled.  Far  Steel.  {D.  K.  Clark.) 
From  .5  Inch  to  8 Inches  in  Width,  one  foot  in  length. 

Width  in  Inches. 


Thick- 

ness. 

•5 

.625  | 

-75  1 

.875 1 

1 | 1.25  | 

Inch. 

Lb. 

Lbs.  j 

Lbs. 

Lbs.  j 

Lbs.  I 

Lbs. 

% 

•425 

•533 

.64 

•743 

•85 

I.06 

/16 

•531 

.665 

.8 

•929 

I.06 

i-33 

% 

.638 

•798 

.96 

1. 11 

1.28 

i-59 

% 

•744 

•93i 

1. 12 

i-3 

I.49 

1.86 

y 

.85 

1.06 

1.28 

1.49 

i-7 

2.13 

y 

1.2 

1.44 

1.67 

1. 91 

2*39 

% 

— 

i*33 

1.6 

1.86 

2.12 

2.66 

% 

— 

— 

1.76 

2.04 

2-34 

2.92 

% 

— 

— 

1.92 

2.23 

2-55 

3-I9 

% 

— 

— 

— 

2.41 

2.76 

3-45 

X 

— 

— 

— 

2.6 

2.98 

3-72 

% 

— 

— 

— 

— 

3-i9 

398 

1 

— 

— 

— 

— 

3-4 

4-25 

Width  in  Inches. 


Thick- 

ness. 

Inch. 

X 

| 


% 


% 

% 


3 1 

3-25 

3-5 

4 1 

»5 1 

5 1 

Lbs. 

Lbs. 

Lbs. 

Lbs.  I 

Lbs.  | 

Lbs.  I 

2.55 

2.76 

2.98 

3-4  j 

3.82 

4.26 

3.19 

3-45 

3-72 

4-25, 

4-78, 

5-32 

3-83' 

4 !4 

4.46 

5-i  1 

5-74! 

6.38 

4.46 

4-83 

5.21 

5-95, 

6.7 

7*44 

5 1 

5-53 

5-95 

6.8 

7.66 

8.5 

5-74 

6.22 

6.69 

7-65 

8.6 

9-56 

6.38 

6.91 

7-44 

8-5 

9-56 

10.6 

7.01 

7.6 

8.18 

9-35 

10.5 

11  *7 

7-65 

8.29 

8.93 

10.2 

n-5 

12.8 

8.29 

1 8.98 

9.67 

11. 1 

12.4 

13.8 

8.93 

i 9-67 

10.4 

11.9 

13-4 

14.9 

9.56110.4 

11.2 

12.8 

14-3 

15-9 

10.2 

In. 1 

11.9 

13.6 

15-3 

17 

Lbs.  I Lbs.  Lbs. 

4-68  5-1  i 5*52 
5.84]  6.38.  6.9 
7,02!  7.66  8.28 
8.181  8.92  9.66 
9.36;  io.2 
10.5  JII.5 
12.8 


■Weight  of  Rolled  Square  Steel. 

„ Tnah  to  6 Inches  Square,  one  foot  in  length. 


Side.  [Weight. 


Inch. 

.125 

.1875 

•25 

•3I25 

•375 

•4375 

•5 

•5625 

.625 

.6875 


Lbs. 

•053 

.119 

.212 

•333 

.478 

.651 

.85 

1.08 

i-33 

1.61 


Side.  | 

Weight. 

Side. 

Weight. 

Side. 

Weight. 

Side. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

•75 

I.92 

1-375 

6.43 

2.125 

15-4 

3-75 

.8125 

2.24 

1-4375 

7-°3 

2.25 

17.2 

4 

•875 

2.6 

1-5- 

7-65 

2-375 

19.2 

4-25 

•9375 

3°6 

1.5625 

8-3 

2-5 

21.2 

4-5 

1 

3-4 

1.625 

8.98 

2.625 

23-5 

4*75 

1.0625 

3-83 

1.6875 

9-79 

2-75 

25-7 

5 

1.125 

4-3 

i*75 

10.4 

2.875 

28.2 

5-25 

1.1875 

4-79 

1.8125 

11. 2 

3 

30.6 

5-5 

1.25 

5-3i 

1-875 

11.9 

3-25 

35-9 

5-75 

i-3I25 

5.86 

2 

13-6 

3-5 

41.6 

6 

Weight. 


Lbs. 

47.8 

54-4 

6l.4 
68-9 
76.7 
85 
93-7 
102*8 
112.4 
122  4 


WEIGHT  OF  ROLLED  STEEL,  SHEET  COPPER,  ETC.  1 3 5 


eiglit  of  Round  Rolled.  Steel. 

From  .125  Inch  to  12  Inches  Diameter . one  foot  in  length. 


Diam. 

Weight. 

Diameter. 

Weight. 

Diameter. 

] Weight. 

Diam. 

Weight. 

Diam. 

Weight. 

Inch. 

Lbs. 

In9. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

.125 

.0417 

•875 

2.04 

I.625 

7-05 

2.875 

22 

5-75 

88.3 

•1875 

•0939 

•9375 

2.35 

I.6875 

7.61 

3 

24.I 

6 

96.I 

•25 

.167 

1 

2.67 

i-75 

8.18 

3.25 

28.3 

6.5 

1 13.2 

•3125 

.26 

1.0625 

3 

1.8125 

8.77 

3-5 

32.7 

7 

130.8 

•375 

•375 

1. 125 

3-38 

1.875 

9-38 

3-75 

34-2 

7-5 

136.8 

•4375 

•5ii 

1.1875 

3-76 

2 

10.7 

4 

42.7 

8 

170.8 

•5 

.667 

1.25 

4.17 

2.125 

12 

4-25 

48.3 

8.5 

193.2 

•5625 

•845 

1-3125 

4.6 

2.25 

13.6 

4-5 

54-6 

9 

218.4 

.625 

1.04 

1-375 

5.05 

2-375 

I5-I 

4-75 

60.3 

9-5 

24I.2 

.6875 

1.27 

1-4375 

5.18 

2.5 

16.7 

5 

66.8 

10 

267.2 

•75 

i-5 

i-5 

6.01 

2.625 

18.4 

5-25 

73-6 

11 

323 

.8125 

1.76 

1.5625 

6.52 

2-75 

20.2 

5-5 

80.8 

12 

352-8 

Weiglit  of  Hexagonal,  Octagonal,  and.  Oval  Steel. 
ONE  FOOT  IN  LENGTH. 


HEXAGONAL. 

OCTAGONAL. 

OVAL. 

Diam. 

Diam. 

Diam. 

Diam. 

over 

Sides. 

Weigh  t.J 

over 

Sides. 

Weight. 

over 

Sides. 

Weight. 

over 

Sides. 

Weight. 

Diam. 
over  Sides. 

Area. 

Weight. 

Inch. 

Lb,. 

Ins. 

Lbs. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Sq.  In. 

Lbs. 

% 

.414! 

I 

2.94 

% 

•396 

I 

2.82 

%x% 

.251 

.853 

A 

•736 

*A 

3-73 

A 

.704 

^A 

3-56 

%xy2 

•344 

1. 17 

% 

I-I5 

% 

4.6 

A 

1. 1 

4.4 

1 xx 

.446 

I.52 

a 

1.66 

I % 

5-57 

% 

1.58 

5-32 

.697 

2-37 

% 

2.25 

*A 

6.63 

A 

2.l6 

6-34 

iXxM 

.884 

3 

"Weiglit  of  a Square  Foot  of  Slieet  Copper. 

Wire  Gauge  of  Wm . Foster  Sf  Co,  (England.) 


Thickness. 

| Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

W.  G. 

Inch. 

Lbs. 

W.  G. 

Inch. 

Lbs. 

W.  G. 

Inch. 

Lbs. 

I 

.306 

14 

II 

.123 

5.65 

21 

•034 

1-55 

2 

.284 

13 

12 

.109 

5 

22 

.029 

i-35 

3 

.262 

12 

13 

.098 

45 

23 

.025 

1. 15 

4 

.24 

II 

14 

.088 

4 

24 

.022 

1 

5 

.222 

10.15 

15 

.076 

3-5 

25 

.019 

.89 

6 

•203 

9-3 

l6 

.065 

3 

26 

.017 

•79 

7 

,l86 

8.5 

17 

•057 

2.6 

27 

•015 

•7 

8 

.l68 

7-7 

l8 

.049 

2.25 

28 

.013 

.62 

9 

•153 

7 

19 

.O44 

2 

29 

.012 

•56 

10 

.138 

6.3 

20 

.038 

i-75 

30 

.Oil 

•5 

'W'eigh.t  of  Composition  SHeatliing  ISTails. 


No. 

Length. 

Number 
in  a 
Pound. 

No. 

Length. 

Number 
in  a 
Pound. 

No. 

Length. 

Number 
in  a 
Pound. 

No. 

Length. 

Number 
in  a 
Pound. 

I 

Inch. 

•75 

29O 

4 

Ins. 

1-125 

201 

*7 

Ins. 
1. 125 

184 

IO 

Ins. 

I.625 

IOI 

2 

•875 

260 

5 

I.25 

I99 

8 

I.25 

168 

II 

i*75 

74 

3 

1 

212 

6 

I 

190 

9 

r*5 

no 

12 

2 

64 

Ij6  WEIGHT  OF  IRON,  STEEL,  COPPER,  ETC. 

Weight  of  Cast  and  Wrought  Iron,  Steel,  Copper,  and 
Brass,  of  a given  Sectional  Area. 

Per  Lineal  Foot. 


Sectional 

Wrought 

Cast  Iron,  j 

Steel. 

Copper.  1 

Lead. 

Brass.  ( 

5 un-metal. 

Area. 

Iron. 

Sq.  Ins. 

.1 

Lbs. 

.336 

Lbs. 

•313 

Lbs. 

•339 

Lbs. 

.385 

Lbs. 

.492 

Lbs. 

•357 

Lbs. 

.38 

.2 

.671 

.626 

.677 

.771 

.984 

.713 

•759 

.3 

1.007 

•939 

1.016 

I.156 

I.476 

I.07 

1. 139 

.4 

1-343 

1.251 

i-355 

1.542 

I.967 

I.427 

I-5I9 

• 5 

1.678 

1.564 

1.694 

I.927 

2.461 

1.783 

1.894 

.6 

2.014 

1.877 

2.032 

2.312 

2-953 

2.I4 

2.279 

.7 

2-35 

2.19 

2.371 

2.698 

3-445 

2.497 

2.658 

.8 

2.685 

2.503 

2.71 

3083 

3-937 

2.853 

3-038 

.Q 

3.021 

2.816 

3-049 

3-469 

4.429 

3.21 

3.418 

I 

3-357 

3.129 

3387 

3-854 

4.922 

3.567 

3.798 

1. 1 

3.692 

3-442 

3-726 

4-24 

5.414 

3-923 

4-177 

1.2 

4.028 

3-754 

4.065 

4.625 

5-906 

4.28 

4-557 

1.3 

4.364 

4.067 

4.404 

5-01 

6.398 

4.636  1 

4-937 

1.4 

4.699 

4-38 

4.742 

5.396 

6.89 

4-993 

5-3I7 

1.5 

5-°35 

4-^93 

5.081 

5-78I 

7-383 

5-35 

5.696 

1.6 

q.371 

5.006 

5-42 

6.167 

7.875 

5.707 

6.076 

1.7 

5.706 

5-3*9 

5-759 

6.552 

8.367 

6.063 

6.456 

1.8 

6.042 

5 ^32 

6.097 

6-937 

8.859 

6.42 

6.836 

1.9 

6.378 

5-945 

6.436 

7-323 

9-351 

6.777 

7-215 

2 

6.714 

6.258 

6-775 

7.708 

9-843 

7-133 

7-595 

2.1 

7.049 

6-57 

7.H4 

8.094 

10.33 

7-49 

7-97 

2.2 

7-385 

6.883 

7-452 

8.474 

10.83 

7-847 

8-35 

2.3 

7. 721 

7.196 

7.791 

8.864 

11.32 

8.203 

8-73 

2.4 

8.056 

7-509 

8.13 

9-25 

11. 81 

8.56 

9.11 

2-5 

8.392 

7.822 

8.469 

9.635 

12.3 

8-9i7 

9.49 

2.6 

8.728 

8.135 

8.807 

10.02 

12.8 

9-273 

9.87 

2.7 

9.063 

8.448 

9.146 

IO.4I 

13.29 

963 

10.25 

2.8 

9-399 

8.76 

9-485 

IO.79 

13-78 

9.98 

10.63 

2.9 

9-734 

9-073 

9.824 

II. l8 

14.27 

10.34 

11. 01 

3 

10.07 

9.386 

10.16 

II.56 

14.76 

10.7 

ii-39 

3.1 

10.41 

9.699 

10.5 

n-95 

15.26 

11.06 

11.77 

3.2 

10.74 

10.01 

10.84 

12.33 

15-75 

11.41 

12.15 

3.3 

11.08 

10.32 

11. 18 

12.72 

16.24 

n-77 

12.53 

3.4 

11.41 

10.64 

11.52 

13.1 

16.73 

12.13 

12.91 

3.5 

n-75 

10.95 

11.86 

13.49 

17.22 

12.48 

13.29 

3.6 

j 12.08 

11.26 

12.19 

13-87 

17.72 

12.84 

13.67 

3.7 

! 12.42 

11.58 

12-53 

14.26 

18.21 

13.2 

14.05 

3-8 

I 12.76 

11.89 

12.87 

14.64 

18.7 

13-55 

14-43 

3.9 

I 13.09 

12.2 

13.21 

15.03 

I9*I9 

I3-9I 

14*01 

4 

1 13.43 

12.51 

13-55 

15.42 

19.69 

14.27 

15- J9 

4*1 

13.76 

12.83 

13.89 

15.8 

20.18 

14.62 

15-57 

4.2 

14.1 

i3-I4 

14.23 

16.19 

20.67 

14.98 

15-95 

4-3 

4.4 

1443 

14-77 

13-45 

13-77 

14.57 

14.91 

16.57 

16.96 

21.16 

21.65 

15-34 

15.69 

16.33 

16.71 

4.5 

15. 11 

14.08 

15.24 

17.34 

22.15 

16.05 

17.09 

4.6 

4*7 

15-44 

15.78 

14-39 

14.7 

15.58 

15.92 

17.73 

18.11 

22.64 

23-13 

16.41 

16.76 

17-47 

17.85 

4.8 

16. 11 

15.02 

16.26 

•18.5 

23.62 

17.12 

18.23 

4.9 

16.45 

I5v33 

16.6 

18.88 

24.12 

17.48 

18.61 

5 

16.78 

15.64 

16.94 

19.27 

24.61 

17*83 

18.99 

IRON  BOILER  TUBES, 


137 


Lap  Welded  Charcoal  Iron.  Boiler  Tubes. 

Standard  Dimensions. 

National  Tube  Works  Company . 


Length  per 


Diam 

iter. 

In- 

i 

a 

<£ 

Circum 

Ex- 

erence. 

In- 

Tran 

Ex- 

sverse Ar 
In- 

eas. 

Sq.l 
of  Su: 
Ex- 

Foot 

rface. 

In- 

Weight 

per 

Foot. 

ternal. 

ternal. 

H 

ternal. 

ternal. 

ternal. 

ternal. 

Metal. 

ternal. 

ternal. 

Ins. 

Ins. 

No. 

Ills. 

Ins. 

Sq.  Ins. 

Sql  Ins. 

Sq.  Ins. 

Feet. 

Feet. 

Lbs. 

I 

.86 

.072 

15 

3-I4 

2.69 

.78 

•57 

.21 

3.82 

446 

.71 

1. 125 

.98 

.072 

15 

3-53 

3.08 

•99 

•76 

.24 

3-39 

3-89 

.8 

1.25 

i.n 

.072 

15 

3-93 

3-47 

1.23 

.96 

.27 

3.06 

3-45 

.89 

I.32 

I-I5 

.083 

14 

4.12 

3-6 

1-35 

1.03 

•32 

2.91 

3-33 

I.08 

1-375 

1. 21 

.083 

14 

4-32 

3-8 

1.48 

I-I5 

•34 

2.78 

3-i6 

1.5 

i-33 

.083 

14 

4.71 

4.19 

1.77 

1.4 

•37 

2-55 

2.86 

1.24 

1.625 

i-43 

.095 

13 

5-i 

4-5i 

2.07 

1.62 

.46 

2-35 

2.66 

i-53 

*•75 

1.56 

.095 

13 

5-5 

4.9 

24 

1. 91 

•49 

2.18 

2-45 

1.66 

1.875 

1.68 

•095 

13 

5-89 

5.29 

2.76 

2.23 

•53 

2.04 

2.27 

1.78 

2 

1. 81 

•095 

13 

6.28 

5-69 

3-14 

2.57 

•57 

1. 91 

2. 11 

1. 91 

2.125 

i-93 

•095 

13 

668 

6.08 

3-55 

2-94 

.61 

1.8 

I-97 

2.04 

2.25 

2.06 

.095 

13 

7.07 

647 

3-98 

3-33 

.64 

*•7 

1.85 

2.16 

2-375 

2.16 

.109 

12 

7.46 

6.78 

4-43 

3-65 

•78 

1.61 

1.77 

2.61 

2-5 

2.28 

.IO9 

12 

7-85 

7-I7 

4 91 

4.09 

.82 

1.53 

1.67 

2-75 

2-75 

2-53 

.109 

12 

8.64 

7-95 

5 94 

| 5-03 

•9 

i-39 

i-5i 

3-04 

2.875 

2.66 

.109 

12 

9 °3 

8.35 

6.49 

5-54 

•95 

i-33 

1.44 

3-i8 

3 

2.78 

.109 

12 

9.42 

8.74 

7.07 

! 6.08 

•99 

1.27 

I-37 

3-33 

3-25 

3.01 

.12 

II 

10.21 

9.46 

8.3 

7* 12 

1. 18 

1. 17 

1.26 

3-96 

3-5 

3.26 

.12 

II 

11 

10.24 

9.62 

8-3S 

1.27 

1.09 

1. 17 

4.28 

3-75 

3-51 

.12 

II 

11.78 

11.03 

11.04 

9-68 

1-37 

1.02 

1.09 

4.6 

4 

3-73 

•134 

IO 

1 12.57 

11.72 

1257 

| 10-94 

1.63 

•95 

1.02 

5-47 

4-25 

3 98 

•134 

IO 

13-35 

12.51 

14.19 

: I2-45 

i-73 

•9 

.96 

5.82 

4-5 

4.23 

-I.34 

lo 

14.14 

13.29 

15.9 

1 I4°Z 

1.84 

.85 

•9 

6.17 

4-75 

448 

-134 

IO 

14.92 

14.08 

17.72 

15.78 

I-94 

.8 

.85 

6.53 

5 

4-7 

.148 

9 

i5-7i 

14.78 

19.63 

1 17-38 

2.26 

•76 

.81 

7-58 

5-25 

4 95 

.148 

9 

16.49 

I5-56 

21.65 

19.27 

2.37 

•73 

•77 

7-97 

5-5 

5-2 

.148 

9 

17  28 

16.35 

23.76 

21.27 

2.49 

•7 

•73 

8.36 

6 

5-67 

.165 

8 

18.85 

17.81 

28.27 

25-25 

3.02 

.64 

•67 

10.16 

7 

6.67 

.165 

8 

21.99 

20.95 

38.48 

34  94 

3-54 

•55 

•57 

11.9 

8 

767 

.165 

8 

25.13 

24.1 

50.27 

46.2 

4.06 

.48 

•50 

13-65 

9 

8.64 

.18 

7 

28.27 

27.14 

63.62 

58.63 

4.99 

.42 

•44 

16.76 

10 

9-59 

.203 

6 

31.42 

30.14 

78  54 

72.29 

6.25 

.38 

•4 

20.99 

11 

10.56 

.22 

5 

34-56 

33-17 

95-03 

87.58 

7-45 

•35 

•36 

25-03 

12 

n-54 

.229 

4-5 

37-7 

36.26 

H3-1 

104.63 

8.47 

•32 

•33 

28.46 

13 

12.52 

.238 

4 

40.84 

39-34 

: 132.73 

123.19 

9-54 

.29 

•3 

32.06 

14 

13  5 

1 .248 

3-5 

! 43.98 

42.42 

15394 

143.22 

IO.7I 

.27 

.28 

36 

15 

1448 

.259 

3 

47.12 

45  5 

176.71 

164.72 

11.99 

•25 

.26 

40.3 

16 

15-43 

.284 

2 

50.26 

48.48 

201.06 

187.04 

14.02 

.24 

•25 

47.11 

17 

16.4 

•3 

1 

5341 

5i-52 

226.98 

211.24 

15-74 

.22 

•23 

52.89 

18 

! 17-32 

•34 

0 

5655 

54-41 

1 254  47 

235.61 

1 18.86 

.21 

.22 

1 63.32 

Note. — In  estimating  effective  heating  or  evaporating  surface  of  Tabes, 
as  heating  liquids  by  steam,  superheating  steam,  or  transferring  heat 
from  one  liquid  or  one  gas  to  another,  mean  surface  of  Tubes  is  to  be 
computed, 

M* 


138 


STEAM,  GAS,  AND  WATER  PIPE. 


Iron  Welded.  Steam,  Gras,  and  Water  Pipe. 

Standard  Dimensions. 

National  Tube  Works  Company. 


Diameter. 


In- 
ternal. i 

Ex-  i 
ternal.  Ii 

Ins. 

Ins. 

.125 

•4 

•25 

•54 

•375 

.67 

•5 

.84 

•75 

1.05 

1 

1*31 

1.25 

1.66 

i-5 

1.9 

2 

2-37 

2*5 

2.87 

3 

3-5 

3-5 

4 

4 

4-5 

4-5 

5 

5 

5-56 

6 

6 62 

7 

7.62 

8 

8.62 

9 

9.62 

10 

10.75 

11 

n-75 

12 

12.75 

13 

14 

14 

i5 

i5 

16 

16 

17 

17 

18 

STEEL  LOCOMOTIVE  TUBES. 

Lap  Welded  Semi-Steel  Locomotive  Tubes 

Standard  Dimensions. 

National  Tube  Works  Company. 


Diameter. 

§ 

Circumference. 

Transverse  Areas. 

Ex- 

ternal. 

In- 

ternal. 

H 

® & 
3 

Ex- 

ternal. 

In- 

ternal. 

Ex- 

ternal. 

In-  I 
ternal. 

Metal. 

Ins. 

I 

Ins. 

•834 

Ins. 

.083 

No. 

14 

Ins. 

3*I42 

Ins. 

2.62 

Sq.  Ins. 

•785 

Sq.  Ins. 

.546 

Sq. Ins 

-239 

1.25 

i-5 

1.75 

I.084 

1-3! 

1.532 

.083 

.095 

.109 

14 

13 

12 

3- 927 

4- 712 

5- 498 

3- 405 
4*II5 

4- 813 

I.227 

I.767 

2.405 

•923 

I.348 

I.843 

•3°4 

.419 

.562 

A 4 Q 

2 

1.782 

.IO9 

12 

6.283 

5-598 

3-I42 

2-494 

•O40 

2.25 

2.032 

.IO9 

12 

7.069 

; 6.384 

3-976 

3-243 

•733 

Q~.Q 

2.5 

2.26 

.12 

II 

7-854 

7-1 

4.909 

4.01 1 

.090 

2-75 

3 

2.51 

2.76 

.12 

.12 

II 

II 

8.639 

9-425 

7.885 

8.67 

5-94 

7.069 

4.948 
1 5-983 

•992 

1.086 

Lengtl 
Sq.  I 
of  Sui 
Ex- 
ternal. 

h per 
root 
rface. 
In- 
ternal. 

Weight 
per  Foot. 

Feet. 

Feet.  | 

Lbs. 

3.82 

4-58  | 

.81 

3-056 

3-524 

I.03 

2.546 

2.916 

I.42 

2.183 

2-493 

I.9I 

1. 91 

2.144 

2.2 

I.698 

1.88 

2.49 

1.528 

1.69 

3-05 

I.389 

1.522 

3-37 

1 1-273 

i 1-384 

3.68 

WEIGHT  OF  LEAD  AND  TIN  PIPE  AND  TIN  PLATES.  1 39 

'W'eiglit  of  Lead,  and  Tin  Lined  3?ipe  per  Foot. 
From  .375  Inch  to  5 Inches  in  Diameter . ( Tatham  Bros.) 


WASTE-PIPE.  1 BLOCK-TIN  PIPE. 


Diam. 

Weight. 

Diam. 

| Weight.  | 

| Diam. 

Weight. 

Diam. 

Weight.  [ 

Diam. 

Weight. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Inch. 

Lb. 

Inch. 

Lbs. 

Ins. 

Lbs. 

i*5 

2 

4 

8 

•375 

•3594 

.625 

•5 

I.25 

I.25 

2 

3 

4-5 

6 

•375 

•375 

.625 

.625 

I.25 

i-5 

3 

3-5 

4-5 

8 

•375 

•5 

•75 

.625 

1-5 

2 

3 

5 

5 

8 

•5 

•375 

•75 

•75 

i-5 

2.5 

4 

5 

5 

10 

•5 

•5 

1 

•9375 

2 

2.5 

4 

6 

5 

12 

•5 

.625 

1 

1.125 

2 

3 

WATER-PIPE. 

From  .375  Inch  to  5 Inches  in  Diameter. 


Diam. 

Thick- 

ness. 

Weight. 

Diam. 

Thick- 

ness. 

Weight. 

Diam. 

Thick- 

ness. 

■Weight. 

Diam. 

Thick- 

ness. 

Weight. 

Inch. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

•375 

.08 

.625 

.625 

•25 

3-5 

I.25 

.19 

4-75 

2-5 

•3*25 

*4 

•375 

.12 

I 

•75 

.1 

1.25 

I.25 

•25 

6 

2-5 

•375 

17 

•375 

.l6 

1.25 

•75 

.12 

*•75 

*•5 

.12 

3 

3 

•1875 

9 

•375 

.19 

1-5 

•75 

.l6 

2.25 

*•5 

.14 

3-5 

3 

•25 

12 

•375 

•34 

2.5 

•75 

.2 

3 

i-5 

*7 

4-25 

3 

•3125 

16 

•5 

.07 

•0545 

•75 

•23 

3-5 

*•5 

.19 

5 

3 

•375 

20 

•5 

.09 

•75 

•75 

•3 

4-75 

*•5 

•23 

6-5 

3?5 

•*875 

9-5 

•5 

.11 

1 

1 

.1 

*•5 

*•5 

•27 

8 

3-5 

•25 

*5 

•5 

•13 

1.25 

1 

.11 

2 

*•75 

•13 

4 

3-5 

•3*25 

18.5 

•5 

.16 

i-75 

1 

.14 

2.5 

*•75 

•*7 

5 

3-5 

•375 

22 

•5 

.19 

2 

1 

•17 

3-25 

*•75 

.21 

6-5 

4 

•1875 

12.5 

•5 

•25 

3 

1 

.21 

4 

*•75 

•27 

8-5 

4 

•25 

16 

.625 

.08 

.0727 

1 

.24 

4*75 

2 

•15 

4-75 

4 

•3*25 

21 

.625 

.09 

1 

1 

•3 

6 

2 

.18 

6 

4 

•375 

25 

.625 

•13 

i-5 

1.25 

.1 

2 

2 

.22 

7 

4-5 

•1875 

14 

.625 

.16 

2 

1.25 

.12 

2-5 

2 

.27 

9 

4-5 

•25 

18 

.625 

.2 

2.5 

1.25 

.14 

3 

2.5 

•1875 

8 

5 

•25 

20 

.625 

.22 

2-75 

1.25 

.16 

3-75 

2-5 

•25 

11 

5 

•375 

3* 

Marks  and  Weight  of*  Tin-plates.  (English.) 


Mark 
or  Brand. 

Plates 
per  Box. 

Dimensions. 

Weight 
per  Box. 

Mark 
or  Brand. 

Plates 
per  Box. 

Dimensions. 

Weight 
per  Box. 

No. 

Ins. 

No. 

No. 

Ins. 

No. 

1 Cor  1 Com. 

225 

13.75X10 

112 

DXXXX 

100 

16.75X12.5 

189 

2 C 

225 

13.25X  9-75 

105 

SDC 

T C V T T 

168 

3 c 

225 

12.75X  9 5 

08 

SDX  . . 

200 

A0  A U 

188 

H C 

225 

13. 75X 10 

I IQ 

SDXX  . 

200 

15  X 1 1 

T C V T T 

209 

230 

H X 

225 

13  75X10 

■ y 
157 

SDXXX 

200 

AH 

15  Xu 

1 X 

225 

13.75X10 

140 

SDXXXX. . . . 

200 

15  XII 

251 

2 X 

225 

I3-25X  9-75 

133 

SDXXXXX. . 

200 

15  Xu 

272 

3 X 

225 

12.75X  9 5 

126 

SDXXXXXX. 

200 

15  X 11 

293 

1 XX 

225 

I3-75Xio 

161 

Leaded  IC. . . 

1 12 

20  X 14 

112 

1 XXX. ..... 

225 

13.75X10 

•182 

“ IX... 

1 12 

20  X14 

140 

I xxxx. ... 

225 

13.75X10 

203 

ICW 

225 

13.75X10 

112 

I xxxxx . . 

225 

13.75X10 

224 

IX  w 

225 

13.75X10 

140 

i xxxxxx. 

225 

13.75X10 

245 

CSDW 

200 

15  Xu 

168 

DC 

100 

l6.  7S  X 12.  K 

08 

CTTW 

DX 

100 

16.75X  12.5 

126 

XIIW 

IOO 

x6-75X  12-5 

105 

126 

DXX 

100 

16.  75X  12.5 

14.7 

TT. .. 

16. 75  X 12.5 

DXXX 

100 

16.75X12.5 

XT  / 

l68 

XTT 

45° 

450 

I3-75X 10 
i3-75Xio 

126 

When  the  plates  are  14  by  20  inches,  there  are  112  in  a box. 


140 


WEIGHT  OF  COPPER  TUBES. 


Weigh-t  of  Seamless  Drawn  Copper  Tubes 
American  T' ube  Works.  (Boston.) 

BY  EXTERNAL  DIAMETER.  ONE  FOOT  IN  LENGTH. 

Stubs’’  W.  G.  From  .25  Inch  to  12  Ins.— f full,  l light. 

13  I 


No.  | 

20 

19 

18 

17 

16  | 

15 

14 

Ins. 

V32  / 

3/64  / 

3/64/ 

V16  l 

l/i6f 

5/64  l 

5/64/ 

Diamet’r. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•25 

.09 

.1 

.12 

•13 

.14 

•15 

•17 

•375 

.14 

.16 

.19 

•23 

.24 

.26 

.29 

•5 

.2 

•23 

•27 

•31 

•34 

•37 

.42 

.625 

•25 

.29 

•34 

•4 

•44 

.48 

•55 

•75 

•3 

•36 

.42 

•49 

•54 

•59 

.67 

.875 

•36 

.42 

49 

.58 

.64 

•7 

.8 

1 

•41 

.48 

•57 

.67 

•74 

.81 

•93 

1-125 

.46 

•55 

.64 

.76 

•83 

.92 

1.05 

1.25 

•52 

.61 

•7i 

.84 

•93 

1.03 

1. 18 

1.375 

•57 

.68 

•79 

•93 

I-°3 

1. 14 

I*3I 

1.5 

.62 

•74 

.86 

1.02 

1 •I3 

1.25 

i-43 

1.625 

.68 

.8 

•94 

1. 11 

1.23 

1.36 

1.56 

i-75 

•73 

.87 

1. 01 

1.2 

i-33 

1.47 

1.69 

1-875 

.78 

•93 

1.09 

1.29 

i-43 

1.58 

1.81 

2 

.84 

1 

1. 16 

i-37 

i-53 

1.69 

1.94 

2.125 

.89 

1.06 

1.24 

1.46 

1.63 

1.8 

2.07 

2.25 

•94 

I*I3 

I*3I 

i-55 

i-73 

1. 91 

2.19 

2375 

1 

1. 19 

i-39 

1.64 

1.82 

2.02 

2.32 

2-5 

1.05 

1.25 

1.46 

i*73 

1.92 

2.13 

2-45 

2.625 

1. 1 

1.32 

i-54 

1.82 

2.02 

2.23 

2-57 

2-75 

1. 16 

1.38 

1.61 

1.9 

2.12 

2-34 

2-7 

2.875 

1. 21 

i-45 

1.68 

1.99 

2.22 

2-45 

2.83 

'X 

1.26 

I-51 

1.76 

2.08 

2.32 

2.56 

2-95 

3-25 

i-37 

1.64 

1. 91 

2.26 

2.52 

2.78 

3-21 

3 5 

1.48 

1.77 

2.06 

2.43 

2.72 

3 

346 

3-75 

1.58 

1.9 

2.21 

2.61 

2.92 

3.22 

3-7i 

4 

1.69 

2.02 

2.36 

2-79 

3-ii 

3-44 

3-97 

4.25 

1.8 

2.15 

2.51 

3-14 

3-31 

3.66 

4.22 

4-5 

1.9 

2.28 

2.65 

3-32 

3-5i 

3.88 

4-47 

4.75 

2.01 

2.41 

2.8 

3-49 

3-7i 

4.1 

4-73 

5 

2.12 

2-54 

2-95 

3-67 

3-9i 

4-32 

4.98 

5.25 

2.23 

2.66 

3-i 

3.85 

4.11 

4-54 

5-23 

5-5 

2-34 

2.79 

3-25 

3.85 

4-3 

4.76 

5-49 

5-75 

2.44 

2.92 

3-4 

4,02 

4-5 

4.98 

5-74 

6 

2-55 

3-°5 

3-55 

4.2 

4-7 

5-2 

5-99 

6.25 

2.66 

3.18 

3-7 

4.38 

4.9 

5-4i 

6.25 

6-5 

2.76 

3-31 

3-85 

4-55 

5-i 

5-63 

6-5 

6-75 

2.87 

3-44 

4 

4-73 

5-3 

5-85 

6-75 

7 

2.98 

3-56 

4-i5 

4.9i 

5-49 

6.07 

7.01 

7.25 

3.09 

3-69 

4*3 

5.09 

5-69 

6.29 

7.26 

7.5 

3.19 

3.82 

4*45 

5.26 

5-89 

6.51 

7-5i 

8 

3 41 

4,08 

4-74 

5.62 

6.29 

6-95 

8.02 

8-5 

3.62 

4-33 

5-04 

5-97 

6.68 

7-39 

8.52 

g 

3-83 

4-59 

5-34 

6-33 

7.08 

7-83 

9°3 

9-5 

4-05 

4-85 

5-^4 

6.68 

7.48 

8.26 

9-54 

10 

4.26 

5-11 

5-94 

7-03 

7.87 

8.7 

10.05 

10.5 

4-47 

5-37 

6 24 

7-39 

8.27 

9-I4 

10.55 

11 

4.69 

5.62 

6-54 

1 7*74 

8.67 

9-58 

11.06 

11.5 

4.9 

5.88 

6.84 

! 8.1 

9.06 

10.02 

11-56 

12 

5-11 

6.13 

7-I3 

18.45 

9.46 

10.45 

1 12.07 

12 


3/32  / j 7/64 


Lbs. 

.18 

•32 

•47 

.61 

.76 

•9 

1.05 
1.19 
i-34 
1.48 

1.63 
i-77  i 

1.92  | 

2.06  j 

2.21 
2.35 
2*5 

2.64 
2.79 

2.93 
3.08 

3.22 

3-37 

3.66 

3-95 

4.24 

4*53 

4.82 

5-11 

5-4 

5-69 

5-98 

6.27 

6.56 

6.85 

7.14 

7 43 
7.72 
8.01 
8.30 

8- 59 
9.17 

9- 75 

10.33 

10.91 

11.49 

12.07 

12.65 

1323 

13.81 


Lbs. 

.19 

•35 

•52 

.69 

•85 

1.02 

1. 18 

I*35 

1.52 

1.68 

1.85 

2.02 

2.18 
2.35 
2.51 

2.68 

2.85 
3.01 

3-lS 
3-35 
3-5i 

3.68 
3-84 

4.18 

4i5I 

4.84 
5-i7 
5-5i 
5-84 

6.17 

6.5 

6.84 

7.17 
7-5 
7-83 

8.17 

8.5 
8.83 
9.16 
9-5 
9-83 

10.49 
11.16 

11.82 

12.49 

1315 

13.82 
14.48 
I5-I5 
15.81 


11 


1/8  1 


Lbs. 

.19 

•37 
•56 
•74 
.92 
1. 11 

1.29 
1.47 
1.65 

1.84 

2.02 

2.2 
2.39 

2.57 
2.75 
2.93 
3-12 
3-3 
3-48 

3- 67 

3.85 

4- 03 
4.22 

4.58 

4- 95 

5- 3i 
5.68 
6.05 
6.41 
6.78 
7-i4 
7-5i 
7.87 
8.24 
8.61 
8.97 
9 34 
9-7 

10.07 
10.44 

10.8 
n-53 
12.26 

13 

13-73 

14.46 

1519 

I5-92 

16.66 

17.29 


WEIGHT  OF  COPPER  TUBES. 


No. 

10 

9 

8 

7 1 

6 

5 

4 

3 

2 

Ins. 

9/64  1 

9/64  / 

n/64  l 

3/16  1 

13/64 

V32  / 

i5/64  / 

r/4  / 

9/32/ 

Diamet’r. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•375 

•4 

.41 

.42 

-44 

— 

— 

— 

— 

— 

•5 

.61 

.64 

.67 

•71 

•73 

•75 

.76 

— 

— 

.625 

.81 

.86 

.92 

•99 

1.04 

1.09 

1. 12 

I*I3 

1. 18 

•75 

1. 01 

1.09 

I-I7 

1.26 

i-35 

1.42 

I.49 

i-53 

I.6l 

.875 

1.22 

I-3I 

1.42 

i-53 

1.66 

1.76 

1.85 

1.92 

2.04 

1 

1.42 

i-54 

1.67 

1.81 

1.97 

2.09 

2.21 

2.32 

2.48 

1. 125 

1.63 

1.78 

i-93 

2.08 

2.28 

2.43 

2.58 

2.71 

2.9I 

1.25 

1.83 

2 

2.18 

2.36 

2-59 

2.76 

2.94 

3-11 

3-34 

x-375 

2.03 

2.22 

2-43 

2.63 

2.9 

3-i 

3-3 

3-5 

3-77 

i-5 

2.24 

2.44 

2.68 

2.91 

3.21 

3-43 

3-67 

3-9 

4.21 

1.625 

2.44 

2.67 

2.93 

3-i8 

3-52 

3-77 

4-03 

4.29 

4.64 

I-75 

2.65 

2.89 

3-i8 

3-45 

3.83 

4.11 

4-39 

4.69 

5-07 

i-875 

2.85 

3.12 

3-44 

3-73 

4.14 

4.44 

4.76 

5.08 

5-5i 

2 

3.06 

3-34 

3-69 

4 

4-45 

4.78 

5.12 

5-48 

5-94 

2.125 

3.26 

3-57 

3-94 

4.28 

4-75 

5-n 

5.48 

5.87 

6-37 

2.25 

3-46 

3-8 

4.19 

4-55 

5.06 

5-45 

5-84 

6.27 

6.81 

2-375 

3-67 

4.02 

4.44 

4.82 

5-37 

5.78 

6.21 

6.66 

7.24 

2-5 

3.87 

4-25 

4.69 

5-i 

5.68 

6.12 

6.57 

7.06 

7.67 

2.625 

4.08 

4-47 

4-95 

5-37 

6 

6-45 

6.93 

7-45 

8.1 

2-75 

4.28 

4-7 

5-2 

5-65 

6-3 

6-79 

7.29 

7-85 

8-54 

2.87s 

4.48 

4.92 

5-45 

5.92 

6 61 

7.12 

7.66 

8.24 

8.97 

3 

4.69 

5-i5 

5-7 

6.2 

6.92 

7.46 

8.02 

8.64 

9.4 

3-25 

5-i 

5-6 

6.2 

6-74 

7-54 

8.13 

8.75 

9-43 

10.27 

3-5 

5-51 

6.05 

6.71 

7.29 

8.16 

8.8 

9-47 

10.22 

11. 14 

3-75 

5-9i 

6-5 

7.21 

7.84 

8.78 

9-47 

10.2 

II.OI 

12 

4 

6.32 

6-95 

7.71 

8-39 

94 

10.14 

10.92 

n.8 

12.87 

4.25 

6-73 

7-4 

8.22 

8.94 

10.02 

10.81 

11.65 

12.59 

13.73 

45 

7.14 

7-85 

8.72 

9.49 

10  64 

11.48 

12.37 

13-38 

14.6 

4-75 

7-55 

8-3 

9.22 

10.04 

11.26 

12.16 

131 

14.17 

15.46 

5 

7.96 

8-75 

9-73 

10.58 

11.88 

12.83 

13.83 

14.96 

1633 

5.25 

8.36 

9.21 

10.23 

11. 13 

12.49 

w 

14.55 

15-75 

17.2 

5-5 

8-77 

9.66 

10.73 

11.68 

13.11 

14.17 

15.28 

16.54 

18.06 

5-75 

9.18 

10. 1 1 

11.24 

12.23 

13-73 

14.84 

16 

17-33 

18.93 

6 

9-59 

10.56 

11.74 

12.78 

14-35 

15-51 

16.73 

18.12 

19.79 

6.25 

10 

II.OI 

12.24 

I3-33 

14.97 

16.18 

17.46 

18.91 

20.66 

6-5 

10.41 

11.46 

12.75 

-3-88 

i5  59 

16.85 

18.18 

19.7 

21-53 

6-75 

10.82 

11.91 

13-25 

14.42 

16.21 

17-52 

18.91 

20.49 

22.39 

7 

11.22 

12.36 

13-75 

14-97 

16.83 

18.19 

19.63 

21.28 

23.26 

7-25 

11.63 

12.81 

14.26 

15-52 

17-45 

18.86 

20.36 

22.07 

24.13 

7-5 

12.04 

13.26 

14.76 

16.07 

18.07 

19-54 

21.08 

22.86 

25 

7-75 

12.45 

I3-7I 

15.26 

16.62 

18.68 

20.21 

21.81 

23-65 

25  86 

8 

12.86 

14.17 

15-77 

17.17 

19-3 

20.88 

22.54 

24.44 

26.72 

8.25 

13-27 

14.62 

16-27 

17.71 

19  92 

21.55 

23.26 

25-23 

27-59 

8.5 

13-67 

15-07 

16.77 

18.26 

20.54 

22.22 

23-99 

26.02 

28.45 

8.75 

14.08 

I5-52 

17.28 

18.81 

21.16 

22.89 

24.71 

26.81 

29.32 

9 

14.49 

15-97 

17.78 

19.36 

21.78 

23-56 

25-44 

27.6 

30.18 

9-25 

14.9 

16.42 

18.28 

I9-9I 

22.4 

24.23 

26.17 

28.39 

31-05 

9-5 

i5-3i 

16.87 

18.79 

20.46 

23.02 

24.9 

26.89 

29.18 

31.92 

9-75 

15-72 

17.32 

19.29' 

21.01 

23.64 

25.57 

27.62 

29.97 

32.78 

10 

16.12 

17-77 

19-79 

21-55 

24.26 

26.24 

28.34 

30.76 

33.65 

10.5 

16.94 

18.68 

20.8 

22.65 

25-5 

27.59 

29.79 

32.34 

35-38 

11 

17.76 

19.58 

21.81 

23-75 

26.73 

28.93 

31-25 

33-92 

37-n 

Ix*5 

18.57 

20.48 

22.81 

24.84 

27.97 

30.27 

32.7 

35-5 

38.84 

12 

*9-39 

21.38 

23.82 

25-94 

29.21 

3I.6l 

34- 1 5 

37.08 

40.58 

141 


19/64/ 

Lbs. 


I.63 

2.09 

2.55 

3 

346 

3- 92 

4- 38 

4- 83 

5- 29 
5-75 
6.21 
6.66 
7.12 

7-57 

8.04 

8.49 

8.95 

9.41 

9.87 

10.78 

n-7 

12.61 

13-53 

14.44 

I5-3^ 

l6.27 

17.19 

I8.I 

1902 

*9-93 

20.85 

21.76 

22.68 

23-59 

24.51 

25.42 

26.34 

27.25 

28.17 

29.08 

30 

30.91 

31-83 

32.74 

33- 66 

34- 57 

35- 49 
37-32 
39-15 
40.98 

I 42.81 


142  WEIGHT  OF  COPPER  AND  BRASS  TUBES,  ETC. 


By  Internal  Diameter. 

Add  following  Units  to  Weights  for  External  Diameter  in  preceding  tables. 


No.  | 

1 

2 

3 ! 

4 | 

5 

6 1 

| 7 l 

8 I 

9 

10 

2.21 

i-97 

1.66 

1-38 

1. 18 

I. Ol 

.78 

.67 

•53 

•43 

No. 

! 11 

12 

13 

14 

15 

16 

17 

00 

rH 

19 

20 

r- 

i -35 

.29 

.22 

•I? 

•13 

.11 

.08 

.06  | 

•05 

•03 

Illustration. — What  is  weight  of  a copper  tube  6 ins.  in  internal  diameter, 
No.  3 gauge,  and  one  foot  in  length? 

By  preceding  table  6 ins.  external,  No.  3 gauge  = 18.12,  and  18.12  1 .66  = 

19.78  lbs.  

WEIGHT  OF  BRASS  TUBES. 

To  Compute  ”VVr eiglit  of  Brass  Tubes. 
American  Tube  Works.  (Boston.) 

Rule. — Deduct  5 per  cent,  from  weight  of  Copper  tubes. 

Example.  — What  is  weight  of  a brags  tube  6 ins.  in  external  diameter,  No.  3 
gauge,  and  one  foot  in  length? 

By  preceding  table  6 ins.  = 18.12,  from  which  deduct  5 per  cent.  = 17.21  lbs. 

By  Internal  Diameter. 

Rule. — Proceed  as  above  for  internal  diameter  of  copper  tube,  and  deduct 
5 per  cent. 

Example.— Weight  of  a copper  tube  6 ins.  internal  diameter,  No.  3 gauge,  and 
1 foot  in  length  = 19.78  lbs. 

Hence,  19.78  — 5 per  cent.  = 18.79  lbs. 

Note.— Diameter  of  Tubes,  as  for  Boilers,  is  given  externally,  and  that  for  Pipes 
internally. 

Weights  of  English  as  given  by  D.  K.  Clark  are  essentially  alike  to  the 
'preceding. 

Brass  Tubes  Corresponding  with,  and  ITitted  for 
Iron  Tubes  or  Pipes. 

American  'F'u.'be  Works.  (Boston.) 


WEIGHT  PER  LINEAL  FOOT. 


Diameter  of  Iron  Pipe. 

Diameter  of  Iron  Pipe. 

Diameter  of  Iron  Pipe. 

Weight. 

Internal. 

External. 

Weight. 

Internal. 

External. 

W eight. 

Internal. 

External. 

Inch. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

.125 

•375 

•25 

I 

1-3125 

1-7 

3 

3-5 

8-3 

•25 

•5625 

•43 

1.25 

I.625 

2-5 

3-5 

4 

IO.9 

•375 

.6875 

•63 

i-5 

1-875 

3 

4 

4-5 

12.7 

•5 

.8125 

•9 

2 

2-375 

4 

5 

5-5 

15-7 

•75 

1.0625 

1.25 

2.5 

2.875 

4.87 

"W eiglit  of  SHeet  Brass. 
one  square  foot.  {Iloltzapjj eV s Gauge.) 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

I Thickness. 

Weight. 

No. 

Inch. 

Lbs. 

No. 

Inch. 

Lbs. 

No. 

Inch. 

Lbs. 

No. 

Inch. 

Lbs. 

3 

•259 

IO.9 

9 

.148 

6.23 

15 

.072 

3-03 

21 

.032 

i-35 

4 

.238 

IO 

10 

•134 

5^4 

16 

.065 

2.74 

22 

.028 

1. 18 

5 

.22 

O.26 

11 

.12 

5-05 

17 

.058 

2.44 

23 

.025 

1.05 

6 

•203 

8.55 

12 

.109 

4-59 

18 

.049 

2.06 

24 

.022 

.926 

7 

.18 

7-58 

13 

•095 

4 

19 

.042 

1.77 

25 

.02 

.842 

8 

.165 

6-95 

14 

.083 

3-49 

20 

•035 

i-47 

WEIGHT  OF  WROUGHT  IRON  TUBES, 


143 


"Weiglit  of  Wrought  Iron  Tubes.  (English.) 

EXTERNAL  DIAMETER.  ONE  FOOT  IN  LENGTH. 


HoltzapjfeVs  Wire-Gauge,  f full , l light. 


No. 

- 

- 

4 

5 1 

6 

7 

8 

9 

Ins. 

• 3125 

.281 

.238 

.22 

.203 

.18 

.165 

. 148 

5/16 

9/32 

*5/64/ 

7/32 

r3/ 64 

3/16  l 

n/64  l 

9/64  / 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

7 

21.9 

19.8 

16.9 

IS.6 

14*5 

12.9 

11. 8 

10.6 

7-5 

23-5 

21.3 

I8.I 

16.8 

15-5 

13.8 

12.7 

11.4 

8 

25.2 

22-7 

19-3 

17.9 

16.6 

14.7 

i3-5 

12,2 

8-5 

26.8 

24.2 

20.6 

19.I 

17.6 

15*7 

14.4 

12.9 

9 

28.4 

25-7 

21.8 

20.2 

18.7 

16.6 

15-3 

13-7 

9-5 

30.1 

27.1 

23.1 

2I.4 

19.8 

17.6 

16. 1 

14-5 

10 

3i-7 

28.6 

24-3 

22.5 

20.8 

18.5 

17 

15-3 

No. 

7 

8 

9 

10 

1 1 

12 

13 

'4 

'5 

Ids. 

.18 

.165 

.148 

•134 

.12 

.109 

•095 

.083 

.072 

3/16  l 

11/64  l 

9/64/ 

9/64  l 

1/8  l 

7/64 

3/32/ 

S/64  / 

5/64  1 

Diaoi. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

1-55 

I.44 

I.32 

1.22 

I. II 

1.02 

•9 

•797 

•7 

1. 125 

I.78 

1.66 

I-5I 

1-39 

I.26 

1. 16 

i-3 

.906 

-794 

I.25 

2.02 

1.88 

1.71 

i-57 

I.42 

Ji3 

I-I5 

I.OI 

.888 

J-375 

2.25 

2.09 

1.9 

i-74 

1.58 

I-45 

1.27 

1. 12 

•983 

i-5 

2.49 

2.31 

2.1 

1.92 

i-73 

i-59 

1.4 

1.23 

1.08 

1.625 

2.72 

2.52 

2.29 

2.09 

1.89 

i-73 

1.52 

i-34 

1. 17 

i-75 

2.96 

2-74 

2.48 

2.27 

2.05 

1.87 

1.65 

i-45 

1.27 

i-875 

3-19 

2.96 

2.68 

2-45 

2.21 

2.02 

1.77 

1.56 

1*36 

2 

3-43 

3-i7 

2.87 

2.62 

2.36 

2.16 

1.9 

1.67 

1-45 

2.125 

3.67 

3-39 

3.06 

2.8 

2.52 

2.3 

2.02 

1.78 

i-55 

2.25 

3-9 

3-6 

3.26 

2.97 

2.68 

2-44 

2.14 

1.88 

1.64 

2-375 

4.14 

3.82 

3-45 

3-i5 

2.83 

2-59 

2.27 

1.99 

1.74 

2-5 

4-37 

4.04 

3-65 

3-32 

2-99 

2-73 

2-39 

2.1 

1.83 

2.625 

4.61 

4.25 

3-84 

3-5 

3-i5 

2.87 

2.52 

2.21 

i-93 

2-75 

4.84 

4-47 

4-03 

3.67 

3-3i 

3.02 

2.64 

2.32 

2.02 

2.875 

5.08 

4.68 

4.23 

3-85 

3-46 

3.16 

2.77 

2-43 

2,11 

3 

5-32 

4.9 

4.42 

4.02 

3.62 

3-3 

2.89 

2-54 

2.21 

3-25 

5-79 

5-33 

4.81 

4-37 

3-94 

3-59 

3-i4 

2-75 

2.4 

3-5 

6.26 

5.76 

5-2 

4.72 

4-25 

3.87 

3- 39 

2.97 

2-59 

3-75 

6-73 

6.19 

5-58 

5-07 

4-57 

4.16 

3-64 

3-i9 

2.77 

4 

7.2 

6.63 

5-97 

5-43 

4.88 

4.44 

3-89 

3-4 

2.96 

4.25 

7.67 

7.06 

6.36 

5-78 

5-2 

4-73 

4-i3 

3.62 

3.15 

4-5 

8.14 

7-49 

6-45 

6.13 

5-5i 

5.01 

4-38 

3-84 

3-34 

4-75 

8.61 

7.91- 

7-i3 

6.48 

5.82 

5-3 

4-63 

4.06 

3-53 

5 

9.08 

8-35 

7-52 

6.83 

6.13 

5.58 

4.88 

4.27 

3-72 

5-25 

9-56 

8.79 

7.91 

7.18 

6.44 

5-87 

5.i3 

4.49 

3-9 

5-5 

10 

9.22 

8-3 

7-53 

6.76 

6.15 

5-38 

4.71 

4.09 

5-75 

10.5 

9^5 

8.68 

7.88 

7.07 

6.44 

5-63 

4-93 

4.28 

6 

11 

IO.  I 

9.07 

8.23 

7-39 

6-73 

5-87 

5-i4 

4-47 

6.25 

11.4 

10.5 

9.46 

8.58 

7-7 

7.01 

6.12 

5.36 

4.66 

6-5 

11.9 

10  9 

9-85 

8-93 

8.02 

7-3 

6-37 

5-58 

4.85 

6-75 

12.4 

11.4 

10.2 

9.28 

8-33 

7-58 

6.62 

5-79 

5-03 

7 

12.9 

11.8 

10.6 

9-63 

8.64 

7.87 

6.87 

6.01 

5.22 

7-25 

13-3 

12.2 

11 

9.99 

8.96 

8.15 

7.12 

6.23 

5*4i 

7-5 

13.8 

12.7 

11.4 

10.3 

9.27 

8.44 

7-37 

6-45 

5-6 

7-75 

14-3 

I3*1 

11.8 

10.7 

9-59 

8.72 

7.62 

6.66 

5-79 

8 

14-7 

13-5 

12.2 

11 

9-9 

9.01 

7.86 

6.88 

5-98 

\ 


144 


WEIGHT  OF  COPPER  TUBES. 


■Weight  of  Seamless  Drawn  Copper  Tubes.  (English.) 
For  Diameters  and  Thicknesses  not  given  in  preceding  Tables.  (D.  K.  Clark.) 


INTERNAL  DIAMETER.  ONE  FOOT  IN  LENGTH. 

HollzapjiJeV s Wire-Gauge,  f full , l light. 
Specific  Weight  = 1.16.  Wrought  Iron  = i. 


No. 

0000  1 

000 

00 

0 

No. 

0000 

000 

00 

0 

•454 

.425 

.38 

•34 

•454 

.425 

•38 

-34 

Ins. 

29/64 

27/64  / 

3/8/ 

n/32 

29/64 

27/64  / 

3/8/ 

n(tm 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•75 



— 

4-5 

5-75 

34-2 

3x-9 

28.3 

25.2 

.875 



• 

5-79 

5.02 

6 

35-6 

33-2 

29-5 

26.2 

1 

8.02 

7*36 

6-37 

5*53 

6-5 

38-4 

35-8 

31.8 

28.3 

1. 125 

8.71 

8 

6.95 

6.05 

7 

41. 1 

38-3 

34-i 

30.3 

1.25 

9.4 

8.65 

7-52 

6-57 

7-5 

43-9 

40.9 

36-4 

32.4 

I*375 

IO.I 

9*3 

8.1 

7.08 

8 

46.6 

43-5 

38-7 

34-5 

1.5 

10.8 

9 94 

8.68 

7.6 

9 

52.1 

48.7 

43-3 

38.6 

1.625 

11. 5 

10.6 

9.26 

8.12 

10 

57-7 

53-8 

47-9 

42.7 

1 .75 

12. 1 

11. 2 

9-83 

8.63 

11 

63.2 

59 

52.5 

46.8 

1.875 

12.8 

11.9 

10.4 

9*I5 

12 

68.7 

64.2 

57-2 

51 

2 

13.5 

12.5 

11 

9.66 

13 

74.2 

69-3 

61.8 

55-1 

2.125 

14.2 

13-3 

1 1. 6 

10.2 

14 

79-7 

74-5 

66.4 

59-2 

2.25 

2-375 

2.5 

14.9 

15-6 

16.3 

13.8 

14- 5 

15- 1 

12. 1 
12.7 
13-3 

10.7 

11.2 

n.7 

15 

16 
x7 

85.2 

90.7 

96*3 

79.6 

84.8 

90 

7i 

75-6 

80.2 

63-4 

67.7 

71.8 

-A 

2.625 

2.75 

17 

17.7 

15.8 

16.4 

x3-9 

14-5 

12.2 

12.8 

18 

19 

101.8 

107.3 

95-i 

100.3 

84.9 

89-5 

76 

80.1 

3 

3.25 

19.1 

20.4 

17.7 

19 

15.6 

16.8 

13.8 

14.8 

20 

21 

112.8 

118.3 

105.5 

110.7 

94.1 

98.7 

04.2 

88.3 

3.5 

21.8 

20.3 

17.9 

15-9 

22 

123.8 

115.8 

103-3 

92-5 

3- 75 
4 

4- 25 
4.5 

23.2 

24.6 

25-9 

27-3 

21.6 

22.9 

24.2 

25*4 

19.1 

20.2 

21.4 

22.5 

16.9 

17.9 

19 

20 

23 

24 

26 

28 

129.3 

134.8 

146 

157-2 

120.9 

126.1 

136.4 

146.7 

107.9 

112.6 

121.8 

I3I 

9O.O 

100.6 

108.8 

117.1 

4.75 

28.7 

26.7 

23-7 

21 

30 

168.4 

I57-1 

140.2 

125.4 

5 

30. 1 

28 

24.8 

22.1 

32 

179.6 

167.4 

149*5 

I33-6 

5-25 

5-5 

31*5 

32.8 

29-3 

1 30-6 

26 

27.1 

23.1 
1 24.1 

34 

36 

190.7 

201.9 

177.7 
1 188 

158.7 

1 167.9 

i4x-9 
i5°- 1 

For  Diameters  from  13  to  24  Inches. 


No. 

1 | 2 

3 

4 

5 

6 

7 

8 

9 

10 

Ins. 

•3 

*9/64/ 

.284 

9/32/ 

•259 

V4/ 

.238 

15/64/ 

.22 

7/32/ 

.203 

13/64 

.18 
3/16  l 

.165 
n/64  1 

.148 

9/64/ 

•134 
9/64  l 

Diam. 

x3 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

Lbs. 

48.5 
52.1 
55-8 
59-4 

63 

66.7 

70.3 

74 

77.6 

81.3 
84.9 

88.6 

Lbs. 

45-8 

49-3 

52.7 

56.2 
59-6 
63.1 
66.5 
70 
73-4 
76.9 

80.3 

83.8 

Lbs. 

41.7 

44.9 
48 
51.2 
54-3 
57-4 
60.6 

63-7 

66.9 
70 
73-2 
76-3 

Lbs. 

38-3 

4I.2 

44.1 
46.9 
49.8 
52.7 
55.6 
58.5 
61.4 

643 

67.2 
70.1 

Lbs. 

35-3 

38 

40.7 

43-4 

46 

48.7 
5i-4 
54 

56.7 
59-4 
62.1 

64.7 

Lbs. 

32.6 

35-i 

37-6 

40 

42.5 

45 

47-4 

49.9 

52.4 

54-9 

57-3 

59-8 

Lbs. 

28.8 

31 

33-2 

35-4 

37-5 

39-7 

41.9 

44.1 

46.3 

48.5 

50.7 

52-9 

Lbs. 

26.4 

28.4 
30-4 

32.4 
34-4 
36-4 

384 

40.4 

42.4 

44.4 

| 46.4 
1 48.5 

Lbs. 

23.6 
254 

27.2 
29 
30.8 

32.6 

34-4 

36.2 
38 
39-8 

41.6 
43-4 

Lbs. 

21.4 

23 

24.6 

26.3 

27.9 

29-5 

31-2 

32.8 

34-4 

36 

37-7 

39-3 

WEIGHT  OF  COPPER  AND  WROUGHT  IRON  TUBES.  I45 


For  Diameters  from  13  to  24  Inches. 


No. 

" 

12 

13 

'4 

15 

J6 

17 

18 

19 

20 

Ins. 

.12 

. 109 

•095 

.083 

.072 

.065 

b 

Ln 

00 

.049 

.042 

•035 

1/8  l 

7/64 

3/32/ 

5/64/ 

5/64  l 

x/i6/ 

x/i6  l 

3/64/ 

3/64  1 

V32  / 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

13 

19.I 

I7‘4 

I5.I 

13.2 

II.4 

IO.3 

9.2 

7-77 

6.65 

5-55 

14 

20.6 

18.7 

16.3 

I4.2 

12.3 

II. I 

9.9 

8.37 

7.16 

5-98 

15 

22.1 

20 

17.4 

15.2 

13.2 

II.9 

10.6 

8.96 

7.67 

6.4 

l6 

23-5 

21.3 

18.6 

l6.2 

14.I 

12.7 

n-3 

9*56 

8.18 

6.82 

17 

25 

22-7 

19.7 

I7.2 

14.9 

13-5 

12. 1 

10.2 

8.69 

7.27 

IS 

26.4 

24 

20.9 

18.2 

15.8 

14*3 

12.7 

10.7 

9.2 

7.69 

19 

27.9 

25-3 

22 

I9.2 

16.7 

I5-1 

13-4 

11  *3 

9.71 

8.12 

20 

29-3 

26.6 

23.2 

20.2 

17.6 

15-9 

14. 1 

11.9 

10.2 

8.54 

21 

30.8 

27.9 

24-3 

21.3 

18.4 

16.6 

14.8 

12.5 

IO.7 

8.96 

22 

32.3 

29-3 

25-5 

22.3 

19-3 

17.4 

15-5 

I3  • 1 

II. 2 

9-39 

23 

33-7 

30.6 

26.7 

23-3 

20.2 

18.2 

16.2 

13-7 

II. 8 

9.81 

24 

35-2 

3i-9 

27.8 

243 

21. 1 

!9 

16.9 

14-3 

12.3 

10.2 

"Weigh,  t of*  Wrought  Iron  Tubes.  (English.) 

For  Diameters  and  Thicknesses  not  given  in  'preceding  Tables.  (D.  K.  Clark.) 

INTERNAL  DIAMETER.  ONE  FOOT  IN  LENGTH. 

HoltzapjfeVs  Wire-Gauge,  f full , l light. 


No. 

4 

5 

1 6 1 

7 

Ins. 

5/8 

9/16 

Thickni 

J/2 

ESS  IN  tl 

7/ 16 

v’CIIES. 

3/8 

5/ 16 

V 4 

.238 

*5/64/ 

.22 

7/32/ 

.203  | 
J3/64  | 

.18 
3/16  l 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

19 

128.5 

II5.2 

102. 1 

89.I 

76.I 

63.2 

50.4 

48 

44-2 

40.8 

36.2 

20 

135 

I2I.I 

IO7.3 

93-6 

80 

66.5 

53 

50.4 

46-5 

42.9 

38 

21 

i4i-5 

127 

112.6 

98.2 

83.9 

69.7 

55-6 

52.9 

48.8 

45-1 

39-9 

22 

148.1 

I32.9 

117.8 

102.8 

87.9 

73 

58.3 

55-4 

51. 1 

47.2 

41.8 

23 

154.6 

138.8 

123. 1 

107.4 

91.8 

76.3 

j 60.9 

57-9 

53-4 

49-3 

43.7 

24 

161.2 

I44.7 

128.3 

112 

95-7 

79.6 

63-5 

60.4 

55-7 

51.5 

45-6 

26 

174-3 

156.5 

138.8 

121.1 

IO3.6 

86.1 

68.7 

65.4 

60.3 

55-7 

49-3 

28 

! 187.4 

168.3 

149.2 

130-3 

III.4 

92.7 

74 

70.4 

64.9 

60 

53-1 

30 

' 200.4 

l8o 

159.7 

139-5 

1 19.3 

99.2 

79.2 

75-4 

69.5 

64.2 

56.8 

32 

! 213.5 

I9I.8 

170.2 

148.6 

I27. 1 

105.7 

84.4 

80.4 

74.I 

68.5 

60.6 

34 

j 226.6 

203.6 

180.6 

157-8 

135 

112.3 

89.7 

85-4 

78.7 

72.8 

64.4 

36 

239-7 

215-4 

191.1 

167 

I42.9 

118.8 

94  9 

90.4 

83.4 

77 

68.1 

No. 

8 

9 

(0 

1 1 

1 2 

13 

'4 

'5 

16 

17 

! '8 

.165 

.148 

• 134 

.12 

. 109 

.095. 

.083 

.072 

.065 

.058 

.049 

Ins. 

11  / 64  l 

9/64/ 

9/64 1 

Vs  l 

7/64 

3/32/ 

5/64/ 

5/64  1 

Vi  6/ 

V16  1 

3/64/ 

Diam. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

19 

33- f 

29.7 

26.9 

24 

21.8 

*9 

16.6 

I4.4 

13 

1 1.6 

9.78 

20 

34-8 

31.2 

28.3 

25-3 

22.9 

20 

17-5 

I5.I 

13.7 

12.2 

IO.3 

21 

36.6 

32.8 

29.7 

26.5 

24.I 

21 

18.3 

15-9 

14-3 

12.8 

10.8 

22 

38.3 

34-3 

31. 1 

27.8 

25.2 

22 

19.2 

16.6 

15 

13-4 

n-3 

23 

40 

35-9 

32.5 

29.I 

26.4 

23 

20.1 

17.4 

15.7 

14 

11.8 

24 

41.8 

37-4 

33.9 

30-3 

27-5 

24 

20.9 

I8.I 

16.4 

14.6 

12.6 

26 

45-2 

40-5 

36.7 

32.8 

29.8 

26 

22.6 

19.7 

17.7 

15.8 

13-4 

28 

48.7 

43-6 

39  5 

35-3 

32.1 

28 

24.4 

21.2 

I9.I 

17 

14.4 

30 

52.1 

46.7 

42.3 

37-8 

34-4 

30 

26.1 

22.7 

20.5 

18.3 

15.4 

32 

55-5 

49.8 

45-i 

40.4 

36.7 

32 

27.9 

24.2 

21.8 

19-5 

16.5 

34 

59 

52.9 

48 

42.9 

39 

34 

29.7 

25.8 

23.2 

20.7 

17.5 

36 

62.4 

56 

50.8 

45-4 

4i.3 

36 

31.4 

27.3 

24.6 

21.9 

18.6 

I46  WEIGHT  OF  IRON,  STEEL,  COPPER,  ETC. 


"Weight  of  a Square  JF'oot  of  "Wronght  and.  Cast 
Iron,  Steel,  Copper,  Lead,  Brass,  and  Zinc  Blates. 
From  .0625  to  1 Inch  in  Thickness, 


Thickness. 

Wrought 

Iron. 

Cast  Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

Gun- 

metal. 

Zinc. 

In^h 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.0625 

2.517 

2.346 

2.54I 

2.89 

3.691 

2.675 

2.848 

2-34 

.125 

5 035 

4-693 

5.081 

5-781 

7.382 

5-35 

5.696 

4.68 

•1875 

7-552 

7-039 

7.622 

8.672 

II.074 

8.025 

8-545 

7.02 

•25 

IO.O7 

9.386 

IO.163 

II.562 

I4-765 

10.7 

n-393 

936 

•3I25 

12.588 

n-733 

12.703 

14-453 

18.456 

13-375 

14.241 

n-7 

•375 

15.106 

14.079 

15  244 

17-344 

22.148 

16.05 

17.089 

1404 

•4375 

17.623 

16.426 

17-785 

20.234 

25-839 

18.725 

19.938 

16.34 

.5 

20.341 

18.773 

20.326 

23-125 

29-53 

21.4 

22.786 

18.72 

•5625 

22.659 

21.119 

22.866 

26.016 

33.222 

24.075 

25-634 

21.06 

.625 

25.176 

23466 

25.407 

28.906 

36-9j3 

26.75 

28.483 

23-4 

.6875 

27.694 

25.812 

27.948 

31-797 

40.604 

29.425 

3i-33i 

25-74 

•75 

30.211 

28.159 

30.488 

34.688 

44.296 

32.1 

34-W9 

28.68 

.8125 

32.729 

30-505 

33.029 

37-578 

47.987 

34-775 

37.027 

30.42 

.875 

35-247 

32-852 

35-57 

40.469 

51.678 

36.656 

39-875 

32.76 

•9375 

37-764 

35-199 

38.11 

43-359 

55-37 

39-331 

42.723 

35-i 

1 

40.282 

37-545 

40.651 

46.25 

59.061 

42.8 

45-572 

37-44 

From  One  Twentieth  Inch  to  Tivo  Inches  in  Thickness. 


Thickness. 

Wrought 

Iron. 

Cast  Iron. 

Steel. 

Copper. 

Lead. 

Brass. 

Uun- 

metal. 

Zinc. 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•°5 

2.OI4 

I.877 

2.033 

2.312 

2-593 

2.14 

2.279 

I.872 

.1 

4.028 

3-754 

4-065 

4.625 

5.906 

4.28 

4-557 

3-744 

.15 

6.042 

5.632 

6.098 

6.938 

8.859 

6.42 

6.836 

5-616 

.2 

8.056 

7.509 

8.13 

9-25 

II.812 

8.56 

9.H4 

7.488 

•25 

IO.O7I 

9.386 

IO.163 

11.562 

14-765 

IO.7 

n-393 

9-36 

•3 

12.085 

11.264 

12.195 

13-875 

17.718 

12.84 

13.672 

11.232 

•35 

14.099 

13.141 

14.228 

16.187 

20.671 

14.98 

15-95 

I3-I°4 

.4 

16.II3 

15.018 

16.26 

18.5 

23.624 

17.12 

18.229 

14.976 

•45 

18.127 

16.895 

18.293 

20.812 

26.577 

19.26 

20.507 

16.848 

.5 

2O.I4I 

18.773 

20.325 

23.125 

29-53 

21.4 

22.786 

18.72 

•55 

22.155 

20.65 

22.358 

25-437 

32.484 

23-54 

25.065 

20.592 

.6 

24.169 

22.527 

24-391 

27-75 

35-437 

25.68 

27-343 

22.464 

•65 

26.183 

24.409 

26.423 

30063 

38.39 

27.82 

29.622 

24.336 

.7 

28.197 

26.281 

28.456 

32.375 

41-343 

29.96 

31-9 

26.208 

•75 

30.211 

28.154 

30.488 

34-687 

44.296 

32.I 

34-179 

28.08 

.8 

32.226 

30-035 

32-521 

37 

47.249 

34-24 

36458 

29.95 

.85 

34-24 

31.912 

34-553 

39-312 

50.202 

36.38 

38.736 

31.824 

•9 

36.254 

33-79 

36.586 

41.625 

53-154 

38.52 

41.015 

33.696 

•95 

38.268 

35-668 

38.628 

43-937 

56.108 

40.66 

43-293 

35.568 

.t 

40.282 

37-545 

40.651 

46.25 

59.061 

42.8 

45-572 

37-44 

1. 125 

45-3I7 

42.238 

45-732 

52.031 

66.443 

48.15 

51.268 

42.12 

1.25 

50.352 

46.93 1 

50.814 

57-813 

73-826 

53-5 

56-965 

46.8 

i-3I25 

52.87 

49.278 

53-354 

60.703 

77.5W 

56.17 

59813 

49.14 

1-375 

55-387 

51.624 

55-895 

63-594 

81.209 

58.85 

62.661 

51.48 

1-4375 

57-9°5 

53-971 

58.436 

66.484 

84-9 

6i.53 

65-51 

53-82 

1.5 

60.422 

56337 

60.976 

69-375 

88.591 

64.2 

68.358 

56.16 

1.5625 

62.94 

58663 

63-517 

72.266 

92.283 

66.88 

71.206 

58.5 

1.625 

65-458 

61.011 

66.058 

75156 

95-974 

69-55 

74-054 

60.84 

i-75 

7°-493 

65.704 

71.139 

S0.93S 

103.356 

74-9 

79-751 

65.52 

1.075 

75-528 

70.397 

76.22 

86.719 

1 10.739 

80.25 

85-447 

70.2 

2 

80.564 

75-09 

81.3 

92-5 

118.122 

85.6 

9I-I44 

74-88 

WEIGHT  OF  CAST  IRON  WATER  PIPES  AND  TUBES.  1 47 


Standard.  Cast  Iron  "Water  Pipes.  {English.) 
For  a Head  of  200  Feet. 


Diameter. 

Thickness. 

Depth 

of 

Socket. 

Thickness 

of 

Socket. 

Packing. 

-*.  1 

TJ 

.“5  os  es  S 

PS 

p.  ! 

1 

P 

I Thickness. 

1 

Depth 

of 

Socket. 

Thickness 

of 

Socket. 

Packing. 

Weight 
per  Yard.* 

Lead 

Joint. 

Ins. 

Inch. 

Ins. 

Inch. 

Inch. 

Lba.  j Lbs. 

Ins. 

Inch. 

Ins. 

Inch. 

Inch. 

Lbs. 

Lbs. 

3 

•3I25 

3-5 

.625 

•25 

36  .8 

8 

•4375 

3-75 

.625 

•375 

113 

3-3 

4 

•3I25 

3 

.625 

•25 

51  , 1.2 

9 

•4375 

3-75 

•75 

•375 

128 

4.6 

5 

•375 

3 

.625 

•375 

61  2 

IQ 

•5 

4 

•75 

•375 

l68 

4.9 

6 

•375 

3-75 

.625 

•375 

75  2-7 

II 

•5 

4 

•75 

•375 

175 

5 3 

7 

•375 

3-75 

.625 

•375 

85  2.9 

12 

.5625  ! 4 

•875 

•375 

213 

5-7 

* Measured  as  laid. 


To  Compute  "W eight  of*  IVIetal  Pipes. 

1)2  __  ^ D and  d representing  external  and  internal  diameters  in  inches , 
and  C coefficient. 

Cast  Iron  2.45.  Wrought  Iron  2.64.  Brass  2.82.  Copper  3.03.  Lead  3.86. 

To  Compute  AW eight  of  jVLotal  Tubes  and  Pipes 
per  Pineal  Foot. 


From  .5  Inch  to  6 Inches  Internal  Diameter. 


Diam. 

Area  of  Plate. 

Diam. 

Area  of  Plate. 

Diam. 

Area  of  Plate. 

Diam. 

Area  of  Plate. 

Ins. 

Sq.  Foot. 

Ins. 

Sq.  Foot. 

Ins. 

Sq.  Feet. 

Ins. 

Sq.  Feet. 

.5 

.1309 

i-3T25 

•3436 

2-75 

.7199 

4-5 

I.I781 

•5625 

•H73 

1-375 

•36 

2 875 

.7526 

4.625 

1.2108 

.625 

.1636 

1-4375 

•3764 

3 

•7854 

4-75 

1.2435 

.6875 

.18 

i-5 

•3927 

3125 

.8l8l 

4875 

I.2763 

•75 

.1964 

1.625 

4254 

3-25 

.8508 

5 

I.309 

.8125 

.2127 

i-75 

.4581 

3-375 

.8836 

5-125 

I-34I7 

.875 

.2291 

i-875 

.4909 

3-5 

•9i63 

5-25 

1-3744 

•9375 

•2454 

2 

•523b 

3-625 

•949 

5-375 

1.4072 

1 

.2618 

2.125 

•5543 

3-75 

.9818 

5-5 

1-4399 

1.0625 

.2782 

2.25 

.587 

4 

1.0472 

5-625 

1.4726 

1.125 

•2945 

2-375 

.6198 

4.125 

1.0799 

5-75 

1.5053 

1.1875 

•3105 

2-5 

•6545 

4-25 

1.1126 

5.875 

i-538i 

125 

.3272 

2.625 

.6872 

4-375 

I-I454 

6 

1.5708 

Application,  of  Ta/ble. 

When  Thickness  of  Metal  is  given  in  Divisions  of  an  Inch. 

To  internal  diameter  of  tube  or  pipe  add  thickness  of  metal ; take 
area  of  the  plate  in  square  feet,  from  table  for  a diameter  equal  to 
sum  of  diameter  and  thickness  of  tube  or  pipe,  and  multiply  it  by 
weight  of  a square  foot  of  metal  for  given  thickness  (see  table,  page 
146),  and  again  by  its  length  in  feet. 

Illustration. — Required  weight  of  10  feet  ot  copper  tube  1 inch  in  diameter  and 
.125  of  an  inch  in  thickness. 

1 4-.  125  = 1. 125  X 3.1416-4- 12^.2945  square  feet  for  1 foot  of  length. 

Weight  of  1 square  foot  of  copper  .125th  of  an  inch  in  thickness,  per  table,  page 
135,  =5.781  lbs. ; then,  .2945  (from  table  above)  X 5.781  X 10  = 17.025  lbs. 

When  Thickness  of  Metal  is  given  in  Numbers  of  a Wire-Gauge. 

To  internal  diameter  of  tube  or  pipe  add  thickness  of  number  from 
table,  pp.  120  or  121 ; multiply  sum  by  3.1416,  divide  product  by  12,  and 
quotient  will  give  area  of  plate  in  square  feet.  Then  proceed  as  before. 


I48  WEIGHT  OF  IRON"  AND  COPPER  PIPES,  BOLTS,  ETC. 

Illustration. — Required  weight  of  10  feet  of  copper  pipe  2 inches  in  diameter 
and  No.  2 American  wire-gauge  in  thickness. 

2-}-. 257 63  X 3. 1416 -r- 12  = 2.25763  X 3.1416-f- 12  = .591  square  feet;  then,  .591 
X 11.6706  (weight  from  table,  page  118)  =6.897  lbs. 

"Weight  of  Riveted  Iron  and.  Copper  Pipes, 
From  5 to  30  Inches  in  Diameter. 

ONE  FOOT  IN  LENGTH. 


Diameter. 

Thickness. 

Iron. 

Copper. 

Diameter. 

Thickness. 

Iron. 

Copper. 

Inch. 

Lbs. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Lbs. 

5 

.125 

7.12 

8.14 

9 

•25 

25.OI 

28.58 

•1875 

10.68 

12.21 

•25 

26.33 

3O.O9 

•25 

14.25 

16.28 

10 

•25 

27-75 

3I-7i 

5-5 

.125 

7.78 

8.89 

10.5 

•25 

29.19 

33-22 

•1875 

11.66 

13-33 

11 

•25 

30-49 

34.85 

•25 

I5-56 

17.78 

12 

•25 

33-I3 

37.86 

6 

.125 

8.44 

9.64 

13 

•25 

35-88 

41 

•1875 

12.65 

14.46 

14 

•25 

38.52 

44.02 

•25 

16.88 

19.29 

i5 

•25 

41.26 

47-15 

6.5 

.125 

9-1 

10.4 

16 

•3I25 

51-57 

58.94 

•1875 

13-65 

15.6 

•25 

43-9 

So- 1 7 

.2  5 

18.2 

20.8 

•3125 

54-87 

62.71 

7 

.125 

9.78 

II. 18 

17 

•25 

46.53 

53-i8 

.1875 

14.68 

16.78 

•3125 

58.17 

66.48 

•25 

19-57 

22.37 

18 

•25 

49.17 

56.2 

7-5 

.125 

10.49 

11.99 

•3I25 

61.47 

70.25 

•1875 

15-73 

17.98 

20 

•3I25 

68.07 

77-79 

•25 

20.89 

23.87 

24 

•3i25 

81 -33 

92-95 

8 

.1875 

16.7 

19.08 

25 

•3i25 

84-57 

96.65 

•25 

22.26 

2544 

28 

•3I25 

94-56 

107-95 

8.5 

•25 

23-59 

26.96 

30 

•3i25 

IOI.I4 

H5-59 

Above  weights  include  laps  of  sheets  for  riveting  and  calking. 

Weights  of  the  rivets  are  not  added,  as  number  per  lineal  foot  of  pipe  depends 
upon  the  distance  they  are  placed  apart,  and  their  diameter  and  length  depend 
upon  thickness  of  metal  of  the  pipe. 


Weight  of  Copper  Rods  or  Bolts, 
From  .125  Inch  to  4 Inches  in  Diameter. 

ONE  FOOT  IN  LENGTH. 


Diameter. 

Weight. 

Diameter. 

W eight. 

Diameter. 

Inch. 

Lbs. 

Ins. 

Lbs. 

Ins. 

.125 

•047 

.8125 

1.998 

1-5 

.1875 

.106 

.875 

2.318 

•5625 

•25 

.189 

•9375 

2.66 

.625 

•3I25 

.296 

1 

3-03 

•75 

•375 

.426 

1.0625 

3-42 

.875 

•4375 

•579 

.125 

3-831 

2 

-5 

-757 

.1875 

4.269 

.125 

•5625 

•958 

•25 

4.723 

.25 

.625 

1.182 

•3I25 

5.21 

•375 

.6875 

t-431 

-375 

5-723 

-5 

•75 

i- 7°3 

•4375 

6.255 

.625 

Weight. 

Diameter. 

Weight. 

Lbs. 

Ins. 

Lbs. 

6.8II 

2.75 

22.891 

7-39 

.875 

25.OI9 

7-993 

3 

27.243 

9.27 

.125 

29-559 

IO.642 

.25 

31.972 

12.108 

•375 

34.481 

13.668 

•5 

37.081 

I5-325 

.625 

39-777 

17-075 

•75 

42.568 

18.916 

.875 

45-455 

20.856 

4 

48.433 

WEIGHT  OF  METALS. 


149 


"Weigh,  t of  Metals  of  a Griven  Sectional  Area. 
From  .1  Square  Inch  to  10  Square  Inches. 

PER  LINEAL  foot.  (. D . K.  Clark.) 


Sect. 

Wrought 

Iron. 

x* 

Cast 

Iron. 

Steel. 

Crass. 

Gun- 

metal. 

Sect. 

W rought 
Iron. 

Cast 

Iron. 

Steel. 

Brass. 

Gun- 

metal. 

Area. 

•9375* 

1.02. 

1.052. 

1.092. 

Area. 

1. 

•9375- 

1.02. 

1.052. 

1.092. 

Sq.Ius. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Sq.Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

.1 

•33 

•31 

•34 

•35 

•36 

5-i 

17 

i5-9 

17-3 

17.9 

18  6 

.2 

.67 

.62 

.68 

•7 

•73 

5-2 

17-3 

16.3 

17.7 

18.2 

18  9 

•3 

1 

•94 

1.02 

!‘°5 

1.09 

5-3 

17.7 

16.6 

18 

18.6 

i93 

•4 

i-33 

1.25I 

1.36 

i-43 

1.46 

5-4 

18 

16.9 

18.4 

18.9 

19.7 

•5 

1.67 

1.56 

i-7 

i-75 

1.82 

5*5 

18.3 

17.2 

18.7 

19*3 

20 

.6 

2 

1.88 

2.04 

2. 11 

2.18 

5-6 

18.7 

17-5 

19 

19.6 

20.4 

•7 

2.33 

2.19 

2.38 

2.46 

2.55 ! 

5-7 

!9 

17.8 

19.4 

20 

20.8 

.8 

2.67 

2.5 

2.72 

2.81 

2.91 

5-8 

19-3 

18.1 

19.7 

20-3 

21. 1 

•9 

3 

2.81 

3.06 

3.16 

3-28 

5-9 

19.7 

18.4 

20.1 

2O.7 

21.5 

1 

3-33 

3-i5 

3 4 

3-51 

3-64 

6 

20 

18.8 

20.4 

21 

21.8 

1. 1 

367 

3-44 

3-74 

3-86 

4 

6.! 

20.3 

19.1 

20.7 

2I.4 

22.2 

1.2 

4 

3 75 

4.08 

4.21 

4-37  j 

6.2 

20.7 

19.4 

21. 1 

2I.7 

22.6 

i-3 

4-33 

4.06 

4.42 

4-56 

4-73 

6-3 

21 

19.7 

21.4 

22.1 

22.9 

1.4 

4.67 

4-38 

4.76 

4-91 

5*1 

6.4 

21.3 

20 

21.8 

22.4 

23-3 

i-5 

5 

4.69 

5-i 

5.26 

546; 

6-5 

21.7 

20.3 

22.1 

22.8 

2.3-7 

1.6 

5-33 

5 

5-44 

56i 

5-82 1 

6.6 

22 

20.6 

22.4 

23.I 

24 

I*7 

567 

5-3i 

5-78 

5-96 

6.19 

6.7 

22.3 

20.9 

22. 8 

23-5 

24.4 

1.8 

6 

5 63 

6.12 

6 31 

6.55 

6.8 

22.7 

21.3 

23.1 

23-9 

24.8 

1.9 

6.33 

5-94 

6.46 

6.66 

6.92  j 

6.9 

23 

21.6 

23-5 

24.2 

25.1 

2 

6.67  , 

6-25  | 

6.8 

7.01 

7.28 

7 

23.3 

21.9 

23.8 

24.6 

25-5 

2.1 

7 

6.56 

7.14 

7-36 

7-64 

7-i 

23.7 

22.2 

24.1 

24.9 

25.8 

2.2 

7 33  ! 

6.88 ! 

7.48 

7.72 

8.01 

7.2 

24 

22.5 

24-5 

25-3 

26.2 

2-3 

7-67  | 

7 19 

7.82 

8 07 

8-37 

7-3 

24-3 

22.8 

24.8 

25.6 

26.6 

2.4 

1 8 

7 5 

8.16 

8.42 

8.74 

7-4 

24.7 

23.1 

25.2 

26 

26.9 

2-5 

! 8,33 

7.81 

8-5 

8.77 

9.1 

7-5 

25 

23-4 

25-5 

26.3 

27-3 

2.6 

1 8.67 

8.13 

884 

9.12 

9.46 

7.6 

25-3 

23.8 

25-9 

26.7 

27.7 

2.7 

i 9 

844 

9.18 

9-47 

9-83 

7-7 

25*7 

24.1 

26.2 

27 

28 

2.8 

! 9 33 

8.75 

952 

9.82 

10.2 

7-8 

26 

24.4 

26.5 

27.4 

28.4 

2.9 

1 9.67 

9 06 

9-86 

10.2 

10.6 

7*9 

26.3 

24.7 

26.9 

27.7 

28.8 

3 

10 

938 

10.2 

10.5 

10.9 

8 

26.7 

25 

27.2 

28.1 

29.1 

3-i 

1 io-3 

969 

io-5 

10.9 

11  *3 

8.1 

27 

25-3 

27-5 

28.4 

29-5 

3-2 

10.7 

10 

10.9 

11. 2 

n.7 

8.2 

27-3 

25.6 

279 

28.8 

29.9 

3-3 

11 

10.3 

1 1. 2 

11.6 

12 

8-3 

27.7 

259 

28  2 

29.1 

30.2 

3-4 

n-3 

10.6 

11.6 

11.9 

12.4 

8:4 

28 

26.3 

28.6 

29*5 

30-6 

3*5 

If  *7 

10.9 

11  *9 

12.3 

12.7 

8-5 

28.3 

26.6 

28.9 

298 

30  9 

3-6 

12 

n-3 

12.2 

12.6 

i3-i 

8.6 

28.7 

26.9 

29.2 

30.2 

3i-3 

3-7 

12.3 

11.6 

12.6 

13 

13-5 

8.7 

29 

27.2 

29.6 

30-5 

317 

3-8 

12.7 

11.9 

12.9 

i3-3 

13.8 

8.8 

29*3 

27-5 

29.9 

30-9 

32 

3-9 

13 

12.2 

13-3 

13-7 

14.2 

8.9 

29.7 

27.8 

30-3 

31,2 

324 

4 

13-3 

12.5 

13-6 

14 

14.6 

9 

30 

28.1 

30.6 

31.6 

32.8 

4.1 

13*7 

12.8 

13-9 

14.4 

14.9 

9.1 

30-3 

28.4 

30-9 

3i-9 

33- 1 

4.2 

14 

I3*1 

14-3 

14.7 

15-3 

9.2 

30-7 

28.8 

3i-3 

32-3 

33-5 

4-3 

14-3 

13-4 

14.6 

i5-i 

15-7 

9-3 

3i 

29.1 

3j*6 

32.6 

33-9 

4.4 

14.7 

13.8 

15 

15-4 

16 

9.4 

3i-3 

29.4 

32 

33 

34-2 

4-5 

15 

14.1 

i5-3 

15.8 

16.4 

9-5 

3i-7 

29.7 

32.3 

33-3 

34-6 

4.6 

15-3 

14.4 

15.6 

16.1 

16.7 

9.6 

32 

30 

32.6 

33-7 

34-9 

4-7 

15-7 

14*7 

16 

16.5 

1 7. 1 

9*7 

32*3 

30-3 

33 

34 

35-3 

4.8 

16 

15 

16.3 

16.8 

17-5 

9.8 

32-7 

30.6 

33-3 

34-4 

35-7 

4.9 

16.3 

x5-3 

16.7 

17.2 

17.8 

9.9 

33 

30-9 

33-7 

34-7 

36 

5 

16.7 

15-6 

17 

17-5 

18.2 

10 

33-3 

3i-3 

34 

35-i 

36.4 

150  LEAD  PIPES. — COPPER  PIPES  AND  COCKS. 

“Weigh,  t of  Lead  Pipe.  (English.) 


ONE  FOOT  IN  LENGTH. 


Diam. 

Thick- 

ness. 

Weight. 

Diam. 

Thick- 

ness. 

Weight.. 

Diam.  j 

Thick- 

ness 

Weight. 

Diam. 

Thick- 

ness. 

Weight 

Inch. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

Ins. 

Inch. 

Lbs. 

•5 

.097 

•93 

I 

.136 

2.4 

i-75 

.166 

5 

3 

•275 

14 

.112 

I.07 

.156 

2.8 

.199 

6 

3-5 

.225 

13 

.124 

1.2 

.2 

3-73 

.228 

7 

•273 

16 

.146 

I.47 

.225 

4.27 

.256 

8 

4 

•257 

17 

.625 

.089 

I 

1.25 

•139 

3 

2 

.178 

6 

•3I25 

20.5 

.IOI 

I*I3 

.l6 

3-5 

.204 

7 

•327 

22 

.121 

1.4 

.18 

4 

.231 

8 

4-25 

•3125 

22.04 

.14 

2 

•193 

4-33 

.266 

9-33 

4-5 

.232 

17 

•75 

.112 

1.6 

i*5 

.156 

4 

2.5 

.2 

8.4 

•295 

22 

.147 

1.87 

.179 

4.67 

.227 

9.6 

•3I25 

23*25 

.l8l 

2.13 

.224 

6 

.261 

11.2 

4-75 

•3I25 

24-45 

.215 

2.4 

•257 

7 

3 

.218 

11. 2 

5 

•3I25 

25.66 

Dimensions  of  Copper  Pipes  and  Composition 
Cocks. 


From  1 Inch  to  23  Inches  in  Diameter. 


•s  4 

Flange  Diameter. 

Thick- 

Bolts. 

s.e-3 

Flange 

Diam. 

Thick- 

Bolts. 

Diar 
Pi; 
and  < 

Pipe. 

Cock. 

ness. 

No.  | 

Diam. 

.5  -a 

0 § 

Pipe. 

ness. 

No. 

Diam. 

Ins. 

Inch. 

Inch. 

Ins. 

Ins. 

Inch. 

Inch. 

I 

3-375 

3-5 

•375 

3 

•5 

9 

12.75 

.625 

9 

.625 

I.25 

3-625 

3-75 

•375 

3 

•5 

9-25 

13-125 

.625 

10 

.625 

1.5 

3-875 

4-25 

•375 

3 

•5 

9-5 

13-375 

.6875 

10 

.625 

1-75 

4.125 

4-375 

•4375 

4 

•5 

9-75 

13-625 

.6875 

10 

.625 

2 

4-375 

4-75 

•4375 

4 

•5 

10 

I3-875 

.6875 

10 

.625 

2.25 

4.625 

5-25 

•4375 

5 

•5 

10.5 

14-5 

.6875 

10 

.625 

2-5 

4-875 

5-5 

•4375 

5 

•5 

11 

15 

.6875 

10 

.625 

■2-75 

5-25 

5-75 

•4375 

5 

•5 

11 -5 

15-625 

•75 

10 

•75 

'Z 

6 

6.25 

•5 

5 

.625 

12 

16.125 

•75 

10 

•75 

3.25 

6.125 

6.625 

•5 

6 

.625 

12.5 

16.625 

•75 

10 

•75 

3-5 

6.375 

6.875 

•5 

6 

.625 

13 

17-25 

•75 

10 

•75 

3*75 

6.625 

7-25 

•5 

6 

.625 

i3-5 

I7-875 

•75 

10 

•75 

4 

6.875 

7-375 

•5 

6 

.625 

14 

i8.375 

•75 

10 

•75 

425 

7-125 

7.625 

•5 

6 

.625 

14-5 

18.875 

•75 

10 

•75 

4-5 

7-375 

8.25 

•5 

6 

•625 

i5 

19-5 

•75 

10 

•75 

4-75 

7*625 

8-5 

•5 

6 

•625 

15-5 

20 

•75 

10 

•75 

5 

8 

9 

•5 

6 

•625 

16 

20.5 

•75 

10 

•75 

5.25 

8.25 

9-25 

•5 

6 

.625 

16.5 

21.125 

•75 

10 

•75 

5.5 

8.5 

9-5 

•5 

6 

•625 

17 

21.625 

•75 

11 

•75 

5.75 

9 

9-875 

•5 

6 

•625 

i7-5 

22.125 

•75 

11 

•75 

6 

9.25 

.625 

8 

•625 

18 

22.75 

•75 

11 

•75 

6 25 

9-75 

.625 

8 

•625 

18.5 

23-25 

•75 

11 

•75 

6.5 

10 

.625 

8 

•625 

19 

23-75 

•75 

12 

•75 

6.75 

10 

.625 

8 

•625 

19-5 

24-375 

•75 

12 

•75 

7 

10.5 

.625 

8 

•625 

20 

24.875 

•75 

12 

•75 

7.25 

10.75 

.625 

8 

•625 

20.5 

25-375 

•75 

13 

•75 

7.5 

11. 125 

.625 

8 

•625 

21 

26 

•75 

13 

•75 

7-75 

n-375 

.625 

8 

•625 

21.5 

26.5 

•75 

13 

•75 

8 

11.625 

.625 

9 

•625 

22 

27 

•75 

13 

•75 

8.25 

12 

.625 

9 

•625 

22.5 

27.625 

•75 

14 

•75 

8.5 

12.25 

.625 

9 

•625 

23 

28.125 

•75 

14 

•75 

8.75 

12.5 

.625 

9 

l -625 

WEIGHT  OF  SHEET  LEAD,  LEAD  AND  TIN  PIPES,  ETC.  I 5 I 


"W^eigh-t  of  Slieet  Lead. 


PER  SQUARE  FOOT. 


Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Thickness. 

Weight. 

Inch. 

Lbs. 

Inch. 

Lbs. 

Inch. 

Lbs. 

Inch. 

Lbs. 

.OI7 

I 

.068 

4 

.118 

7 

.169 

IO 

.034 

2 

.085 

5 

•135 

8 

.186 

II 

.051 

3 

.IOI 

6 

.152 

9 

.203 

12 

"Weigh/t  of  Tin  Pipe. 

ONE  FOOT  IN  LENGTH. 


Diam. 

External. 

THICK 

inch. 

:ness. 

% inch. 

Diam. 

External. 

THICK 

inch. 

:ness. 

% inch. 

Diam. 

External. 

THICKN. 

3^  inch. 

Diam. 

External. 

Inch. 

Lb. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Ins. 

.25 

.148 

— 

1.25 

I.O95 

I-4I7 

2.25 

5-04 

3-25 

•5 

•384 

.472 

i-5 

I.328 

1.732 

2.5 

5-67 

3-5 

•75 

.62 

.787 

i-75 

I.564 

2.O47 

2-75 

6-3 

3-75 

1 

.856 

I.  IO3 

2 

1.802 

2.362 

3 

6-93 

4 

THICK??. 
3^  inch. 

Lbs. 

7-56 
8.19 
8 82 
945 


Diameter. 


Ins. 

•375 

•5 

.625 

•75 

1 

1.25 

i-5 


Light  Weights. 

For  Si 

50  feet  and  under. 

apply  of  Water  Head 
51  to  250  feet. 

* 

251  to  500  feet. 

Lbs. 

Lbs. 

Lbs. 

I. 

,bs. 

Lbs. 

Lbs. 

I 

1-5 

2 

2-5 

to 

4 

3 

to 

4-5 

3-5 

to 

5 

2 

2.5 

3 

3*5 

u 

5 

4 

u 

6 

4-5 

7 

3 

3-5 

4 

4-5 

u 

7 

5-25 

u 

8 

6 

9 

3-5 

4 

4-5 

5-5 

' u 

8 

6 

9 

7 

|| 

10 

4-5 

5 

5-5 

7-25 

IQ 

8 

“ 

11 

9 

u‘ 

12 

6.5 

7 

8 

9 

u 

12.5 

10 

“ 

14 

12 

16 

8 

9 

10 

11 

u 

16 

12.5 

» 

18 

14 

21 

11 

13 

— 

16 

a 

23 

18.5 

u 

26 

21 

U 

30 

Dimensions  and.  "Weiglit  of  Sh.eet  Zinc.  (Vielle-Montagne. | 

PER  SQUARE  FOOT. 


No. 

Thickness. 

2X.5  metres ; 
area,  1 square  metre. 

6.56X1.64  feet ; area, 
10.76  square  feet. 

2X  .65  metres ; 
area,  1.3  sq.  metres. 

6.56X2.13  feet ; area, 
13.99  square  feet. 

2X.8  metres ; 
area,  1.6  sq.  metres. 

6 56X2.62  ft. ; area, 
17.22  square  feet. 

Weight. 

Millim. 

Inch. 

Kilom. 

Lbs. 

Kilom. 

Lbs. 

Kilom. 

Lbs. 

Lbs. 

9 

•41 

.Ol6l 

2.9 

6-39 

3-7 

8.16 

4.6 

IO.I4 

•589 

10 

•5i 

.0201 

3-45 

7.61 

4-45 

9.81 

5-5 

12.12 

•704 

11 

.6 

.0236 

4-05 

8-93 

5-3 

11.68 

6-5 

14-33 

.832 

12 

.69 

.0272 

4-65 

IO.25 

6.1 

13-45 

7-5 

16.53 

.96 

13 

.78 

.0307 

5-3 

11.68 

6.9 

15.21 

8-5 

18.74 

I.088 

14 

.87 

.0343 

5-95 

I3-1? 

7-7 

16.94 

9-5 

20.94 

I.2l6 

15 

.96 

.0378 

6.55 

14.44 

8-55 

18.85 

10.5 

23-15 

i-344 

16 

1. 1 

•0433 

7-5 

16.53 

9-75 

21.5 

12 

26.46 

i-536 

17 

1.23 

.0485 

8-45 

18.63 

10.95 

24.14 

13-5 

29.97 

1.74 

18 

1.36 

•0536 

9-35 

20.61 

12.2 

26.9 

15 

33-07 

1.92 

19 

1.48 

.0583 

10.3 

22.71 

13-4 

29.54 

16.5 

3638 

2.112 

20 

1.66 

.0654 

11.25 

24.8 

14.6 

32.19 

18 

39.68 

2.304 

21 

1.85 

.0729 

12.5 

27.56 

16.25 

35-82 

20 

44.09 

2.56 

22 

2.02 

•0795 

13-75 

30-3i 

17.9 

39-46 

22 

48.5 

2.816 

23 

2.19 

.0862 

15 

33  07 

19-5 

42.99 

24 

52.91 

3-073 

-24 

2-37 

•0933 

16.25 

35-82 

21.1 

46.52 

26 

57-32 

3-329 

25 

2.52 

.0992 

17*5 

38.58 

22.75 

50.15 

28 

61.73 

3-585 

26 

2.66 

.IO47 

18.8 

41.44 

24.4 

53-79 

31 

68.34 

3-969 

152  WEIGHT  OF  SHEET  ZINC. SPIKES,  HORSESHOES. 


Ta"ble— (Continued). 

Special  Sizes  for  Sheathing  Ships. 


No. 

Thickness. 

Dimension 1 
1. 15  X .35  metres  ; 
area,  .402  sq.  metre. 

3.77  X 1. 15  feet ; area, 
4.33  sq.  feet. 

j of  Sheets. 

1.3  X .4  metres  ; 
area,  .52  sq.  metre. 

4.26  X 1. 31  feet ; area, 
5.6  sq.  feet. 

Weight 

per 

Sq.  Foot. 

Mi  Him. 

Inch. 

Kilom. 

Lbs. 

Kilom. 

Lbs 

Lbs. 

15 

.96 

.0378 

2.65 

5-84 

3-4 

7-5 

1-344 

16 

1. 1 

•0433 

3 

6.6l 

3-9 

8.6 

i-S36 

17 

I.23 

.0485 

34 

7-5 

4.4 

9-7 

1.74 

18 

I.36 

•0536 

3-75 

8.27 

4.9 

10.8 

1.92 

19 

I.48 

•0583 

4-15 

9-I5 

5-35 

11.79 

2.112 

20 

1.66 

.0654 

4-55 

10.03 

5-85 

12.9 

2.304 

Note.— A deviation  of  25  dekagrammes,  or  about  half  a pound,  more  or  less,  from 
the  proper  weight  of  each  number  of  sheet,  is  allowed. 

Nos.  1 to  9 are  employed  for  perforated  articles,  as  sieves,  and  for  articles  de 
Paris.  Nos.  10  to  12  are  used  in  manufacture  of  lamps,  lanterns,  and  tin-ware  gen- 
erally, and  for  stamped  ornaments.  The  last  numbers  are  used  for  lining  reservoirs, 
and  for  baths  and  pumps. 


Skip  and  Railroad  Spikes. 
DIMENSIONS  AND  NUMBER  PER  POUND.  (P.  C.  Page,  MOSS .) 
Sliip  Spikes. 


X In.  Sq.  | 

« In.  Sq. 

I In.  Sq. 

Y In.  Sq. 

In.  Sq.  | 

% In.  Sq. 

Y In.Sq. 

.fl 

•S’a 

.3 

R-3 

a 

•Sr 

5 

C ”3 

,3 

.C-O 

.R-3 

4 

e-i 

ti 

R 

3 

’ . s 

0 0 

R 

►3 

S 9 

1 ^ 

o'  | 

Zn*  ■ 

be 

q 

71 

A 

O O 
^ P-l  1 

J 

° ° 

c? 

0) 

° © 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

| 

Ins. 

Ins. 

3 

19 

3 

IO 

4 

5-4 

5 

3-4 

6 

2.2  ! 

8 

1.4 

IO 

.8 

3-5 

15.8 

3-5 

9.6 

4-5 

5 

5-5 

3-i 

6-5 

.2 

9 

1.2 

15 

.6 

4 

I3.2 

4 

8 

5 

4.6 

6 

3 

7 

1.9  j 

10 

1. 1 

— 

— 

4-5 

12.2 

4-5 

6 

5-5 

4.2 

6-5 

2.8 

7-5 

1,8 

11 

I 

— 

— 

5 

10.2 

5 

5-8 

6 

4 

7 

2.6 

8 

J-7 

— 

— 

— 

— 

— 

— 

6 

5-2 

6.5 

3-2 

7-5 

2.4 

8-5 

1.6 

— 

— 

— 

— 

8 

2.2 

9 

i-5 

— 

— 

— 

— 

10 

1.4 

— 

— 

— 

— 

Railroad  Spikes 5 inch  square  X 5.5  ins.  2 per  lb. 

“ “ 5625  “ u x 5.5  “ 1.6  “ 

Spikes  and  Horseshoes. 


length  and  number  per  pound.  (Z7.  Burden , Troy , N.  F.) 


Length. 

Boat  £ 

c . 

— 

0 -3 

Spikes. 

Ul 

R 

►3 

No.  in 
Lb. 

Length. 

No.  in  c/3 
§ 

*3, 

Length,  jjf 

No.  in 
Lb. 

Hook  Hea 
Length. 

d. 

c . 

— x> 

Horse 

"Sc 

3 

shoes. 

R . 
""  3 

6 *-3 
£ 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

3 

i7-5 

6-5 

4.78 

4 

8 

7-5 

2.5 

4 X.375 

5-55 

I 

.84 

3-5 

14.68 

7 

3.62 

4-5 

6-5 

8 

I.74 

i 4-5  X. 4375 

4.14 

2 

•75' 

4 

12-57 

7-5 

3-37 

5 

4-37 

8-5 

I.63 

5 X .5 

2.52 

3 

•65 

4-5 

9-2 

8 

2-95 

5-5 

4-3 

9 

i-55 

5-5  X. 5 

2.41 

4 

•56 

5 

7.2 

8-5 

2.9 

6 

4.2 

10 

I-I5 

! 5.5  x. 5625 

1.87 

5 

•39 

5-5 

6-3 

9 

2.1 

6-5 

3-77 

— 

— 

6 X .5625 

I-72 

— 

— 

6 

4-97 

10 

1.98 

7 

2-75 

— 

— 

! 6 X .625 

1 I,38 

— 

— 

CAST  IRON  AND  LEAD  BALLS. NAILS.  I 53 


Wei glit  and.  Volume  of  Cast  Iron  and  Lead  Balls. 
From  1 Inch  to  20  Inches  in  Diameter . 


Diameter. 

Volume. 

Cast  Iron. 

Lead. 

; Diameter. 

Volume. 

Cast  Iron.  | 

Lead. 

Ins. 

Cube  Ins. 

Lbs. 

Lbs. 

Ins. 

Cube  Ins. 

Lbs. 

Lbs. 

I 

.523 

.136 

.215 

9 

381.703 

99-51 

156.553 

i-5 

I.767 

.461 

.725 

9-5 

448.92 

1 17.034 

184.121 

2 

4.189 

I.092 

I.718 

TO 

523-599 

136.502 

214.749 

2.5 

8.l8l 

2.133 

3-355 

10.5 

606.132 

158.043 

248.587 

3 

14-137 

3*685 

5-798 

II 

696.91 

181.765 

285.832 

3-5 

22.449 

5-852 

9.207 

11 -5 

796.33 

207.635 

326.591 

4 

33-51 

8.736 

13-744 

12 

904.778 

235.876 

371.096 

4-5 

47-713 

12.439 

19.569 

12.5 

IO22.656 

266.647 

419.512 

5 

65  45 

17.063 

26.843 

13 

II50.346 

299.623 

471.806 

5-5 

87.114 

22.721 

35-729 

14 

1436.754 

374.563 

589.273 

6 

1 13.097 

29.484 

46.385 

15 

I767.I45 

460.696 

724.781 

6-5 

143-793 

37-453 

58.976 

16 

2144.66 

559- 1][4 

879.616 

7 

179-594 

46.82 

73-659 

17 

2572.44 

670.717 

1055.066 

7-5 

220.893 

57-587 

90.598 

18 

3053.627 

796.082 

1252.422 

8 

268.082 

69.889 

109.952 

19 

359 1 -363 

936.271 

1472.97 

8.5 

321.555 

83.84 

I3T.883 

20 

4188.79 

1092.02 

1717.995 

Note. — To  compute  weight  of  balls  of  other  metals,  multiply  weight  given  in 
table  by  following  multipliers: 

For  Wrought  Iron 1.067.  I Brass .1.12. 

Steel 1.088.  I Gun-metal 1.165. 


Wei glit  and  Diameter  of  Cast  Iron  Balls. 


Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

I 

I.94 

12 

4-45 

50 

7.16 

224 

II. 8 

1344 

21.44 

2 

2-45 

14 

4.68 

56 

7-43 

336 

13-51 

1568 

22.57 

3 

2.8 

l6 

4.89 

60 

7-6 

448 

I4.87 

1792 

23.6 

4 

3.08 

18 

5-09 

70 

8.01 

560 

16.02 

2016 

24.54 

,5 

3-32 

20 

5-27 

80 

8-37 

672 

17.02 

2240 

25.42 

6 

3-53 

25 

5.68 

90 

8.71 

784 

I7.9I 

2800 

27.38 

7 

3-72 

28 

5-9 

IOO 

9.02 

896 

18.73 

3360 

29.I 

8 

3.89 

30 

6.04 

112 

9-37 

1008 

19.48 

3920 

30.64 

9 

4.04 

40 

6.64 

l68 

10.72 

1120 

20.17 

4480 

32.03 

B eng tli  of  Horseshoe  IS'ails. 
By  Fumhers. 

No.  5 1.5  Ins.  I No.  7 

“ 6 1.75  “ I “ 8 


1.875  Ins.  I No.  9 2.25  Ins. 

2 “ | “10 2.5  “ 


Lengths  of  Iron  UNTails,  and  Number  in  a Ifb. 


Size. | 

L’gth. 

No. 

Size. 

L’gth. 

No. 

Size. 

L’gth. 

No. 

I Size. 

L’gth. 

No. 

Size. 

L’gth. 

No. 

3 d) 

Ins. 

1-25 

420 

5d. 

Ins. 

i-75 

220 

8 d. 

Ins. 

2-5 

IOO 

126?. 

Ins. 

3-25 

52 

30  cl 

Ins. 

4 

24 

4 

1 1-5 

270 

6 

2 

175 

10 

3 

65 

20  | 

3-5 

28 

40 

4.25 

20 

154 


NAILS,  SPIKES,  TACKS,  ETC, 


"W^ronght  Iron  Cut  Nails,  Tacks,  Spikes,  etc* 

( Cumberland  Nail  and  Iron  Co.) 

Lengths  and  Number  per  Lb, 


Ordinary- 

Size.  Length. 


3 fine 

3 

4 

5 

6 

7 

8 


20 

30 

40 

50 

6o 


4 

5 

6 

8 


6<* 

7 

8 


No.  per  Lb.  Size. 


Finishing. 

Length.  No.  per  Lb. 


Ins. 

.875 

I.0625 

I.0625 

1-375 

i-75 

2 

2.25 

2.5 

2.75 

3-5 

3- 75 

4- 25 

4- 75 
5 

5- 5 

Light 

1-375 

1- 75 
2 

Brads. 

2 

2- 5 

2- 75 

3- 125 

Fence 

2 

2.25 
2.5 
2.75 

3 


716 

588 

448 

336 

216 

166 

118 

94 

72 

50 

32 

20 

17 

14 


373 

272 

196 

163 

96 

74 

50 

96 

66 

56 

50 

40 


6^ 

8 

10 

12 

20 

30 

40 

WH 

WHL 


6<i 

7 

8 


Ins. 

I-375 

i-75 

2 

2.5 

3 

3-625 

3-875 

Core. 


384 

256 

204 

102 

80 

65 

46 


2- 5 
2 333 

3- 125 

3- 75 

4- 25 
4-75 
2-5 
225 

Clinch. 


143 

68 

60 

42 

25 

18 

14 

69 

72 


2.25 
2-5 

2- 75 
3 

3- 25 

Slate. 

1.625 

1-4375 

I-75 


152 

133 

92 

72 

60 

43 

288 

244 

187 

146 


Shingle. 

Size.  ] Length.  No.  per  Lb. 


1 OZ. 

i-5 

2 

2.5 

3 

4 
6 
8 


14 

16 

18 

20 


Ins. 

i-75 

2.5 

2.75 

3 

Tacks, 

.125 

•1875 

•25 

•3125 

•375 

•4375 

•5625 

.625 

.6875 

•75 

.8125 

.875 

•9375 


178 

74 

60 

52 

16000 
10  666 
8 000 
6 400 

5 333 
4000 
2 666 
2 000 
1 600 
1333 
1 143 
1 000 
888 
800 


Boat. 


Ins. 

I-5 

i 

3- 5 

4 

4- 5 

5 

5- 5 

6 


No.  per  Lb. 


Spikes. 


206 

19 

15 

13 

10 

9 

7 


Bailroad  Spikes. 
Number  in  a Keg  of  150  lbs. 


Length. 

No. 

Length. 

No. 

Length. 

No. 

Length. 

No. 

No. 

3 X -375 

930 

Ins. 

3-5  X .4375 

675 

Ins. 

4 X .5 

450 

Ins. 

S X .3625 

300 

3-5  X .375 

89O 

4 X .4375 

540 

4-5  X .5 

400 

5-5  X .5623 

! 280 

4 X .375 

76O 

4-5  X .4375 

510 

5X  .5 

340 

1 

5.5  x .5625  standard  for  a gauge  of  4 feet  8.5  ins. 


Skip  ancl  Boat  Spikes. 
Number  in  a Keg  of  150  lbs. 


Length. 

No. 

Length. 

No. 

Length. 

No. 

Length. 

No. 

Ins. 

4 X.25 

1650 

Ins. 

5 X. 3125 

930 

Ins. 

8X.375 

455 

Ins. 

10  X. 4375 

270 

4-5  X. 25 

1464 

6x  *3I25 

868 

9X.375 

424 

8x-5 

256 

5 X.25 

1380 

7X.3125 

662 

10  x. 375 

390 

9X.5 

24O 

6 X.25 

1292 

6X-375 

570 

8 X. 4375 

384 

10  X. 5 

222 

7 X.25 

Il6l 

7X.375 

482 

9 X. 4375 

300 

11X.5 

203 

VARIOUS  METALS, 


155 


'W'eigh.t  of*  VTaricms  Metals. 
Per  Cube  Inch  and  Foot. 


Spec. 

W’ght 

Ins, 

Weight 

Specific 

W’ght 

Ins. 

Weight 

Mktals. 

Gravi- 

in an 

in  a 

in  a 

Metals. 

Gravi- 

in an 

in  a 

in  a 

ty. 

Inch. 

Lb. 

Foot. 

ty- 

Inch. 

Lb. 

Foot. 

Wrought-iron 

Lb.  ' 

Lbs. 

Lb. 

Lb. 

plates 

7734 

.2797 

3-57 

483-38 

Brass,  rolled. 

8 217 

.2972 

3-37 

5I3-6 

“ wire. 

7774 

.2812 

3.55 

485.87 

“ cast... 

8 080 

.2922 

3-42 

505 

Cast  iron 

7209 

.2607 

3-84 

450.54 

Lead,  rolled . 

II 340 

.4101 

2.44 

708.73 

Steel  plates. . 

7804 

.2823 

3-54 

487.8 

Tin,  cast 

7 292 

• 2673 

3-74 

462 

“ wire... 

7847 

.2838 

3-52 

490.45 

Zinc,  rolled. . 

7 188 

.26 

3-85 

449.28 

Copper,  | . . . 
rolled  ( . . . 

8697 

8880 

.3146 
• 3212 

3-19 

3- 

543-6 

555 

Alumini-  ) 
um,  cast  ) 
Silver 

2 560 

.0926 

10.8 

160 

Gun-metal,) 

0 

u~> 

ts 

OO 

•3I^5 

3.16 

546.875 

10480 

. 7701 

2.64 

655 

cast f 

* O/y  A 

English.  (D.  K.  Clark.) 


Wrought  iron 

Cast  iron 

Steel 

Copper  plates 
Gun-metal. . . 


7.698 

7.217 

7.852 

8.805 

8.404 


.278 
.26 
.283 
• 318 

•304 


3-6 

480 

Tin 

7.409 

.268 

3-74 

3-84 

45o 

Zinc 

7.008 

.253 

3-95 

3-53 

489.6 

Lead 

11.418 

.412 

2.43 

3-15 

549 

Brass,  cast. . . 

8.099 

.292 

3-42 

2.02 

524 

“ wire.. 

8.548 

.308 

3-24 

462 

437 

712 

505 

533 


WROUGHT  AND  CAST  IRON. 

To  Compute  Weight  of'  Wrought  01*  Oast  Iron. 

Rule.— Ascertain  number  of  cube  inches  in  piece;  multiply  sum  by  .2816*  for 
wrought  iron  and  .2607*  for  cast,  and  product  will  give  weight  in  pounds. 

Or,  for  cast  iron  multiply  weight  of  pattern,  if  of  pine,  by  from  18  to  20,  accord- 
ing to  its  degree  of  dryness. 

Example.— What  is  weight  of  a cube  of  wrought  iron  10  inches  square  by  15 
inches  in  length  ? 

10  X 10  X 15  X .2816  = 422.4  lbs. 

COPPER. 

To  Compute  Weight  of  Copper. 

Rule.— Ascertain  number  of  cube  inches  in  piece;  multiply  sum  by  .32118* 
and  product  will  give  weight  in  pounds. 

Sheathing  and.  Braziers’  Sheets. 

For  dimensions  and  weights  see  Measures  and  Weights,  pages  118-121,  131,  142. 


LEAD. 

To  Compute  Weight  of  Lead. 

Rule. — Ascertain  number  of  cube  inches  in  piece;  multiply  sum  by  .41015  * 
and  product  will  give  weight  in  pounds. 

Example. — What  is  weight  of  a leaden  pipe  12  feet  long,  3.75  inches  in  diameter 
and  1 inch  thick? 

BlJ  Pule  in  Mensuration  of  Surfaces , to  ascertain  Area  of  Cylindrical  Rings. 
Area  of  (3.75  -f- 1 -|- 1)  — 25.967 
“ “ 3.75  —11. 044 

^ 00  Difference,  14.923  {area  ofnng)  x 144  (12  feet)  = 2148.912 

X .410 15  = 881.376  lbs.  n * 

BRASS. 

To  Compute  Weight  of  Ordinary  Brass  Castings. 
Rule.— Ascertain  number  of  cube  inches  in  piece;  multiply  sum  by  .2022  * and 
product  will  give  weight  in  pounds.  * y ’ 


crnvitv  «f  tL°Ltt«ib,e  !inch  a8  Eere  5ivei?  are  for  the  ordinary  metals;  when,  however,  the  specific 

& rnites\re9g1vrem°n  “ kn°Wn> the  we*hfc  of  a tube  * » ^uld 


DIMENSIONS  AND  WEIGHTS  OF  BOLTS  AND  NUTS. 


Dimensions  and  Weights  of  Wrought  Iron  Bolts 
and.  JNTvits. 

SQUARE  AND  HEXAGONAL  HEADS  AND  NUTS. 

rtoagh,  and  from  .25  Inch  to  4 Inches  in  Diameter , 
Square  Head  and  USTirt. 


Diameter 
of  Bolt. 


Ins. 

.25 

.3125 

•375 

•4375 

•5 

.5625 

.625 

.6875 

•75 

.8125 

.875 

1 

1.125 

1.25 

1-375 

i-5 

1.625 

i-75 

1.875 


Width. 
Head.  I Nut. 


Ins. 

.36 

•45 

•54 

.63 

.72 

.82 

•91 

1 

1.09 
1. 18 
1.27 
1.45 
1.63 

1.81 
1.99 

2.17 
2.36 

2- 54 

2.72 

2.9 

3.08 
3.26 

3- 44 

3.62 

3.81 

3- 99 

4.17 

4- 35 

4.71 

5- 07 

5-44 

5.8 


Diagonal. 
Head. 


Ins. 

•49 

.58 

.67 

.76 

.84 

•94 

1.03 

1. 12 

1. 21 
i-3 
i-39 
i.57 

1- 75 

1.94 

2.12 

2- 3 
2.48 
2.66 

2.84 

3°2 

3.21 

3- 39 
3-57 
3-75 
3-93 
4.11 
4.29 

447 

4.84 
5-2 
556 
5-92 


Ins. 

•51 

.64 

.76 

.89 
1.02 
1. 16 
I.29 
1. 41 

1- 54 
1.67 
1.8 
2.05 

2- 3 
2.56 
2.81 
3*07 

3- 34 
3-59 

3- 85 

4- 1 
4*35 
4.61 
4.86 
5.12 

5- 49 
5-64 
5-9 
6.15 
6.66 
7-i7 
7.69 
8.2 


Depth. 
Head.  I Nut. 


Ins. 

.69 

.82 

•95 
1.07 
1. 19 

1- 33 

1.46 

1.58 

1.71 

1.84 

1.96 

2.22 

2.47 

2- 74 
3 

3- 25 

3*51 

3- 76 
4.02 
4.27 

4- 54 

4- 79 

5- 05 
5-3 
5-56 
5.81 
6.07 
6.32 
6.84 

7*35 

7.86 

8.37 


Ins. 

•25 

•3 

•34 

•4 

.44 

.48 

•53 

.58 

•63 

.67 

.72 

.81 

•9 

1 

1. 1 

1. 18 

1.28 

i-37 

1.46 

1.56 

1.65 

i-75 

1.84 

1.94 

2.03 

2.12 

2.22 

2.31 

2.5 

2.68 

2.87 

3.06 


.25 

•3I25 

•375 

•4375 

•5 

•5625 

.625 

.6875 

•75 

.8125 

•875 

1 

1. 125 

1.25 

i-375 

i-5 

1.625 

i-75 

1.875 


Weight. 


Head 
and  Nut. 


, Bolt 
jper  Inch. 


2.125 

2.25 

2.375 

2-5 

2.625 

2- 75 

2.875 

3 

325 

3- 5 
3-75 

4 


Lbs. 

.024 

•043 

.068 

.104 

•145 

.204 

•273 

•356 

•454 

•565 

.696 

1.013 

1.416 

1.923 

2.543 

3- 234 

4- 105 
5.087 
6.182 
7-491 
8.936 

10.543 

12.335 

14-359 

16.549 

18.897 

21-545 

24.464 

30.922 

38-391 

47.168 

56.882 


Threads 
per  Inch. 


Lbs. 

.014 

.022 

.031 

.042 

.055 

•07 

.086 

.104 

.124 

.145 

.168 

.22 

.278 

•344 

.416 

•495 

.581 

.674 

•773 

.88 

•993 

i-ii3 

1.24 

1-375 

i-5i5 

1.663 

1.818 

1.979 

2.323 

2.694 

3093 

3-518 


20 

18 

16 

14 

13 


7 

7 

6 

6 

5.5 

5 

5 

4-5 

4-5 

4-5 

4-375 

4-25 

4 

4 

3-75 

3-5 

3-5 

3-25 

3 

3 


2.125 
2.25 
2.375 

2.5 

2.625 
2.75 
2.875 

3 

3-25 
3-5 
3-75 

4 j"-'  o w 

Finished. — Deduct  .0625  from  diameters  of  bolts  and  depths  of  all  heads 

and  nuts. 

Screws  with  square  threads  have  but  one  half  number  of  threads  of  those 
with  triangular  threads. 

Note.— The  loss  of  tensile  strength  of  a bolt  by  cutting  of  thread  is,  for  one  of  1.25 
ins.  diameter,  8 per  cent.  The  safe  stress  or  capacity  of  a wrought  iron  boit  and  nut 
may  be  taken  at  5000  lbs.  per  square  inch. 

Preceding  width,  depth,  etc.,  are  for  work  to  exact  dimensions,  whether 
forged  or  finished. 

To  Compute  Weiglit  of  a I3olt  and  Nut. 
Operation. — Ascertain  from  table  weight  of  head  and  nut  for  given  di- 
ameter of  bolt,  and  add  thereto  weight  of  bolt  per  inch  of  its  length,  multi- 
plied by  full  length  of  its  body  from  inside  of  its  head  to  end. 

Note.— Length  of  a bolt  and  nut  for  measurement , as  such,  is  taken  from  inside 
of  head  to  inside  of  nut,  or  its  greatest  capacity  when  in  position. 


DIMENSIONS  AND  WEIGHTS  OF  BOLTS  AND  NUTS. 


157 


Illustration. —A  wrought  iron  bolt  and  nut  with  a square  head  and  nut  is  1 inch 
in  diameter  and  10  inches  in  length;  what  is  its  weight? 

Weight  of  head  and  nut 1.013  lbs. 

“ bolt  per  inch  of  length  .22  X 10  = 2.2  “ 

3.213  “ 


Hexagonal  Head  ancl  ISTnt. 


Diameter 
of  Bolt. 

Wi 

Head. 

1th. 

Nut. 

Diag 

Head. 

onal. 

Nut. 

Di 

Head. 

epth. 

Nut. 

Weig 
Head 
and  Nut. 

;ht. 

Bolt 

per  Inch. 

Threads 
per  Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

lbs. 

Lbs. 

No. 

•25 

•375 

•5 

•43 

.58 

•25 

•25 

.022 

.014 

20 

•3125 

•4375 

.5625 

•5 

•65 

•3 

•3125 

.037 

.022 

18 

•375 

.5625 

.6875 

.65 

•79 

•34 

•375 

.062 

.031 

l6 

•4375 

.625 

•75 

•72 

.87 

•4 

•4375 

.094 

.O42 

14 

•5 

•75 

•875 

.87 

1 

•44 

•5 

•134 

•055 

13 

•5625 

.8125 

•9375 

•94 

1.08 

.48 

•5625 

.18 

.07 

12 

.625 

•9375 

1.0625 

I.08 

1.23 

•53 

.625 

•249 

.086 

II 

.6875 

1 

1. 125 

1. 16 

i-3 

.58 

•6875 

.318 

.IO4 

II 

•75 

1. 125 

1-2.5 

i-3 

1.44 

•63 

•75 

•413 

.124 

IO 

.8125 

1.25 

1-375 

1.44 

i-59 

•67 

.8125 

.522 

•145 

IO 

.875 

1-3125 

1-4375 

1.52 

1.66 

•72 

.875 

•639 

.168 

9 

1 

i-5 

1.625 

i-73 

1.88 

.81 

1 

•931 

.22 

8 

1. 125 

1.6875 

1.8125 

1-95 

2.09 

•9 

1.125 

I.299 

.278 

7 

1.25 

1-875 

2 

2.17 

2.31 

1 

1.25 

1-759 

•344 

7 

1-375 

2 

2.1875 

2.31 

2-53 

1. 1 

1-375 

2.263 

.416 

6 

i-5 

2.25 

2-375 

2.6 

2.74 

1. 18 

i-5 

2.958 

•495 

6 

1.625 

2.4375 

2.5625 

2.81 

2.96 

1.28 

1.625 

3-741 

.581 

5-5 

i-75 

2.625 

2-75 

3-03 

3-i8 

i-37 

I-75 

4-654 

•674 

5 

1-875 

2.8125 

2-9375 

3-25 

3-39 

1.46 

i-875 

5-675 

•773 

5 

2 

3 

3-125 

3-46 

3.61 

1.56 

2 

6.854 

.88 

4-5 

2.125 

3-1875 

3-3125 

3.68 

3-83 

1.65 

2.125 

8.163 

•993 

4-5 

2.25 

3-375 

3-5 

3-9 

4.04 

i-75  | 

2.25 

9.658 

1-113 

4-5 

2-375 

3-5625 

3-6875 

4.11 

4.26 

1.84 

2-375 

11.263 

1.24 

4-375 

2-5 

3-75 

3-875 

4-33 

4-47 

1.94 

2.5 

I3-I49 

i-375 

4-25 

2.625 

3-9375 

4.0625 

4-55 

4.69 

2.03 

2.625 

I5.I5 

i-5i5 

4 

2.75 

4-125 

4-25 

4-77 

4.91 

2.12 

2-75 

17.285 

1.663 

4 

2.875 

4-3125 

4-4375 

4.99 

5.12 

2.22 

2.875 

I9.75I 

1.818 

3-75 

3 

4-5 

4.625 

5-2 

5-34 

2.31 

3 

22.378 

1.979 

3-5 

3-25 

4-875 

5 

5-63 

5-77 

2-5 

3-25 

28.258 

2.323 

3-5 

3-5 

5-25 

5-375 

6.06 

6.21 

2.68 

3-5 

35-o8i 

2.694 

3-25 

3-75 

5-625 

5-75 

6-5 

6.64 

2.87 

3-75 

43-I78 

3-093 

3 

4 

6 

6.125 

6.93 

7.07 

3.06 

4 

51.942 

3-518 

3 

Finished. — Deduct  .0625  from  diameters  of  bolts  and  depths  of  all  heads 
and  nuts. 

For  Wood  or  Carpentry, 

Head  and  Nut  (Square),  1.75  diameter  of  bolt.  Depth  of  Head,  .75,  and 
of  Nut , .9. 

Washer. — Thickness,  .35  to  .4  of  diameter  of  bolt,  on  Pine  3.5  diameter, 
and  Oak  2.5. 

English. 

Molesworth  gives  following  elements  of  Thread  of  Bolts : 

Angle  of  thread , 550.  Depth  of  thread  = Pitch  of  screw. 

Number  of  threads  per  Inch.  — Square , half  number  of  those  in  angular 
threads. 

Depth  of  thread . — .64  pitch  for  angular  and  .475  for  square  threads. 

0 


158  DIMENSIONS  AND  WEIGHTS  OF  BOLTS  AND  NUTS. 
ITrencli  Standard  Bolts  and  iNoxts.  (Armengautf s.) 


HEXAGONAL  HEADS  AND  NUTS. 


Equ 

Diamete 
of  Bolt. 

late 

X 

0 . 

«I 

** 

Threads  cl* 

per  Inch.  ^ 

iangu 

Thick 

Head. 

lar  1 

ness. 

Nut. 

Breadth  ^ 

across  Flats. 

a 

l 

Safe 

Tensile 

Stress. 

S 

Diameter 
of  Bolt. 

quare 

c -a' 

f,2 

Threads  ^ 

per  Inch. 

— — os 

ad. 

Is? 

Safe 

Tensile 

Stress. 

Min. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Ins. 

Lbs. 

Mm. 

Ins. 

Ins. 

No. 

Ins. 

Lbs. 

5 

.2 

.13 

18.I 

.24 

.2 

•55 

44 

20 

•79 

.O72 

6.57 

1.82 

717 

7-5 

•3 

.22 

16 

•3 

•3 

.68 

99 

25 

.98 

.081 

5-97 

2.01 

I I42 

IO 

•39 

•31 

14.I 

.38 

•39 

.88 

178 

30 

1. 18 

•093 

5-4 

2.22 

1635 

12.5 

•49 

•39 

12.7 

•44 

•49 

1.04 

277 

35 

1.38 

.1 

4-93 

2.4I 

2 2l8 

15 

•59 

.48 

ii-5 

•52 

•59 

1.2 

400 

40 

i-57 

.106 

4-53 

2.63 

2 912 

I7-5 

.69 

.58 

10.6 

.58 

.69 

1.4 

545 

45 

1.77 

.114 

4.2 

2.85 

3^4 

20 

•79 

.66 

9.8 

.66 

•79 

i-5 

713 

50 

1.97 

.128 

3-91 

3-07 

4547 

22.5 

.89 

.76 

9.1 

.72 

.89 

1.68 

902 

55 

2.17 

•13 

3-65 

3-3 

5288 

25 

.98 

.84 

8-5 

.8 

.98 

1.84 

1 120 

60 

2.36 

.14 

343 

3-5 

6540 

3° 

1. 18 

1.02 

7-5 

•94 

1. 18 

2.16 

1635 

65 

2.56 

•15 

3-23 

3-7 

7 660 

35 

1.38 

1.2 

6.7 

1.08 

1.38 

2.48 

2 218 

70. 

2.76 

.158 

3.06 

3-92 

8893 

4° 

1.58 

1.4 

6 

1.22 

1.58 

2.8 

2912 

75 

2-95 

.166 

2. 92 

4-i3 

10  214 

45 

1.77 

1.56 

5-5 

1.36 

1.77 

3-2 

3674 

80 

3-i5 

.174 

2.76 

4-36 

11 603 

50 

1.97 

1-74 

5-i 

i-5 

1.97 

3-44 

4 547 

85 

3-35 

.183 

2.63 

4-58 

13 100 

55 

2.17 

1.92 

4-7 

1.64 

2.17 

3-76 

5288 

90 

3-54 

.192 

2.51 

4.78 

14  794 

60 

2.36 

2.08 

4.4 

1.74 

2.36 

4.08 

6540 

95 

3-74 

.2 

2.41 

5 

16352 

65 

2.56 

2.26 

4.1 

1.92 

2.56 

4.4 

7 660 

100 

3-94 

.209 

2.31 

5.22 

18  144 

70 

2.76 

2.44 

3-8 

2.06 

2.76 

4-7 

8893 

105 

4-i3 

.22 

2.22 

5*43 

20000 

75 

2-95 

2.6 

3-5 

2.2 

2-95 

5 

10214 

no 

4-33 

.226 

2.13 

5.66 

21950 

80 

13-15 

2.78 

1 3*4 

2-34 

3-i5 

5-35 

11 468 

115 

4-53 

•23 

2.06 

5-87 

23990 

English.  Bolts  and  HSTnts.  (Whitworth's.) 

Hexagonal  Heads  and  Nuts,  and.  Triangular  Threads. 


Diame 

Bolt. 

S’ 

Base  of  r* 
Thread. 

Threads 
per  Inch. 

De 

Head. 

ipth. 

Nut.- 

Width 

of 

Head 

and 

Nut. 

Diam 

Bolt. 

eter. 

Base 

of 

Thread. 

Threads 
per  Inch. 

Dep 

Head. 

th. 

Nut. 

Width 

of 

Head 

and 

Nut. 

— Ins 

Inch. 

No. 

Inch. 

Ins. 

Ins. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

Ins. 

.125 

•093 

40 

.109 

.125 

•338 

1.25 

I.067 

7 

I.O94 

1.25 

2.048 

•1875 

.134 

24 

.164 

•1875 

•448 

1-375 

I.l6l 

6 

1.203 

1-375 

2.215 

.2187 

•25 

.186 

24 

20 

.219 

•25 

•525 

i-5 

1.625 

1.286 

I.369 

6 

5 

i*312 

1.422 

i-5 

1.625 

2.413 

2.576 

.3125 

•375 

.241 

•295 

l8 

l6 

•273 

.328 

•3125 

•375 

.601 

•709 

i*75 

i-875 

I.494 

1-59 

5 

4-5 

I-531 

1.641 

i-75 

1.875 

2.758 

3.018 

•4375 

•346 

14 

.383 

•4375 

.82 

2 

1-715 

4-5 

i-75 

2 

3-T49 

.5 

•393 

12 

•437 

•5 

•9*9 

2.125 

1.84 

4-5 

1.859 

2.125 

3-337 

•5625 

•456 

12 

•492 

•5625 

1. 01 1 

2.25 

i-93 

4 

1.969 

2.25 

3-546 

.625 

.508 

II 

•547 

.625 

I.IOI 

2-375 

2.055 

4 

2.078 

2-375 

3-75 

.6875 

•571 

II 

.601 

.6875 

1. 201 

2-5 

2.18 

4 

2.187 

2-5 

3-894 

•75 

.622 

IO 

.656 

•75 

1.301 

2.625 

2.305 

4 

2.297 

2.625 

4.049 

.8125 

.684 

IO 

•711 

.8125 

i*39 

2-75 

2.384 

3-5 

2.406 

2-75 

4.181 

.875 

•733 

9 

.766 

.875 

1.479 

2.875 

2.509 

3-5 

2.516 

2.875 

4-346 

•9375 

1 

•795 

•84 

9 

8 

.82 

.875 

•9375 

1 

i-574 

1.67 

3 

3-25 

2.634 

2.84 

3-5 

3-25 

2.625 

3 

4-531 

1. 125 

I.942 

7 

.984 

1. 125 

1.86 

3-5 

3.06 

3-25 

1 ~ 

RETENTION  OF  SPIKES  AND  NAILS. 


159 


Sqxiare  Heads  and.  N'nts.  ( Whitworth's .) 


Diameter. 

Threads 
per  Inch. 

Diameter. 

Threads 
per  Inch. 

Diameter. 

Threads, 
per  Inch. 

Bolt. 

Base  of 
Thread. 

Bolt. 

Base  of 
Thread. 

Bolt. 

Base  of 
Thread. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

No. 

Ins. 

Ins. 

No. 

3-75 

3-25 

3 

4-5 

3-875 

2.875 

5-25 

4-4375 

2.625 

4 

3-5 

3 

4-75 

4.0625 

2-75 

5-5 

4.625 

2.625 

425 

3-75 

2.875 

5 

4-25 

2-75 

6 

4-875 

2-5 

"Weight  of  Heads  and  Nuts  in  Lbs.  ( Molesworth .) 
Hexagonal , 1.07  D 3.  Square , 1.35  3 D 3.  D representing  diameter  of  bolt 
in  inches. 


Fieteii.tiven.ess  of  NV' rou.glrt  Iron  Spikes  and.  UNTails. 
Deduced  from  Experiments  of  Johnson  and  Bevan. 

SPIKES. 


43 

Vi 

O O .J 

Spike. 

Wood. 

1 

a, 

g-s 

ill 

0 3 £ 

HI 

Remakks. 

ea 

O 

Qm 

0^73 

as  is 

Ins. 

Ins. 

Ins. 

Lbs. 

Square 

Hemlock) 

•39 

3-5 

1297 

1.58 

Seasoned  in  part. 

V 

it  >H 

Chestnut 

•37 

•38 

3-5 

1873 

2.l6 

Unseasoned. 

It  * 

Yellow  pine 

•375 

•375 

3-375 

2052 

2-37 

Seasoned. 

it  * 

White  oak 

•375 

•375 

3-375 

39IQ 

4-52 

a 

It 

Locust 

•4 

•4 

3-5 

5967 

6-33 

a 

Flat  narrow. . 

Chestnut 

•39 

•25 

3-5 

2223 

3-93 

Unseasoned. 

“ “ 

White  oak 

•39 

•25 

3-5 

3990 

7-05 

Seasoned. 

Locust 

•39 

•25 

3-5 

5673 

9-32 

ti 

“ broad . . 

Chestnut 

•539 

.288 

3-5 

2394 

2.66 

Unseasoned. 

it  n 

White  oak 

•539 

.288 

3-5 

533° 

5-7i 

Seasoned. 

it  u 

Locust 

•539 

.288 

3-5 

7040 

7.84 

ti 

Square)  £• 

Hemlock) 

•4 

•39 

3-5 

1638 

i-75 

Seasoned  in  part. 

Chestnut) 

•4 

•39 

3-5 

1790 

1.81 

Unseasoned. 

“ J 

Locust) 

•4 

•39 

3-5 

3990 

4.17 

Seasoned  in  part. 

Round  and) 
grooved. . J 

Ash 

Diam.  .5 

3-5 

2052 

2.21 

Seasoned. 

u 

U 

U 

•5 

3-5 

2451 

2.41 

U 

it 

White  oak 

U 

.48 

3-5 

3876 

3-2 

u 

* Burden’s  patent.  t Soaked  in  water  after  the  spikes  were  driven. 


NAILS. 


Depth  of 
Insertion. 

Force  required  to  draw  it. 

Pressure  required 

Nail. 

Length. 

Pine. 

Hemlock. 

Elm. 

Oak. 

Beech. 

to  force  them 
into  Pine. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Sixpenny 

2 

I 

187 

312 

327 

507 

667 

235 

tt 

2 

i-5 

327 

539 

571 

675 

889 

400 

a 

2 

2 

530 

857 

899 

1394 

1834 

6lO 

G-eneral  Remarks. 

With  a given  breadth  of  face,  a decrease  of  depth  will  increase  retention. 
In  soft  woods,  a blunt  - pointed  spike  forces  the  fibres  downwards  and 
backwards  so  as  to  leave  the  fibres  longitudinally  in  contact  with  the  faces 
of  the  spike. 


l6o  ANGLES  AND  DISTANCES. DISTANCES  AND  ANGLES. 

To  obtain  greatest  effect,  fibres  of  the  wood  should  press  faces  of  the  spike 
in  direction  of  their  length;  thus,  a round  blunt  bolt,  driven  into  a hole  of 
a less  diameter,  has  a retention  equal  to  that  of  any  other  form,  when  who  y 

drThe’  retei^tlon^of^ a spike,  whether  square  or  flat,  in  unseasoned  chestnut, 
from  two  to  four  inches  in  length  of  insertion,  is  about  800  lbs.  per  square 
inch  of  the  two  surfaces  which  laterally  compress  the  faces  of  the  spike. 

When  wood  was  soaked  in  water,  after  spikes  were  driven,  order : of  their 
retentive  power  was  Locust,  White  oak,  Chestnut,  Hemlock,  and  1 ellow  Pme. 


Threads  per  Inch . . . 


1 -I2S  1 

•25  I 

•375  I 

•5  I 

•75  I 1 I 

j 1.25  I 1-5  I 1-75  1 

1 28 1 

19  1 

19  1 

14  1 

14  1 11  | 

| 11  1 11  1 11  1 

ANGLES  AND  DISTANCES. 

Angles  and.  Distances  corresponding  to  Opening 
. ® Rule  of  Two  Feet. 


of  a 


Angle.  | 

Distance. 

Angle. 

O 

Ins. 

0 

I 

.2 

19 

2 

• .42 

20 

3 

.63 

21 

4 

.84 

22 

5 

1.05 

23 

6 

I.26 

24 

7 

1.47 

25 

8 

I.67 

26 

9 

1.88 

27 

10 

2.09 

28 

11 

2-3 

29 

12 

2.51 

30 

13 

2.72 

31 

14 

2.92 

32 

i5 

3-i3 

33 

16 

3-34 

34 

17 

3-55 

35 

18 

3-75 

36 

Distances 

and 

1 Distance 


Distance. 


Ins. 

7.61 

7.81 

8.01 

8.2 

8.4 

8.6 

8.8 

8.99 

9.18 

938 

9-57 

9.76 

9-95 

10.14 

10.33 

10.52 

10.71 

10.9 


Angle. 


55 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 


Distance. 


Ins. 

11.08 
II.27 
11-45 
II.64 
11.82 
12 

I2.l8 

12.36 

12.54 

12.72 

12.9 
I3.O7 
13-25 

1342 

13.59 

13.77 

13.94 

i4.II 


Angle.  | Distance. 


73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 


Ins. 

I4.28 

I4.44 

14.61 

14.78 

14.94 

i5.II 

15.27 

15-43 

15.59 

1575 

15-9 

16.06 

16.21 

16.37 

16.52 

16.67 

16.82 

16.97 


Rule  of  Two  Feet. 


to  Opening  of  a 


Distance. 


Ins. 

.25 

•375 

•5 

.625 

•75 

.875 

1 

1.25 
i-5 

1- 75 

2 

2.25 

2.5 

2- 75 


Angle. 


Distance. 


1. 12 

I. 48 
2.24 
2-59 
3*35 
4.12 
448 
5.58 
7*1 
8.22 

9-34 

10.46 

II. 58 

13.1 


Ins. 

3 

3-25 

3.5 

3- 75 

4 

4.25 

4- 5 

4- 75 

5 

5.25 

5- 5 
5-75 

6 

6.25 


Angle. 


Distance. 


14.22 

15.34 

16.46 

17.58 

19.11 
20.24 
21.37 
22.5 
24.4 
25.16 
26.3 
2744 

28.58 

30.12 


Ins. 

6.5 

6- 75 

7 

7.25 

7- 5 
7-75 

8 

8.25 

8.5 
8.75 
9 

9.25 
95 
9-75 


Angle. 


3I26 

32- 4 

33- 54 
35.09 
36.24 
37*4 

38.56 

40.12 

41.28 

42.46 

44.2 

45-2 

46.38 

47.56 


Distance. 


Ins. 

10 

10.25 

10- 5 

10.75 

11 

11.25 

11- 5 
n-75 

12 

12.25 
12.5 

12.75 

13 

13.25 


Angle. 


49*  x4 
50.34 
51.54 
53.14 
54*34 
55-54 
57.i6 

58.38 

60 

61.23 

62.46 

64.1 

65-36 

67.02 


Distance.  ] Angle. 


Ins. 

13- 5 

13.75 

14 

14.25 

14- 5 
14-75 

15 

15.25 

15.5 
15*75 

16 

16.25 

16.5 

16.75 


68.28 

69-54 

71.22 

72-5 

74.2 
75.5 

77*22 

78-54 

80.28 

82.2 
83-36 
85.14 

86.52 

88.32 


WIRE  ROPE. 


161 


WIRE  ROPE. 

Wire  rope  of  same  strength  as  new  Hemp  rope  will  run  on  sheaves 
of  like  diameter ; but  greater  diameter  of  sheaves,  less  the  wear.  Short 
bends  should  be  avoided,  and  wear  increases  with  the  speed.  Adhesion 
is  same  as  that  of  hemp  rope.  It  should  not  be  coiled,  but  should  be 
wound  as  upon  a reel. 

When  substituting  wire  rope  for  hemp,  it  is  well  to  allow  for  former 
same  weight  per  foot  which  experience  has  approved  of  for  latter.  As 
a general  rule,  one  wire  rope  will  outlast  three  of  hemp*.  To  guard 
against  rust,  stationary  rope  should  be  coated  once  a year  with  linseed- 
oil,  or  well  painted  or  tarred.  Running  rope  in  use  does  not  require 
any  protection. 

Where  great  pliability  is  required,  centre  or  core  of  rope  should  be 
of  hemp. 

Annealing  wire,  in  rendering  it  more  pliable  than  when  unannealed, 
reduces  its  elasticity  and  consequent  strength  from  25  to  50  per  cent. 

Running  rope  is  made  of  finer  wire  than  standing  rope. 

For  safe  working  load,  deduct  one  fifth  to  one  seventh  of  ultimate 
strength,  according  to  speed  and  vibration.  It  is  better  to  increase  load 
than  speed,  as  it  increases  wear. 

Standing  rigging  of  a vessel  of  wire  rope  is  one  fourth  less  in  weight 
than  when  of  hemp. 

Rope  of  19  wires  to  a strand  is  more  pliable  than  one  of  7 and  12 
wires,  and  hence  it  is  better  suited  to  operation  over  small  drums,  for 
hoisting,  etc. 

Ultimate  strength  of  iron  ropes  is  4480  lbs.  for  each  pound  in  weight 
per  fathom,  and  for  galvanized  steel  6720  lbs. 

Strength  per  square  inch  of  section  of  a rope  is  about  53  per  cent,  of 
an  equal  section  of  solid  metal  of  same  tensile  strength  per  square  inch. 

Steel  ropes  may  be  one  third  less  in  weight  than  iron  for  same  load. 
Their  durability  is  much  greater,  especially  when  required  to  run  rapidly 
over  sheaves.  Hemp  should  be  one  third  heavier  than  iron. 

Steel  wire  No.  14  W.  G.  = .083  inch,  weight  2 lbs.  per  yard,  will  bear  a 
stress  of  2000  lbs. 

The  combined  sectional  area  of  the  wires  in  a cable  is  to  the  area  of  the 
cable  as  1 to  1.3.  Hence,  to  ascertain  areas  of  the  wires  in  a cable  multiply 
diameter  by  .77,  and  for  areas  of  the  voids,  multiply  area  of  cable  by  .23. 

In  short  transmissions,  it  is  necessary  to  connect  rope  quite  taut,  and  an 
additional  diameter  of  two  numbers  of  rope  must  be  given  to  it. 

In  long  transmissions,  when  there  is  an  insufficiency  of  height  to  admit 
of  a proper  deflection  of  rope,  and  it  becomes  necessary  to  connect  it  very 
taut,  an  additional  diameter  of  one  number  of  rope  must  be  given  to  it. 

When  distance  exceeds  350  feet,  transmission  should  be  divided  into  two 
or  more  equal  lengths  by  aid  of  intermediate  wheels. 

Rope  Nos.  7 and  8 (. Roebling's ) are  made  with  Nos.  1 and  2 as  strands, 
and  twisting  six  of  them  around  a hemp  centre. 

Results  of  an  Experiment  with  Galvanized  Wire. 

A strand  of  2-inch  wire  rope  broke  with  a strain  of  13  564  lbs.,  and  a 
piece  of  a like  rope,  when  galvanized,  withstood  a strain  of  14  796  lbs.  be- 
fore breaking. 

0* 


1 62 


WIRE  ROPES. 


Elements  of  Running  and  Standing  Wire  Rope. 
J.  A.  Roebling's  Sons  Co.,  New  York. 

19  Wires  in  a Strand. 


Diam. 

Circum.  j 

Iro> 

Breaking 

Weight. 

r. 

Safe 

Load. 

Circum. 
of  Hemp 
Rope. 

Weight 

per 

Foot. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

• 5 

1-5 

6960 

I OOO 

3-5 

•35 

.5625 

I.625 

• 8 480 

I 500 

4 

•44  1 

.625 

2 

10  260 

2 5OO 

4-5 

•7 

•75 

2.25 

17  280 

3 500 

5 

; .88 

.875 

2.75 

23  000 

5 ooo 

6 

1.2 

1 

3-I25 

32  000 

6000 

7 

i-58 

1. 125 

3-5 

40  000 

8 ooo 

8 

2 

1.25 

4 

54000 

II  ooo 

9-5 

2.5 

i-5 

4-375 

70  000 

14  ooo 

io-75 

3-65 

1.625 

5 

88  000 

18  ooo 

12 

4*1 

x-75 

5-5 

108  000 

22  OOO 

13 

5-25 

2 

6 

130  000 

26  OOO 

14-5 

6.3 

2.25 

6.75 

148  000 

30000 

15-5 

8 It 

Break  mg 
Weight. 

Lbs. 

II  OOO 
13  OOO 
18  OOO 
26  OOO 
40  OOO 
48  OOO 
60000 
78  OOO 

IIO  OOO 

128  OOO 
156  OOO 

200  OOO 

260000 


Cast  Steel. 

Circum. 
of  Hemp 
Rope. 


Safe 

Load. 


Lbs. 

2 OOO 

3 000 
4 OOO 
6006 
8 000 

10000 
12  000 
16  000 
22  000 
26  000 
34000 
40000 
44  000 


Wires  in  a Strand. 


Diam. 


Cast  Steel. 

Circum. 
of  Hemp 
Rope. 


Safe 

Load. 


Lbs. 

I 050 

1 25O 

2 OOO 

2 4OO 
2 500 

3 500 

45OO 
6000 
7 000 
10000 
13  000 
16  000 
20  000 
25  000 
32  000 


Weight 

per 

Foot. 


Ins. 

2.375 

3 

3- 75 

4- I25 

4- 75 
5 

5- 5 

6- 5 

7- 25 

8- 5 
10 

10.75 

12 

13 

15 


Lbs. 

.125 
.16 
.19 
.23 
•31 
.41 
.51 
.68 
.86 
1. 12 
i.5 
1.82 
2.28 

2.77 

3- 37 


r A 021;  72  OOO  i 10  cue  /O  0*0/  ^ 

Note. -When  made  with  wire  centre  instead  of  hemp,  weight  is  ,o  per  cent.  more. 


Galvanized  Charcoal  Iron  Wire. 

Rigging  and  Derrick  Grays. 

12  Wires  in  a Strand. 


"Vessels’ 


Circum. 

Circum. 
of  Hemp 
Rope. 

Breaking 

Weight. 

Ins. 

Ins. 

Lbs. 

3 

6 

24OOO 

3-25 

6-5 

28000 

3-5 

7 

32  OOO 

3-75 

7-5 

40  OOO 

4 

8 

460OO 

4-25 

8.5 

52  OOO 

Safe 

Load. 

Lbs. 
6000 
7 OOO 
8000 
10000 
II 500 
13000 


Weight 

Circum.  1 

Breaking 

Safe 

per 

Foot. 

Circum. 

of  Hemp 
Rope. 

W eight. 

Load. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Lbs. 

1-33 

4-5 

9 

60  000 

15000 

x-58 

4-75 

9-5 

66000 

1.6  500 

I.83 

5. 

10 

7OOOO 

17  500 

2 

5-25 

10.5 

80  000 

20  000 

2.46 

5-5 

11 

86000 

21  500 

2.66 

6 

12 

IOO  OOO 

25000 

I Weight 

I Pttr 

F oot. 


Lbs. 

3 

3- 46 
366 
4.12 
4.46 

4- 83 


WIRE  ROPES  AND  CABLES. 


163 


Galvanized  Charcoal  Iron. 
"Vessels’  .Rigging  and  Derrick  Grays. 
J.  A.  Roebling's  Sons  Co .,  New  York . 

7 Wires  in  a Strand. 


Circum. 

Circum. 
of  Hemp 
Rope. 

Breaking 

Weight. 

Safe 

Load. 

Weight 

per 

Foot. 

Circum. 

Circum. 
of  Hemp 
Rope. 

Breaking 

Weight. 

Safe 

Load. 

Weight 

Foot. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

I 

2 

4000 

I OOO 

.125 

3-5 

7 

32  000 

8000 

I.79 

1.25 

2.5 

5000 

I 250 

•25 

3-75 

7-5 

40  000 

10  OOO 

2 

i-5 

3 

7000 

1750 

•334 

4 

8 

460OO 

II 500 

2.46 

i-75 

3-5 

IO  OOO  | 

I 2500 

.427 

4-25 

8-5 

52  000 

13000 

2.67 

2 

4 

14000 

3 500 

.583 

4-5 

9 

60  000 

15000 

3 

2.25 

4-5 

16000 

4000 

.708 

4-75 

9-5 

66000 

16500 

3-42 

2-5 

5 

18000 

4 500 

•875 

5 

10 

70  000 

17500 

3.66 

2-75 

5-5 

20  000 

5000 

1. 17 

5-25 

10.5 

80  000 

20  000 

4.08 

3 

6 

24OOO 

6000 

i*33 

5-5 

11 

86000 

21  50O 

4.41 

3-25 

6-5 

28  000 

7 000 

1.58 

6 

12 

IOO  OOO 

25  000 

4.81 

Grange,  'Weight,  and  Length  of  Iron  "Wire. 


No. 

6/0 

5/o 

4/0 

3/0 

2/0 

1/0 


Diann. 

Weight 
per  100 
Feet. 

= 

% -a 
— c 
ro  3 

S3 

| Area. 

Gauge. 

Diam. 

Weight 
per  100 
Feet. 

Weight 
of  one 
Mile. 

63  lbs. 
Bundle. 

Area. 

Inch. 

Lbs. 

Lbs. 

Feet. 

j Sq.  Inch. 

No. 

Inch. 

Lbs. 

Lbs. 

Feet. 

Sq. Inch. 

.46 

56.I 

2962 

1 12 

.166  19 

3:6 

•063 

1.05 

55 

6000 

.003  II 7 

•43 

49.OI 

2588 

129 

.145  22 

17 

•054 

•77 

41 

8 182 

.002  29 

•393 

4O.94 

2l62 

154 

.121304 

18 

•047 

.58 

31 

IO862 

.OOI  734 

.362 

34-73 

1834 

l8l 

.102  921 

19 

.041 

•45 

24 

I4OOO 

.OOI  32 

•33 1 

29.04 

1533 

217 

.086  O49 

20 

•035 

•32 

17 

19687 

.OOO  962 

•307 

27.66 

1460 

228 

.O74O23 

21 

.032 

•27 

14 

23333 

.OOO  804 

.283 

21.23 

1 121 

296 

.062  9OI 

22 

.028 

.21 

II 

30  000 

.000  61 5 

.263 

18.34 

968 

343 

•054  325 

23 

.025 

•175 

9.24 

360OO 

.000  491 

.244 

I5-78 

833 

399 

.046  759 

24 

•023 

.14 

7-39 

45  000 

.000415 

.225 

13-39 

707 

470 

i -039  76 

25 

.02 

.116 

6.124 

54  310 

.000314 

.207 

ir-35 

599 

555 

•033653 

26 

.018 

•093 

4.91 

67  742 

.000  254 

.192 

9-73 

5i4 

647 

.028952 

27 

.OI7 

•083 

4.382 

75  903 

.000227 

.177 

8.03 

439 

759 

.024  605 

28 

.Ol6 

•074 

3907 

85  135 

•OOO  201 

.162 

6.96 

367 

905 

.020612 

29 

.015 

.061 

3.22 

IO3  278 

.OOO  176 

.148 

5.08 

306 

1086  j 

.017  203 

30 

.OI4 

•054 

2.851 

1 16  666 

.OOO  I54 

•i35 

4-83 

255 

r3°4 

•014  3L3 

31 

•0135 

•05 

2.64 

126000 

.OOO  I33 

.12 

3.82 

202 

1649 

.011  309 

32 

•013 

.046 

2.428 

136956 

.OOO  I32 

.105 

2.92 

154 

2158; 

.008  659 

33 

.Oil 

•037 

I-953 

170270 

•OOO  O95 

,092 

2.24 

118 

2813 

.006647 

34 

.OI 

•03 

1.584 

210000 

.OOOO78 

.08 

1.69 

89 

3728: 

.005  026 

35 

•OO95 

•025 

1.32 

252  000 

.OOO  07I 

.072 

i-37 

72 

4598: 

.004071 

36 

.009 

.021 

1.161 

286  363 

.OOO064 

Galvanized  Steel  Cables  for  Suspension  Bridges. 


Diameter. 

Ultimate 

Strength. 

Weight 
per  Foot. 

Diameter. 

Ultimate 

Strength. 

Weight 
per  Foot. 

Diameter. 

I Ultimate 
Strength. 

Weight 
per  Foot. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

i-5 

130  000 

3-7 

1-875 

200  OOO 

5-8 

2-375 

360OOO 

IO 

1.625 

150  OOO 

4-35 

2 

220  000 

6-5 

2.5 

400  000 

II.3 

i-75 

19OOOO 

5-6 

2.25 

310000 

8.64 

2.625 

44OOOO 

13 

IRON,  STEEL,  AND  HEMP  HOPE. 

"Weight  and.  Strength,  of  Single  Strand  and  Cable 
laid  Fence  "Wire.  [F.  Morton  & Co.) 

Length 
per  iooo  lbs* 

Of  a Of 

Rope. 


Strands. 

No. 

Single  Wire 
of  equal 
Diameter. 

No. 

No. 

Inch. 

3 

2A 

8 

.159 

4 

2 

7 

.174 

7 

I 

6 

.191 

7 

O 

5 

.209  1 

Strand. 


Feet. 
20  090 
14  730 
13  I25 


Feet. 

15  270 
12  79O 
IO  580 

8928 


1 

No. 

Single  Wire 
of  equal 
Diameter. 

Len 
per  io< 
Of  a 
Strand. 

gth 
x>  lbs. 
Of 

Rope. 

No. 

Inch. 

Feet. 

Feet. 

I OO 

4 

.229 

83OO 

7366 

3/0 

3 

•25 

8036 

6228 

U/O 

2 

•274 

7500 

1 5/0 

1 

•3 

5090 

4286 

No.  and  diameter  of  wire  is  that  of  Ryland  s Bros.,  pp.  122  4. 

Hemp,  Iron,  and.  Steel.  (K.  f bewail  <&  Co.) 

ROUND. 


HEMP. 

| Weight 

Circumference.  per  1 

Foot. 

Ins. 

Lbs. 

2.75 

•33 

3-75 

.66 

4-5 

.83 

5-5 

1. 16 

6 

i-5 

6.5 

1.66 

7 

2 

7-5 

2.33 

8 

2.66 

8.5 

3 

9-5 

3.66 

IO 

4-33 

11 

5 

12 

5.66 

Dimensions. 

4 X .5 

3-33] 

5 Xi.25 

4 

5-5  X 1.375 

4-33 

5-75XI.5 

4.66 

6 X 1.5 

5 

7 X 1.875 

6 

8.25X2.125 

6.66 

8.5  X2.25 

7-5 

9 X2.5 

8.33 

9-5  X 2-375 

: 9-I6 

10  X2.5 

10 

IRON. 

cumference.  J 

Weigh  J 
per  I 
Foot. 

STEEL. 

I1 

Circumference. 

Weight 

£o\. 

Ins. 

Lbs. 

Ins. 

Lbs. 

I 

.16 

— 

— 

i-5 

•25 

I 

.16 

1.625 

•33 

— 

i-75 

.42 

i-5 

•25 

1.875 

•5 

— 

— 

2 

.58 

1.625 

•33 

2.125 

.66 

1-75 

.42 

2.25 

•75 

— 

2.375 

.83 

1.875 

•5 

2-5 

.92 

— 

2.625 

1 

2 

.58 

2.75 

1.08 

2.125 

.66 

2.875 

1. 16 

2.25 

•75 

3 

1.25 

— 

— 

3I25 

i.33 

2.375 

.83 

3*25 

1.41 

— 

3.375 

i.5 

2.5 

.92 

3-5 

1.66 

2.625 

1 

3-625 

1.83 

2.75 

1.08 

3-75 

2 

— 

3.875 

2.16 

3-25 

1*33 

4 

2-33 

— 

— 

4.25 

2.5 

3*375 

i-5 

4*375 

2.66 

— 

— 

4-5 

3 

3.5 

1.66 

4.625 

333 

1 3-75 

2 

Tensile  Strength. 

Ultimate 
Strength. 


Lbs. 

672 

I 008 

1344 

1 680 

2 016 
2 352 
2 688 
3024 

3360 

3696 

4032 

4368 

4704 

5040 

5 376 
5672 
6048 

6 720 

7 392 
8064 
8736 
9408 

10080 
10  752 
12096 
13440 


FLAT. 


Dimensions. 

2.25  x .5 

2.5  x .5 

2.75  X.625 

3 X .625 

3.25  X.625 

3.5  X.625 

3- 75  X.6875 

4 X .6875 

4- 25  X.75 
4-5  X .75 
4.625X.75 


Dimensions. 


1.85  I 

— 

— 

4928 

2.16 

— 

— 

5824 

2-5 

— 

— 

6 720 

2.66 

2 x .5 

1.66 

7 168 

3 

2.25  x. 5 

1.83 

8064 

3*  33 

2.25  X- 5 

2 

8960 

3.66 

2.5  x .5 

2.16 

9850 

4.16 

2-75  X- 375 

2-5 

11  200 

4.66 

3 X .375 

2.66 

12544 

5-33 

3.25  X. 375 

3 

14  336 

5.66  1 

3-5  X.375 

3*2  3 

I5232 

Lbs. 

4480 
6 720 
8960 
11  200 
13440 
15  680 
17920 
20  160 
22400 
24640 
26680 
29  120 

3*36° 

33600 
36840 
38  080 
40320 
44800 
49  280 
53  76° 
58240 
62  720 
67  200 
71  680 
80640 
89600 


44800 
51520 
60480 
62  720 
71 680 
80640 
89600 
100800 
112000 
125440 
134400 


ROPES  AND  CHAINS. 


165 


From  preceding  tables  following  results  are  determined: 


Ultimate  Strength 

Safe  Load 

per  Lb.  Weight  per 

per  Lb.  Weight  per 

per  Square  of  Circum- 

Foot. 

Foot. 

ference  in  Inches. 

Lbs. 

Lbs. 

Lbs. 

Hemp 

15  OOO 

4550 

IOO 

Iron 

22  OOO 

4500 

600 

Steel 

( 3OOOO 

f 6000 

r 1000 

(45  500 

\ 8000 

(1300 

ROUND  AND  FLAT  MINING  ROPES. 
(MM.  Harmegnies , Dumont  <&  Co.,  Anzin,  France.) 
For  a Depth  of  400  Metres  or  440  Yards, 


Round. 

Flat. 

No. 

Diameter. 

Weight 
per  Foot. 

Safe  Load. 

No.  of 
Strands. 

Width. 

Thick- 

ness. 

Weight 
per  Foot. 

Safe  Load. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Lbs. 

17 

♦51 

2.l6 

560 

9 

2.4 

•55 

2 

3 360 

16 

•59 

1.66 

1120 

6. 

2.8 

•59 

2.13 

4032 

15 

•63 

1.26 

1680 

/ 

3-2 

•63 

2.66 

4 480 

14 

•7i 

1 

2240 

3*2 

.67 

3 

5600 

13 

.83 

.83 

3360 

! 3-5 

•79 

3-33 

6 720 

12 

.98 

.66 

4480 

f 4-3 

.67 

3.66 

7840 

II 

1. 1 

•5 

5600 

0 

3-9 

.83 

4 

8 960 

IO 

i-3 

•33 

6720 

8 

4-7 

•79 

4-33 

10080 

8 

5-i 

.87 

5-33 

11  200 

Ropes  and.  Ch.ai.ns  of  Equal  Strength. 


Diameter 

of 

Iron  Chain. 

CIRCI 

Hemp 

Rope. 

UMFEREJs 

Crucible 

Steel 

Rope. 

CE. 

Charcoal 

lion 

Rope. 

Steel 

Rope. 

WEI 

Iron 

Rope. 

GET  PER  ] 

Hemp 

Rope. 

700T. 

Iron 

Chain. 

Safe 

Load. 

Ins. 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs, 

Lbs. 

Tons. 

.218  75 

2.75 

— 

I 

— 

.14 

•34 

•5 

•3 

•25 

3 

— 

1. 18 

.21 

.46 

•65 

•4 

.281  25 

3-5 

I 

i-39 

•17 

.28 

.67 

.81 

•5 

•312  5 

4-25 

I.26 

i-57 

•25 

•33 

•75 

.96 

.6 

•375 

4-5 

i-45 

1.77 

•3 

•45 

•S3 

1.38 

.8 

•437  5 

5 

i-57 

1.97 

•35 

•57 

1. 16 

I.76 

1 

.468  75 

5-5 

1.77 

2.19 

•45 

•7 

1.2 

2.2 

1-3 

•5 

5-75 

1.96 

2.36 

•59 

•83 

1.6 

2.63 

1.5 

.625 

6-75 

2.36 

2-75 

.85 

1.08 

2 

4-2i 

2-3 

.6875 

7-75 

2-75 

3-I4 

1. 1 

i-43 

2.65 

4-83 

3-i 

•75 

8-75 

2-95 

3-53 

1.28 

1.8 

3-35 

5-75 

3-8 

•875 

9-75 

3-i4 

3-93 

i-45 

2-3 

4.6 

7-5 

4.8 

•937  5 

10.5 

3-53 

4-32 

1.83 

2.94 

4.92 

9-33 

5-9 

1.062  5 

“"•75 

3-93 

4.71 

2-33 

3-56 

5.83 

10.6 

7 

1. 125 

“•75 

4-32 

5-i 

2.98 

4 

6.2 

11.9 

8.2 

1.25 

14-75 

4.71 

5-5 

3-58 

4.8 

8.7 

14-5 

9-5 

1-375 

15-25 

4.81 

5-89 

3-65 

5-6 

9 

17.6 

11 

i-5 

I5-75 

5-i 

6.28 

4.04 

6.3 

10. 1 

20 

!2.5 

1.625 

17-75 

5-8 

7.07 

5.65 

7-95 

13-7 

22.3 

15-9 

i-75 

19-5 

6-35 

7.85 

6-5 

9.81 

16.4 

24-3 

19.6 

By  experiments  of  U.  S.  Navy,  hemp  rope  of  this  circumference  has  a breaking 
weight  0/71 309  lbs.,  and  a wire  rope  of  5.34  ins.  has  equivalent  strength. 


1 66  WEIGHT,  STRESS,  AND  TENSION  OF  ROPES. 


Circum- 

ference. 


Weight  of  Hemp  and.  Wire  Rope. 

In  Lbs.  per  Fathom. 

Hemp.  Wire. 


(Molesworth.) 


Common. 


Good. 


Ins. 

1 

i-5 

i-75 

2 

2.25 

2*5 

2.75 

3 

3-25 

3-5 

3-75 

4 


Lbs. 

.18 

.41 

•55 

.72 

.91 

1.13 

1.36 

1.62 

1.9 

2.21 

2.53 

2.88 


Lbs. 

.24 

•54 

•74 

.96 

1.22 

i-5 

1.82 

2.16 

2.54 

2.94 

3-38 

3-84 


Iron. 


Steel. 


Lbs. 

.87 

I.96 

2.66 
3-48 
4.4 
5-44 
6.58 
7-83 
9-I9 

10.66 
12.23 
13.92 


Lbs. 

.89 

2 

2- 73 

3- 56 

4- 5i 

5- 56 

6- 73 
8.01 

9*4 

10.9 

12.52 

14.24 


Circum- 

ference. 

Heu 

Common. 

4P. 

Good. 

Ins. 

Lbs. 

Lbs. 

5 

4-5 

6 

5-5 

5-45 

7.26 

6 

6.48 

8.64 

6-5 

7.61 

10.14 

7 

8.82 

11.76 

7-5 

10.13 

13,5 

8 

II.52 

15-3^ 

8-5 

13-05 

17-34 

9 

14.58 

19.44 

10 

18 

24 

12 

26 

34-56 

15 

4O.52 

54 

To  Compute  Stress  upon  a Rope  set  at  an  Inclination. 

Rule.— Multiply  sine  of  angle  of  elevation  by  strain  in  lbs.,  add  an  allow- 
ance for  rolling  friction  and  weight  of  rope,  and  multiply  by  factor  of  safety . 

Factor  of  safety. -Vox  standing  rope  4,  for  running  5,  and  for  inclined 
planes  from  5 to  7. 

an^=.^ 

wldchis  tUelddld  rolb^Tric^rand'weightof  rop^assurnedm  be  n;  hence, 

3 Factor  of  safety  assumed  at  6,  consequently  1375  X 6 =;  8250  lbs. , capacity  or  break- 
inq  weight  or  stress  of  rope.  „ A . 

By  table,  page  162,  8200  lbs.  is  breaking  weight  of  a wire  rope  of  7 strands,  .625 
inch  in  diam. 

To  Compute  Tension  of  a Rope. 

= L r representing  velocity  of  rope  in  feet  per  minute,  H>  horses' power, 
and  t tension  in  lbs. 

Illustration -Assume  wheel  7 feet  in  diameter,  revolution  140  per  minute,  and 
BP  as  per  preceding  table,  29.6. 

Then  29-6  X 33ooo__  976800  _ m 

7 X 3-1416  X r4o  3079 

To  Compute  Operative  Reflection,  of  a Rope. 

1)2  w X)  representing  distance  between  centres  of  wheels  or  drums  m 

feet^J weight  of  rope  in  feet  per  lb .,  t tension , or  power  required  to  produce 
required  power  or  tension  of  rope  when  at  rest , and  d deflection  m feet. 

Illustration. -Take  elements  of  preceding  case:  diam.  of  wire  rope  of  7 strands 
= . 5625  inch,  and  by  table,  page  162,  w = .41  lb. , and  D — 300  teet. 

3002  X -41  


Then 


10.7  X 3*7-2 


: 10.87  feet. 


Capacity. — At  the  Falls  of  the  river  Rhine  there  is  a wire  rope  in  operation 
that  transmits  the  power  of  600  horses  for  a distance  exceeding  one  mi  . 


TRANSMISSION  OF  POWER  AND  EQUIVALENT  BELT.  1 6/ 


Endless  Ropes. 

Wire  Ropes,  when  practicable  and  proper  for  application,  can  be  used  for 
transmission  of  power  at  a less  cost  than  belting  or  shafting. 


Transmission  of  Power. 


Diameter 
of  Wheel. 

Revolu- 
tions per 
Minute. 

Diameter 
of  Rope. 

Horse 

Power. 

Diameter 
of  Wheel. 

Revolu- 
tions per 
Minute. 

Diameter 
of  Rope. 

Horse 

Power. 

Diameter 
of  Wheel. 

Revolu- 
tions per 
Minute. 

Diameter 
of  Rope. 

Horse 

Power. 

Feet. 

4 

80 

Ins. 

•375 

3-3 

Feet. 

7 

IOO 

Ins. 

.5625 

21. 1 

Feet. 

II 

I40 

Ins. 

.6875 

I32.I 

4 

IOO 

•375 

4.1 

7 

140 

.5625 

29.6 

12 

80 

•75 

99-3 

4 

120 

•375 

5 

8 

80 

.625 

22 

12 

IOO 

•75 

1 24. 1 

4 

14O 

•375 

5-8 

8 

IOO 

.625 

27-5 

12 

I40 

•75 

173-7 

5 

80 

•4375 

6.9 

8 

140 

.625 

38.5 

13 

80 

•75 

122.6 

5 

IOO 

•4375 

8.6 

9 

80 

.625 

4i-5 

13 

IOO 

•75 

153-2 

5 

120 

•4375 

10.3 

9 

IOO 

.625 

5i-9 

A3 

120 

•75 

183.9 

5 

140 

•4375 

12.1 

9 

140 

.625 

72.6 

14 

80 

.875 

148 

6 

80 

•5 

10.7 

10 

80 

.6875 

58.4 

14 

IOO 

.875 

176 

6 

IOO 

•5 

13-4 

10 

IOO 

.6875 

73 

14 

120 

.875 

222 

6 

120 

•5 

16.1 

10 

140 

.6875 

102.2 

15 

80 

.875 

217 

6 

140 

•5 

18.7 

11 

80 

.6875 

75-5 

15 

IOO 

•875 

259 

7 

80 

•5625 

16.9 

11 

IOO 

.6875 

94.4 

15 

120 

.875 

300 

Wire  Rope  and  Ecini valent  Belt. 

Li  substituting  wire  rope  for  an  ordinary  flat  belt,  the  diameter  is  deter- 
mined by  rule  in  practice  for  estimating  power  transmitted  by  a belt — viz., 

One  horse  power  for  every  70  square  feet  of  running  belt  surface  per 
minute.  Thus,  a belt  15  inches  wide  running  at  rate  of  1400  feet  per  min- 
ute, its  power  would  be  equal  to  (1400  X 15)  — (70  X 12)  = 25  horses’  power. 

The  same  result  is  obtained  by  the  use  of  a wire  rope  .5625  inch  in  diam- 
eter, running  over  a wheel  6 feet  in  diameter,  making  130  revolutions  per 
minute. 

Average  life  of  iron  wire  rope  with  good  care  is  from  3 to  5 years,  and 
that  of  steel  rope  is  greater.  Wear  increases  rapidly  with  velocity. 

Greneral  Notes. — Kemp  and  Wire  Ropes. 

White  Hope,  2 inches  in  circumference,  of  different  manufactures,  parted  at 
a stress  of  from  4413  to  6160  lbs. 

Specimens  of  Italian,  Russian,  and  French  manufacture  parted  with  an 
average  stress  of  5128  lbs.  = 1633  lbs.  per  square  inch  of  rope. 

Bearing  capacity  of  a hemp  rope  is  proportional  to  its  thickness,  number 
of  its  strands,  slackness  with  which  they  are  twisted,  and  quality  of  the 
hemp. 

Hemp  and  Wire  Ropes. — Ultimate  Strength  is  2240  lbs.  per  lb.  per  fathom 
for  round  hemp,  4480  lbs.  for  iron,  and  6720  to  7840  lbs.  for  steel. 

Working  Load  is  336  lbs.  per  lb.  weight  per  fathom  for  round  hemp,  672 
lbs.  for  iron,  and  1120  lbs.  for  steel. 

Or,  .83  times  square  of  circumference  in  inches  for  round  hemp,  5 times 
square  of  circumference  for  iron,  and  9 times  square  of  circumference  for 
steel.  (D.  K.  Clark.) 

Steel  Ropes  may  be  one  third  less  in  weight  than  iron  for  like  working 
load,  and  Hemp  Ropes  should  be  one  third  heavier  than  iron  for  like  work- 
ing load. 


1 68 


ropes  and  chains. 


IRON  WIRE  AND  UNITED  STATES  NAVY  HEMP  ROPE. 
Wire  6 Strands , Hemp  Core.  Rope  4 Strands. 

WIRE. 


Circumference. 


4-937 

4-375 

3-5 

3-i87 

2.75 

2.5 

2-375 


Nominal.j 


4.9 

4-5 

3-36 

2.98 

2.68 

2.45 

2.4 

2.06 


Core. 

Ins. 

2*35 

2.25 

i-57 

i-57 

1.27 

1.17 

.78 

.78 

.78 

•39 


Wires. 


No. 

108 

108 

114 

114 

114 

114 

114 

114 

42 

114 


Breaking 
Weight.  | 

Circui 

Actual. 

Lbs. 

Ins. 

187400 

1 12 

104  050 

11 

65409 

! 10.5 

55  316 

10 

34  480 

9 5 

28  606 

1 9 

21  846 

8*5 

15692 

8 

15718 

7‘5 

IO925 

II  7 

HEMP. 


Lbs. 

75  966 
77  633 
76933 
70  533 
58  766 
56  466 
42  866 
38  500 

40  OCX) 

32  166 


Weight  and  Strength,  of  Stud-lmk  Cham  Cable. 
(English.) 

Dimensions. 

Width 
of 

Link. 


Dimensions. 


Diarn. 
of  each 
Side. 


Ins. 

-4375 

•5 

.5625 

.625 

.6875 

•75 

.875 


Length 

of 

Link. 


1. 125 

1.25 

1-375 


Tns. 

2.625 

3 

3-375 

3- 75 

4- i25 

4- 5 

5- 25 
6 

6.75 
!'  7-5 
; 8.25 


Width 

of 

Link. 


Ins. 

i-575 

1.8 

2.025 

2.25 
2.475 
2.7 
3-i5 

3- 6 

4- 05 
I 4-5 

4-95 


Weight 


Lbs. 

11 -3 
13-4 

17.2 
21 
25-4 

30.2 
41.2 
53-8 

69 

84 

101.6 


Tons. 

3- 5 

4- 5 

5- 5 
7 

8*5 

10.125 
13-75 
18 

22.75 

28.125 

34 


Dimensions. 
Diam. 
of  each 


Side. 


Ins. 

i-5 

1.625 

i-75 

1.875 


2.125 

2.25 

2-375 

2.5 

2.75 


Length 

of 

Link. 


Ins. 

9 

9-75 

10.5 

11.25 


Weight 

per 

Fathom. 


12 


12- 75 

13- 5 
14.25 
15 
16.5 


Ins. 

54 

5- 85 

6- 3 

6- 75 

7.2 

7- 65 

8.1 

8- 55 
9 

9- 9 


Admiralty 
Proof-stress 
(adopted  by 
Lloyds’). 


Lbs. 

121 

142 

164.6 
189 
215 
242.8 
276.2 
303-2 

336 

406.6 


Tons. 

40.5 
47-5 
55-125 

63-25 

72 

81.25 

9i-i25 

101.5 

112.5 
136.125 


No™  I. -Safe  Working-stress  is  taken  at  half  Proof-stress,  3.82  tons  per  sq.  inch 
"^r«  and  Safe  Working  - stress  for  close-link  chains  are  respectively  i 

two-thirds  of  those  of  stud-link  chains.  . g/rnioth 

, Proof-stress  averages  72  per  cent,  ultimate  strength,  and  Ultimate  Strength. 
avlmg^?ioteper  Iquafe  inVof  section  of  rod  or  one  s.de  of  a hnk. 

Weight  of  close-link  chain  is  about  three  times  weight  of  bar  from  which 
it  is  made,  for  enual  lengths.  . 

Karl  von  Olt  comparing  weight,  cost,  and  strength  of  the  three  materials, 

of  cost  of  chains. 

Load  of  Cliains.  ( ilolesworth ). 

Diameter 


Safe  Worliiiv 


Diameter 
of  Iron. 


ROPES  AND  CHAINS. 


I69 


Breaking  Strain  and  Proof  of  Chain  Cables. 


Diam. 
of  Chain. 

Breaking 

Strain. 

Diam. 
of  Chain. 

Breaking 

Strain. 

Diam. 
of  Chain. 

Breaking 

Strain. 

Diam. 
of  Chain. 

Breaking 

Strain. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

I 

67  700 

I.1875 

92  940 

i-5 

143  IOO 

2 

243  180 

I.0625 

75640 

I.25 

102  l6o 

1.625 

165  920 

2.125 

272  580 

I.I25 

84  IOO 

i-375 

12 1 84O 

i-75 

2l6  120 

2.25 

303  280 

Proof-stress  is  50  per  cent,  of  estimated  strength  of  weakest  link  and  46 
per  cent,  of  strongest. 

Comparison  of  Wire  Ropes  and-  Tarred  Hemp  Hope, 
Hawsers,  and  Cables. 


COARSE  LAID. 


FINE  LAID. 


Diam- 

eter. 

Circum. 

Safe 

Load. 

Three 
Strands.  ^ 
0 

-0 

Four  “ 
Strands. 

a 

Three  ^ 
Strands.  «. 

Cables. 

Is 

Diam- 

eter. 

Safe 

Load. 

Four  0 
Strands. 

Haws’rs. 

2 2 
rd  05 

Cables. 

2 -o 
t.  a 
jz  as 
Hi 

C/3 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

•25 

l’78 

425 

I.25 

— 

— 

— 

•5 

1875 

3.12 

2.87 

— 

•3125 

690 

2-43 

2.25 

3-32 

— 

•5625 

2 420 

3-56 

3-25 

4.87 

•375 

1.25 

825 

2.68 

III5 

3-5 

— 

.625 

2900 

3-93 

3.62 

5-25 

•5 

i-375 

1 600 

2.87 

3-87 

— 

-75 

4320 

4.81 

4-37 

6-37 

:l875 

x-75 

2 800 

3.81 

3-5 

5.18 

— 

.875 

5700 

5-5 

5 

7.25 

2.125 

3800 

4-75 

4-25 

6.12 

— 

1 

8200 

n 

8.81 

6.25 

8-75 

•8  75 

2-375 

2.625 

4400 
6 150 

5- 25 

6.12 

4.87 

5-75 

l 

8.62 

8 

1-125 

1.25 

10  IOO 
13600 

£.06 

9-5 

11 

I 

3 

8 400 

6.62 

6.12 

8.62 

\:LS 

17500 
21  800 

10 

9-75 

12.5 

1.25 

3-75 

1340° 

8.81 

8.5 

10.93 

10.93 

11. 18 

10.93 

— 

1-375 

4-25 

16  800 

9.87 

9-56 

12.25 

12.12 

27000 

12.5 

12.12 

— 

l\l5 

4-625 

20  160 

io-75 

IO-5 

13 

13. 12 

1-875 

32  500 

— 

— 

— 

5 

24600 

— 

ix. 87 

11  56 

n-75 

2 

37000 

— 

— 

— 

In  above  table,  determination  of  circumference  of  rope,  etc.,  is  based  upon 
Breaking  Weight  or  Tensile  resistance  of  wire  being  reduced  by  one  fourth, 
and  ultimate  resistances  of  rope,  etc.,  are  reduced  one  third. 


Result  of  Experiments  upon  Wire  Hope  at  XT.  S.  Navy 
Yard,  Washington.  (J  A.  Roeblingys  Sons.) 


Circumference. 

Se- 

Weight 

Breaking 

Circumference. 

£ « c 

p a> 

1U-S 

Breaking 

Actual. 

Nom- 

inal. 

*li 

ll* 

/oot. 

Weight. 

Actual. 

Nom- 

inal. 

jT  « 5s 

.553 

Weight. 

Ins. 

Ins. 

No, 

No. 

Lbs. 

Lbs. 

Ins. 

Ins. 

No. 

No. 

Lbs. 

Lbs. 

4-9375 

4.9 

J9 

II 

3-14 

65  409 

2-375 

2.4 

7 

13 

.14 

15718 

4-375 

4-5 

19 

13 

2.15 

55  3*6 

2.1875 

2.12 

7 

14 

.11 

14478 

3-9375 

3-91 

19 

14 

2.0875 

44420 

2 

2.06 

19 

19 

.1 

IO925 

3-5 

3-36 

19 

14 

1-1525 

34840 

1-9375 

1.9 

7 

14 

.1 

IO  Il8 

3-1875 

2.98 

19 

15 

1.09 

28606 

i-75 

1.85 

7 

17 

.07 

7 880 

2-75 

2.68 

19 

17 

1.0275 

21  846 

1-4375 

1-45 

19 

20 

.06 

5687 

2.6875 

2.56 

7 

13 

1.0225 

l88lO 

1.3125 

1-31 

7 

18 

•05 

4428 

2-5 

2-45 

19 

18 

.14 

15692 

1. 125 

I.II 

7 

19 

-035 

3 729 

To  Compute  Circumference  of  Wire  Hope  with  Hemp 
Core,  of  Corresponding  Strength  to  Hemp  Hope,  and 
of  Hemp  Hope  to  Circumference  of  "Wire  Hope. 

Rule  i. — Multiply  square  of  circumference  of  hemp  rope  by  .223  for  iron 
wire  and  .12  for  steel,  and  extract  square  root  of  product. 

2. — Multiply  square  of  circumference  of  hemp-core  wire  rope  by  4.5  for 
iron  wire  and  8.4  for  steel  wire. 

Example.— What  are  the  circumferences  of  an  iron  and  steel  wire  rope  corre- 
sponding to  one  of  hemp-core,  having  a circumference  of  8 ins.  ? 


V82x  .223  = 3. 78  ins.  iron , and  V82  X . 12  = 2. 77  ins.  steel. 

P 


170 


ropes,  hawsers,  and  cables. 


ROPES,  HAWSERS,  AND  CABLES. 

Rones  of  hemp  fibres  are  laid  with  three  or  four  strands  of  twisted  fibres, 
and  are  made  up  to  a circumference  of  12  ins.,  and  those  of  four  strands  up 
to  8 ins.  are  fully  16  per  cent,  stronger  than  those  of  three  strands. 

Hawsers  are  laid  with  three  or  four  strands  of  rope.  Cables  are  laid  with 
but  three  strands  of  rope.  Hawsers  and  Cables,  from  having  a less  Propor- 
tionate number  of  fibres,  and  from  the  irregularity  of  the  resistance  ot  their 
fibres  in  consequence  of  the  twisting  of  them,  have  less  strength  than  ropes, 
difference  varying  from  35  to  45  per  cent.,  being  greatest  with  least  circum- 
ference, and  those  of  three  strands  up  to  12  ins.  are  fully  10  per  cent,  strong- 
er than  those  having  four  strands. 

Tarred  ropes,  hawsers,  etc.,  have  25  per  cent,  less  strength  than  white 
ropes ; this  is  in  consequence  of  the  injury  fibres  receive  from  the  high  tem- 
perature of  the  tar,  viz.  290°. 

Tarred  hemp  and  Manila  ropes  are  of  about  equal  strength,  and  have  from 
25  to  30  per  cent,  less  strength  than  white  ropes. 

White  ropes  are  more  durable  than  tarred. 

The  greater  degree  of  twisting  given  to  fibres  of  a rope,  etc.,  less  its 
strength,  as  exterior,  alone  resists  greater  portion  of  strain. 

Ultimate  strength  of  ropes  varies  from  7000  to  12000  lbs.  per  square  inch 
of  section,  according  as  they  are  wetted,  tarred,  or  dry.  One  sixth  oi  ulti- 
mate strength  is  a safe  working  load  = 1166  to  2000  lbs.  per  square  inch. 


Units  for  computing  Safe  Strain  tliat  may  he  Horne  by 
USTevv  Liopes,  Hawsers,  and  CaUles.  (U.  S.  Navy.) 


Descrip- 

tion. 

Circumference. 

Wh 
3 strands. 

Rope 

ite. 

4 strands. 

s. 

Tar 
3 9tr’ds. 

red. 

4 str’ds. 

Haw: 

White. 
3 str?ds. 

3ERS. 

Tarred. 
3 str’ds. 

Cab: 

White. 

3 str’ds. 

LE9. 
Tarred. 
3 str’ds. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

White 

2-5 

to 

6 

II40 

1330 

— 

— 

600 

— 

“ 

— 

u 

6 

u 

8 

IO9O 

1260 

— 

— 

570 

— 

510 

it 

8 

U 

12 

1045 

880 

— 

— 

530 

— 

530 

— 

u 

12 

u 

18 

. — 

— 

— 

550 

— 

550 

— 

a 

18 

u 

26 

— 

— 

— 

— 

— 

— 

560 

— 

Tarred 

2-5 

u 

5 

— 

— 

855 

1005 

— 

460 

— 

— 

u 

5 

a 

8 

— 

— 

825 

94O 

— 

480 

— 

— 

ii 

8 

u 

12 

— 

— 

780 

820 

- — : . 

505 

— 

505 

u 

12 

11 

18 

— 

— 

— 

— 

— 

— 

— 

525 

a 

18 

u 

26 

— 

— 

— 

— 

— ’ 

— 

— 

550 

Manila 

2-5 

a 

6 

8lO 

950 

— , 

— 

440 

— 

— 

u 

6 

u 

12 

760 

835 

— 

— 

46s 

— 

510 

u 

12 

u 

18 

— 

— 

— 

— 

— 

— 

535 

— 

u 

18 

u 

26 

— 

' 

— 

— 

1 "T  • 

560 

— 

Illustration. — What  weight  can  be  borne  with  safety  by  a Manila  rope  of  3 
strands,  having  a circumference  of  6 inches?  ( See  Rule, page  167.) 

6 2 X 760  = 27  360  lbs. 

When  it  is  required  to  ascertain  weight  or  strain  that  can  be  borne  by 
ropes , etc.,  in  general  me,  preceding  Units  should  be  reduced  from  one  third  , 
to  two  thirds,  in  order  to  meet  their  condition  or  reduction  of  their  strength 
by  chafing  and  exposure  to  weather.  Molesworth’s  table  is  based  upon  a 
reduction  of  three  fourths. 

•Illustration. — What  weight  can  be  borne  by  a tarred  hawser  of  3 strands,  10 
inches  in  circumference,  in  general  use  ? 

io2  X (505  — 505  -5-  3)  = 100  X 366-67  = 33  667  lbs. 


ROPES,  HAWSERS,  AND  CABLES. 


171 


Destructive  Strength,  of  Tarred  Hemp  Hopes 
(D.  K.  Clark.) 


109. 

3 

3- 5 

4 

4- 5 

5 


Diam. 

Reg 

Common 

Cold. 

ister. 

Russian 

Warm. 

Circum. 

Diam. 

Reg 

Common 

Cold. 

iater. 

Russian 

Warm. 

Ins. 

Lbs. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Lbs. 

•95 

7 390 

8620 

5-5 

I*75 

24800 

29  120 

1. 11 

II  200 

II  760 

6 

1.91 

28985 

33  150 

1.27 

13  IOO 

15  340 

6-5 

2.07 

34  030 

40550 

i*43 

16330 

I944O 

7 

2.24 

40  320 

47041 

i-59 

19580 

23990 

8 

2-54 

52480 

61 420 

Specimens  furnished  by  National  Association  of  Rope  and  Twine  Spinners , 
As  tested  by  Mr.  Kirkaldy. 


Rope. 

Circum- 

ference. 

Weight 
per  Lb. 

Extreme 

Strength. 

Breaking 

Weight 

perlb.per 

Fathom. 

Extensk 
at  Stra 
pe 

1000  lbs. 

>n  in  50  in 
3s  per  lb. 
r Fathom 
2000  lbs. 

s.  Length 

Weight 

of 

3000  lbs. 

Russian  rope ...  48  thr’ds. 
Machine  yam. . . 50  “ 

Hand-spun  yam,  51  “ 

Ins. 

5.26 

5*37 

5-39 

Lbs. 

.926 

.891 

1.006 

Lbs. 

II  088 
II  514 
18278 

Lbs. 

1933 

2152 

3024 

Ins. 

5-29 

4-53 

4.46 

Ins. 

6.56 

5-9i 

Ins. 

6.63 

Breaking  Strength  of  Tarred  Hemp  Hopes. 


Ins. 

3 

3*5 

4 

4-5 

5 


Ins. 

•95 
1. 11 
1.27 
i-43 


Old  Method. 
Common 
Hemp. 


Lbs. 

5 056 
7466 
8 780 
10300 


I*59  I3328 


Lbs. 

6248 
8668 
10460 
12432 
15  859 


By  Register. 


Cold. 


Lbs. 

7 392 

II  200 

13  IO4 
16330 
20496 


Warm. 


Lbs. 

8624 
II  760 
17810 
19443 
23990 


Ins. 

1- 75 
1.91 
2.07 
2.24 

2- 54 


Old  Method. 
Common 
Hemp. 


Lbs. 

15  456 
18  144 
20518 
22938 
26  680 


Best 

Russian. 


Lbs. 
18414 
21  6lO 
23  6lO 

27462 
32  032 


(Mr.  Glynn.) 
By  Register. 
Cold.  Warm. 


Lbs. 

24  797 
28986 
34630 
40320 

52483 


Lbs. 

29  120 
33150 
40  544 
47  040 
6l  42O 


To  Compute  Strain  that  may-  he  home  with  safety  by 
new  Hopes,  Hawsers,  and  Cables. 

Deduced  from  experiments  of  Russian  Government  upon  relative  strength 
of  different  Circumferences  of  Ropes , Hawsers , etc. 

U.  S.  Navy  test  is  4200  lbs.  for  a White  rope  of  three  strands  of  best  Riga 
hemp,  of  1.75  inches  in  circumference  (=  17000  tbs.  per  square  inch  of  fibre), 
but  in  preceding  table  (page  166)  14000  lbs.  is  taken  as  unit  of  strain  that 
may  be  borne  with  safety. 

Rule.— Square  circumference  of  rope,  hawser,  etc.,  and  multiply  it  by 
Units  in  table. 

To  Compute  Circumference  of  a Hope,  Hawser,  or  Cable 
for  a Given  Strain. 

Rule.— Divide  strain  in  pounds  by  appropriate  units  in  preceding  table, 
and  square  root  of  product  will  give  circumference  of  rope,  etc.,  in  ins. 

Example  i. — Stress  to  be  borne  in  safety  is  165  550  lbs. ; what  should  bo  circum- 
ference of  a tarred  cable  to  withstand  it  ? 

165  552  4-  55°  — 301,  and  y/301  = 17. 35  ins. 

2.— What  should  be  circumference  of  a Manila  cable  to  withstand  a strain,  in 
general  use , of  149  336  lbs.  ? 

Assuming  circumference  to  exceed  18  ins.,  unit  = s6o. 

*49  336  4-  (560  — 560  -4-  3)  = 400,  and  y/400  = 20  ins. 


ROPES,  hawsers,  and  cables. 

1 7Z 

XTrixTcrGlPPS*  gtlld  O 3/13 A 0 S • 

To  Compute  Weight  of  it  by  appropriate  unit  in 

Ru  le.  Square  f rJXct  wi'u  fihe  per  f oof  in  lbs.: 

following  table,  and  product  wtj 

3-strand  Hemp °32  ■°iI  '°3‘  f.JjJSSd  tarred  Hemp,  .048  — ~ 

3-strand  tarred  Hemp,  .042  .04  - 4 * trand  Manila °35  -°34  -34 

istrand  Manila 032  .03  • 3 1 of  material. 

fathoms?  ^ x Q34_  3.4)  and  120  X 6 X 3-4  = 2+48  lbs' 

■ ^ ,d  Strength-  of  Hemp  and  Wire  Ropes. 
Weigh  ai  ( Molesworth .)  n 

_w  C2fc  = L;  C2x  = S;  andv/jt=C- 
values  of  y , ar,  asd  k,  ,}  k 


V 1 

* 

k | 

HOPES.  1 

y 

— — - 

.131 

.117 

Warm  register,  hemp 
Manila  hawser 1 

1 -x77 
•155 

• 235 

.22 

•037 

.87 

.207 

•15 

.025 

1 -89 

— 1 

.6 

.1 

| Steel  • • 

Hawser,  hemp 

Tarred  hawser,  hemp 
“ cable,  u 

cold  register,  ~ - " Qr  Wire  Rope 

l0CTr0fSSlgB^  (H.  S.  Navy-) 

Rule. — To  length  of  ®astdb^e 

*— * <0M>  “4  ■,“™ 

of  product.  . 

For  Mizzen,  take  .74  of  Fore  and  Mam.  masl  of  a TesSel 

,6  < . 

Head “ 7$~  “ 

Breadth  of  beam,  45  feet. 

58  + 45^4S  = ,58iandv(3.58X^  = V.oM6  = »o,iina 

Then  if  circumference  of  *“  ir0U 

wire  "rope  ofT'h'fanda  steel  rope  of  3.25+  >“s-  Washington,  gave  for  flex- 

for\ensile  strength  a like  loss  of 

°ent'  of  Hemp  Rope  and  Iron  and  Steel 

Relative  Dimension^  ^ope.  (U.  S.  Navy.) 

Circumference  in  Inches. 

o e ne  11  n-75  x3-5  ‘5 

. - r.25  6.5  7-75  8-5  9-5  1 6 7 

Hemp.  2.5  3-125  4 4-5  5 5 ^ 4 4-5  5 5 5 5.2S 

Iron  . . 125  x-625  J ell  i.87s  »»5  2*5  2*75  3-25  3-5 

Steel..  .875  X-I25  *-5  I>C)2:>  7b 


ANCHORS,  CABLES,  ETC. 


*73 


ANCHORS,  CABLES,  ETC. 

Anchors,  Chains,  etc.,  for  a Given  Tonnage. 
( American  Shipmasters’  Association.) 

SAILS. 


I Tonnage 
computed  ns 
per  Rule.* 

Bo 

With- 

out 

Stock. 

wers. 

Admi 

ralty 

Test. 

Anciioi 

Inc 

Stream 

IS. 

luding  S 
. J Kedge 

tock. 

2d 

Kedge 

| 

Diameter 

Ce 

"So 

C 

[AIN  CAB! 

Admi- 

ralty 

Test. 

le.— Six 
Weig 

Stud. 

m. 

;ht  per  F 

Short 

Link. 

athom. 

Eng- 

lish.f 

75 

100 

125 

150 

175 

200 

250 

3°o 

350 

400 

45o 

500 

600 

700 

800 

900 

1000 

1200 

1400 

1600 

1800 

2000 

2500 

3000 

Lbs. 

6l6 

728 

840 

952 

IO36 

1120 

1288 

1456 

1624 

1848 

I9°4 

2016 

2352 

2688 

3024 

3248 

3584 

3808 

4032 

4256 

4480 

4704 

5040 

5376 

Tons. 

7 

8 

9 

10 

11 

12 

13 

14 
15-5 

17 

18.5 

20 

22 

24 

26 

28 

295 

3i 

32.5 

34 

35-5 

37 

39 

41 

Lbs. 

168 

196 

224 

280 

336 

392 

448 

504 

560 

616 

672 

784 

896 

1008 

1120 

I232 

*344 

J456 

1568 

1680 

1792 

1904 

2128 

2353 

Lbs. 

84 
1 12 
112 
I40 
l68 
I96 
224 
252 
280 
308 
336 
392 
448 
504 
560 

6l6 

672 

738 

784 

840 

896 

952 

1120 

1232 

Lbs. 

1 12 
126 
I4° 
154 
168 
196 
224 
252 
280 
308 
336 
364 
392 
420 
448 
504 
560 
6l6  j 

Ins. 

.8125 

•875 

•9375 

1 

1.062=5 
1. 125 

i-i875 

1 1.25 
1-3125 
I*3I25 
I,375 
*•4375 
i-5 

*•5625 

1.625 
i*6875 
I*75 
i-875 
1-9375 

2 
2 

2.0625 
2.125 
2.1875 

Paths. 

90 

105 

105 

120 

120 

120 

135 

135 

150 

150 

165 

165 

l8o 

l8o 

l8o 

l8o 

180 

180 

180 

180 

ISO 

l8o 

l8o 

180 

Tons. 

II 

*3 

15 

17-5 

20 

22.5 

25 

28 

3i 

3i 

37 

40 

44 

47 

5i 

55 

59 

63 

67 

72 

72 

81 

86 

96  | 

Lbs. 

40 

44 

5i 

59 

66 

75 

82 

9i 

100 

100 

115 

120 

132 

145 

156 

162 

i75 

189 

205 

219 

240 

Lbs. 

42 

48 

55 

63 

70 

79 

88 

98 

106 

106 

118 

35 

48 

54 

68 

84 

102 

122 

T4  3 

166 

191 

217 

244 

t Brown,  Lennox,  & Co. 


To  Compute  Tonnage. 


^ 10  icv/uinmt 

Hawsers  and  Warps  to  be  90  fathoms  in  length. 


Sh.roo.ds. 

a tonnage  of  7S’ ia' 

greater  thanfoTsquIre-riggeci?01'1  '25  t0  1 mch  in  diameter  progressively 


174 


ANCHORS,  CABLES,  ETC. 


(. American  Shipmasters1  Association.') 
STEAM. 


[ Tonnage 
computed 
as  per  Rule 
preceding. 

With- 
out a 
Stock.  | 

era. 

gjjs 

NCHORS 

Inclu 

g 

03 

ding  Stc 
80 
£ 

ck. 

•sf 

Diam- 

eter. 

c 

u 

c 

!hain  C 
< ~ 

ABLE.— STI 

Diam. 

Stream. 

[ID. 

Weigl 

03 

lit  per  ] 

1| 

O’. 

Fath. 

fc£  rJ 

ws 

Lbs. 

Tons. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

Faths. 

Tons. 

Ins. 

Lbs. 

Lbs. 

IOO 

336 

4.9 

1 12 

— 

— 

.6875 

105 

8.1 

•5 

— 

— 

25 

150 

448 

6.4 

I96 

— 

— 

.8125 

120 

11.9 

•5625 

40 

42 

35 

200 

6l6 

7.6 

224 

— 

— 

•875 

120 

13-8 

•5625 

44 

48 

— 

250 

672 

8.2 

280 

— 

— 

•9375 

120 

15-8 

.625 

5i 

55 

48 

3°° 

8l2 

9-5 

308 

— 

— 

1 

120 

18 

.625 

59 

63 

54 

35o 

924 

10.4 

336 

— 

— 

1.0625 

120 

20.3 

.6875 

66 

70 

— 

400 

1120 

12 

532 

252 

— 

1. 125 

135 

22.8 

.6875 

75 

79 

68 

450 

1344 

13-9 

560 

280 

— 

1.1875 

135 

25-4 

•75 

82 

88 

— 

500 

1512 

15.2 

672 

336 

— 

1,25 

150 

28.1 

•75 

9i 

98 

84 

600 

1708 

16.7 

738 

364 

— 

1-3125 

150 

3i 

.8125 

100 

106 

— 

700 

1876 

18 

784 

392 

— 

1-375 

165 

34 

.8125 

115 

118 

104 

800 

2026 

19 

896 

448 

224 

1-4375 

165 

37-2 

.875 

120 

— 

— 

900 

2352 

21.6 

1008 

504 

252 

i-5 

l8o 

40-5 

•875 

132 

— 

122 

1000 

2632 

23-5 

1120 

560 

280 

1-5625 

l8o 

44 

•9375 

145 

— 

— 

1200 

2856 

25.2 

II76 

588 

308 

1.625 

l8o 

47-5 

•9375 

156 

— 

143 

1400 

3iq8 

26.9 

1232 

6l6 

308 

1.6875 

l8o 

51.2 

1 

162 

— 

— 

1600 

336o 

28.6 

1344 

672 

336 

i-75 

l8o 

55-i 

1 

175 

— 

166 

1800 

3534 

30.1 

1456 

738 

364 

1.8125 

l8o 

59-i 

1.0625 

189 

— 

— 

2000 

3808 

31.6 

1512 

. 766 

364 

1-875 

l8o 

633 

1.0625 

205 

— 

191 

2300 

4088 

33-4 

1568 

784 

392 

1-9375 

l8o 

67.6 

1-125 

215 

— 

— 

2600 

4256 

34-5 

1624 

8l2 

392 

2 

27O 

72 

1-125 

240 

— 

217 

3000 

4480 

35-7 

1680 

840 

420 

2.0625 

27O 

76.6 

1.1875 

— 

— 

— 

3500 

4592 

37 

1792 

896 

476 

2.125 

27O 

81.3 

1.1875 

— 

— 

244 

4000 

4816 

38 

i960 

952 

504 

2.1875 

27O 

86.1 

i 1-25 

— 

— 

— 

4500 

5040 

39-2 

2128 

1064 

532 

2.25 

27O 

91. 1 

ji-25 

— 

— 

— 

5000 

5264 

41 

2352 

1120 

560 

2.3125 

27O 

96 

1 1-3125 

— 

— 

— 

* Brown,  Lennox,  & Co. 


ANCHORS  AND  KEDGES. 

(U.  S.  Navy.) 

To  Compute  "Weight  of  a Bower  Anchor  for  a Vessel 
of  a given  Character  and.  Rate. 

Rule. — Multiply  approximate  displacement  in  tons,  by  unit  in  following 
table,  and  product  will  give  weight  in  lbs.,  inclusive  of  stock. 


TJ  nits  to  determine  Weights  and  IN'  umber  of  A.ncliors 
or  Kedges. 


Displacement 
of  Vessel  in 
Tons. 

Unit. 

Bower. 

Sheet. 

Stream. 

Kedge. 

Displacement 
of  Vessel  in 
Tons. 

Unit. 

Bower. 

Sheet. 

Kedge. 

Over  3700 

i-75 

2 

2 

I 

4 

Over  1500  . . 

2-5 

2 

2 

3 

“ 2400 

2 

2 

2 

I 

3 

“ 900  . . 

2-75 

2 

I 

3 

“ 1900 

2.25 

2 

2 

I 

3 

900  and  under 

3 

2 

I 

2 

Example. — Tonnage  of  a bark-rigged  steamer  is  1500. 

1500  x_2. 5 = 3750  lbs. , weight  of  anchor. 

Bower  and  Sheet  Anchors  should  be  alike  in  weight. 

Stream  Anchors  and  Kedges  are  proportional  to  weight  of  bowers.  Thus, 
Stream.  Anchor  .25  weight.  Kedges.  — If  1,  .125  weight;  if  2,  .16  and  .1 
weight;  if  3,  .16,  .125,  and  .1  weight. 


ANCHORS,  CABLES,  ETC. — TONNAGE. 


i75 

To  Compute  Diameter  ot  a Chain.  Cable  correspondin°' 
to  a Given  Weight  of  Anchor. 

(U.  S.  Navy.) 

Rule.— Cut  off  the  two  right-hand  figures  of  the  anchor’s  weight  in  lbs., 
multiply  square  root  of  remainder  by  4,  and  result  will  give  diameter  of 
chain  in  sixteenths  of  an  inch. 

Example. — The  weight  of  an  anchor  is  2500  lbs. 

V 25.00  X 4 = 20  sixteenths  — 1.25  ins. 

Note.— Diam.  of  a messenger  should  be. 66  that  of  the  cable  to  which  it  is  applied. 
Lengths  of  Chain  Cables  for  each  Anchor. 

(V.  S.  Navy.) 


Weight  of  Anchor. 

Bower. 

Sheet. 

Stream. 

Weight  of  Anchor. 

Bower, 

Sheet. 

Stream. 

Lbs. 

Under  800 
Over  800 
“ 1200 

“ 1600 

Fathoms. 

60 

90 

90 

105 

Fathoms. 

60 

90 

90 

105  1 

Fathoms. 

60 

60 

75 

75 

Lbs. 

Over  2000 
u 3000 
“ 5°°° 

“ 75oo 

Fathoms. 

120 

120 

120 

135 

Fathoms. 

120 

120 

120 

135 

Fathoms. 

90 

90 

105 

105 

ANCHORS. 

jFi  om  Experiments  of  a Joint  Committee  of  Representatives  of  Ship- 
owners and  Admiralty  of  Great  Britain. 

An  anchor  of  ordinary  or  Admiralty  pattern,  Trotman  or  Porter’s  im- 
proved (pivot  fluke),  Honiball,  Porter’s,  Aylin’s,  Rodgers’s,  Mitcheson’s,  and 
Lennox  s,  each  weighing,  inclusive  of  stock,  27000  lbs.,  withstood  without 
injury  a proof  strain  of  45000  lbs. 

Breaking  weights  between  a Porter  and  Admiralty  anchor,  as  tested  at 
Woolwich  Dock-yard,  were  as  43  to  14. 

Comparative  Resistance  to  Dragging. 

Trotman  s dragged  Aylin’s,  Honiball’s  Mitcheson’s  and  Lennox’s ; Aylin’s 
mid  Mitcheson’s  dragged  Rodgers’s ; and  Rodgers’s  and  Lennox’s  dragged 


TONNAGE  OF  VESSELS. 

To  Compute  Tonnage  of  Vessels. 

For  Laws  of  United  States  of  America,  with  amendments  of  1882  relative 
to  Steam-vessels,  see  Mechanics’  Tables,  with  rule  and  illustrated  diagrams 
by  Chas.  H.  Haswell,  3d  edition,  Harper  & Bros.,  New  York,  1878. 

Rnglish  Registered.  Tonnage.  ( New  Measurement.) 

*?,ivide  length  of  upper  deck  between  after-part  of  stem  and  fore-part  of  stern- 
post  into  6 equal  parts,  and  note  foremost,  middle,  and  aftermost  points  of  division 

SrUrendeifS  at  thaesei three  pointS  in  feet  and  tentlls  of  a f00t;  also  dePths  from 
under  side  ot  upper  deck  to  ceiling  of  limber-strake;  or  in  case  of  a break  in  the 

pPPherHde?,k^frf°m  a Iin®  fetched  in  continuation  of  the  deck.  For  breadths,  divide 
each  depth  into  5 equal  parts,  and  measure  the  inside  breadths  at  following  points 
;;7AQ  ;2  and'8  fr°m,uPPcr  deck  of  foremost  and  aftermost  depths;  and  from 
fmm  off  from.  uPP®r  deck  of  amidship  depth.  Take  length  at  half  amidship  depth 
from  after-part  of  stem  to  fore  part  of  stern-post  p 

‘Tf?  a™idship  depth  add  foremost  and  aftermost  depths  for  sum  of 
depths  and  add  together  foremost  upper  and  lower  breadths,  3 times  upper  breadth 
with  lower  breadth  at  amidship,  and  upper  and  twice  lower  breadth  at  after  division 
ior  sum  oj  oreadths. 

bv^'^hiT^r  SUm  of  depths  sum  of  breadths,  and  length,  and  divide  product 
°y  35°°,  which  will  give  number  of  tons.  1 

If  the  vessel  has  a poop  or  half-deck,  or  a break  in  upper  deck,  measure  inside 

^bulSd  bmnhtni  “I’h  hei,ghht  °f  6Uch  part  thereof  as  ma>'  be  included  within 
tne  bulkhead,  multiply  these  three  measurements  together,  divide  product  bv  02  a 

and  quotient  will  give  number  of  tons  to  be  added  to  result  as  above  ascertained.'4’ 


TONNAGE  OF  VESSELS. 


176 


For  Open  Vessels.  —Depths  are  to  be  taken  from  upper  edge  of  upper  strake. 

For  Steam  Vessels.—  Tonnage  due  to  engine-room  is  deducted  from  total  tonnage 
computed  by  above  rule.  To  determine  this,  measure  inside  of  the  engine-room 
from  foremost  to  aftermost  bulkhead;  then  multiply  this  length  by  amidship  depth 
of  vessel,  and  product  by  inside  amidship  breadth  at  .4  of  depth  from  deck,  and 
divide  final  product  by  92.4. 

The  volume  of  the  poop,  deck-houses,  and  other  permanently  enclosed  spaces, 
available  for  cargo  or  passengers,  is  to  be  measured  and  included  in  the  tonnage, 
but  following  deductions  are  allowed,  the  remainder  being  the  Register  tonnage. 

Deductions.—  Houses  for  the  shelter  of  passengers  only;  space  allotted  to  crew 
(12  square  feet  in  surface  and  72  cube  feet  in  volume  for  each  person);  and  space 
occupied  by  propelling  power. 

Approximate  H vile. 

Gross  Register.— Tonnage  of  a vessel  expresses  her  entire  cubical  volume  in  tons 
of  100  cube  feet  each,  and  is  ascertained  by  following  formula  : 

— Gross  tonnage,  and  L B—  c = Register  tonnage.  L representing  length 

IOO  100 

of  keel  between  perpendiculars,  B breadth  of  vessel,  and,  D depth  of  hold,  all  in  feet. 


13  nil  tiers’  Measurement. 

(Lr.6B)XBX:5 A^Tonnage, 

94 

Fore-perpendicular  is  taken  at  fore  part  of  stem  at  height  of  upper  deck. 

Aft-perpendicular  is  taken  at  back  of  stern-post  at  height  of  upper  deck. 

In  three-deckers,  middle  deck  is  taken  instead  of  upper  deck. 

Breadth  is  taken  as  extreme  breadth  at  height  of  the  wales,  subtracting  differ- 
ence between  thickness  of  wales  and  bottom  plank.  Deductions  to  be  made  for 
rake  of  stem  and  stern. 

Iron  Vessels.  (Girth  + B)  ead—\  x length  ==  Gross  tonnage. 

10000  \ 2 / 

Length  measured  on  upper  deck,  between  outside  of  outer  plank  at  stem  and 
the  after-side  of  stern-post  and  rabbet  of  stern-post,  at  point  where  counter-plank 
crosses  it  Girth  measured  by  a chain  passed  under  bottom  from  upper  deck  at 
extreme  breadth,  on  one  side,  to  corresponding  point  on  the  other. 


Register  tonnage  = - 
follows : 

Ships  of  usual  form 

Clippers  and  Steamers 


LxBxD 


X C.  C representing  a coefficient  for  vessels  as 


( 2 decks. . . .6<$ 

(3  “ .68 


I Yachts  above  60  tons 5 

I Small  vessels  { A";": 


Units  for  Measurement  and.  Dead-weiglit  Cargoes. 

(C.  Mackrow , M.  S.  N.  A.) 

To  Compute  Approximately  for  an  Average  Length  of  Voijage  the  Measure- 
ment Cargo , at  40  feet  per  Ton , which  a Vessel  can  carry. 

Rule. — Multiply  number  of  register  tons  by  unit  1.875,  and  product  will 
give  approximate  measurement  cargo. 

To  Compute  Approximately  Deacl-iveight  Cargo  in  Tons  which  a J essel  can 
carry  on  an  Average  Length  of  Voyage. 

Rule.— Multiply  number  of  register  tons  by  1.5,  and  product  will  give 
approximate  dead-weight  cargo  required. 

With  regard  to  cargoes  of  coasters  and  colliers,  as  ascertained  above,  about 
10  per  cent,  may  be  added  to  said  results,  while  about  10  per  cent,  may  be 
deducted  in  cases  of  larger  vessels  on  longer  voyages. 


TONNAGE  OF  VESSELS. 


177 

In  case  of  measurement  cargoes  of  steam-vessels,  spaces  occupied  by  ma- 
chinery , fuel,  and  passenger  cabins  under  the  deck  must  be  deducted  from 
space  or  tonnage  under  deck  before  application  of  measurement  unit  thereto! 

J.  J,n  case ' °l  dead-weight  cargoes,  weight  of  machinery,  water  in  boilers,  and 

appiStofd^dgwLr  dead  weight,as  ascertoincd  above  ljy 
sdectio“”r  f°r  Pr°ViSi0nS’  St°reS’  CtC-  are  allowed  - 

To  Ascertain  Weight  of  Cargo  for  an  Average  Length  of  Voyage.  ( Moorsom. ) 

ton^d"Ct‘ra^-°f  SpacCS°^  Pafenger  accommodations  from  net  register 
tonnage,  and  multiply  remainder  by  1.5. 

Average  space  for  each  ton  weight  of  cargo  on  such  a voyage  67  cube  feet. 

Freight  Tonnage  or  Measurement  Cargo. 

anfi Cargo  is  40  cube  feet  of  sPace  for  cargo, 
and  it  is  about  1.875  times  net  register  tonnage  less  that  for  passenger  space! 

Royal  Thames  Yacht  Club. 

sSSrS 

All  fractional  parts  of  a ton  are  to  be  considered  as  a ton 
Measurements  to  be  taken  either  above  or  below  main  wales. 

L — BxBx.S  B 

— — Tons.  L representing  length  and  B breadth , in  feet 

Corinthian  and  New  Thames  Yacht  Clnb. 

board;  multiply  length  breadth  ^nlfde^nh  Tule»f  n^dePth  to  toP  of  covering 

tient  will  give  tonnage  ’ ’ “d  depth  divide  result  by  200,  and  quo" 

LXBXI) 

— - '=  Tons. 

200 

Snez  Canal  Tonnage. 

all^co°ve^  sTchaisdnooDbfIOW  and  uppermost  deck, 

cook^deck,  and  wheel  houses,  and  all^K^ 

spaces  for  steward  for  crew’  not  including 

except  captain;  galleys,  cook  houses  etcUtn’«5Pd  accommodations  for  officers, 
spaces  above  uppermost  dec^d^S  f°,r  crTew’  aud  Closed 

SpCL“esPrS 

£ ST  -r  d,  or  owner 

exceed^o^er  cent  oTgro^toimage!0™  “USt  deduction  tor  Propelling  power 


I7S 


"WORKS  OF  MAGNITUDE. 

WORKS  OF  MAGNITUDE. 

.American. 

Aqueducts,  Roads,  and  Railroads. 

** " 

38.134  idles  in  length  er  of  masonry  g feet  in  diameter.  Stone  arch 

Aqueduct , Washington.  y 25  feet  rise, 

over  Cabin  John’s  Creek,  220  feet  spa  , si-  5 Cumberland  to  Illinois  Town, 

National  m”li^  Macadamized  for  a width  of  30  feet. 

^^mS^Chicago  to  Cairo,  length  365  miles,  Centralia  to  Bun- 
leith  344  miles,  total  709  miles. 


Bridges. 

Suspension  Bridge,  Niagara  River.-Wire,  Span  ro42  feet  ro  ins. 

Suspension  Bridge , iVeie 

ins. ; of  each  land  span  930  feet , Jsq  feet;  width  85  feet;  number 

approach  1562  feet  6 ins. ; % ins  . each9 consisting  of  6300  parallel  steel 
of  cables  4;  diameter  of banned  to  a solid  cylinder;  ultimate  strength 
wires  No.  7 gauge,  closely  laid  P foundation  below  high  water,  Brooklyn, 

of  each  cable  n 200  tons;  depth  01  to  t i;ne  iaovc;q  feet;  towers  at  roof 

45  feet— New  York  78  feet;  »*;  clear  h,eight 

course  136X53  feef?  total  height  . , * -0o  joe  feet;  height  of  floor 

of  bridge  in  centre  of  riyer  span  abov  .ggrade  of  roadway  3 feet  in  100;  anchor- 

Iron  Pipe  Bridge  over  [tnch'thick"0  These  pipes 

conveying  themselves  over  the  great  span,  but  support 

a street  road  and  railway.  Md.-3  ns  each  375 

Iron  Bridge  over  Kentucky  River  near  Shakers  Ferry, 
feet,  and  275-5  feet  above  oww  ^ ^ Western  Railroad  across  the  Kinzua.- 
Of^ror? lengthei^o^St ; ^central  span  30.  feet  in  height. 

Jron  Truss.  —Cincinnati  and  Southern  Railway,  over  Ohio  River,  $ feet. 
Foreign* 

Pyramids,  Statues,  etc. 

• pyramid  of  Clteops,  Egypt.-l^1^®®  Snld^engfh  568^  *fl  ™Sle 

w g 527  00 

built  2170  years  B.C.  . . -Rronze*  height  of  horse  17  feet;  of  man 

“>*> » wide> and  17  h,gh’wclBhing 

hfeZ  New  Vork  Harbor.  - jo  ^ " 

- **  "ference  87 

feet;  face  8.5  feet;  circumference  of  thumb  3.5  me  . 

Colossus  of  Rhodes. — Height,  105  feet. 

Bridge.  . 

. m r , t>  .w„„  of  iron  with  a double  lino  of  Railway,  964  feet  in 
Britannia  Tubulat  Bi  idge.  ’ ri,  weight  3658  tons, 

length,  with  two  approaches  of  230  feel  each-  w e g 3 5 


WORKS  OF  MAGNITUDE. 


179 


NT  oiiolith.s. 

Obelisk  at  Karnak , Egypt.—  Of  granite,  108  feet  10  ins. : pedestal  13  feet  2 ins  • 
height  400  tons.  J * 

Obelisk  in  Central  Park , N.  Y.-Of  granite,  68  feet  n ins.;  weight  168  tons. 

raids  of  Egyptlry’  WashiDSton— Some  stones  of,  are  heavier  than  any  in  the  Pyra- 

Steam  Hammers. 

. workshops  of  Herr  Krupp,  at  Essen,  there  is  a steam  hammer  weighing  so  tons 
i°^.3  metres;  and  at  Creusot  there  is  a hammer  weighing  between  7? 
and  80  tons  having  a fall  of  5 metres.  s g cen  75 


Crane. 

tons!  CreUSOt  there  is  a steam  crane  havin®  a capacity  to  lift  and  revolve  with  150 
Chimney's. 

J.  Townsend’s  chemical  works,  Glasgow,  diameter  at  foundation  so  feet-  at  ton 
12  feet  8 ins. ; height  from  foundation  488  feet;  from  ground  474  feet.  % t0p 

New  York  Steam  Heating  Co.,  220  feet  in  height. 


Filial*. 

At  a gate  near  Delhi  is  a wrought-iron  pillar  having  diameters  of  t * ?r>c< 
feet  in  its  height  above  ground  and  ,,  inSPat  its I«“eXmted ^rt,m  the  re- 

font0fTfXCaVatl^>np  at  its  base  to  be  60  feet  in  lei3gth  or  height  and  to  weigh  n 
tons.  Its  period  of  structure  is  assigned  to  the  3d  or  4th  century  A.D.  S 7 

Roofs. 

Midland  Railway  Station, London.  240  ft.  I Union  Railway  Station  Glasgow  Tr.-  ft 
Imperial  Riding-School,  Moscow.  235  “ | Grand  Central  Station,  N.  Y.&. . . ! 200  “* 


Diameters  of  Domes 


Domes.  I 

Feet,  j 

Domes. 

Feet.  ’I 

Domes. 

Feet. 

Capitol,  Washington 
Glasgow  W.  Railw’y  | 

I24-75  I 
198  | 

St.  Paul’s,  London . 
St.  Peter’s,  Rome. . 

XI2 

*39  1 

Midl’nd  Rail’y,  Lon. 
Great  North’n,  Eng. 

240 

210 

Tunnels. 

Feet. 

Tunnels. 

Blaizy 

13455 

4280 

25031 

y'Vf _____  _ _ 

Blue  Ridge. . . 

crunpowcicr.  Mu.  • . 
Sutro. . 4 . 

Hoosac 

Semmering . . . 

| Feet. 

365oo 


Thames  and  Medway,  n 880  feet. 


Tunnels. 


Nerthe. 

Nochistongo. . . . 

_ _ RiquiVel 

Weebawken,  4000  feet. 


*5  153 

21  659 
18623 


Mont  Cenis  7.5  miles  242  yards,  rises  1 in  45,  and  descends  r in  2000. 


- - 1 1 1 iu  2000. 


mum  grade  2.7  feet  per  100.^ 

Miscellaneous. 

Fortress  Monroe , Old  Point  Comfort,  Ya.  —Largest  fortress. 


60 ^tiS:~Spa“  °Ver  r‘Ver  KiStnah  between  B"  Sectanagran, 


Deer  Park , Copenhagen.  — 4200  acres. 


Alfred!”1*  C°Ue9e ’ EDgland—  Largest.  University;  said  to  have  been  founded  by 


Rome-Widtb  °f  «6  feet;  of  the  cross  25.  feet;  total 


len^tmtso^Ues.25  feet  baSe’  15  at  top;  height>  with  a parapet  of  5 feet,  20  feet; 


of  disctogeTf^glltons^wday.1  deptb’  temPerature  of  water  99o;  volume 


I l80  BELLS,  CHURCHES,  COLUMNS,  TOWERS,  ETC. 


Bells. 


Y 

Lbs. 

TV'eiglits  of  Bells 

Bells.  I Lbs. 

Bells. 

Lbs. 

120000 

Oxford,  “ Great 

Tom,”  Eng 

Olmutz,  Bohemia. 
Rouen,  France. .. 
St.  Paul’s,  Eng. . . 
St.  Ivan’s,  Moscow 

17024 
40  320 
40000 
42  000 
127  830 

St.  Peter’s,  Rome. 

Yipnna 

18000 
40  200 

35620 

24080 

13000 

10233 
28  560 
443  772 
30  800 
28670 

Westm’ster,  “Big 
Ben,”  England. 
York 

State  House, Phila. 

Pekin 

Lewiston,  Me. . . 

Montreal,  Can. . . 

Moscow,  Russia. 

Erfurt,  Saxony. . 

Notre  Dame,  Paris| 

Rangoon,  Burmah,  201 600  lbs. 

Capacity  of  Principal  Churches  and  Opera  Houses. 
Estimating  a person  to  occupy  an  Area  0/19.7  Ins.  Squaie. 

C lvu.rcli.es . 


St.  Peter’s 54  000 

Milan  Cathedral 37  000 

St.  Paul’s,  Rome 32000 

St.  Paul’s,  London 25  600 

St.  Petronio,  Bologna 24400 

Florence  Cathedral 24300 

Antwerp  Cathedral 24000 

St.  Sophia’s,  Constantinople 23000 


St.  John,  Lateran 22900 

Notre  Dame,  Paris 21 000 

Pisa  Cathedral 13000 

St.  Stephen’s,  Vienna 12400 

St.  Dominic’s,  Bologna 12000 

St.  Peter’s,  Bologna 11  400 

Cathedral  of  Sienna 11 000 

St.  Mark’s,  Venice 7 000 


Opera  Houses  and.  Theatres 


Carlo  Felice,  Genoa 2560 

Opera  House,  Munich 2370 

Alexander,  St.  Petersburg 2332 

San  Carlos,  Naples 2240 

Imperial,  St.  Petersburg 2160 

La  Scala,  Milan 2113 

Academy  of  Paris 2092 


Teatro  del  Liceo,  Barcelona 409° 

Covent  Garden,  London 2684 

Opera  House,  Berlin 1636 

New  York  Academy 2526 

“ “ Windsor 34°° 

Philadelphia  Academy 3124 

Chicago  “ 3°°° 


Heights  of  Columns,  Towers,  Homes,  Spires, 


Locations. 


CHIMNEYS. 

Townsend’s Glasgow. , 

St.  Rollox “ 

Musprat’S. Liverpool. 

GasWorks Edinburgh 

New  England  Glass  Co . Boston . 
Steam  Heating  Co New  York. 

COLUMNS. 

Alexander St.  Peters’g 

Bunker  Hill Mass 

City London... 

July Paris. . . . 

Napoleon _“•••• 

Nelson’s Dublin. . 

Nelson’s London . 

Place  Vendome Paris 

Pompey’s  Pillar Egypt... 

Trajan J^me . 7 

Washington Wash  gton 

York London  . . . 

TOWERS  AND  DOMES. 

Babel 

Balbec •••••••  7 ‘ ' 

Capitol Wash  gton 

Cathedral Antwerp  . . 

u Cologne... 

u Cremona.. 

<1  **  Escurial. . 


etc. 

Feet. 


TOWERS  AND  DOMES. 

Cathedral Florence  . . 

u Magdeb’rg 

a Milan 

<c  ***] Petersburg 

Leaning P*s.a 

Porcelain China 

St.  Mark’s Venice... 

St.  Nicholas Hamburg. 

St.  Paul's London  .. 

St.  Stephen Vienna. . . 

Strasburg 

Utrecht ; 

Votive  Church. Vienna. . . 

spires. 

Cathedral ^ ew  York . 

Strasburg. 

Grace  Church New  York. 

Freiburg 

Salisbury • • • 

St.  John’s ^ew  York. 

St.  Paul’s ‘ 

St.  Mary’s Lubeck  . . . 

St.  Peter’s Rome..... 

Trinity  Church New  York. 

Balustrade  of  Notre 

Dame..... Paris. 

Towers  of  ditto “ 

Hotel  des  Invalides. . . “ 


39°-5 

339-9 

438 

363 

188 

200 

328 

473 

355-J 

443-8 

486 

464 

3*4-9 


325 

465-9 

216  1 

410 

450 

210 

200 

4°4 

469-5 

286 

216 

232.9 

344 


BRIDGES,  CANALS,  BREAKWATERS,  ETC. 


1 8 1 


Areas  of  Lakes  in.  Europe,  Asia,  and.  Africa. 


Lakes. 

Mi?e’s. 

Lakes. 

IVLles. 

Lakes. 

Geneva 

400 

Dembia,  Abyssinia. 
Loch  Lomond 

13  000 
27 

I Lough  Neagh,  Irel’d 
Tonting,  China. . . . 

Tchad,  Africa 

11  600 

Lengths  of  Bridges. 


Bridges. 

j Feet. 

Bridges. 

Feet. 

Bridges. 

Feet. 

Avignon 

1710 

Lyons 

1560 

Potomac 

530° 

Badajoz 

1874 

Menai 

1050 

Riga 

2600 

Belfast 

2500 

N.  Y.  and  Brook- ) 

St.  Lawrence  Riv’r 

9144 

Blackfriars 

995 

lyn  spans  and  [ 

5989 

Strasburg 

339° 

Boston 

3483 

approaches ) 

Vauxhall 

860 

London 

95o 

Pont  St.  Esprit. . . 

3060 

Westminster 

1223 

Lengths  of  Spans  of  Bridges. 


Bridges. 

Feet. 

Bridges. 

Feet. 

| , Bridges. 

Feet. 

Britannia 

000 

0_CO^ 

Niag’a  at  the  Falls 
“ at  Queens- 
town  

1268 

1040 

I Schuylkill 

Southwark 

j Wheeling 

340 

240 

1010 

Conway 

Menai 

Canals. 

Lengths.  — Lake  Erie  to  Albany  352  miles;  Chesapeake  and  Ohio  307;  Schuylkill 
108;  Delaware  and  Hudson  109;  Rideau  132;  London  to  Liverpool  265  - Caledonia 
25;  Liverpool  and  Leeds  127.5;  Rhone  to  Rhine  203. 

Capacity  of  Locks  of  Erie  240  tons,  and  of  Welland  1500. 

Welland^  26. 77  miles.  Lake  Erie  to  Montreal  via  Canal  70.5;  Lake  and  River 
375  miles. 

Montreal  to  Kingston.— Canal  120  miles;  River  126.25.  Suez,  see  page  183. 

Breakwaters. 

Delaware.  Average  depth  of  water  29.4  feet  below  low- water  level;  range  of  tide 
6^66  feet;  Outer  slope  45°;  Inner  slopes  1.5,5,  3,  and  1.3  to  1;  length  of  base  172.12 

Plymouth.  Outer  slopes  1.75  to  1 from  bottom  to  7 feet  6 ins.  below  low- water 
line;  4 to  1 to  low-water  line;  16  to  1 to  4 feet  6 ins.  above  low- water  line*  c to  1 
tokigh  water;  Inner  slope  1. 5 to  1 above  low-water  line ; 2 to  1 below  low- water  line 
Depth  oi  water  at  high  tide  46.5  feet  ; at  low  tide  30  feet. 

Body  of  breakwater  cased  with  large  squared  stones  cramped  together. 

frn^rw?nnTt^epthf*0f^ig.h  w?ter  58  feet;  of  low  water  5*  feet;  Outer  slopes  i to  i 
horn  bottom  to  20  feet  below  low  water;  2 to  1 to  12  feet  below  low  water*  6 to  1 
to  low- water  line;  4 to  1 to  high-water  line;  Inner  slope  1.25  to  1 
Body  of  breakwater,  rubble,  with  crest  wall  of  ashlar. 

Dover.  Depth  of  high-water  line  61  feet;  of  low- water  line  42  feet 
rootsfe^rup^re^h”  W°CkS  ^ With  tetter  3 -d.es  to  the 

Marseilles.—  Depth  of  water  33  feet;  Outer  casing  of  beton  25.5  tons  each  • average 
thickness  of  casing  from  x 4 to  20  feet;  slope  1 to  1 from  bottom  to  water  line*  2% 
to  1 above  water-line;  all  other  slopes  .33  to  1;  Inner  casing  of  first-class  rubbie 
!nf  2 tt0  5,tons  w?\"ht)>  about  12  feet  thick;  Hearting,  second-class  rubble 

(of  stones  .5  to  2 tons  weight),  about  6 feet  thick;  Nucleus,  of  quarry  rubbish. 
Algiers.  Depth  of  water  50  feet;  rubble  base  carried  up  to  33  feet  from  surface  of 

Saso'Ai T 1 °fla^bdon  blocks  25.5to..s\\ch;  slopes  of  rabble 
base  1 to  1 , Outei  slope  of  beton  blocks  1.25  to  1 ; Inner  slope  of  beton  blocks  1 to  1. 

f Said  (Suez  Canal). — Concrete  blocks,  10  cubic  metres  each,  composed  of  1 

forby  mnnlh«  w °f  SaUd’  miXGd  With  Sea  water;  4 days  in  the  molded  dried 
bemnTfinr^  before  being  put  in  position.  In  some  instances  the  composition  of 
ballasting  k '33  ,me  °F  cemeut  t0  ,66  sand  and  broken  stone,  about  the  size  of 

:^hbl*  °\  B!oc7c  FilBn9  - Proportion  of  interstices  to  volume  of  breakwater  fin- 
rubble  of  * t0  5 tons,  .25;  second-class  rubble  of  .5  to  2 tons  2* 
third-class  rubble,  quarry  chips,  etc.,  .16;  beton  blocks,  15  to  25  tons  33  ’ ' 

Noth. — For  force  of  water,  see  Waves  of  the  Sea,  page  853. 

Q 


1 82  LAKES,  OCEANS,  SEAS,  MOUNTAINS,  ETC. 

Areas,  Depths,  and  Heights  of  Great  Northern  hakes 
of  TJ nited.  States. 


Lakes. 

Length. 

Breadth. 

Mean  Depth. 

Height 
above  Sea. 

Area. 

Erie 

Miles. 

250 

Miles. 

80 

Feet. 

200 

Feet. 

564 

Sq.  Miles. 
6000 

Hnuon 

200 

160 

120 

574 

20000 

Michigan 

360 

180 

109 

900 

587 

20000 

Ontario 

65 

500 

234 

6000 

Superior* 

400 

160 

288 

635 

32000 

Elevation  Above  Tide-water  at  Albany.  — 'Late  Erie  570.6  feet;  Hudson  River 
2.46  feet. 

Mean  Depths  and  Areas  of  the  Oceans  and  Seas. 
{Herr  Krummel) 


' 

Fathoms. 

Atlantic 

2013 

Archipelago 

487 

Azof. 

— 

Baltic  Sea 

36 

1 Black  Sea 

— 

a Behring’s  Straits 

550 

c Caspian  Sea 

— 

' China  (East)  Sea 

66 

c Dead  Sea 

— 

4 English  Channel,  etc. 

47 

Area 

Sq.  Miles. 


I 

Fathoms. 

Gulf  of  Mexico 

IOOI 

“ “ St.  Lawrence 

160 

Indian 

1829 

Japan 

1200 

Mediterranean 

729 

North  Sea 

48 

North  Ice  Sea. 

845 

Persian  Gulf 

20 

Pacific 

3887 

j Red  Sea 

243 

29  5H275 
3 046  600 
8800 
159  690 
1 56  000 
864  555 
120000 
472  210 
37° 

78  416 

Mean  depth  of  Ocean  surrounding  land  1877  fathoms  = 2. 19  miles. 

land  = 1377  feet,  or  one  eighth  that  of  Ocean. 

Heiahts  of  Mountains,  Volcanoes,  and  Passes 
above  Level  of*  Sea. 


Area 

Sq.  Miles. 


I 765910 
101  075 
28369  595 
383  205 
I 109  230 
210  505 
5 264  600 
90  100 
60  343  690 
170820 


Mountains. 


| Feet. 


EUROPE. 

Azores  Pico 

Barthelemy,  France 
Ben  Lomond. ..... 

Ben  Nevis 

Elbrus, Caucasus. . . 
Guadarama,  Spain. 

Hecla 

Ida 

Jungfrau,  Switz’d. 

Mont  Blanc 

“ Cenis 

Mont  d’  Or,  France 
Mulahassen,Gren’a. 

Nepbin,  Ireland 

Olympus 

Parnassus 

Plynlimmon,  Wales. 
The  Cylinder,  Pyr. . 
Wetterhorn 


Ararat 

Caucasus 

Dhawalagheri 

Geta,  Java 

Mount  Lebanon. . . 


Mountains. 


7613 

7 365 
3240 
4380 

17776 

8 520 
5147 
4960 

13  725 
15  797 
6 780 
6 510 

11  663 
2634 
6510 
6000 
2463 

10930 

12  154 


17  100 

16433 

28077 
8 500 
12000 


MountEverest  (Him- 
alaya, highest) 
Mount  Libanus. 

Petcha 

Sinai 


AFRICA. 


of 


Atlas 

Compass,  Cape 
Good  Hope.. 
Dianai  Peak,  St  He- 
lena  

Kilimanjaro .... 
Ruivo,  Madeira 
Teneriffe  Peak. 


29003 
9 523 
15000 
7496 


10400 

10000 

2 700 
20000 
5 160 
12  300 


AMERICA. 
Aconcagua  (highest 

in  America) 

Blue  Mount,  Jam’a. 

Catskill 

Chimborazo.. 

Correde,  Potosi  .... 
Crows’  Nest,  High- 
lands, N.  Y. . . 
Great  Peak,  New 

Mexico 

Mauna  Rauh,Owy’e 


23910 
8000 
3804 
21  441 
16036 

1 370 

19788 

18400 


Mountains. 


Mount  Pitt 

Mount  Washington. 
Nevado  de  Sorata. . 

Orizaba 

Potosi 

Sierra  Nevada 

Tahiti 

White  Mountains . . 

VOLCANOES. 

Cotopaxi 

Etna 

Hecla 

Popocatapetl 

Sahama 

St.  Helen’s,  Oregon. 
Vesuvius 


96$ 
25  248 
18879 
18000 
15  700 
IO895 
6 230 


18887 
IO874 
5 000 
17784 
22  350 
3320 
3 930 


PASSES. 

Cordilleras j 

Mont  Cenis 

“ Cervis. .. ... . 

Pont  d’  Or 

St. Bernard,  Great. 
“ Little.. 

St.  Gothard 

Simplon 


13525 

15225 
6778 
11 100 
9843 
8 172 

7 x92 

6808 

6578 


CANAL  LOCKS,  ELEVATIONS,  AND  RIVERS. 
Dimensions  of  Canal  Locks. — (U.  S.) 


Albemarle  and 
Chesapeake. . 
Black  River, 
Crook’d  L’ke, 
Chenango, 
Chemung, 
and  Genesee  I 

Valley J 

Chesapeake  and ) 
Delaware j 


90 


Length  of 
Canal. 


Miles. 

14 

f 77 
8 

97 

33 

113-75 


Champlain 

Cayuga  and  ) 

Seneca j 

Delaware  and  ) 

Raritan j 

Dismal  Swamp. . . 

Erie 

Falls  of  Ohio,  Ky. 

Oswego 

Welland,  Canada. . 


18 

24 

i7-5 

18 

80 

18 

45 


Feet. 

5 

7 

7 

5-5 

7 

2-60 

4 

14 


183 


Length 

of 

CanaL 

Miles. 

66.75 

24-75 

43 

44 
352 

38 

28 


Length  of  vessel  that  can  be  transported  is  somewhat  less  than  lengths  of  locks. 

Suez  Canal.  — Width  196  to  328  feet  at  surface,  72  at  bottom,  and  26  deep 
Length  99  miles.  ’ 


Hei.gh.ts  of  obtained  Elevations,  and  various  -Places 
and  Points  above  the  Sea. 

Locations. 


Aconcagua,  Chili . . 
Antisana,  highest 
established  eleva- 
tion (Farmhouse)  . . 
Bal  lo  on  ( G a y Lussac) 
“ (Green,  1837) 
“ (Glaisherand 

Coxwell) 

Brazil,  Quito,  and 
Mexico  plains. . 

Condor’s  flight 

Eagle’s 


Feet. 


23910 


*3  434 
22  900 
27  000 

37000 
6000 
8 000 
29  500 
16  500 


Everest,  Himalaya.  | 29003 


Locations. 


Geneva  city 

Geneva  Lake... 

Gibraltar 

Humboldt’s  highest 

elevation 

Tsthmus  of  Darien.. 
Jungfrau,  Switz’d. . 
La  Paz,  Bolivia. . . . 
Laguna,  Teneliffe. . . 

London,  city 

Madrid 

Mexico,  city  of 

Mont  Blanc,  Alps.. . 


Feet. 


1 220 
1 096 

1 439 

19  400 

645 

13725 

12  225 

2 OOO 

64 
2 200 
7 525 
J5  797 


Locations. 


Mont  Rosa,  Alps . . 

Mount  Adams 

Mount  Katahdin . . . 

Mount  Pitt 

Mount  Washington. 

Paris,  city 

Pont  d’  Oro,  Pyr’s. . 
Posthouse,  Ap. , Peru 

Potosi,  Bolivia 

Quito 

St.  Bernard’s  Mon’y 

Vegetation 

White  Mountain . . . 


Feet. 


15  155 
5 930 

5 36a 
9 549 
6426 

ii5 
9843 
I4  377 
13223 
13  5oo 
8 040 
17000 

6 230 


Lengths  of  Rivers. 


Rivers. 

Miles. 

Rivers. 

Miles. 

Rivers. 

Miles. 

EUROPE. 

Ganges 

1514 

3040 

1800 

176 

2762 

1160 

Kansas 

Hoang  Ho 

La  Platte 

850 

Danube 

1800 

Indus 

Maekenyie 

Dnieper 

1243 

400 

1035 

780 

Jordan  

2440 

Douro 

Lena 

Missouri 

*35° 

Dwina 

Tigris  . , 

3°3° 

1480 

Elbe 

Yenesei  and  Se- 

Ohio and  Allegheny 
Potnm  n r*. 

Garonne 

442 

545 

420 

760 

5io 

450 

250 

tin 

lenga 

358o 

Red 

420 

Loire 

Yang-Tse 

1520 

Po 

33*4 

n , 

2300 

Rhine 

AFRICA. 

irwio  (jrnin&o..  •••••• 

St.  Lawrence 

1800 

Rhone 

Gambia 

2172 

620 

Seine 

Niger 

Tennessee. ........ 

Shannon 

Nile 

2400 

790 

Tagus 

4000 

SOUTH  AMERICA. 

Thames 

220 

NORTH  AMERICA. 

Tiber 

190 

630 

2400 

Arkansas 

p,  ' 

4000 

Vistula 

Colorado 

2070 

1050 

lSSGCJUIDO  . • • 

520 

900 

1600 

Volga,  Russia 

Columbia 

Orinoco 

ASIA. 

Connecticut 

410 

420 

Platte 

Delaware 

Rio  Madeira 

2300 

Amoor 

2500 

1786 

Hudson  and  Mo- 
hawk 

2300 

Euphrates 

325  | 

Rio  Negro. ........ 

TTrn  orn  o xr 

1650 

1100 

1 84 


SEA  DEPTHS,  BUILDING  STONES,  ETC. 


Large  Trees  in.  California. 

“ Keystone  State.  ”— Calavera  Grove,  is  325  feet  in  height. 

« Father  of  the  Forest.” -Felled,  is  385  feet  in  length,  and  a man  on  horseback 
can  ride  erect  90  feet  inside  of  its  trunk. 

« Mother  of  the  Forest”- Is  315  feet  in  height,  84  feet  in  circumference  (26.75  feet 
in  dfameter)  inside  of  its  bark,  and  is  computed  to  contain  537  000  feet  of  sound  1 
inch  lumber.  Sea  -Deptlls. 


| Feet.  I 

Feet. 

Feet. 

Baltic  Sea 1 

Adriatic 

English  Channel. . . 
Straits  of  Gibraltar. 1 
Eastward  of  “ 

120 

13° 

| 3°° 

too 
1 3000 

Coast  of  Spain 

West  of  St.  Helena. 
Tortugas  to  Cuba  . . 

Gulf  of  Florida 

Off  Cape  Florida. . . 

6 000 
27  000 
4 200 
3720 
1950 

Off  Cape  Canaveral. 

“ Charleston 

“ Cape  Hatteras. . 

“ Cape  Henry 

“ Sandy  Hook 

26  000  fe 

2400 

4200 

3120 

4200 

2400 

et. 

“ “ Pacific 

250  miles  off  Cape  Cod,  no  bottom  at  7800  feet. 

Cascades  and  “W aterfalls. 


29000 


Arve,  Savoy. . . 
Cascade,  Alps  . 


Cataracts  of  the  Nile. 


Chachia,  Asia 

Foyers,  Scotland  . . 

Garisha,  India 

Gavarny,  Pyrenees 


Feet. 


1600 

2400 

3° 

34 

40 

362 

197 

1000 

1260 


Genesee,  N.  Y 100 

Lidford,  England 1 100 

Lulea,  Sweden 1 600 

Mohawk 68 

(5° 

Missouri | ]8o 

(94 
250 
800 


Location. 


Niagara 

Great  Fall 

Passaic 

Potomac 

Ribbon,  Yosemite) 

Valley ) 

Ruican,  Norway  — 
Staubbach,  Switz'd. . 
Tendon,  France... 


164 

152 

74 

74 

3300 

800 

798 

125 


Montmorenci  . . . 

Nant  d’Apresias. 

Yosemite  Valley 2600  feet. 

E^aixsion  and  Contraction  of  Building  Stones  for  eacli 
Degree  of  Temperature.  {Lieut  W.  H.  C Bartlett,  U.  S.  E.) 

For  One  Inch. 

Sandstone 000  009  532 

Whitepine 00000255 


For  One  Inch.  1 

Granite 000004825 

Marble 000005668  | 


Resistance  of  Stones,  etc.. 


to  tlie  Effects  of  Freezing. 

Various  experiments  show  that  the  power  of  stones,  etc.,  to  resist  effects  of  freez- 
ing is  a fair  exponent  of  that  to  resist  compression. 

Magnetic  Bearings  of  ISTew  York. 

The  Avenues  of  the  City  of  New  York  bear  28°  50'  30"  East  of  North. 

Filters  for  Waterworks. 

x square  yard  of  filter  for  each  840  U.  S.  and  700  Imp’l  gallons  in  24 
hours • formed  of  2.5  feet  of  fine  sand  or  gravel  and  6 inches  of  common 
sand  or  shells. 

Led  off  by  perforated  pipes  laid  in  lowest  stratum. 

Distances  between  ISTew  York,  Boston,  Bliiladelpkia, 
Baltimore,  and  Western  Cities  of  TJ. 

Assuming  Boston  as  standard,  New  York  averages  .2  per  cent,  nearer  to  these 
cities,  Philadelphia  18  per  cent.,  and  Baltimore  22  per  cent. 

Between  New  York  and  Chicago  the  line  of  the  Penns>dyania  Railroad  8 47  ™^ 
shorter  than  that  by  the  Erie  and  its  connections,  50  miles  sbofter 
N.  Y.  Central  and  Hudson  River  and  its  connections,  and  114  miles  shorter  than  that 
by  the  Baltimore  and  Ohio  and  its  connections. 

For  Distances  between  these  and  other  cities  of  the  U.  S.,  see  page  88. 


WEATHER-PLANTS,  ANTIDOTES,  ETC. 


i85 

"Weather- foretelling  iPlaxits.  ( Hanneman .) 

If  Rain  is  imminent. — Chickweed,*  Stellaria  media ; its  flowers  droop 
and  do  not  open.  Crowfoot  anemone,  Anemone  ranunculoides ; its  blossoms 
close.  Bladder  Ketmia,  Hibiscus  trionum ; its  blossoms  do  not  open.  Thistle, 
Carduus  acaulis;  its  flowers  close.  Clover,  Trifolium  pratense , and  its  allied 
kinds,  and  Whitlow  grass,  Draba  verna ; all  droop  their  leaves.  Nipple- 
wort, Lampsana  communis ; its  blossoms  will  not  close  for  the  night.  Yel- 
low Bedstraw,  Galium  verum ; it  swells,  and  exhales  strongly ; and  Birch, 
Betula  alba , exhales  and  scents  the  air. 

Indications  of  Rain. — Marigold,  Calendula  pluvialis ; when  its  flowers  do 
not  open  by  7 A.  M.  Hog  Thistle,  Sonchus  arvensis  and  oleraceus ; when  its 
blossoms  open. 

Rain  of  short  duration.— Chickweed,  Stellaria  media ; if  its  leaves  open 
but  partially. 

if  cloudy. — Wind-flower,  or  Wood  Anemone,  A nernone  memorasa ; its 
flowers  droop. 

Termination  of  Rain.  — Clover,  Trifolium  pratense ; if  it  contracts  its 
leaves.  Birdweed  and  Pimpernel,  Convolvulus  and  Anagallis  arvensis;  if 
they  spread  their  leaves. 

Uniform  Weather. — Marigold,  Calendula  pluvialis ; if  its  flowers  open  early 
in  the  A.  M.  and  remain  open  until  4 l\  M. 

Clear  Weather. — Wind-flower,  or  Wood  Anemone,  Anemone  memorasa ; 
if  it  bears  its  flowers  erect.  Hog  Thistle,  Sonchus  arvensis  and  oleraceus ; 
if  the  heads  of  its  blossoms  close  at  and  remain  closed  during  the  night. 

Treatment  and.  -Antidotes  to  Severe  Ordinary  Poisons. 
Antidotes  in  very  small  doses. 

Chloroform  and  Ether. — Cold  affusions  on  head  and  neck,  and  ammonia 
to  nostrils.  Antidote. — Camphor,  petroleum,  sulphur. 

Toadstools. — (Inedible  mushroom).  Antidote.—  Same  as  for  chloroform. 
Arsenic  or  Fly  Powder. — Emetic ; after  free  vomiting  give  calcined  mag- 
nesia freely.  If  poison  has  passed  out  of  stomach,  give  castor  oil. 

Antidote.—  Camphor,  nux  vomica,  ipecacuanha. 

Acetate  of  Lead  (Sugar  of  lead).  — Mustard  emetic,  followed  by  salts, 
Large  draughts  of  milk  with  white  of  eggs. 

Antidote. — Alum,  sulphuric  acid  alike  to  lemonade,  belladonna,  strychnine. 
Corrosive  Sublimate  (Bug  poison).  — White  of  eggs  in  1 quart  of  cold 
water,  give  cupful  every  two  minutes.  Induce  vomiting  without  aid  of 
emetics.  Soapsuds  and  wheat  flour  is  a substitute  for  white  of  eggs. 

Antidote. — Nitric  acid,  camphor,  opium,  sulphate  of  zinc. 

. Phosphorus  Matches— Rat  Paste. — Two  teaspoonfuls  of  calcined  magne- 
sia, followed  by  mucilaginous  drinks.  Antidote. — Camphor,  coffee,  nux  vomica. 

Carbonic  Acid  (Charcoal  fumes),  Chlorine , Nitrous  Oxide , or  Ordinaiy 
Lu.?.— Fresh  air,  artificial  respiration,  ammonia,  ether,  or  vapor  of  hot  water. 
Antidote. — Camphor,  coffee,  nux  vomica. 

Belladonna  (Nightshade).  — Emetic  and  stomach  pump,  morphine  and 
strong  coffee.  Antidote.— Camphor. 

Opium.  Stomach  pump  or  emetic  of  sulphate  of  zinc,  20  or  30  grains,  or 
mustard  or  salt.  Keep  patient  in  motion.  Cold  water  to  head  and  chest. 
Antidote.  Strong  coffee  freely  and  by  injection,  camphor,  ether,  and  nux  vomica. 

Strychnine  (Nux  vomica). — Stomach  pump  or  emetic,  chloroform,  cam- 
phor, animal  charcoal,  lard,  or  fat. 

Antidote.  Wine,  coffee,  camphor,  opium  freely,  and  alcohol  in  small  doses. 

Vegetable  Poisons.— As  a rule,  an  emetic  of  mustard  and  drink  freely  of 
warm  water.  J 


* Spreads  its  leaves  about  9 A.  M.,  and  they  remain  open  until  noon. 

Q* 


veterinary. 


1 86 


"V'eterinary. 

u „„  Cnthartic  Ball Cape  Aloes,  6 to  io  drs. ; Castile  Soap,  i dr. ; Spirit 

of  wSrf  dr"  Simp  to  form  a baPll.  If  Calomel  is  required,  add  from  20  to  50 
grains.  ’ Daring  its  operation,  feed  upon  mashes  and  give  plenty  of  water. 

geP3adrtsleiDx(;aaid1 giveTn  aPquartof  g4rueT  For  Calves,1 on^third  will  be  sufficient. 

, „ , nnimn  t dr  • Gineer  2 drs.;  Allspice,  3 drs.,  and  Caraway  Seeds,  <1 

drs  °?all  powdered.  ’ Make  into  a ball  with  sirup,  or  give  as  a drench  m gruel. 

cordial 

strong  Ale° or’  (frae?,  r pint.  Give  every  morning  till  purging  ceases.  For  Sheep 

2 A Iterative  — Ethiop’s  Mineral,  .5  oz. ; Cream  of  Tartar,  1 oz. ; Nitre,  2 drs.  Divide 
into  from  16  to  24  doses,  one  morning  and  evening  in  all  cutaneous  di.ea.es. 

Diuretic  Ball- Hard  Soap  and  Turpentine,  each  4 drs. ; Oil  of  Jumper,  20  drop  , 

^SSS^SBaaaisossas^^ 
5rrt  wsssasaai  ssa&vami  «s 

Tallow,  each  , lb. ; Turpentine  .5  lb.  Melt  and  mix. 

Repeat’every  5 houra’till it  o3pSs°“  °f 
* * — fuI  or  two  of 

common  salt.  Give  twice  a week  if  required.  . 

Distemper  Powder  - Antimonial  Powder, ^^3,  0^4  ^5, 

STay'^K1 -5  gr.  to  , gr.  of  Digitalis,  and  every  3 or 

J'Tfrd"^  . 

ei^ris  : 

eral  days  between  each  dose. 

Age  of  Horses. 

To  Ascertain  a Horse’s  Age. 

A foal  of  six  months  has  six  grinders  in  each  jaw,  three  in  each  side,  and  also  six 

nippers  or  front  teeth,  with*Tn  front' Weth  begin  to  decrease,  and  he  l.as  four 
At  age  of  one  Jf  sffieCaone  of  permanent  and  remainder  of  milk  set. 
grinders  upon  each  side  1 first  milk  grinders  above  and  below,  and  front 

At  age  of  two  years  be  Jos  8 horses  of  eight  years  of  age.  j 

casts  his  two  front  uppers,  and  in  a i 
short  time  after  the  ‘«  ““t  and  about  four  ^ a half  his  nippers  < 

ar^Vormanent^by8 re^lactog  ^ ^remaining  two  corner  teeth ; tushes  then  appear,  . 
anlthflvt  Thorne  ha^^ bt'tushes,  and  there  is  a black-colored  cavity  in  centre  of  all  ' 

cavity  is  obliterated  in  the  two  front  lower  nippers 
At  six,  this  matK  ; c ; y and  tushes  blunted;  and  at  eight,  that 

At  seven,  cavities  of  next  two  are  ini  be  a ed  cavities  in  nippers 


DISTANCES,  POPULATION,  DEOWNING,  ETC.  1 87 


Distances  between  ^Principal  Cities  of  ICast  and.  West. 
In  Miles. 


Cities. 


Burlington,  la. 

Chicago 

Cincinnati 

Cleveland 

Columbus,  0. . 

Detroit 

Indianapolis  . . 
Kansas  City. . . 


j Boston. 

New 

York. 

Phila- 

delphia. 

[ Balti- 
more. 

Cities. 

Boston. 

New 

York. 

Phila- 

delphia. 

Balti- 

more. 

1216 

1106 

1030 

995 

Louisville 

1161 

870 

794 

706 

1009 

900 

823 

802 

Memphis 

I438 

1247 

1171 

1083 

927 

743 

667 

576 

Milwaukee 

998 

947 

908 

887 

671 

580 

504 

483 

Omaha 

1503 

1393 

I3W 

1294 

807 

623 

547 

512 

St.  Joseph 

1478 

1356 

1280 

1223 

724 

673 

682 

661 

St.  Louis 

1212 

1050 

973 

917 

951 

810 

735 

700 

St.  Paul 

1418 

1308 

1232 

1211 

1487 

1324 

1248 

1192 

Toledo 

784 

693 

617 

596 

Population  of  Principal  Cities  (1SS2). 


London 3832440 

Paris 2225910 

Berlin 1222500 

New  York 1206299 

Vienna 1103  no 

St.  Petersburg. . . 876  570 

Philadelphia 847  170 

Moscow 611970 

Constantinople . . 600  000 

Chicago 583  185 

Brooklyn 566663 

Hamburg 410 120 

Naples 403  no 

Lyons 372  890 

Madrid 367  280 

Boston 362  839 

Buda-Pesth 360  580 


Marseilles 357530 

St.  Louis  350518 

Warsaw 339400 

Baltimore 332313 

Milan 321440 

Amsterdam 317  010 

Rome 300470 

Lisbon.. 246300 

Palermo 244990 

Copenhagen 234850 

San  Francisco. . . . 233959 

Munich 230200 

Cincinnati 225  139 

Bucharest 221  800 

Dresden 220820 

New  Orleans 216190 

Florence 169000 


Stockholm 168770 

Brussels 161  820 

Cleveland 160146 

Pittsburgh 156389 

Buffalo 155  134 

Antwerp 150650 

Washington 147293 

Cologne 144770 

Frankfort 136820 

Newark 136508 

Venice 132830 

Louisville 123758 

Jersey  City 120722 

Detroit 116340 

Milwaukee.. 115587 

Providence 104857 

Rouen 104  010 


Treatment  of  Drowning  Persons. 

Practice  adopted  by  Board  of  ITealtli,  Ne w York. 

Place  patient  face  downward,  with  one  of  his  wrists  under  his  forehead.  Cleanse 
his  mouth.  If  he  does  not  breathe,  turn  him  on  his  back  with  shoulders  raised  on 
a support.  Grasp  tongue  gently  but  firmly  with  fingers  covered  with  end  of  a hand- 
kerchief or  cloth,  draw  it  out  beyond  lips,  and  retain  it  in  this  position. 

To  Produce  and  Imitate  Movements  of  Breathing. — Raise  patient’s  extended  arms 
upward  to  sides  of  his  head,  pull  them  steadily,  firmly,  slowly,  outwards.  Turn 
down  elbows  by  patient’s  sides,  and  bring  arms  closely  and  firmly  across  pit  of 
stomach,  and  press  them  and  sides  and  front  of  chest  gently  but  strongly  for  a mo- 
ment, then  quickly  begin  to  repeat  first  movement. 

Let  these  two  movements  be  made  very  deliberately  and  without  ceasing  until 
patient  breathes,  and  let  the  two  movements  be  repeated  about  twelve  or  lifteen 
times  in  a minute,  but  not  more  rapidly,  bearing  in  mind  that  to  thoroughly  fill  the 
lungs  with  air  is  the  object  of  first  or  upward  and  outward  movement,  and  to  expel 
as  much  air  as  practicable  is  object  of  second  or  downward  motion  and  pressure 
This  artificial  respiration  should  be  maintained  for  forty  minutes  or  more  when  the 
patient  appears  not  to  breathe;  and  after  natural  breathing  begins,  let  same  motion 
be  very  gently  continued,  and  give  proper  stimulants  in  intervals. 

What  Else  is  to  he  Done , and  What  is  Not  to  he  Done , while  the  Movements  arc 
being  Made. — If  help  and  blankets  are  at  hand,  have  body  stripped  wrapped  in 
blankets,  but  not  allow  movements  to  be  stopped.  Briskly  rub  feet  and  legs,  press- 
ing them  firmly  and  rubbing  upward,  while  the  movements  of  the  arms  and  chest 
are  in  progress.  Apply  hartshorn,  or  like  stimulus,  or  a feather  within  the  nostrils 
occasionally,  and  sprinkle  or  lightly  dash  cold  water  upon  face  and  neck.  The 
legs  and  feet  should  be  rubbed  and  wrapped  in  hot  blankets,  if  blue  or  cold  or  if 
weather  is  cold.  ’ 


What  to  Do  when  Patient  Begins  to  Breathe. — Give  stimulants  by  teaspoonful  two 
or  three  times  a minute,  until  beating  of  pulse  can  be  felt  at  wrist,  but  be  careful 
and  not  give  more  of  stimulant  than  is  necessary.  Warmth  should  be  kept  up  in 
teet  and  legs,  and  as  soon  as  patient  breathes  naturally,  let  him  be  carefully  removed 
to  an  enclosure,  and  placed  in  bed,  under  medical 


MISCELLANEOUS  ELEMENTS. 


1 88 


MISCELLANEOUS  ELEMENTS. 

Earth. 

Polar  diameter  7890.3  miles.  Mean  density  or  specific  gravity  of  mass  5.672.  Mass 
5 37° Zo  000  000  io9^,  000  tons.  Apparent  diameter  as  seen  from  Sun  .7  seconds. 

Sun. 

Heat  of  Sun  equal  to  322  794  thermal  units  per  minute  for  each  sq.  foot  of  pho- 

t0Dhimeter  of  Su^S^ooo  miles,  tangential  velocity  1.25  miles  per  second  or  4.41 
times  greater  than  that  of  the  Earth. 

Distance  from  Earth  91.5  to  92  millions  of  miles. 

Mason  and.  Dixon’s  Dine. 

39O  ^3'  26. 2>"  N.  mean  latitude.  68.895  miles. 


Divisions. 

Area. 

Population. 

Divisions. 

Area. 

Population. 

America. 

Europe 

Sq.  Miles. 
14  491  000 
3 760000 
16  313000 
10936  000 

95  495  5oo 
315  929000 
834  707  000 
2°5  679  OOO 

Ocean  ica. . . 
Greenland  \ 
Iceland  j 

Total. 

Sq.  Miles. 
4 500  OOO 

4031  060 
82000 

Africa 

50000000 

M55  923  500 

Austria  ) 
Hungary ) 

China 

France. . . 


, . 38  000  000 

.434626000 
37  000  000 


Countries. 

Germany 43900000  India,  British  ..240298000 

Great  Britain. .34000000  Canada 3839000 

(Russia.. 66000000  Mexico 9485000 

(Territories  ..  .22 000000  Brazil 11106000 

( United  States 50000000  I (Turkey.......  8866000 

(Indians 300000  | ( ‘ in  Asia.  .16  320 000 

About  one  thirtieth  of  whole  population  are  born  every  year,  and  nearly  an  equal 
Dumber  die6 in  same  time;  making  about  one  birth  and  one  death  per  second. 
Earlier  authority  estimated  population  at  1 288000000,  divided  as  follow  s. 


Caucasians 360  000  000 

Mongolians 552000000 

Ethiopians 190  000  000 

Asiatics 60000000 


Mohammedans . 190  000  000 

Pagans 300  000  000 

Catholics  1 250000000 
Rom.  & Greek ) 


Malays  and  , 177000000 
Indo-Amer’s  ) 11 

Protestants 80000000 

Israelites 5000000 

Descent  of  Western  Divers. 

Slope  of  rivers  flowing  into  Mississippi  from  East  is  about  3 inches  per  mile; 

1 DM ean^descTn t* of* Oh k)  River  from  Pittsburgh  to  Mississippi,  975  miles,  is  about  5.2 
inK ^per  mfle;  and  that  of  Mississippi  to  Gulf  of  Mexico,  1x80  miles,  about  2.8 

inches.  Transmission  of  Horse  Dower. 

T nn?est  and  perhaps  most  successful,  wire  rope  transmission  is  one  at 
han«?en  at  Falls  of  the  Rhine.  Here,  power  of  a number  of  turbines,  amounting 
to  over’ 600  IP,  is  conveyed  across  the  stream,  and  thence  a mile  to  a town,  where  it 

1 S Mm  toes  *0?  *Fa  km*  Sweden,  a power  of  over  100  horses  is  transmitted  in  like 
manner  for  a distance  of  three  miles. 

Acids. 

Acetic  Acid  (Vinegar),  acid  of  Malt  beer.  etc.  Tartaric  Acid,  acid  of  Grape  tin ns. 
Lactic  Acid,  acid  of  Milk , Millet  beer,  and  Cider. 

IVf  anures. 

Relative  Fertilizing  Properties  of  Various  Manures. 

Peruvian  Guano  . . . . i | Horse 048  | Farm-yard 0298 

Human,  mixed 069  | Swine 044  | Cow • * ‘ 59  w 

Or,  1 lb.  guano  = 14.5  human,  21  horse,  22.5  swine,  33.5  farm-yard,  and  38.5  cow. 

Relative  Value , Covered  and  Uncovered , on  an  Acre  of  Ground. 

Cowed  1 1 tons  1665  lbs.  potatoes,  61  lbs.  wheat,  215  lbs.  straw 

Uncovered.'.'.'. 7 “ *397  “ “ 6l*5  “ ‘ 156 


MISCELLANEOUS  ELEMENTS, 


I89 


Yield,  of*  Oil  of*  Several  Seeds. 

PerCent.  I Per  Cent.  I Per  Cent.  I Per  Cent.  I PerCent 

Poppy. . 56  to  63  I Castor . . 25  I Sunflower.  15  | Hemp.  14  to  25  I Linseed.  n to  22 

Thickness  of*  Walls  of*  Buildings.  (English.)  ( Molesworlh .) 


Outer  Walls. 

Maximum 
Height 
of  Wall. 

Width 

of 

Footings. 

Ground 

Floor. 

Mi 

ist 

Floor. 

inimum 

2d 

Floor. 

Width  < 
3*1 

Floor. 

>f  Walls 
4th 
Floor. 

5 th 
Floor. 

I 6th 
| Floor. 

Feet. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins 

Ins. 

“ l~  g ' 

1st  class 

dwelling. 

85 

38.5 

21.5 

21.5 

i7-5 

17-5 

17-5 

13 

I3 

2d  u 

70 

30-5 

17-5 

*7-5 

17-5 

13 

13 

13 

3d  “ 

52 

30.5 

I7-5 

13 

13 

13 

13 



4th  “ 

38 

21.5 

13 

13 

8.5 

8.5 

— 

— 

Party 

Walls. 

1st  class 

dwelling. 

85 

38.5 

21.5 

21.5 

i7-5 

17-5 

17-5 

13 

13 

2d  “ 

“ 

70 

30-5 

17-5 

17-5 

17-5 

13 

13 

1.3 

3d  “ 

“ 

52 

30-5 

I7*5 

13 

13 

13 

8.5 



4 th  “ 

u 

38 

21.5 

13 

8-5 

8.5 

8.5 

— 

— 

by  half  a brick. 

Warehouses  Mw-^m 

Width 

1st  Class.  of  Wall. 
For  a height  of  36  feet  from  ins. 
topmost  ceiling 17.5 


Warehouses 

2d  Class.  of  Wall. 
For  a height  of  22  feet  below  ins. 


topmost  ceiling 


13 


For  a height  of  40  feet  lower . . 21.5  ; For  a height  of  36  feet  lower  . . 17, 


24  feet  lower . . 26 
For  footings 43.5 

3d  Class. 

For  a height  of  28  feet  below 

topmost  ceiling 13 

For  a height  of  16  feet  lower  . . 17.5 
For  footings. 


1 7-5 

8 feet  lower  . . 21.5 
For  footings 34.5 

dtli  Class. 

For  a height  of  9 feet  below 

topmost  ceiling 8.5 

For  a height  of  13  feet  below . . 13 
30.5  ; For  footings 21.5 


Wooden  Roofs.  (English.) 


Span 
in  Feet. 

Principal 

Beam. 

Tie  Beam. 

King 

Posts. 

Queen 

Posts. 

Small 

Queens. 

Straining 

Beam. 

Struts. 

20 

4X4 

9X4 

4X4 

— 

— 

— 

3 

X 3 

25 

5X4 

10  X 5 

5x5 

— 

■f— 

— 

5 

X 3 

30 

6x4 

11  X 6 

6X6 

— 

— 

— 

6 

X 3 

35 

5X4 

11  X 4 

— 

4X4 

— 

7X4 

4 

X 2 

45 

6X5 

13  X 6 

— 

6x6 

— 

7X6 

5 

X 3 

50 

8x6 

13X8 

— 

8x8 

8X4 

9X6 

5 

X 3 

55 

8x7 

14  x 9 

— 

9X8 

9X4 

IO  x 6 

5-5 

X 3 

60 

8x8 

15  X 10 

— 

10  X 8 

IO  X 4 

11  X 6 

6 

X 3 

Mineral  Constituents  absorbed  or  removed  fr, 
Acre  of  Soil  b y several  Crops.  (Johnson.) 


om  an 


Crops. 

Wheat, 

25 

bushels. 

Barley, 

40 

bushels. 

Turnips, 
20  tons. 

Hay, 
1.5  tons. 

Crops. 

Wheat, 

25 

bushels.  - 

Barley, 

40 

bushels. 

Turnips, 
20  tons. 

Hay, 
1.5  tons. 

Potassa 

Lbs. 

29.6 

Lbs. 

Lbs. 

47.1 

8.2 

Lbs. 

Q ry 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Soda 

I7-5 

JO.  ^ 

oUipnunc  I 
Acid. . . } • ‘ 

fib  Torino 

10.6 

2.7 

z3-3 

9.  2 

Lime 

3 

12.9 

10.6 

5-2 

W 

9.2 

29.9 

19.7 

7-i 

46-3 

44-5 

7.  I 

16 

129.5 

2.4 

Magnesia 

Silica.  . . 

118. 1 

3-6 

247.8 

4.I 

78.2 

Oxide  of  Iron. 
Phosphoric  ) 

2.6 

20.6 

2. 1 

/•  1 
.6 

Alumina. 

Acid ] 

25.8 

i5- 1 

Total 

210 

213 

423 

209 

190 


miscellaneous  elements. 


Average  Quantity  of  Tannin  in  Several  Substances, 
( Morjit . ) 


Catechu.  Per  Cent. 

Bombay 55 

Bengal 44 

Kino 75 

Nutgalls. 

Aleppo 65 

Chinese 

Oak. 

Old,  inner  bark 


Oak.  Per  Cent. 

Young,  inner  b’k  15.2 
“ entire  b’k.  6 

“ spring-  ) 
cut  bark ) 

“ root  bark.  8.9 

Chestnut. 

Amer.  rose,  bark  8 

14.2  Horse,  “ 2 

Sassafras , root  bark  58 
Alder  hark 36  per  cent. 


69 


Sumac.  Per  Cent. 

Sicily  and  Malaga  16 

Virginia 10 

Carolina 5 

Willow. 

Inner  bark 16 

Weeping 16 

Sycamore  bark 16 

Tan  shrub  “ 13 

Cherry-tree 24 


To  Convert  Chemical  Formulae  into  a Mathematical 
Expression. 

1?ULE Multiply  together  equivalent  and  exponent  of  each  substance,  and  product 

Win  give  proportion  A compound  by  weight.  Divide  .ooo  by  sum  of  ttor  praduete, 
and  multiply  this  quotient  by  each  of  these  products,  and  products  w ill  give  re 
spective  proportion  of  each  part  by  weight  in  1000. 

Example. — Chemical  formula  for  alcohol  is  CAH602.  Required  their  propor- 
tional parts  by  weight  in  1000? 

Ca  Carbon  ==  6. 1 X 4 = 24-  4 ) 

. He  Hydrogen  = 1X6=  6 /X21.55 

02  Oxygen  = 8X2  = 16 ) 

1000  -f-  46.4  =21.55 

Symbols  and. 


525-82 

129.3 

344-8 

999.92 


by  weight. 


Elementary  Bodies,  with,  their 
Equivalents. 


Body. 


| Symb.  | Equiv. 


Aluminium.. . 
Antimony.. . . 

Arsenic 

Barium 

Bismuth. .... 

Boron ... 

Bromine. . . . . 

Cadmium 

Calcium 

Carbon 

Chlorine 

Chromium. . . 

Cobalt 

Columbium. . 

Copper 

Fluorine 

Glucinum  — 

Analysis 

Body. 


13-7 

64.6 
37-7 

68.6 
7i-5 
11 

78.4 
55-8 

20.5 
6.1 

35-5 
26.2 
29-5 
184. 8 
3i-7 
18.7 
6.9 


Body. 


| Symb. 


Equiv. 


Gold 

Hydrogen 

Iodine 

Iridium 

Iron 

Lead 

Lithium 

Magnesium . . 
Manganese. . . 

Mercury 

Molybdenum. 

Nickel 

Nitrogen 

Osmium 

Oxygen 

Palladium — 
Phosphorus. . 


196.6 

1 

126.5 

98- 5 

28 

io3-7 

7 

12.7 

26 

200 

47-9 

29-5 

14.2 

99- 7 

8 

53-3 

15-9 


Body. 

Symb. 

Equiv. 

Platinum 

Pt 

98.8 

Potassium . . . 

K 

39-2 

Rhodium 

R 

52.2 

Selenium .... 

Se 

40 

Silicon 

Si 

22 

Silver 

Ag 

108.3 

Sodium 

Na 

23-5 

Strontium.... 

Sr 

43-8 

Sulphur 

S 

16. 1 

Tellurium 

Te 

64.2 

Tin 

Sn 

58-9 

Titanium 

Ti 

24-5 

Tungsten  . . . . 

W 

¥ 

Uranium  . . . . 

U 

60 

Yttrium 

Y 

32 

Zinc 

Zn 

32-3 

Zirconium . . . 

Zr 

34 

of  certain  Organic  Substances  "by  Weight. 


Albumen 

Alcohol 

Atmospheric  air 

Camphor 

Caoutchouc .... 

Casein 

Fibrin 

Gelatine 

Gum 

Hordein 

Lignin 


Hydro 

gen. 


52-9 

52- 7 

73-4 

87.2 
59*8 

53- 4 
47-9 
42.7 

44.2 
52.5 


7-5 

12.9 

10.7 

12.8 
7-4 
7 

l9 

6.4 

6.4 

5-7 


Oxy- 

gen. 


Nitro- 

gen. 


23-9 

34-4 

77  . 

15.6 

11. 4 

19.7 
27.2 

50.9 

47.6 

41.8 


15-7 

23 


21.4 

19.9 

17 

1.8 


Morphine 

Narcotine 

Oil,  Castor 

Linseed. . . . 
Spermaceti. 

Quinine 

Starch 

Strychnine 

Sugar 

Tannin 

Urea 


Hydro- 

gen 


72.3 
65 
74 
76 

78  Q 

75-8 

44.2 

76.4 

42.2 
52.6 
18.9 


6.4 
5-5 
10.3 
ii-3 
11. 8 

7-5 

6.7 

6.7 

6.6 

3-8 

9-7 


Oxy- 

gen. 


16.3 

27 

i5-7 

12.7 

10.2 

8.6 

49.1 

11. 1 

43- 6 

26.2 


8.1 

7s 


45-2 


MISCELLANEOUS  ELEMENTS, 


191 


Dilution  Per  Cent.  Necessary  to  Reduce  Spiritnons 
Diqnors. 

Water  to  be  added  to  100  volumes  of  spirit  when  of  following  strength: 


Strength 

Required. 

90 

85 

80 

75 

70 

65 

60 

55 

50 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

Per  cent. 

85 

5-9 

— 

— 

— 

— 

— 







80 

12.5 

6-3 

— 

— 

— 









75 

20 

13-3 

6-7 

— 

— 

— 







70 

28.6 

21.4 

14-3 

7- 1 

— 









65 

38.5 

30.8 

23.1 

i5-4 

7-7 

— 







60 

50 

41.7 

33-3 

25 

16.7 

8-3 

— 





55 

63.6 

54-5 

45-5 

36.4 

27.4 

18.2 

9.1 

— 



50 

80 

7° 

60 

50 

40 

30 

20 

10 



40 

125 

112.5 

100 

87-5 

75 

62.5 

50 

37-5 

25 

30  • 

200 

183-3 

166.7 

150 

133-3 

116.7 

loo 

83*3 

66.7 

Illustration.— 100  volumes  of  spirituous  liquor  having  00  per  cent,  of  spirit  con- 
tains: alcohol  90,  water  10,  — 100. 

To  reduce  it  to  30  per  cent,  there  is  required  200  volumes  of  water. 


Hence  200  -f- 10  5=  210,  and  — = — = 30  spirit’  or  3Q  per  cent 
210  70  70  water,  0 * 


Proportion  of  Alcohol  Per  Cent. 

In  100  Parts  of  Spirit , by  Weight  or  Volume , at  6o°. 


Alcohol. 

Specific 

Gravity. 

Alcohol. 

Specific 
G ravity . 

Alcohol. 

Specific 

Gravity. 

Alcohol. 

Specific 

Gravity. 

0 

I 

20 

.972 

50 

.918 

80 

.848 

5 

.991 

30 

•958 

60 

.896 

90 

.823 

10 

.984 

40 

•94 

70 

.872 

100 

•794 

In  100  Parts  of  Alcohol  and  Water , by  Weight,  at  6o°. 


Alcohol. 

Specific 

Gravity. 

Alcohol. 

Specific 

Gravity. 

Alcohol. 

Specific 

Gravity. 

Alcohol. 

Specific 

Gravity. 

0 

1 

1.99 

•99  6 

5.01 

.991 

7-99 

.987 

•53 

•999 

3.02 

•994 

6.02 

•99 

9-°5 

.985 

1.02 

.998 

4.02 

•993 

7.02 

.988 

10. 07 

.984 

Tid.es  of  Atlaiatic  and  IPacific  Oceans  at  Isthmus  of 
Panama.  (Totten.) 

Atlantic , Navy  Bay.—  Highest  tide  1.5  feet;  lowest . 63  feet. 

Pacific , Panama  Bay. — Highest  tide  17.72  to  21.3  feet ; lowest  9. 7 feet. 


State. 


JSq. 


Areas  of  TJ.  S.  Coal  Fields. 


. Miles. 


Illinois 

Virginia 

Pennsylvania* . . . 
Kentucky 


44000 
21 000 
I5  437 
13500 


State. 


Ohio 

Indiana. . . 
Missourif. 
Michigan!. 


* Bituminous  and  Anthracite. 


Sq.  Miles. 


11  900 
7 700 
6000 
5000 


Tennessee  . . 

Alabama 

Maryland. . . 

Georgia 

t Anthracite. 


England 

France 106.5' 

Holland  ) 0 

102° 


Belgium  ' 


Denmark  1 

Sweden  [ 

. . 99. 50 

Norway  ) 

Russia 

Egypt. . 
Africa. . 
Asia  . . . 
Suez. . . 


England. . .—  50  ] Denmark ) 
Holland  ) 0 Sweden  [ 

Belgium  j | Norway  ) 


Sq.  Miles. 

4300 

3400 

550 

150 


Extremes  of  Heat  in  Various  Countries. 

Greece......  105° 

Italy 1040 

Spain 1020 

Tunis 112.5' 

Germany 103o  | Manilla u3.5o 

Extremes  of  temperature  upon  the  Earth  240°. 

Extremes  of  Cold  in.  Various  Countries. 


-67  c 


I France —24° 

Russia  . . . . — 46° 
I Germany. .— 320 


. 116.  i° 
• 133-4° 
. 120° 

. 1 26. 5° 


Italy —10  o 

Fort  Reliance,  N.  A.  .—70° 
Semipalatinsk,  “ ..—76° 


192 


MISCELLANEOUS  ELEMENTS. 


NT e aii.  Temperatures  of  Various  Localities. 

Polar  Regions. . 36° 
Globe 500 


London 5*° 

Edinburgh 410 


I Rome 

. . . 60°  I 

I Poles — 130  I 

1 Equator 

...  82°  | 

| Torrid  Zone.  75°! 

Line  of  Perpetual  Congelation,  or  Snow  Line. 


Latitude. 


15 

20 

25 


Height. 


Feet. 
14764 
14  760 
13  478 
12  557 


Latitude. 


30 

35 

40 

45 


Height. 


Feet. 

11484 

10287 

9000 

7670 


Latitude. 


50 

55 

60 

65 


Height. 


Feet. 

6334 

5020 

3818 

2230 


70 

75 

80 

85 


Height. 


Feet. 

1278 

1016 

451 

327 


and  in  Iceland  3084  feet. 


At  the  Equator  it  is  15260  feet;  at  the  Alps  8120  feet; 

At  Polar  Regions  ice  is  constant  at  surface  of  the  Earth. 

Limits  of  Vegetation  in  Temperate  Zone. 

The  Vine  ceases  to  grow  at  about  2300  feet  above  level  of  the  sea,  Indian  Corn  at 
2800,  Oak  at  3350,  Walnut  at  3600,  Ash  at  4800,  Yellow  Pine  at  6200,  and  Fir  at  6700. 


Periods  of  Gres  tat  ion  and  Number  of  Young. 

— ’ ' Weeks. 

9 


Weeks, 
Elephant.  100 


Horse 

Camel 

Ass 


No. 

1 

Cow 

Weeks. 

No. 

1 

Sheep . . 

Weeks. 
. 21 

No. 

2 

Buffalo . 

. 40 

1 

Goat . . . 

, . 22 

2 

1 

Stag. . . . 

• 36 

1 

Beaver , 

..  17 

3 

1 

Bear  . . . 

• 30 

2 

Pig 

..  17 

12 

1 

Deer  . . . 

. 24  | 2 

Wolf . . 

. . 10 

5 

Rabbit. . 

• 4 

6 

Guinea  Pig.  3 

3 

No. 

6 

5 

6 
8 
6 


Periods  of  Incubation  of  Birds. 

Swau  42  days-  Parrot,  40  days;  Goose  and  Pheasant,  35  days;  Duck  Turkey,  and 
Peafowl,  28  days;  Hens  of  all  gallinaceous  birds,  21  days;  Pigeon  and  Canaiy,  14 
days.  Temperature  of  incubation  is  1040. 

Ages  of  Animals,  etc. 

Bear  20;  Cow,  20;  Deer,  20;  Rhinoceros,  20;  Swine,  20;  Wolf,  20,  Cat,  15,  Fox,  15, 
Dog,  15;  Sheep,  10;  Hare,  Rabbit,  and  Squirrel,  7. 

Relative  Weights  of  Brain. 

Man,  154.33;  Mammifers,  29.88;  Birds,  26.22;  Reptiles,  4.2;  Fish,  1. 

Buoyancy  of  Casks. 

Buoyancy  of  a c ask  in  fresh  water  in  lbs.  = »-97  «mes  volume  of  it  in  U.  S.  gal- 
Ions  and  10  times  in  Imperial  gallons,  less  weight  of  cask. 

Transportation  of  Horses  and  Cattle. 

Space  re9dir®d  dfei»°is  HbrSHors°  s^Hay^  15  lte!;  ■ 

Beeves,  Haj^  18  lbs. ; Water,  6 gallons. 

Rock  and  Earth  Excavation  and  Embankment. 

Number  of  Cube  Feet  of  various  Earths  in  a Ton. 

^ n nA  . cHv  18.6  1 Clay  with  Gravel 14.4^ 

Loose  Earth 24  | uay ' - ~ 1 * 


Coarse  Sand 18.6  | Earth  with  Gravel. . . 17.8  | Common  Soil. .......  15.6 

The  volume  of  Earth  and  Sand  in  embankment  exceeds  that  in  a primary  ex- 
cavation  in  following  proportions: 

Clay  and  Earth  will  subside  about  .12. 


MISCELLANEOUS  ELEMENTS. 


193 


Hills  or  Plants  in  an  Area  of  One  Acre. 

From  1 to  40  feet  apart  from  centres . 


Feet  apart. 

No. 

Feet  apart. 

No. 

Feet  apart. 

No. 

Feet  apart.  | No. 

1 

43  560 

5 

1742 

9 

538 

16 

171 

*•5 

19360 

55 

1440 

9-5 

482 

17 

151 

2 

10890 

6 

1210 

10 

435 

18 

135 

2-5 

6969 

6.5 

1031 

10.5 

361 

20 

108 

3 

4 840 

7 

889 

12 

302 

25 

69 

3-5 

3 556 

7-5 

775 

13 

258 

30 

48 

4 

2 722 

8 

680 

14 

223 

35 

35 

4-5 

2151 

8-5 

692 

i5 

193 

40 

27 

Number  of  several  Seeds  in  a 33ushel,  and  Number  j>er 
Square  Foot  per  Acre. 


Timothy. 
Clover . . . 


No. 

Sq.  Foot. 

No. 

Sq.  Foot. 

41  823  360 
16  400  960 

960 

Rye 

888  390 
556  290 

20.4 

12.8 

376 

Wheat 

"V  olumes. 

Permanent  gases,  as  air,  etc.,  are  diminished  in  their  volume  in  a ratio  direct 
with  that  of  pressure  applied  to  them.  With  vapor,  as  steam,  etc.  this  rule  is 
varied  in  consequence  of  presence  of  the  temperature  of  vaporization 


Minerals. 


Talc 1 

Gypsum 2 

Mica 2.5 

Carbonate  of  lime.  3 


Carats. 

Mattam 367 

Grand  Mogul* 270.0 

Orlotf. 194.25 

Florentine,  brilliant . 139.5 
Crown  of  Portugal. . .138.5 
* Rough  900. 


re  Hardness  of  some  Minei 

1 Barytes 

• 3-5 

Opal 6 

Fluor-spar. . . , 

• 4 

Quartz 7 

| Feldspar 

. 6 

Tourmalin 7 

| Lapis  Lazuli 

. 6 

Garnet 7.5 

Emerald 8 

Topaz 8 

Ruby g 

Diamond 10 


“Weight  of  Diamonds. 

Carats. 

Regent  or  Pitt 136.75  I 

Star  of  the  Southf. . 125 

Koh-i-NoorJ 106.06 

Piggott 82.25  I 

Napac 78.625! 

t Rough  254.5.  • 


Carats. 

Dresden 76.5 

Sancy 53.5 

Eugenie,  brilliant . 51 

Hope  (blue) 48.5 

Polar  Star 40.25 

% Originally  793. 


Heat  of  the  Sun. 

artowNgrton 3138740°!  Waterston ,6000000° 

Capt.  John  Ericsson 4 909  86o°  J Soret 104432230 

Sundry  others  ranging  from  25200  to  183600°.  a j 

Moon. — Distance  of  Moon  from  Earth  237000  miles. 

Frigorific  Mixture. 

Lowest  temperature  yet  procured.  Faraday  obtained  166°  by  evaporation  of  a 
mixture  of  solid  carbonic  acid  and  sulphuric  ether. 

Current  of  Rivers. 

A fall  of . 1 of  an  inch  in  a mile  will  produce  a current  in  rivers. 

Sandstones. 

condi tion16S  8ands*one  erec^  England  in  12th  century  are  yet  in  good 
Canal  Transportation. 

. Erie  and  Hudson  River.— From  Buffalo  to  New  York  405  miles  cost  of 

transportation  2.46  mills  per  ton  (inclusive  of  tolls)  per  mile.  Transportation  of 
vheat  costs  when  it  reaches  New  York  4.72  cents  per  bushel,  and  .61  cents  ner 
bushel  for  elevating  and  trimming.  5 uu  .01  cents  per 

♦ ~rEr)e . Canal.  —Four  mules  will  tow  230  tons  of  freight  down  and  IOo 

of  69q  macs1  V°  V mg  a pen°d  °f  30  days’  at  a cost  of  8 cents  Pcr  miIe  for  a course 

R 


194 


MISCELLANEOUS  ELEMENTS. 


Matter. 

Unit  of  the  Physicist  is  a molecule,  and  a mass  of  matter  is  composed  of  them, 

having  same  physical  properties  as  parent  mass.  . .. 

It  exists  in  three  forms,  known  as  solid,  liquid,  and  gaseous.  Solids  have  mdi- 
vidualitv  of  form,  and  they  press  downward  alone.  Liquids  have  not  individuality 
of  fornAxcept  in  spherical  form  of  a drop,  and  they  press  downward  and  sideward. 
Gases  are  wholly  deficient  in  form,  expanding  in  all  directions,  and  consequently 
thev  uress  upward,  downward,  and  sideward. 

Liquids  are  compressible  to  a very  moderate  degree.  Water  has  been  forced 
through  pores  of  silver,  and  it  may  be  compressed  by  a pressure  of  one  pound  pei 
square  inch  to  the  3300000th  part  of  its  volume. 

Gases  may  be  liquefied  by  pressure  or  by  reduction  of  their  temperature. 

Combustible  matter  (as  coal)  may  be  burned,  a structure  (as  a - bouse)  may  be 
destroyed  as  such,  and  the  fluid  (of  an  ink)  may  be  evaporated,  yet  the  matter  of 
which  coal  and  house  were  composed,  although  dissipated,  exists,  and  the  water 
and  coloring  matter  of  the  ink  are  yet  in  existence. 

Spaces  between  the  particles  of  a body  are  termed  pores.  . 

All  matter  is  porous.  Polished  marble  will  absorb  moisture,  as  evidenced  in  its 
discoloration  by  presence  of  a colored  fluid,  as  ink,  etc. 

Silica  is  the  base  of  the  mineral  world,  and  Carbon  of  the  organized. 

M.iivu.teness  of  NIatter. 

A piece  of  metal,  stone,  or  earth,  divided  to  a powder,  a particle  of  it,  however 
minute,  is  yet  a piece  of  the  original  material  from  which  it  was  separated,  retain- 
ing its  identity,  and  is  termed  a molecule.  , 

It  is  estimated  there  are  120000000  corpuscles  in  a drop  of  blood  of  the  musk-deer 

Thread  of  a spider’s  web  is  of  a cable  form,  is  but  one  sixth  diameter  of  a fibre  of 
silk,  and  4 miles  of  it  is  estimated  to  have  a weight  of  but  1 grain.  . 

One  imperial  gallon  (277.24  cube  ins.)  of  wTater  will  be  colored  by  mixture  therein 
of  a grain  of  carmine  or  indigo. 

A grain  of  platinum  can  be  drawn  out  the  length  of  a mile. 

Film  of  a soap- and- water  bubble  is  estimated  to  be  but  the  300000th  part  ot  an 
inch  in  thickness.  , . . -T. 

It  is  computed  that  it  would  require  12000  of  the  insect  knowTn  as  the  twilignt 
monad  to  fill  up  a line  one  inch  in  length.  . 

A drop  of  water,  or  a minute  volume  of  gas,  however  much  expanded— even  to 
the  volume  of  the  Earth— would  present  distinct  molecules. 

Gold  leaf  is  the  280000th  part  of  an  inch  in  thickness. 

A thread  of  silk  is  2500th  of  an  inch  in  diameter.  , .. 

A cube  inch  of  chalk  in  some  places  in  vicinity  of  Paris  contains  100000  ol  shells 
of  the  foraminifera.  „ _ . 

There  are  animalcules  so  small  that  it  requires  75  000  000  of  them  to  weign  a gram. 


"Velocity,  Weight,  and.  Volume  of  Molecules. 

Velocity.—  Collisions  among  the  particles  of  Hydrogen  are  estimated  to  occur  at 
the  rate  of  17  million-million-million  per  second,  and  in  Oxygen  less  than  half  this 
number. 

Weight— A million-million-million-million  molecules  of  Hydrogen  are  estimated 
to  weigh  but  60  grains. 

Volume. — 19  million-million-million  molecules  of  Hydrogen  have  a volume  of  .061 
cube  ins.  Diameter.—  Five  millions  in  a line  would  measure  but  .1  inch. 

i 

Cliarcoal,  .AJLcoliol. 

Charcoal  as  yet  has  not  been  liquefied,  nor  has  Alcohol  been  solidified. 

Metals. 

Metals  have  five  degrees  of  lustre — splendent , shining , glistening , glimmering , and 
dull.  . . 

All  metals  can  be  vaporized,  or  exist  as  a gas,  by  application  to  them  of  their  ap- 
propriate temperature  of  conversion. 

Repeated  hammering  of  a metal  renders  it  brittle ; reheating  it  restores  its  tenacity. 

Repeated  melting  of  iron  renders  it  harder,  and  up  to  twelfth  time  it  becomes 
stronger. 

Platinum  is  the  most  ductile  of  all  metals. 


MISCELLANEOUS  ELEMENTS. 


195 


Impenetrability. 

Impenetrability  expresses  the  inability  of  two  or  more  bodies  to  occupy  same 
space  at  same  time. 

A mixture  of  two  or  more  fluids  may  compose  a less  volume  than  that  due  to  sum 
of  their  original  volume,  in  consequence  of  a denser  or  closer  occupation  of  their 
molecules.  This  is  evident  in  the  mixture  of  alcohol  and  water  in  the  proportion 
of  16.5  volumes  of  former  to  25  of  latter,  when  there  is  a loss  of  one  volume. 

Elasticity. 

Elasticity  is  the  term  for  the  capacity  of  a body  to  recover  its  former  volume 
after  being  subjected  to  compression  by  percussion  or  deflection. 

Glass,  ivory,  and  steel  are  the  most  elastic  of  all  bodies,  and  clay  and  putty  are 
illustrations  of  bodies  almost  devoid  of  elasticity.  Caoutchouc  (India  rubber)  is  but 
moderately  elastic^  it  possesses  contractility,  however,  in  a great  degree. 

Momentum. 

Momentum  is  quantity  of  motion,  and  is  product  of  mass  and  its  velocity.  Thus, 
the  momentum  of  a cannon-ball  is  product  of  its  velocity  in  feet  per  second  and  its 
weight,  and  is  denominated  foot-pounds. 

A foot-pound  is  the  power  that  will  raise  one  pound  one  foot. 

Sot*  nd. 

Velocity  of  sound  is  proportionate  to  its  volume;  thus,  report  of  a blast  with  2000 
lbs.  of  powder  passed  967  feet  in  one  second,  and  one  of  1200  lbs.  1210  feet.  It  passes 
in  water  with  a velocity  of  4708  feet  per  second.  Conversation  in  a low  tone  has 
been  maintained  through  cast-iron  water  pipes  for  a distance  of  3120  feet,  and  its 
velocity  is  from  4 to  16  times  greater  in  metals  and  wood  than  air. 

Eight. 

Sun’s  rays  have  a velocity  of  185  000  miles  per  second,  equal  to  7.  k times  around 
the  Earth.  ' _ 

Color  Blindness 

Is  absence  of  elementary  sensation  corresponding  to  red. 

Luminous  Point. 

To  produce  a visual  circle,  a luminous  point  must  have  a velocity  of  10  feet  in  a 
second,  the  diameter  not  exceeding  15  ins. 

All  solid  bodies  become  luminous  at  800  degrees  of  heat. 

AX  i rage. 

When  air  near  to  surface  of  Earth  becomes  so  highly  heated,  as  upon  a sandy 
plain,  that  its  density  within  a defined  distance  from  it  increases  upwards,  a line 
of  vision  directed  obliquely  downwards  will  be'  rendered  by  refraction,  gradually 
increasing,  more  and  more  nearly  horizontal  as  it  advances,  until  its  direction  is  so 
great  as  to  produce  a total  reflection,  and  the  reflected  ray  then,  by  successive  re- 
fractions, is  gradually  elevated  until  it  meets  the  eye  of  the  observer. 

Looming  is  inverted  mirage,  frequently  seen  over  calm  water,  and  is  effect  of 
lower  or  surface  stratum  of  air  being  colder  than  that  above  it. 

Snow  Flakes. 

96  forms  of  snow  flakes  have  been  observed. 

AX  el  ted  Snow 

Produces  from  .25  to  .125  of  its  bulk  in  water. 

Strength,  of  Ice. 

Two  inches  thick  will  support  men  in  single  file  on  planks  6 feet  apart;  4 inches 
will  support  cavalry,  light  guns,  and  carts;  and  6 inches  wagons  drawn  by  horses. 

Temperature. 

Sulphuric  acid  and  water  produce  a much  greater  proportionate  contraction  than 
alcohol  and  wrater.  Both  of  these  mixtures,  however  low  their  temperature  pro- 
duce heat  which  is  in  a direct  proportion  to  their  diminution  in  volume.  ’ 
year  ^ deptl1  of  45  feet>  the  temPerature  of  the  Earth  is  uniform  throughout  the 

Temperature  of  Earth  increases  about  1°  for  every  50  to  60  feet  of  depth,  and  its 
crust  is  estimated  at  30  miles. 

A body  at  Equator  weighs  two  hundred  and  eighty-nine  parts  less  than  at  the  Poles. 


196 


MISCELLANEOUS  ELEMENTS, 


Colors  for  Drawings. 


Material. 

Colors. 

Mateiial. 

Colors. 

Brass  .... 

Brick 

Cast  Iron. 

P|qV 

Gamboge. 

Carmine. 

Neutral  tint. 

Burned  umber. 

“ “ light. 

Sepia  with  dark  spots. 
Lake  and  Bur’d  Sienna. 

Granite 

Lead 

Indian  Ink,  light. 

“ “ and  Prussian  blue. 

Light  blue  and  Lake. 

Cobalt  or  Verdigris. 

( Burned  Sienna,  deep  and  light, 
\ for  dark  and  light  wood. 
Prussian  Blue,  light. 

Steel 

Water 

y 

Earth 

Concrete  . 
Copper. . . 

Woods 

Wr’ght  Iron. 

Dird.s  and.  Insects, — (J/.  De  Lacy.) 

Elements  of  Flight—  Resistance  of  air  to  a body  in  motion  is  in  ratio  of  surface 
of  body  and  as  square  of  its  velocity. 

Wing  Surface—  Extent  or  area  of  winged  surface  is  in  an  inverse  ratio  to  weight 
of  bird  or  insect. 

A Stag-beetle  weighs  460  times  more  than  a Gnat,  and  has  but  one  fourteenth  of 
its  wing  surface;  150  times  more  than  a Lady  Bird  (bug),  and  has  but  one  fifth. 

An  Australian  Crane  weighs  339  times  more  than  a sparrow,  and  has  but  one  sev- 
enth- 3000000  times  more  than  a Gnat,  and  has  but  one  hundred  and  fortieth.  A 
Stork  weighs  eight  times  more  than  a Pigeon,  and  has  but  one  half.  A Pigeon 
weighs  ten  times  more  than  a Sparrow,  and  has  but  one  half;  97  000  times  more  than 
a Gnat,  and  has  but  one  fortieth. 

A resisting  surface  of  30  sq.  yards  will  enable  a man  of  ordinary  weight  to  descend 
safely  from  a great  elevation. 

Strength  of  Insects.  —Insects  are  relatively  strongest  of  all  animals.  A Cricket 
can  leap  80  times  its  length,  and  a Flea  200  times. 

Application  for  Stings  and  Burns. 

Sting  of  Insects.  —Ammonia,  or  Soda  moistened  with  water,  and  applied  as  a paste. 
Burns. — Hot  alcohol  or  turpentine,  and  afterwards  bathed  with  lime  water  and 
sweet  oii.  Cold  water  not  to  be  applied. 

To  Preserve  jVIeat. 

Meat  of  any  kind  may  be  preserved  in  a temperature  of  from  8o°  to  ioo°,  for  a 
period  of  ten  days,  after  it  has  been  soaked  in  a solution  of  1 pint  of  salt  dissolved 
in  4 gallons  of  cold  water  and  .5  gallon  of  a solution  of  bisulphate  of  calcium. 

By  repeating  this  process,  preservation  may  be  extended  by  addition  of  a solution 
of  gelatin  or  white  of  an  egg  to  the  salt  and  water. 

To  Detect  Starch,  in  [Milk. 

Add  a few  drops  of  acetic  acid  to  a small  quantity  of  milk ; boil  it,  and 
after  it  has  cooled  filter  the  whey.  If  starch  is  present,  a drop  of  iodine  j 
solution  will  produce  a blue  tint. 

This  process  is  so  delicate  that  it  will  show  the  presence  of  a milligram  of  starch 
in  a cube  centimeter  of  whey  (1  grain  of  starch  in  2.16  fluid-ounces). 

Detaining  "Walls  of  Iron  Diles. 

Sheet  Piles. — 7 feet  from  centres,  18  ins.  in  width  and  2 ins.  in  thickness,  strength-  j 
ened  with  2 ribs  8 ins.  in  depth.  { 

Plates.— 7 feet  in  length  by  5 feet  in  width  and  1 inch  in  thickness,  with  one 
diagonal  feather  1 by  6 ins. 

Tie-rods  2 ins.  in  diameter 

Stone  Sawing. 

Diamond  Stone  Sawing.— {Emerson.)  Alabama  marble  6 feet  X 2.5  feet  in  22  min- 
utes — 41  sq.  feet  per  hour. 

"Wood.  Sawing. 

7722  feet  of  poplar,  board  measure,  from  9 round  logs  in  1 hour.  Engine  12  ins 
diameter  by  24  ins.  stroke. 


MISCELLANEOUS  ELEMENTS. 


197 


Cost  of  Dredging. 

Actual  cost , if  on  an  extended  ivorJc,  inclusive  of  Delivery , if  dredging  into  or  on  a 
vessel  alongside  of  dredger.  — ( Trautwine. ) 

Labor  at  $ 1 per  day  and  Repairs  of  Plant  included. 


Depth. 

Cents. 

Depth. 

Cents. 

Depth. 

Cents. 

Depth. 

Cents. 

Feet. 

Cube  Yards. 

Feet. 

Cube  Yards. 

Feet. 

Cube  Yards. 

Feet. 

Cube  Yards. 

10 

6 

20 

8 

25 

10 

35 

18 

15 

7 

22 

. 9 

30 

13 

40 

25 

Discharge  of  Scows  or  Camels.-^ Towing  .25  mile  4 cents  per  cube  yard,  .5  mile  6 
cents,  .75  mile  8 cents,  and  1 mile  10  cents. 

Note.  — A Scow  is  a flat-bottomed  vessel  or  boat.  A Camel  is  a shallow,  flat- 
bottomed  and  decked  vessel,  designed  for  the  transportation  of  heavy  freight  or  the 
sustaining  of  attached  bodies,  as  a vessel,  by  its  buoyancy. 

Dredging. 

A steam  dredge  will  raise  6 cube  yards,  or  8.5  tons,  per  hour  per  IP. 

jVIetal  Boring  and  Turning. 

Boring. — Cast  iron. — Divide  25  by  the  diameter  of  the  cylinder  in  inches  for  the 
revolutions  per  minute. 

Wrought  iron. — The  speed  is  one  fourth  to  one  fifth  greater  than  for  cast  iron. 

Brass.— The  speed  is  about  twice  that  for  cast  iron. 

Turning. — Cast  iron.—  The  speed  is  twice  that  of  boring. 

Wrought  iron.— The  speed  is  one  fourth  to  one  fifth  greater  than  that  for  cast  iron. 

Brass.— The  speed  is  twice  that  of  boring. 

Vertical  boring.— The  speed  may  be  twice  that  of  horizontal  boring. 

The  feed  depends  upon  the  stability  of  the  machine  and  depth  of  the  cut. 

"Well  Boring. 

At  Coventry,  Eng.,  750000  galls,  of  water  per  day  are  obtained  by  two  borings  of 
6 and  8 ins.,  at  depths  of  200  and  300  feet. 

At  Liverpool,  Eng.,  3000000  galls,  of  water  per  day  are  obtained  by  a bore  6 ins 
in  diameter  and  161  feet  in  depth. 

This  large  yield  is  ascribed  to  the  existence  of  a fault near  to  it,  and  extending  t® 
a depth  of  484  feet.  ° 

At  Kentish  Town,  Eng.  , a well  is  bored  to  the  depth  of  1302  feet. 

At  Passy,  France,  a well  with  a bore  of  1 meter  in  diameter  is  sunk  to  a depth  of 
1804  feet,  and  for  a diameter  of  2 feet  4 ins.  it  is  further  sunk  to  a depth  of  100  feet 
10  ins.,  or  1903  feet  10  ins.,  from  which  a yield  of  5 582  000  galls,  of  water  are  obtained 
per  day. 

Tempering  Boring  Instruments. 

Heat  the  tool  to  a blood-red  heat;  hammer  it  until  it  is  nearly  cold-  reheat  it  to 
a blood-red  heat,  and  plunge  it  into  a mixture  of  2 oz.  each  of  vitriol,  soda  sal-am- 
moniac, and  spirits  of  nitre,  1 oz.  of  oil  of  vitriol,  .5  oz.  of  saltpetre,  and  -2  sails  of 
water,  retaining  it  there  until  it  is  cool.  J 

Circular  Saws. 

Revolutions  per  Minute.— % ins.  4500,  10  ins.  3600,  and  36  ins.  1000. 

Masonry. 

Concrete  or  Beton  should  be  thrown,  or  let  fall  from  a height  of  at  least  10  feet 
or  well  beaten  down.  1 

The  average  weight  of  brickwork  in  mortar  is  about  102  lbs.  per  cube  foot. 


Blastering. 

,?la?tSrers’  work  openings,  as  doors,  windows,  etc.,  are  com- 
at  ?n,f-  ha!iof  their  areas>  and  cornices  are  measured  upon  their  extreme 

edges,  including  that  cut  off  by  mitring. 

Gr  la  zing. 

In  Glaziers’  work,  oval  and  round  windows  are  measured  as  squares 
R* 


1 98 


MISCELLANEOUS  ELEMENTS. 


Corn  Measure. 

Two  cube  feet  of  corn  in  ear  will  make  a bushel  of  corn  when  shelled. 

Tenacity  of  Iron  Bolts  in  Woods. 

Diameter  1.125  ins.  and  12  ins.  in  length  required  for  Hemlock  8 tons,  and  for 
Pine  6 tons  to  withdraw  them. 

Length  of  Gun  Barrels.  (C.  T.  Coathupe.) 

The  length  of  the  barrel  of  a gun,  to  shoot  well,  measured  from  vent-hole,  should 
not  be  less  than  44  times  diameter  of  its  bore,  nor  more  than  47. 

Hay  and.  Straw. 

Hay,  loose.  5 lbs.  per  cube  foot.  Ordinarily  pressed,  as  in  a stack  or  mow,  8 lbs. 
Close  pressed,  as  in  a bale,  12  to  14  lbs.  , 

Ordinarily  pressed,  as  in  a wagon  load,  450  to  500  cube  feet  will  weigh  a ton. 

Straw  in  a bale  10  to  12  lbs.  per  cube  foot. 

jS’atixral  Bowers. 

$Mn The  power  or  work  performed  by  the  Sun’s  evaporation  is  estimated  at 

Niaaaj-a Volume  of  water  discharged  over  the  falls  is  estimated  at  33000000 

tons  per  hour,  and  the  entire  fall  from  Lake  Erie  at  Buffalo  to  Lake  Ontario  is  323.35 
feet. 

"Velocity  of*  Stars. 

According  to  computation  of  Mr.  Trautwine  a Star  passes  a range  in  3'  55-91"  less 
time  each  day. 

Service  Train  of  a Quartermaster. 

Quartermaster’s  train  of  an  army  averages  i wagon  to  every  24  me“:  an?  ® r"'eI1’ 
equipped  army  in  the  field,  with  artillery,  cavalry,  and  trains,  requires  1 horse  or 
mule,  upon  the  average,  to  every  2 men. 

Tides. 

The  difference  in  time  between  high  water  averages  about  49  minutes  each  day. 
Atlantic  and  Pacific  Oceans.—  Rise  and  fall  of  tide  in  Atlantic  at  Aspmwall  2 feet, 
in  Pacific  at  Panama  24  feet. 

Dimensions  of  Drawings  for  Batents. 

United  States,  8.5  X 12  inches. 

Eatitxxde. 

One  minute  of  latitude,  mean  level  of  Sea,  nearly  6076  feet  = 1.1508  Statute  miles. 

Artesian  "Well. 

White  Plains,  Nev.,  Depth  2500  feet. 

Foirndation  Biles. 

A pile  if  driven  to  a fair  refusal  by  a ram  of  1 ton,  falling  30  feet,  will  bear  1 ton 
vreight  for  each  sq.  foot  of  its  external  or  frictional  surface,  or  a safe  load  of  750  lbs. 
per  sq.  foot  of  surface. 

N Earth.. 

Density  of  its  mass  5.67. 

Tripolith. 

A new  building  material,  compounded  of  Coke,  Sulphate  of  Lime,  and  Oxide  of 
Iron.  It  has  increased  tensile  strength  after  exposure  to  the  air,  being  much  in 
excess  of  that  of  lime  and  cement. 

Gras  and  Electric  Eight. 

Gas  light  of  16  candle  power  costs  5 cent  per  hour;  Electric,  4.15  cents. 

Niagara. 

Discovered,  1678.  Falls  have  receded  76  feet  in  175  years.  Height,  American 
Falls,  164  feet;  Horseshoe,  158  feet. 


BRIDGES. — U.  S.  ENSIGNS,  PENNANTS,  AND  FLAGS.  199 


Suspension.  Bridges. 
Lengths  of  Spans  in  Feet. 


You-Mau,  China 330 

Schuylkill  (Phila.) 342 

Hammersmith,  Eng.  422 

Pesth  (Danube) 660 

New  York  and  Brooklyn,  930,  1595.5,  and  930;  clear  height  of  Bridge  above  high 
water  at  qo°,  135  feet. 


Niagara 

Lewistown  and  Queenstown . 

Cincinnati 

Niagara  Falls. 


822 

1040 

1057 

1280 


U.  S.  ENSIGN,  PENNANTS,  AND  FLAGS. 

Ensign. — Head  (Depth,  or  Hoist). — Ten  nineteenths  of  its  length. 

Field— Thirteen  horizontal  stripes  of  equal  breadth,  alternately  red  and 
white,  beginning  with  red. 

Union. — A blue  field  in  upper  quarter,  next  the  head,  .4  of  length  of  field, 
and  seven  stripes  in  depth,  with  white  stars  ranged  in  equidistant,  horizon- 
tal, and  vertical  lines,  equal  in  number  to  number  of  States  of  the  Union. 

Pennants  (Narrow). — Head. — 6.24  ins.  to  a length  of  70  feet;  5.04  ins.  to  a 
length  of  40  feet;  4.2  ins.  to  a length  of  35  feet.  Night , 3.6  ins.  to  a length  of  20 
feet,  and  3 ins.  to  a length  of  9 feet.—  Boat,  2.52  ins.  to  a length  of  6 feet. 

Union.— A.  blue  field  at  head,  one  fourth  the  length,  with  13  white  stars  in  a hori- 
zontal line.  Field. — A red  and  white  stripe  uniformly  tapered  to  a point,  red  up- 
permost. Night  and  Boat  Pennants. — Union  to  have  but  7 stars. 

Union  Jack. — Alike  to  the  Union  of  an  Ensign  in  dimensions  and  stars. 

Flags. — President. — Rectangle,  with  arms  of  the  U.  S.  in  centre  flf 
a blue  field. 

Secretary-  of  Navy. — Rectangle,  with  a vertical  white  foul  anchor 
in  centre  of  a blue  field. 

Admiral. — Rectangle,  with  4 white  stars  in  centre  of  a blue  field,  set  as 
a square. 

Vice-Admiral. — Same  as  Admiral’s,  with  3 white  stars  set  as  an  equi- 
lateral triangle. 

Rear-Admiral. — Same  as  Admiral’s,  with  two  white  stars  set  vertical. 

If  two  or  more  rear-admirals  in  command  afloat  should  meet,  their  seniority  is  to 
be  indicated  respectively  by  a Blue  flag,  a Red  with  White  stars,  and  a White  with 
Blue  stars,  and  another  or  all  others,  a White  flag  with  Blue  stars. 

Commodore’s  ( Broad  Pennant). — Blue,  Red,  or  White,  according  to 
rank,  with  one  star  in  centre  of  field,  being  white  in  blue  and  red  pennants, 
and  blue  in  white. 

Swallow-tailed,  angle  at  tail,  bisected  by  a line  drawn  at  a right  angle  from  centre 
of  depth  or  hoist,  and  at  a distance  from  head  of  three  fifths  of  length  of  pennant; 
the  lower  side  rectangular  with  head  or  hoist;  upper  side  tapered,  running  the  width 
of  pennant  at  the  tails  .1  the  hoist.  Head. — .6  length.  Fly  1.66 -hoist. 

Divisional  Marks.  — Triangle,  1st  Blue,  2d  Red,  3d  White.  Blue 
vertical.  Reserve  Division. — Yellow  Red  vertical.  Division  mark  is  worn 
by  Commander  of  a division  of  a squadron  at  mizzen,  when  not  authorized 
to  wear  Broad  Pennant  of  a Commander  or  Flag  of  an  Admiral.  Fly  .8  hoist. 

Signal  Numbers. — Fly  i. 25  hoist.  Signal  Pennants , Fly  4.6  hoist . 
Repeaters  1.89  hoist. 

International , Signal  Number , Square , Signal  Pennants.  Fly  .3  hoist . 


200 


ANIMAL  FOOD, 


Alimentary  Principles. 

Primary  division  of  Food  is  into  Organic  and  Inorganic. 

Organic  is  subdivided  into  Nitrogenous  and  Non-Nitrogenous ; Inorganic 
is  composed  of  water  and  various  saline  principles.  The  former  elements 
are  destined  for  growth  and  maintenance  of  the  body,  and  are  termed  “ plas- 
tic elements  of  nutrition.”  The  latter  are  designed  for  undergoing  oxidation, 
and  thus  become  source  of  heat,  and  are  termed  “ elements  ot  respiration,  or 
“Calorificient.”  , 

Although  Fat  is  non-nitrogenous,  it  is  so  mixed  with  nitrogenous  matter  that  it 
becomes  a nutrient  as  well  as  a calorificient. 

Alimentary  Principles.  — i.  Water;  2.  Sugar;  3.  Gum;  4.  Starch;  5-  P^tine, 
6.  Acetic  Acid;  7.  Alcohol;  8.  Oil  or  Fat.  Vegetable  and  Animal— 9.  Albumen, 
10.  Fibrine;  11.  Caseine;  12.  Gluten;  13.  Gelatine;  14.  Chloride  of  Sodium. 

These  alimentary  principles,  by  their  mixture  or  union,  form  our  ordinary  foods, 
which  bv  way  of  distinction,  may  be  denominated  compound  aliments  ; thus  meat 
is  composed  of  fibrine,  albumen,  gelatine,  fat,  etc. ; wheat  consists  of  starch,  gluten, 
sugar,  gum,  etc. 

Analysis  of  IVIeats,  Pish,  ‘Vegetable  s,  etc. 

Ash,  etc. 


Food. 

Water. 

Nitro-  I 
genous 
Matter,  j 

Fat.  | 

Sal'.ne  ^ 
Matter. 

Non-Nitro- 

genous 

Matter. 

Sugar.  | 

Cellu-  A 
lose.  Al 

Arrowroot 

18 

— 

>-  I 

— 

82 

— 

— 

Barley  Meal 

15 

6-3 

2.4 

2 

•69.4 

4.9 

— 

Beans,  White 

9.9 

255 

2.8 

55-7 

' 

2.9 

Beef,  roast 

54 

27.6 

15-45 

2 95 

— 

fat 

5i 

14.8 

29.8 

44 

— 

. lean 

72 

*9-3 

3-6 

5‘i 

— 

salt 

49.1 

29.6 

.2 

21. 1 

— 

— 

Beer  and  Porter. . . . 

91 

.1 

— 

.2 

— 

Buckwheat 

13 

13- 1 

3 

•4 

64-5 

3-5 

Blitter  and  Fats — 

15 

— 

83 

2 

— 

— 

Cabbage 

91 

2 

•5 

•7 

5-8 

Carrots 

83 

i-3 

.2 

1 

7-4 

0. 1 

Cheese 

36.8 

33-5 

24-3 

5-4 

— 

— 

Corn  Meal 

14 

11. 1 

#8. 1 

*•7 

57*6 

•4 

5-9 

Cream 

66 

2.7 

26.7 

1.8 

— 

2.8 

— 

Egg 

74 

14 

10.5 

i-5 

— 

yolk 

52 

16 

3°-7 

*•3 

— 

Fish,  white  flesh. . . 

78 

18.1 

2.9 

1 

— 

1 

— 

Eels 

75 

9.9 

13.8 

i-3 

— 

Lobster,  flesh. 

76.6 

19.17 

1.17 

1.8 

1.26 

— • 

Oysters 

80.39 

14.01 

1.52 

2.7 

1.38 

Liver,  Calf’s 

72-33 

20.55 

5-58 

i-54 

— 

Milk,  Cow’s 

86 

4-1 

3-9 

.8 

— 

5-2 

Mutton,  fat 

53 

12.4 

3Ii 

3-5 

— 

Oatmeal 

i5 

12.6 

5-6 

3 

58.4 

5-4 

’ 

Oats 

21 

14.4 

5-5 

— 

48.2 

— 

7.6 

Parsnips 

82 

1. 1 

•5 

1 

9.6 

5-8 

Peas 

i5 

23 

2. 1 

2.5 

50.2 

2 

3- 1 

Pork,  fat 

39 

9.8 

48.9 

23 

— 

Bacon,  dry. . . 

15 

8.8 

73-3 

2.9 

— 

Potatoes 

75 

2.1 

.2 

•7 

16.8 

3-2 

1 

Poultry 

74 

21 

3-8 

1.2 

— 

Rice 

i3 

6-3 

•7 

•5 

78. 1 

•4 

Rye  Meal 

i5 

8 

2 

1.8 

69-5 

3-7 

Sugar 

5 

— 

— 

— 

95 

Tripe 

68 

13.2 

16.4 

2.4 

Turnips 

91 

1.2 

— 

.6 

4-3 

2. 1 

Veal 

63 

16.5 

15.8 

4-7 

— 

■ 

Wheat  Flour 

15 

10.8 

2 

i-7 

61. 1 

4.2 

3.5 

Bread* 

. 37 

8.1 

1.6 

2-3 

45-4 

3- 6 

- | 

Bran 

■1  i3 

18 

6 

— 

60 

— 

3-2 


2.5 


3-3 


x*7 


— 2 


ing  most.  100  lbs.  flour  yield  130  lbs.  bread. 


ANIMAL  FOOD. 


201 


Analysis  of  Different  Foods 

In  their  Natural  Condition. 


Ni- 

trates. 

'Carbon- 

ates. 

Phos- 

phates. 

Water. 

Ni- 

trates. 

Apples 

5 

10 

! 

84 

Milk  of  cow. . 

5 

Barley 

1 7 

6o.  c 

3.  ^ 

Mutton  . 

12.5 

Beans 

2 A. 

y 0 
C7.  7 

3.  ^ 

14. 8 

Oats 

Beef 

T 
I C 

J/  / 

j D 
c 

eo 

Parsnips 

x7 

Buckwheat  . . 

8.6 

75-4 

J 

1.8 

14.2 

Pork 

9.2 

10 

Cabbage  

4 

5 

1 

9° 

Potatoes  

2.4 

Chicken 

19 

3-5 

4-5 

73 

“ sweet 

1.5 

Corn,North’n 

12 

73 

1 

14 

Rice 

6.5 

“ South’n 

35 

48 

3 

!4 

Turnips 

5 

Cucumbers. . . 

*•5 

1 

.5 

97 

Veal 

16 

Lamb 

11 

35-5 

3-5 

50 

Wheat 

i5 

8 

40 

66.4 
7 

50 

22.5 

28.4 
79-5 

4 

16.5 
69.2 


4-5 

3 

1 

*•5 

•9 

2.6 
• 5 
•5 

4-5 

1.6 


86 

43 

13.6 

82.8 

38.5 

74.2 
67-5 
i3-5 
9°-5 

63 

14.2 


Nitrates— Are  that  class  which  supplies  waste  of  muscle. 

Carbonates— Are  that  class  which  supplies  lungs  with  fuel,  and  thus  furnishes  heat 
to  the  system,  and  supplies  fat  or  adipose  substances. 

Phosphates— Are  that  class  which  supplies  bones,  brains,  and  nerves  and  gives 
vital  power,  both  muscular  and  mental. 

From  above  it  appears,  that  Southern  corn  produces  most  muscle  and  least  fat 
and  contains  enough  of  phosphate's  to  give  vital  power  to  brain,  and  make  bones 
strong.  Mutton  is  the  meat  which  should  be  eaten  with  Southern  corn. 

The  nitrates  in  all  the  fine  bread  which  a man  can  eat  will  not  sustain  life  beyond 
fifty  da)rs ; but  others,  fed  on  unbolted  flour  bread,  would  continue  to  thrive  for  an 
indefinite  period.  It  is  immaterial  whether  the  general  quantity  of  food  be  reduced 
too  low,  or  whether  either  of  the  muscle-making  or  heat-producing  principles  be 
withdrawn  while  the  other  is  fully  supplied.  In  either  case  the  effect  will  be  the 
same.  A man  will  become  weak,  dwindle  away  and  die,  sooner  or  later  according  to 
the  deficiency;  and  if  food  is  eaten  which  is  deficient  in  either  principle,  the  appe- 
tite will  demand  it  in  quantity  till  the  deficient  element  is  supplied.  AH  food  be- 
yond the  amount  necessary  to  supply  the  principle  that  is  not  deficient,  is  not  only 
wasted,  but  burdens  the  system  with  efforts  to  dispose  of  it. 

Analysis  of  Fruits. 


Apple,  white. 

Apricot,  average 

Blackberry 

Cherry,  red 

sour 

black 

Currant,  red 

Gooseberry,  red 

yellow 

Grape,  white 

Peach,  Dutch 

Pear,  red 

Plum,  yellow  gage. . . . 

large  “ 

black  blue 

“ red 

Italian,  sweet. . . 

Raspberry,  wild 

Strawberry,  “ 

Banana. 


Water. 

| Sugar. 

Acid. 

Albumi- 
nous sub- 
stances. 

Insoluble 

matter. 

Pec  to  us 
Sub- 
stances. 

85 

7.6 

1 

.22 

1.83 

3.88 

83-5 

1.8 

1. 1 

•51 

4-7 

7-55 

86.4 

4.44 

1. 19 

•51 

5.26 

1.72 

75-4 

13  1 

•35 

•9 

5-83 

3-73 

80.5 

8.77 

1.28 

.83 

5-91 

2.07 

79*7 

10.7 

•56 

I 

6. 04 

1-33 

85-4 

5-6 

i-7 

•36 

3-74 

2.4 

85.6 

8 

i-35 

•44 

2.92 

1.26 

85-4 

7 

1.2 

.46 

3-17 

2.4 

80 

13-78 

1 

•83 

2.48 

1.44 

85 

1.58 

.61 

.46 

5-49 

6.4 

83-5 

7-5 

.07 

•25 

3-54 

4.8 

80.8 

2.96 

.96 

.48 

3-98 

10.48 

79-7 

3-4 

.87 

•4 

3-9i 

IJ*3 

88. 7 

2 

1.27 

•4 

6.86 

•23 

8s  3 

2.25 

I-33 

•43 

4-23 

5-85 

81.3 

6-73 

.84 

.83 

4.01 

5-63 

83-9 

3-6 

2 

•55 

8-37 

1.28 

87 

4 

i-5 

.6. 

5-5 

•4 

73-9  1 

Sugar,  Pectin,  Salt,  Acid 

, etc.,  26.1. 

Sugar  and.  Water  in  Various  Products  not  Included  in 
tire  Tat>le.  ( Per  Cent.) 

Water. 


„ Sugar. 

Sugar,  crude 95 

Molasses 77 

Buttermilk 6.4 


Molasses. . 


23 

Lean  beef. 72 

Buttermilk 88 


Water. 

Cabbage g1 

Ale  and  Beer 

Coffee  and  Tea 100 


202 


ANIMAL  FOOD, 


Relative  Values  of  Foods  or  Assimilating  Quality  to 
make  an  Equal  Quantity  of  Rlesh.  in  Cattle  or  Sheep. 
[Ewart.) 


Turnips 

Carrots 

Beets 

Parsnips  and  Swedes  . . . 
Meadow  grass  in  bloom. 

Vetches,  pods  open 

Potatoes  at  maturity 

Oat  straw,  cut  green  . . . . 

Bean  or  Vetch  straw 

Meadow  hay 

Vetch  “ 

Linseed  cake 


Cattle. 

Sheep. 

Article. 

Cattle. 

Sheep. 

800 

400 

Wheat  bran 

45 

105 

630 

— 

Corn  and  Barley  meal . . 

35 

600 

300 

Oatmeal 

34 

— 

600 

200 

Beanmeal 

33 

— 

400 

— 

Peameal 

32 

— 

360 

90 

Cabbage  

500 

280 

200 

Pea  straw 

— 

200 

125 

— 

Rye  bran  . 

— 

109 

200 

Oats 

— 

70 

100 

100 

Buckwheat 

— . 

65 

9° 

— 

Barley 

— 

60 

50 

— 

Pease  or  Beans 

— 

54 

Note.  — When  these  values  express  weight  in  lbs.,  then  such  food  will  produce 
about  4 to  5 lbs.  beef  or  mutton. 

Nutritive  Constituents  and  Values  of  Rood,  in  G-rains 
per  Round. 


Food. 


Bakers’  Bread. . . . . 

Barley  Meal 

Beef. 

Beer  and  Porter. . . 
Bullock’s  Liver  . . . 

Buttermilk 

Carrots 

Cheddar  Cheese . . . 

Cocoa 

Dry  Bacon 

Fat  Pork 

Flour,  Seconds 

Fresh  Butter 

Green  Bacon 

Green  Vegetables. . 

Indian  Meal 

Lard 

Molasses 


Carbon.  Nitrogen. 


1975 

2563 

1854 

274 

934 

387 

508 

3344 

3934 

59S7 

4ii3 

2700 

6456 

5426 

420 

3016 

4819 

2395 


68 

184 

1 

204 

44 

14 

306 

140 

,9J 

116 

76 

120 


Food. 

Carbon. 

! Mutton 

Now  Milk 

1900 

cqq 

Oatmeal 

2831 

USA 

Pfi r«n i ps  

Pearl  Barley 

JJT 

2660 

Potatoes 

769 

T AlC 

Red  Herrings 

Rino  

RyD  Meal 

2693 

4585 

1947 

438 

2698 

Salt,  Rutter 

Skim  Cheese 

Skimmed  Milk 

Split  Pease 

Suet, 

Sugar  

4710 

2Qg:  C 

Turnips 

*yoo 

263 

I KA. 

Whev 

Whitefish 

871  1 

189 

44 

136 


217 

68 

86 

483 

43 

248 


13 

13 

*95 


The  Full  Daily  Diet  of  a man  is  held  to  be  12  oz.  bread,  8 oz.  potatoes, 
6 oz.  meat,  4 oz.  boiled  rice  with  milk,  .375  pint  of  broth  or  pea  soup,  1 pint 
milk,  and  1 pint  of  beer. 

Nutritive  Values  and  Constituents  of  jVIilh. — ( Payen .) 


Animal. 

Nitrogenous 
Matter  and 
insoluble 
Salts. 

Butter. 

Lactic 

and 

soluble 

Salts. 

Water. 

Animal. 

Nitrogenous 
Matter  and. 
insoluble 
Salts. 

Butter. 

Lactic 

and 

soluble 

Salts. 

Water. 

Goat. . . . 

4-5 

41 

5-8 

85.6 

Ass 

*•7 

1.4 

6.4 

9°-5 

Cow 

4-55 

3-7 

5-35 

86.4 

Mare . . . 

1.62 

.2 

8-75 

89-43 

Woman. 

3-35 

3-34 

3-77 

89-54 

Ewe 

4.68 

4.2 

5-5 

85.62 

Weigh. t of  some  Different  Roods  required  to  furnish 
1220  Grains  of  Nitrogenous  Nlatter. 

Lbs. 

Barley  Meal..  2.9 

Milk 42 

Potatoes 8.3 

Parsnips 15-9 


Cheese 

Lbs. 

Meat,  fat 

Lbs. 

i-3 

Bacon,  fat. 

Lbs. 
....  1.8 

Pease 

Oatmeal 

i-5 

Bread 

Meat,  lean 

• -9 

Corn  Meal 

1.6 

Rye  Meal. , 

Fish,  White . . 

. 1 

Wheat  Flour. . 

i-7 

Rice 

....  2.8 

Turnips,  15.9  lbs. 

; Beer  or  Porter, 

158.6  lbs. 

ANIMAL  FOOD, 


203 


[Proportion  of  Sugar  and  Acid  in  Various  Fruits. 
(Fresenius.) 


Fruit. 

Sugar. 

Acid. 

Fruit. 

Sugar. 

Per  Cent. 

Per  Cent. 

Per  Cent. 

Apple 

8.4 

.8 

Plum. . 

Apricot 

1.8 

1. 1 

Prune. . . . 

6-3 

Blackberry 

4.4 

1.2 

Raspberry. 

Currants 

6.1 

2 

Red  Pear 

4 

Gooseberry 

7. 2 

I.5 

Sour  Cherry 

7*5 

0 0 

Grape 

14. 9 

.7 

Strawberry 

0. 0 

Mulberry 

Q.  2 

/ 

I.  0 

Sweet,  Cherry 

5-7 

T-.  p 

Peach 

1.6 

•7 

Whortleberry 

10. 0 
5.8 

Acid. 


Per  Cent. 
i-3 
•9 
i-5 


.1 


i-3 

i-3 

.6 

x-3 


Proportion  of  Oil  in  Various  Air-dry  Seeds.  ( Berjot .) 


Beechnut 

Hemp 

Watermelon 


Mustard  . 

Flax 

Peanut . 


Almond.. 


40 

Colza 1 4° 

(45 


Orange 40 

Poppy 1 4o 

Analysis  of  different  Articles  of  Food,  with.  Reference 
only  to  their  Properties  for  giving  Heat  and  Strength. 

( Payen .)  In  100  Parts. 


Alcohol 

Barley 

Beans 

Beef,  meat 

Beer,  strong. . 
Bread,  stale. . . 
Buckwheat. . . 

Butter 

Carrots 

Caviare 

Cheese, Chest’r 

Chocolate 

Cod-fish,  salt’d 


Car- 

bon. 

Nitro- 

gen. 

Substances. 

Car- 

bon. 

Nitro- 

gen. 

Substances. 

Car- 

bon. 

Nitro- 

gen. 

52 

40 

42 

1 9 
4-5 

Coffee 

Corn 

Eels 

9 

44 

30-05 

x3-5 

34 

1. 1 
x-7 

2 

Oil,  Olive 

Oysters 

Pen  se 

98 

7.18 

2.13 

3-66 

11 

3 

Eggs 

I.Q 

Potatoes 

44 

4-5 

.08 

Figs,  dried 

•i.y 

.92 

Rice 

41 

•33 

1.8 

28 

42-5 

1.07 

2.2 

Herring,  salt- 
ed  

23 

15.68 
10. 06 

3. 1 1 

Rye  Flour 

Salmon 

4i 

16 

29 

x-75 

83 

5-5 

.64 

•31 

Liver,  Calf’s. . 
Lobster 

3-93 

2.93 

3-74 

.66 

Sardines 

Tea 

2.09 

6 

27.41 

4.49 

4-x3 

Mackerel 

19.26 

8 

Truffes. 

2.1 

.2 

41.04 

Milk,  Cow’s. . . 

Wheat 

9-45 

41 

x-35 

3 

58 

16 

1-52 

5-02 

Nuts. 

Oatmeal 

10.65 

44 

x-4 

i-95 

“ Flour.. 
Wine 

38.5 

4 

1.64 

.015 

nitrogenous  matter  is  obtained. 


5.5,  and  equivalent  amount  of 


Human  and  Animal  Sustenance. 

Least  Quantity  of  Food  required  to  Sustain  Life.  (E.  Smith , M.D.) 

Carbon.  Hydrogen. 

Grs. 

, 3900  j ’ 180}  Mean>  I9°- 

An  adult  man,  for  his  daily  sustenance,  requires  about  1220  grs.  nitrog- 
enous matter  or  200  of  nitrogen,  and  bread  contains  8.1  per  cent,  of  it. 

1220 


Grs. 

Adult  Man,  4300)  ,r 
Adult  Woman,  3900)  Mean» 


Hence,  — .15  062  grains  which  -4-  7000  in  alb.  —2  lbs.  2.43  oz.  of  bread. 


These  quantities  and  proportions  are  also  contained  in  about  16  lbs.  of 
turnips. 

i3TofUnitrogenle  of  nutritive  values>  PaSe  202,  turnips  have  263  grains  of  carbon  and 


HenCe’^  and  ~ = i6-35  lbs.  for  the  necessary  carbon  and  15.4  lbs.  for  the 
nitrogen. 


[Relative  Value  of  Foods  compared  with  IOO  lbs.  of 
Grood  Flay. 


m Lb9- 

Clover,  green. . 400 
Corn,  green  ...  275 
Wheat  straw  . . 374 


Lbs. 

Corn 59 

Linseed  cake  . . 69 
Wheat  bran 105 


ANIMAL  FOOD. 


204 

WeigKt  of  Articles  of  Food,  required  to  "be  consumed  in 
tbe  Tinman  system  to  develop  a power  equal  to  rais- 
ing 140  lbs.  to  a height  of  IO  OOO  feet.  ( Frankland .) 

Weight. 


Substances.  [Weight,  j 

Substances. 

Weight.  ] 

Substances. 

Lbs.  I 

•553 
•555  1 
.67  | 

.693 

•797' 

•97 

1.156 
1.281 
1.287 
1 311  I 

"Pin**  . . f 

Lbs. 

I-34I 

1-379 

i-5°5 

Salt  Beef 

Cod-liver  oil.. 

T)  oof*  fo  t 

Tsin  glass 

Veal,  lean 

Sugar,  lump 

Porter 

T)  n f fnr 

Cream 

2.062 
2. 209 
2-345 
2.826 
3.001 

Potatoes 

Egg  boiled 

Fish 

TT'n  f of  Povlr 

Rrpari  

Apples 

X uX  01  r 01  Iv  ......  • 

Salt  Pork 

Milk 

Ham,  lean,  boiled. . 

Egg,  white  of. 

Mackerel 

3.124 

3.461 

Carrots 

Arrowroot.  ....... 

Wheat  flour 

Ale,  bottled 

Cabbage 

Lbs. 

3-65 

4*3 

4.615 

5.068 

6.316 

7- 8i5 
8.021 

8- 745 
9.685 
12.02 


Relative  Value  of  Various  Foods  as  Productive  of  Force 
when.  Oxidized  in  the  Body. 


Cabbage. 

Carrots.. 


White  of  Egg.. 

Milk 

Apples 

Ale 

Fisli 

Potatoes  


1 

Porter 

2.6 

Egg, hard  boil’d 

5-4 

1.2 

Veal,  lean..... 

2.8 

Cream 

5-9 

1.2 

Salt  Beef 

3-3 

Egg,  yolk 

7- 9 

1.4 

Poultry 

3-3 

Sugar 

8 

1.5 

Lean  Beef. 

3-4 

Isinglass 

8.7 

1.5 

Mackerel 

3.8 

Rice 

8.9 

1.8 

Ham,  lean 

4 

Pea  Meal 

9- 

1.9 

Salt  Pork 

4-3 

Wheat  Flour .. 

9.1 

2.4 

Bread,  crumb. . 

5-i 

Arrowroot 

9-3 

Oatmeal 9.3 

Cheese 10.4 

Fat  of  Pork.  12.4 

Cocoa 16.3 

Pemmican . . 16.9 

Butter 17.3 

Bacon 17.94 

Fat  of  Beef. . 21.6 
Cod-liver  Oil.  21.7 


Nutritions  Properties  of  different  Vegetables  and  Oil- 
cake,  compared  with  each  other  in  Quantities. 


Rye.. 

Bran, 


wheat 


Corn . . . 
Barley . 


2.5 

irs 

3 

3 


Clover  hay — 

Cabbage 

18 

Hay 

Wheat  straw.. 

, 26 

Potatoes 

Barley  “ 

26 

“ old.. 

. . 20 

Oat  “ 

27-5 

Carrots  

i7-5 

Turnips 

■ 30  ; 

Oil-cake 1 

Pease  and  Beans  1.5 
Wheat,  flour. . . 2 
“ grain..  2.5 

Oats 2.5 

Illustration.— 1 lb.  of  oil-cake  is  equal  to  18  lbs.  of  cabbage. 

Volume  of  Oxygen  required  to  Oxidize  100  parts  of  following  Foods  as  con- 
sumed in  the  Body. 

Grape  Sugar..  106  | Starch i2o  | Albumen 150  | Fat 293 

Hence  assuming  capacity  for  oxidation  as  a measure,  albumen  has  half  value  of 
fat  asTfeXroducinl  element,  and  a greater  value  than  either  starch  or  sugar. 

Proportion  of  Alcohol  in  IOO  Parts  of  following  Liquors. 
(Brande. ) 


Small  Beer. ..  1 and  1.08 

Porter 3-5  and.  5.26 

Cider 5 2 and  9.8 

Brown  Stout.  5.5  and  6.8 
Ale 6.87  and  10 

Hermitage,  red 12.32 

Champagne 12.61 

Amontillado 12.63 

Frontignac 12.89 

Barsac 13-86 

Lisbon 18.94 

Lachryraa 19.7 

Teneriffe 19-79 

Currant  Wine 20.55 

Madeira 22.27  < 

Port 23 

Rhenish 7-58 

Moselle 8.7 

Johannisberger 8.71 

Elder  Wine 8.79 

Claret  ordinaire 8.99 

Champagne  Burg’dy,  14.57 

White  Port 15 

Bordeaux i5- 1 

Malmsey 16.4 

Sherry I7*17 

Sherry,  old 23.86  ' 

Marsala 25.09 

Raisin  Wine 25.12 

Madeira,  Sercial 27.4 

Cape  Madeira 29.51 

Tokay 9-33 

AT 0 1 0 rro  17.  2 

Gin Si-6 

Rudesheimer 10.72 

Marcobrunner n-6 

Gooseberry  Wine  ...  11.84 

Hockheimer 12.03 

Vin  de  Grave 12.08 

Alba  Flora 17-26 

Hermitage,  white . . . 17.43 

Cape  Muscat 18.25 

Constantia,  red 18.92 

Brandy 53-39 

Rum 53-68 

Irish  Whiskey 53-9 

j Scotch  Whiskey 54.32 

ANIMAL  FOOD. 


205 


Proportion  of  Food  Appropriated  and  Expended  "by 
following  Animals. 


Oxen. 

Sheep. 

Swine. 

Proportion  appropriated 

“ in  manure 

8 • 

17.6 

3i-9 

16.9 

“ respired 

60. 1 

65-5 

100  100  100 


Specific  (Gravity  of  Alifk  and  Percentage  of  Cream,  etc. 


Milk. 

Specific 

Gravity. 

Volume 

of 

Cream. 

Volume 

of 

Curd. 

Specific 
Gravity  when 
skimmed. 

Milk,  pure* 

1030 

12 

6-3 

1032 

“ xo  per  cent,  water. 

1027 

10.5 

5-6 

1029 

“ 20  “ “ “ 

1024 

8.5 

4.9 

1026 

“ 30  “ “ “ 

1021 

6 

4.2 

1023 

* For  a method  of  testing  the  purity  of  milk,  see  Pavv  on  Food  (Philadelphia,  1874),  page  196. 


Note. — The  average  proportion  of  cream  is  10,  or  10  per  cent. 


Proportion  Per  cent,  of  Starch,  in  sundry  ‘Vegetables. 


Arrowroot. . . 

, . 82  I Wheat  flour. . 

. 66.3  I 

I Oatmeal . . . . 

Potatoes. . . 

..  18.8 

Rice 

, . 79. 1 1 Corn  meal . . . 

. 64.7  1 

Pease 

■•••  55-4  1 

Turnips 

..  5-1 

Composition  of  Cheese  of  Different  Conn  tries. — ( Payen .) 


Fat. 

Nitrogen. 

Salt. 

Water. 

Fat. 

Nitrogen. 

Salt. 

Water. 

Neufchatel. . 

18.74 

2.28 

4-25 

61.87 

Chester 

25.41 

5-56 

4.78 

30-39 

Parmesan  . . 

21.68 

5-48 

7.09 

30-3I 

Gruydres  . . . 

28.4 

5-4 

4.29 

32-05 

Brie 

24.83 

2-39 

5-63 

53-99 

Marolles 

28.73 

3-73 

5-93 

40.07 

Holland  .... 

25.06 

4.1 

6.21 

41.41 

Roquefort. . . 

32.31 

5.07 

4-45 

26.53 

Nntritive  Equivalents.  Computed  from  Amount  of  Ni- 
trogen in  Snbstances  -when  Dried.  Human  AI ilk  at  1. 


Rice 

Potatoes.. 

Corn 

Rye 

Wheat . . . 


Oats. 


00  00 

Bread,  White. 
Milk,  Cows’  . . 

1.42 
2. 37 

Cheese 

Eel 

3-31 
a.  n 

Lamb 

Egg,  White. . . 
Lobster. ..... 

Ppa.se 

2. 

Mussel 

5.28 

5-7 

7-56 

7.  79 

1.06 

Lentils 

jy 

2.76 
3- °5 
3-05 
3-2 

Liver,  Ox  ... . 

Yeal 

T.  JO 

Egg,  Yolk 

Oysters 

Pigeon 

Beef 

1.  x y 

Mutton 

Pork 

! 1.38 

Beans 

Salmon 

7.76 

Ham 

Herring,  9.14. 


8-33 

8-45 

8-59 

8-73 

8.8 

8-93 

9.1 


Thermometric  Power  and  JVCechanical  Energy  of  IO 
Grains  of  Various  Snbstances  in  their  N atnral  Con- 
dition, when  Oxidized  in  the  Animal  Body  into  Car- 
bonic -A.cid,  Water,  and  Urea. — ( Franktand .) 


Substance. 

Water 

raised 

i°. 

Lifted 
1 foot 
high. 

Substance. 

Water 

raised 

i°. 

Lifted 
1 foot 
high. 

Substance. 

Water 

raised 

i°. 

Lifted 
1 foot 
high. 

Ale,  Bass’s  . . 

Apples 

Arrowroot. . . 

Lbs. 

1.99 

Lbs. 

i-54 

Cheese 

Lbs. 
11. 2 

Lbs. 

8.65 

Mackerel — 

Lbs. 

4.14 

Lbs. 

3-2 

1.48 
10. 06 

1.29 
7.7  7 

Cocoa-nibs  . . 
Cod-liver  oil. 

17 

11 

7-3 
18. 12 

Milk 

Oatmeal 

1.64 
10. 1 

1.25 

7.8 

Beef,  lean . . . 

3.66 

2.83 

Egg,  h’d  boil. 

5-86 

4-53 

Pea  meal .... 

9-57 

7-49 

Bread 

5-52 

4.26 

“ yolk 

8.5 

6.56 

Potatoes  .... 

2.56 

1.99 

Butter 

18.68 

14.42 

“ white... 

1.48 

1. 14 

Porter 

2.77 

2. 19 

Cabbage 

1.08 

.83 

Flour,  wheat. 

9.87 

7.62 

Rice,  ground. 

9-52 

7-45 

Carrots 

i-33 

1.03 

Ham,  boiled . 

4-3 

3-32 

Sugar,  grape. 

8. 42 

6.51 

s 


206 


ANIMAL  FOOD. 


Digestion. 

Time  required,  for  Digestion  of  several  Articles  of  Food. 
• (Beaumont,  M.D.) 


Apple,  sweet  and  mellow  .... 

sour  and  mellow 

sour  and  hard 

Barley,  boiled 

Bean,  boiled 

Bean  and  Green  Corn,  boiled . 

Beef,  roasted  rare 

roasted  dry 

Steak,  broiled 

boiled 

boiled,  with  mustard,  etc. 

Tendon,  boiled 

44  fried 

old  salted,  boiled 

Beet,  boiled 

Bread,  Corn,  baked 

Wheat,  baked,  fresh  . . . 

Butter,  melted 

Cabbage,  crude 

crude,  vinegar 


crude,  vin’r,  boiled 


Carrot,  boiled 

Cartilage,  boiled 

Cheese,  old  and  strong. 
Chicken,  fricasseed.  . . 
Custard,  baked 


Duck,  roasted 

Dumpling,  Apple,  boiled. 

Egg- 


whipped  

boiled  hard 

44  soft 

fried 

Fish,  Cod  or  Flounder,  fried  . . 
Cod,  cured,  boiled. ..... 

Salmon,  salt’d  and  boil’d 
Trout,  boiled  or  fried. . . 

Fowl,  boiled  or  roasted 

Goose,  roasted 

Gelatine,  boiled 


h.  m. 

1 50 

2 

2 50 
2 

2 30 

3 45 
3 

3 30 
3 

2 45 

3 30 
5 30 

4 

4 15 
3 45 
3 i5 
3 30 

3 30 
2 30 

2 

4 

4 30 

3 15 

4 15 

3 30 
2 45 

2 45 

4 

4 30 

3 

2 

1 30 

3 30 
3 

3 30 
3 30 


1 30 
4 

3 

2 30 


Heart,  Animal,  fried 

Lamb,  boiled 

Liver,  Beef’s,  boiled 

Meat  and  Vegetables,  hashed  . 

Milk,  boiled  or  fresh j 

Mutton,  roasted 

broiled  or  boiled  .... 

Oyster 

roasted 

stewed 

Parsnip,  boiled 

Pig,  sucking,  roasted 

Feet,  soured,  boiled 

Pork,  fat  and  lean,  roasted  . . . 
recently  salted,  boiled  . . 
“ u fried . . . 

44  “ broiled  . 

44  44  raw.  . . . 

Potato,  boiled 

baked 

roasted 

Rice,  boiled 

Sago,  boiled 

Sausage,  Pork,  broiled 

Soup,  Barley 

Beef  and  Vegetables  . . . 

Chicken - 

Mutton  or  Oyster 

Sponge-cake,  baked 

Suet,  Beef,  boiled 

Mutton,  boiled 

Tapioca,  boiled 

Tripe,  soured 

Turkey,  roasted  | Domestic  . 

boiled 

Turnip,  boiled 

Veal,  roasted 

fried 

Brain,  boiled 

Venison  Steak,  broiled 


2 30 


2 30 


i5 

i5 

55 

i5 

30 

30 

30 


5 15 
4 30 
4 15 
3 15 
3 

3 30 
3 20 

2 30 
1 

1 45 

3 20 

1 30 

4 
3 

3 30 

2 30 

5 30 

4 30 


2 18 
2 30 

2 25 

3 30 

4 

4 50 
1 45 
1 35 


General  Notes. 

The  per-centage  of  loss  in  the  cooking  of  meats  is  as  follows:  Boiling  23;  Baking 
31 ; Roasting  34. 

Potatoes  possess  anti-scorbutic  power  in  a greater  degree  than  any  other  of  the 
succulent  vegetables. 

The  average  yearly  consumption  of  wheat  and  wheat  flour  in  Great  Britain  is  5.5 
bushels  per  capita  of  its  population. 

The  daily  ration  of  an  Esquimaux  is  so  lbs.  of  flesh  and  blubber.-(Sir  John  Rost.) 


ANIMAL  FOOD. 


207 


An  adult  healthy  man,  according  to  Dr.  Edward  Smith,  requires  daily  of 

Phosphoric  acid  from  . . 32  to  79  grains.  Potash 27  to  107  grains. 

( Chlorine 51  *75  “ Soda 80  “ 171  “ 

(Or  of  common  salt 85  u 291  “ Lime 2.3  “ 6.3  “ 

and  of  Magnesia  2.5  to  3 grains. 

A common  fowl's  egg  contains  120  grains  of  Carbon  and  17.75  of  Nitrogen. 
An  ordinary  working-man  requires  for  his  daily  sustenance 

Starch 66 

Salts 04 

4-535 


Oxygen i-47 

Albuminous  matter 305 

Fat 


= 7.23  lbs.  avoirdupois. 

Milk.—  If  the  milk  of  an  animal  is  taken  at  three  immediately  successive  periods, 
that  which  is  first  received  will  not  be  as  rich  in  milk-fat  as  the  last. 

In  a Devon  cow,  milked  in  this  manner,  the  first  milk  gave  but  1.166  per  cent,  of 
fat,  and  the  last,  or  that  known  as  strippings, 5 ’ 5.81  per  cent. 

Relative  Rieliiiess  of  Alillc  of  Several  ^Aiaiinals. 
Human  Millc=.  1. 


Cow. 

Milk-fat. 
1.66 

Casein. 

1.38 

Sugar. 

.69 

Ass 

Milk-fat. 
5 

Casein. 

•33 

Sugar. 

•94 

Mare 

I-I9 

•75 

•94 

Sheep. . . . 

2.1 

.72 

Goat 

1.04 

.69 

Camel 

i-4 

— 

.96 

The  condensation  of  milk  reduces  it  to  about  one  third  of  its  original  volume. 

A Farm  of  second-rate  quality,  properly  cultivated,  will  sustain  100  head  of  cattle 
per  100  acres,  besides  laboring-stock  (employed  in  cultivation  of  farm),  and  swine. 
— (Ewart.) 

Thus,  calves  25 ; do.  1 year  25 ; do.  2 years  25 ; cows  25. 

Cane  Sugar  (Saccharose) — Is  insoluble  in  absolute  alcohol,  and  in  diluted  alcohol 
it  is  soluble  only  in  proportion  to  its  weakness.  Loaf  sugar,  as  a rule,  is  chemically 
pure. 

Beet  Root  Sugar— Contains  85  to  96  per  cent,  of  cane  sugar,  1.6  to  5.1  of  organic 
matter,  and  2 to  4. 3 of  water. 

Honey — Contains  32  per  cent,  of  sugar  (levulose),  25.5  of  water,  27.9  of  dextrine, 
and  14.6  of  other  matter,  as  mannite,  wax,  pollen,  and  insoluble  matter. 

Molasses— Contains  47  per  cent,  of  cane  sugar,  20. 4 of  fruit  sugar,  2.6  of  salts  2.7 
extractive  and  coloring  matter,  and  27. 3 of  water. 

Flour. — Tests  of  flour,  see  A.  W.  Blyth,  London,  1882,  page  152. 

Bread.  — Wheat  loses  of  water  after  1 day  7.71  per  cent.,  3 days  8.86,  and  7 days 
14.05  per  cent. 

Sago. — 2.5  lbs.  per  day  will  support  a healthy  man. 

Fig  Contains  nearly  as  much  gluten  as  wheat  bread  (as  6 to  7),  and  in  starch  and 
sugar  it  is  16  per  cent,  richer. 

Gooseberry  (dry)— Is  as  nutritious  as  wheat  bread. 

Watermelon , Vegetable  marrow , and  Cucumber — Contain  94,  95,  and  97  per  cent, 
of  water  respectively.  * 

Onion  (dry)— Contains  25  to  30  per  cent,  of  gluten.  Potato  containing  but  5. 

Cabbage,  Cauliflower , Broccoli,  and  Leaves  are  generally  rich  in  gluten,  while  the 
potato  is  poor. 


Ratio  of  Flesh-formers  of  Tubers. 
Per  Cent. 


Tubers. 

Flesh- 

formers. 

Starch, 

etc. 

Ratio  to 
Heat-giv’rs. 

Tubers. 

Flesh- 

formers. 

Starch, 

etc. 

Ratio  to 
Heat-giv’rs. 

Beet  root 

Turnip*. 

Carrot 

Potato 

•4 

•5 

•5 

1.2 

13-4 

4 

5 

18 

1:30 
1:8 
1 : 10 
1 : 16 

Parsnip 

Onion 

Sweet  Potato. 
Yam 

1.2 

i-5 

i-5 

2.2 

8.7 

4.8 

20.2 

16.3 

1 : 10 
3-5 

1:13 

7-5 

208  gravity  of  bodies.— geavity  and  weight. 

GRAVITY  OF  BODIES. 

Gravity  acts  equally  on  all  bodies  at  equal  distances  from  Earth’s 
centre ; its  force  diminishes  as  distance  increases,  and  increases  as  dis- 
tance  diminishes. 

Gravitating  forces  of  bodies  are  to  each  other, 

1.  Directly  as  their  masses. 

2.  Inversely  as  squares  of  their  distances. 

Gravity  of  a body,  or  its  weight  above  Earth’s  surface,  decreases  as 
square  of  its  distance  from  Earth’s  centre  in  semi-diameters  of  Earth. 

Illustration  i —If  a body  weighs  900  lbs.  at  surface  of  the  Earth,  what  will  it 
weigh  2000  miles  above  surface  ?-Earth’s  semi-diameter  is  3963  miles  (say  4000). 

Then  2000  -f  4000  = 6000  = 1.5  semi-diam's , and  900  -^-1.5—  — — 4°°  L0S' 

Inversely , If  a body  weighs  400  lbs.  at  2000  miles  above  Earth’s  surface,  what  will 
it  weigh  at  surface? 

400  X 1.5  =90°  lbs. 

2.  — A body  at  Earth’s  surface  weighs  360  lbs. ; how  high  must  it  be  elevated  to 
weigh  40  lbs.? 

352  = 9 semi-diameters , if  gravity  acted  directly;  but  as  it  is  inversely  as  square 
of  the  distance,  then  -fq  = 3 semi-diameters  = 3 X 4000  = 12000  miles. 

3. _To  what  height  must  a body  be  raised  to  lose  half  its  weight? 

As  ,/i  : y/2  ::  4000  : 5656  = as  square  root  of  one  semi-diameter  is  to  square  root 
of  two  semi- diameters,  so  is  one  semi-diameter  to  distance  required. 

Hence  5656  — 4000  = 1656  = distance  from  Earth's  surface. 

Diameters  of  two  Globes  being  equal , and  their  densities  different , weight 
of  a body  on  their  surfaces  will  be  as  their  densities. 

Their  densities  being  equal  and  their  diameters  different , weight  of  them 
will  be  as  their  diameters. 

Diameters  and  densities  being  different , weight  will  be  as  their  product. 
Illustration.— If  a body  weighs  10  lbs.  at  surface  of  Earth,  what  will  it -weigh  at 
surface  of  Sun,  densities  being  392  and  100,  and  diameters  8000  and  883000  miles . 

883000  X 100  -4-  8000  X 392  = 28. 157  = quotient  of  product  of  diameter  of  Sun  and 
its  density , and  product  of  diameter  of  Earth  and  its  density. 

Then  28.157  X 10  = 281.57  lbs. 

Note.— Gravity  of  a body  is  .00346  less  at  Equator  than  at  Poles. 

SPECIFIC  GRAVITY  AND  WEIGHT. 

Specific  Gravity  or  Weight  of  a body  is  the  proportion  it  bears  to  the 
weight  of  another  body  of  known  density  or  of  equal  volume,  and  which  is 

adopted  as  a standard.  . , , , 

If  a body  float  on  a fluid,  the  part  immersed  is  to  whole  body  as  specmc 

gravity  of  body  is  to  specific  gravity  of  fluid. 

When  a body  is  immersed  in  a fluid,  it  loses  such  a portion  of  its  own 
weight  as  is  equal  to  that  of  the  fluid  it  displaces.  1 

An  immersed  body,  ascending  or  descending  in  a fluid,  h^  a force  equa 
to  difference  between  its  own  weight  and  weight  of  its  bulk  of  the  fluid,  less 
resistance  of  the  fluid  to  its  passage.  . , « . ..p 

Water  is  well  adapted  for  standard  of  gravity ; and  as  a cube  foot  of  it 
at  62°  F.  weighs  997.68  ounces  avoirdupois,  its  weight  is  taicen  as  the  unit, 
or  approximately  1000. 


SPECIFIC  GRAVITY  AND  WEIGHT. 


209 


French  standard  temperature  for  comparison  of  density  of  solid  bodies 
and  determination  of  their  specific  gravities,  is  that  of  maximum  density  of 
water,  at  40  C.  or  39. i°  F.,  and  for  gases  and  vapors  under  one  atmosphere  or 
.76  centimeters  of  mercury  is  320  F.  or  o°  C.,  and  specific  gravity  of  a body 
is  expressed  by  weight  in  kilogrammes  of  a cube  decimeter  of  that  body. 

Densities  of  metals  vary  greatly. 

Potassium,  Sodium,  Barium,  and  Lithium  are  lighter  than  water.  Mercury 
is  heaviest  liquid  and  Platinum  heaviest  metal.  Volcanic  scoriae  is  lighter 
than  water. 


Pomegranate  and  Lignum-vitae  are  heaviest  of  woods.  Pearl  is  heaviest 
of  animal  substances,  and  Flax  and  Cotton  are  heaviest  of  vegetable  sub- 
stances, former  weighing  nearly  twice  as  much  as  water. 

Zircon  is  heaviest  of  precious  stones,  being  4.5  times  heavier  than  water. 
Garnet  is  4 times  heavier,  Diamond  3.5  times,  and  Opal,  lightest  of  all,  is  but 
twice  as  heavy  as  water. 

To  Ascertain  Specific  (Gravity  of  a Solid.  Body-  Heavier 
tli an.  W ater. 


Rule.— Weigh  it  both  in  and  out  of  water,  and  note  difference  ; then,  as 
weight  lost  in  water  is  to  whole  weight,  so  is  1000  to  specific  gravity  of  body. 

^ w x 1000  ~ TTT  7 

Dr,  — ^ = G,  W and  w representing  weights  out  and  m icater , and  G 
specific  gravity. 

Example. — What  is  specific  gravity  of  a stone  which  weighs  in  air  1=;  lbs  in 
water  10  lbs.?  ’ 


15  — 10  = 5;  then  5 : 15  1000  : 3000  Spec.  Grav. 


To  Ascertain  Specific  Gravity  of  a Body  lig-liter  tlian 
Water. 

Rule.— Annex  to  lighter  body  one  that  is  heavier  than  water,  or  fluid 
used  ; 'weigh  piece  added  and  compound  mass  separately,  both  in  and  out  of 
water,  or  fluid ; ascertain  how  much  each  loses,  by  subtracting  its  weight 
from  its  weight  in  air,  and  subtract  less  of  these  differences  from  greater. 

Then,  as  last  remainder  is  to  weight  of  light  body  in  air,  so  is  1000  to 
specific  gravity  of  body. 

Example.— What  is  specific  gravity  of  a piece  of  wood  that  weighs  20  lbs.  in  air- 
annexed  to  it  is  a piece  of  metal  that  weighs  24  lbs.  in  air  and  21  lbs.  in  water,  and 
the  two  pieces  in  water  weigh  8 lbs.? 

20  -f-  24  — 8 = 44  — 8 = 36  — loss  of  compound  mass  in  water  ; 

24  — 21  = 3 = loss  of  heavy  body  in  water. 

33  : 20  : *.  1000  : 606  = 24  Spec.  Grav. 


To  Ascertain  Specific  (Gravity  of  a-  Blnid. 

Rule.— Take  a body  of  known  specific  gravity,  weigh  it  in  and  out  of 
the  fluid ; then,  as  weight  of  body  is  to  loss  of  weight,  so  is  specific  gravity 
of  body  to  that  of  fluid. 

Example.  — What  is  specific  gravity  of  a fluid  in  which  a piece  of  copper  (spec, 
grav.  =9000)  weighs  70  lbs.  in,  and  80  lbs.  out  of  it  ? 

80  : 80  — 70  = 10  : ; 9000  : 1125  Spec.  Grav. 

To  .Ascertain  Specific  (Gravity  of  a Solid.  Body  'wliich. 
is  soluble  in  Water. 

Rule.— Weigh  it  in  a liquid  in  which  it  is  not  soluble,  divide  its  weight 
out  of  the  liquid  by  loss  of  its  weight  in  the  liquid,  and  multiply  quotient 
by  specific  gravity  of  liquid ; the  product  is  specific  gravity. 

Example.— What  is  specific  gravity  of  a piece  of  clay,  which  weighs  15  lbs.  in  air 
and  5 lbs.  in  a liquid  of  a specific  gravity  of  1500,  in  which  it  is  insoluble  ? 

15-i-zoX  1500  = 2250  Spec.  Grav 

s* 


210 


SPECIFIC  GRAVITY  AND  WEIGHT. 


SOLIDS. 


Substances.  ( 

Specific  ( 
Gravity. 

Weight 
)f  a Cube 
Inch. 

Metals. 

2560 
2670 
7700 
6 712 

5763 

470 

Lb. 

.0926 

.0906 

.2785 

.2428 

“ wrought.... 

“ Bronze 

.2084 

.017 

9823 

2000 

•3553 

.0723 

Brass. 

Sheet,  cop.  75,  zinc  25. 
Yellow  66,  “ 34. 

Muntz  “ 60,  “ 40. 

8450 
8300 
8 200 
8 380 

.3056 

.2997 

.2966 

.3026 

8 100 

.2930 

8 214 

.2972 

3000 
8750 
8 217 
8832 
8 700 

8060 

7 39° 

8 650 

.1085 

.3i65 

.2972 

•3i94 

.2929 

.291 

.2668 

3129 

“ ’ ordinary  mean  . 

“ cop.  84,  tin  16  . . 
“ “ 81,  “ 19  . . 

“ small  bells,  cop. 

35,  tin  65.... 
“ cop.  21,  tin  74  . . 

Cadmium 

Calcium 

1 580 
5900 
8 098 
8 600 

•057 

Chromium. ............ 

•2134 

Cinnabar 

.2929 

Cobalt 

• 3111 

Columbium ............ 

6000 

.217 

Copper,  cast. ........... 

8788 
8698 
8880 
8880 
19258 
19361 
17486 
i5  7°9 
18680 

3179 

u plates 

•3I46 

“ wire  and  bolts.. 

“ ordinary  mean. 

.32x2 

.3212 

.6965 

.7003 

Gold  pure  , r 

it  Jjap'impred 

“ 22  carats  fine 

“ 20  “ u 

Iridium 

•6325 

.5682 

.6756 

it  hammered 

23000 
7308 
6900 
7500 
7207 
7217 
7065 
7 218 
, 7788 

■ 7 774 

3 7704 

, 7 698 

• 7 54° 

, 7 808 

. 8 140 

• 7 744 

• 11  352 

.8319 

.264 

.2491 

.2707 

Iron,  Cast,  gun  metal. . . 
44  minimum 

u maximum 

“ ordinary  mean 

* * mean  F-ug 

.2607 

.2609 

•2555 

.2611 

.2817 

.2811 

“ cast,  hot  blast 

“ cold  ‘‘  

“ Wrought  bars 

<(  “ wire 

“ “ rolled  plates 

“ “ average  — 

“ “ Eng.  rails  . . 

“ “ Lowmoor. . , 

n “ pure 

.2787 

.2779 

.2722 

.2819 

.2938 

.2801 

.4106 

“ ordinary  mean — 
Lead  cast  

tt  yollftd 

. 11388 

.4119 

Lithium  

. 59° 

1 .0213 

Magnesium 

• 1 75° 

1 .0633 

Manganese 

. 800c 

1 .2894 

Mercury  <jr>o 

. is6a2 

: -S66i 

“ +320 

. 13  59s  1 *49lS 

Substances.  ( 

Specific  , 
gravity. 

Weight 
of  a Cube 
Inch. 

Metals. 

Mercury  6o° 

“ 212° . ••• 

13  569 

Lb. 

.4908 

13  370 

.4836 

Molybdenum 

8 600 

• 3m 

Nickel 

8 800 

.3i83 

“ cast 

8279 

.2994 

Osmium 

10000 

.3613 

Palladium 

11  350 

.4105 

Platinum,  hammered . . . 

20337 

•7356 

“ native 

16  000 

•57s7 

“ rolled 

22  069 

.7982 

Potassium,  590 

865 

.0313 

Red  lead 

8940 

•324 

Rhodium 

10650 

.3852 

Rubidium 

1 520 

.055 

Ruthenium 

8 600 

.3111 

Selenium 

4 5oo 

.1627 

Silver,  pure,  cast 

10474 

.3788 

“ “ hammered. 

10511 

.3802 

Sodium 

970 

.0351 

Steel,  minimum 

7 7oo 

.2785 

“ maximum 

7900 

.2857 

“ plates,  mean 

7 806 

.2823 

“ soft 

7 s33 

.2833 

“ temper’d  andhard- 
ened 

7 818 

.2828 

“ wire 

7 847 

.2838 

“ blistered 

7823 

.283 

u crucible 

7842 

.2836 

“ cast 

7 M 

.2839 

“ Bessemer 

7 s52 

.284 

“ ordinary  mean 

7 s34 

.2916 

Strontium 

2540 

.0918 

Tellurium 

6 no 

.221 

Thalium 

1 1 850 

.4286 

Tin,  Cornish,  hammered. 

7 39° 

.2673 

“ “ pure 

7291 

.2637 

Titanium 

5 300 

.1917 

Tungsten 

Uranium 

Wolfram 

Zinc,  cast 

“ rolled 

Woods  (Dry) 

Alder 

Apple 

Ash 

Bamboo 

Bay  tree 

Beech 

Birch 

Blackwood,  India  . . 
Boxwood,  Brazil — 
“ France... 

“ Holland. . 

Bullet-wood 

Butternut 


17  000 
18330 

7ll9 

6 861 
7191 


.0149 
.6629 
• 2575 
.2482 
.26 


800 

793 

845 

690 

400 


Cube 

Foot. 

50 

49.562 
52.812 
43- 125 
25 


822 

51-375 

852 

53-25 

690 

43-I25 

567 

35-437 

720 

45 

898 

56.125 

1 031 

04-437 

1 328 

s3 

912 

57 

928 

5s 

376 

23-5 

SPECIFIC  GRAVITY  AND  WEiGHT. 


21 1 


Substances. 


Woods  {Dry). 

Campeachy 

Cedar 

“ Indian 

Charcoal,  pine 

“ fresh  burned. 

“ oak 

“ • soft  wood  . . . 

“ triturated... 

Cherry 

Chestnut,  sweet 

Citron 

Cocoa  

Cork 

Cypress,  Spanish 

Dog- wood 

Ebony,  American 

u Indian 

Elder 

Elm \ 


“ rock 

Erroul,  India 

Filbert 

Fir,  Norway  Spruce. 

“ Dantzic 

Fustic 

Greenlieart  or  Sipiri. 

Gum,  blue 

“ water 

Hackmatack 

Hawthorn 

Hazel 

Hemlock 

Hickory,  pig-nut 

“ shell-bark. . 

Holly 

Iron -wood. 

Jasmine 

Juniper 

Khair,  India 

Lancewood,  mean. . . 

Larch 


Lemon 

Lignum-vitse 

Lime 

Linden 

Locust 

Logwood 

Mahogany 

“ Honduras. 

“ Spanish... 

Maple 

“ bird’s-eye 

Mastic 

Mulberry 

Oak,  African 

11  Canadian 

il  Dantzic. 


Weight 
Specific  0f  a Cube 
Gravity.  Foot. 


9X3 

56i 

i3x5 

441 

380 

i573 

280 

1380 

7X5 

610 

726 

1040 

240 

644 

756 

i33i 

1209 

695 

570 

671 

800 

1014 

600 

512 

582 

970 

io55 

843 

1000 

592 

910 

860 

368 

792 

690 

760 

990 

770 

566 

1171 

720 

544 

560 

703 

650 

x333 

804 

604 

728 

9*3 

720 

1063 

560 
852 
750 
576 

849 

561 

897 
823 
872 
759 
* U. 


57-062 

35.062 
82.157 
27.562 
23-75 
98.312 
*7-5 

86. 25 

44.687 

38.125 
45-375 
65 

*5 

40.25 
47-25 
83.187 
75  562 
43-437 
35  625 
4i  937 

5o 

63-375 

37-5 

32 

36.375 

60.625 

65-95 

52.687 
62.5 
37 

56.875 
53-75 
23 
49*5 
43-125 
47-5 

61.875 

48.125 
35-375 
73-187 
45 

34 

35 

43-937 

40.625 
83-312 

50.25 
37-75 
45-5 
57  062 
45 

66.437 

35 

53- 25 

46.875 

36 

53.062 

35.062 

56.062 
51-437 

54- 5 
47-437 


Woods  {Dry). 


Oak,  English. , 


* ‘ green 

“ heart,  60  years 

“ live,  green 

“ “ seasoned 

“ white 

Olive 

Orange 

Pear 

Persimmon 

Plum 

Pine,  pitch 

“ red 

“ white 

“ yellow 

“•  Norway.. 

Pomegranate 

Poon 

Poplar 

“ white 

Quince 

Rosewood 

Sassafras 

Satinwood 

Spruce 

Sycamore 

Tamarack 


. Weight 
Specific  of  a Cube 
Gravity.  Foot. 


Teak  (African  oak) 


Walnut. 


black. 


Willow . 


Yew,  Dutch. . . 
Spanish. 


( Well  Seasoned.*) 

Ash 

Beech  

Cherry 

Cypress 

Hickory,  red 

Mahogany,  St. Domingo. 

Pine,  white . 

yellow 

Poplar 

White  Oak,  upland 

“ u James  River 

Stones,  Earths, 
etc. 

Alabaster,  white 

“ yellow 

Alum 

Amber 

Ambergris 

Asbestos,  starry 

Asphalte 

Barytes,  sulphate  . . . . j 
Beton,  N.  Y.  St.Con’g  Co. 


858 

932 

1146 

1170 

1260 

1068 

860 

680 

705 

661 

710 

785 

660 

590 

554 

461 

740 

*354 

580 

383 

529 

7°5 

728 

482 

885 

500 

623 

383 

657 

980 

671 

500 

486 

585 

788 

807 


722 

624 

606 

441 

838 

720 

473 

54i 

587 

687 

759 


2730 

2699 

i7x4 

1078 

866 

3073 

2250 

4000 

4865 

2305 


Lbs. 

53-625 

58.25 

71.625 
73-125 

78.75 

66.75 
53-75 
42.5 

44.062 
4i-3x2 

44- 375 

49.062 

41.25 
36.875 
34-625 
28.812 

46.25 

84.625 

36.25 
23-937 

33.062 

44.062 

45- 5 
30.125 
55-312 

31-25 

38.937 

23-937 

41.062 

61.25 
4X  937 
3x-25 
30-375 
36.562 
49-25 
50.437 


45-125 

39  n 

37-875 

27.562 

52.375 

45 

29.562 
33-8i2 
36.687 
42-937 
42-437 


170.625 
168.687 
107. 125 

67-375 

192.062 

140.625 
250 

304.062 
144. 06 


212 


SPECIFIC  GRAVITY  AND  WEIGHT. 


Specific 

Gravity. 


Stones,  Eartlis, 
etc. 

Basalt | 

Bitumen,  red 

“ brown 

Borax 

Brick | 

“ pressed 

“ fire 

“ work  in  cement. . . 

“ “ u mortar 

Carbon 

Cement,  Portland  . 

“ Roman... 

Chalk 

Clay 


•{ 


with  gravel. . 
Coal,  Anthracite. , 


Borneo 

Cannel 

Caking 

Cherry 

Chili 

Derbyshire  . . 

Lancaster 

Maryland 

Newcastle  . . . 
Rive  de  Gier. 

Scotch 


Weight 
of  a Cube 
Foot. 


“ Splint 

“ Wales,  mean. . . 

Coke 

“ Nat’l,  Va 

Concrete,  in  cement. 

“ mean 

Earth  * common  soil, dry 

“ loose 

“ moist  sand 

“ mold,  fresh 

“ rammed 

“ rough  sand 

“ with  gravel ... . 

“ Potters’ 

“ light  vegetable. . 

Emery 

Feldspar 

Flint,  black 

“ white 

Fluorine 

Fuel,  Warlich’s 

“ Lignite 

Glass,  bottle - 

“ Crown 


“ flint.. 


2740 

2864 

1160 

830 

1714 

1367 

1900 

2400 

2201 

1800 

1600 

2000 

35°° 

130° 

1560 

1520 

2784 

1930 

2480 

x35o 

1436 

1640 

1290 

1238 

1318 

1277 

1276 

1290 

1292 

1273 

1355 

1270 

1300 

1259 

1300 

1302 

1315 

1000 

746 

2200 

2000 

1216 

1500 

2050 

2050 

1600 

1920 

2020 

1900 

1400 

4000 

2600 

2582 

2594 

1320 

1150 

130° 

2732 

2487 

2933 

3200 


Substances. 


Specific 

Gravity. 


171.25 
179 
72-3 
51-7 
107. 125 
85-437 

118.75 
50 

137.562 

12.5 
100 

125 

218.75 
81.25 
97-25 
95 

174  . 
120.625 

i55 
84-375 
89-75 

102.5 
80.625 
77-375 

82.375 

79.812 

79-75 

80.625 

80.75 

79.562 

84.687 

79-375 

81.25 

78.687 

81.25 
8i-375 
82. 187 

62.5 
46.64 

137-5 

125 

76 

93-75 

128.125 

28.125 
100 
120 

126.25 
i8.75 
87-5 

250 

162.5 

161.375 

162.125 
82.5 
7i-875 
81.25 

170-75 
155-437 
183.312 
196 


Weight 
of  a Cube 
Foot. 


Stones,  Eartlis, 
etc. 

Glass,  green 

optical 

white •• 

window 

soluble 

Gniess,  common 

Granite,  Egyptian  red. . 

14  Patapsco 

“ Quincy 

u Scotch 

“ Susquehanna 

“ “ gray 

Graphite 

Gravel,  common 

Grindstone 

Gypsum,  opaque  . . . 
Hone,  white,  razor  . 

Hornblende 

Iodine 


Lava,  Vesuvius 

Lias 

Lime,  quick 

“ hydraulic 

Limestone,  white  . . . 

“ green... 
Magnesia,  carbonate 

Magnetic  ore 

Marble,  Adelaide  . . . 

“ African 

“ Biscayan,  black. 

“ Carrara 

“ - common... 

“ Egyptian. . . 

“ French 

“ Italian,  white. . 

“ Parian 

“ Vermont,  white. 

“ Silesian 

Marl,  mean 

“ tough 

Masonry,  rubble 

Granite. . . 
Limestone. . . 
Sandstone. 

Brick 

“ rough  work 


Mica 

Millstone  . 


Quartz. 


Mortar  . 
Mud. 


wet  and  fluid 

u u u pressed, 

Nitre 

Oyster-shell 

Paving-stone 

Peat,  Irish,  light .... 
“ “ dense.... 

“ very  “ .... 


2642 

3450 

2892 

2642 

1250 

270 

2654 

2640 

2652 

2625 

2704 

2800 

2200 

1749 

2143 

168 

2876 

3540 

4940 

1710 

2810 

1350 

804 

2745 

3*56 

3180 

2400 

5094 

2715 

2708 

2695 

27x6 

2686 

2668 

2649 
2708 
2838 

2650 
2730 

*75° 

2340 

2050 

2640 

2640 

2160 

2240 

1600 

2800 

2484 

1260 

1384 

1750 

1630 
1782 
1920 
1900 
2092 
2416 
278 
562 

675 


165.125 

215.625 

180.75 

165.125 

78.125 

5-875 

65-875 

65 

65-75 

64.062 

69 

i75 
37-5 
109. 312 
133-937 
35-5 
179-75 
221.25 


06.875 

75-625 

46-875 

50.25 
171.562 
197-25 

98.75 

150  . 
3W-6 
69.687 

69.25 
168.437 

169.75 

167.875 

166.75 
65.562 

169.25 
177-375 
i65-57 
170.625 

109.375 

146.25 
128.125 

165 

165 

135 

140 

100 

175 

155-25 

78-75 

86.5 

109.375 

101.875 

112 

120 

118.75 

i3°-75 

151 

17-375 

35-125 

42.187 


* Specific  gravity  of  earth  is  estimated  at  from  1520  to  2200. 


SPECIFIC  GRAVITY  AND  WEIGHT. 


213 


SUBSTANCES. 

Specific 

Gravity. 

Weight 
of  a Cube 
Foot. 

Stones,  Eartlis, 
etc. 

Peat,  black j 

1058 

1329 

Lbs. 

66. 125 
83.062 
110.625 
73-5 
212.5 

87-5 

Plaster  of  Paris j 

“ “ “ dry 

Plumbago 

1176 

3400 

1400 

2100 

Porcelain,  China 

14^7^ 

2765 

665 

172  8l2 

66  187 

Quartz 

8940 

558.75 
68.062 
1 70. 937 
123.812 

Too  TOC 

Resin 

Rock,  crystal  

070  C 

Rotten-stone 

IQ8i 

Salt,  common 

“ rock 

2200 

137-5 

130.625 

112.5 
io4-375 

87 

97-5 

88.75 

103.66 

107.25 
106.33 
137-5 
139.81 
198.125 

51-875 

140.625 

162.5 
167 

181.25 
*74 

Saltpetre 

2090 

1800 

1670 

1392 

1560 

Sand,  coarse 

“ common 

“ damp  and  loose. . . 
li  dried  u “ ... 
“ dry 

“ mortar,  Ft.  Rich m’d 
“ “ Brooklyn.. 

“ silicious 

1659 

1716 

1701 

Sandstone,  mean 

“ Sydney 

Schorl 

2237 

Scoria,  volcanic  

3*7° 

830 

Sewer  pipe,  mean 

Shale 

2600 

2672 

2900 

2784 

Slate j 

“ purple 

Smalt 

Soapstone 

2440 

2730 

979  c 

I52-  5 

170.625 

170.937 

168.312 

169 

212.  5 

Spar,  calcareous 

“ Feld,  blue 

z / DJ 

2693 

2704 

“ “ green 

‘‘  Fluor 

Specular  ore 

3400 

c OC  T 

Stalactite 

320.  IO7 

150-937 

122.562 

165 

164.062 
169  ] 
156.875  j 
I29-75 
157-5 
144-75 
165.687 
172 

144 

148 

186 

168 

127.062 

Stone,  Bath,  Engl 

2415 

1961 

2640 

2625 

2704 

2510 

2076 

“ Blue  Hill 

“ Bluestone  (basalt) 

“ Breakneck,  N.Y. . 

“ Bristol,  Engl 

“ Caen,  Normandy. 
“ common 

“ Craigleith,  Scotl. . 

“ Kentish  rag,  . 

“ Kip’s  Bay,  N.Y. . 

11  Norfolk  (Parlia- 

ment House). . . 
“ Portland,  Engl... 

“ Staten  Isl’d,  N.Y. 

“ Sullivan  Co.,  “ 

Sulphur,  native 

2316 

2651 

2759 

2304 

2368 

2976 

2688 

2033 

1952 

1815 

2720 

Terra  Cotta 

Tile 

ir3-437 
170  1 

Trap 

Substances. 


Grranite. 

(Gen' l Gill  more,  U.  S.  A.) 

Duluth,  Minn.,  dark 

Fall  River,  Mass.,  gray. . 
Garrison’s,  N.  Y.  “ .. 
Jersey  City,  N.  J.,  soap.. 
Keene,  N.  H.,  bluish  gray 

Maine 

Millstone  Ft.,  Conn 

New  London,  “ 

Quincy,  Mass.,  light 

Richmond,  Va 

“ “ gray 

Staten  Island,  N.  Y 

Westchester  Co.,  N.  Y.. 
Westerly,  R.  I.,  gray 

Limestone. 
(Gen' l Gillmore , U.  S.  A. ) 
Bardstown,  Ky. , dark  . . 

Caen,  France 

Canajoharie,  N.  Y 

Cooper  Co. , Mo. , d’k  drab 
Erie  Co.,  N.  Y , blue.... 

Garrison’s,  N.  Y 

Glens’  Falls,  “ 

Joliet,  111.,  white 

Kingston,  N.  Y. 

Lake  Champlain,  N.  Y. . 
Lime  Island,  Mich.,  drab 
Marblehead,  Ohio,  white 
Marquette,  Mich.,  drab  . 
Sturgeon  Bay,  Wis.,  blu- 
i ish  drab 

ZVTar'ble. 

1 (Gen' l Gillmore , U.  S.  A. ) 

Dorset,  Vt 

East  Chester,  N.  Y 

Italian,  common 

Mill  Creek,  111.,  drab.... 
North  Bay,  Wis.,  “ 

Sandstone. 
(Gen' l Gillmore , U.  S.  A.) 

Albion,  N.Y.,  brown 

Belleville,  N.  J.,  gray. . . 

Berea.  Ohio,  drab 

Cleveland,  u olive  green 
Edinb’h,Sc’tl.,CraigIeith 
Fond  du  Lac,  Wis.,  purple 
Fontenac, Minn., l’g’t  buff 
Haverstraw,  N.  Y.,  red. . 

Kasota,  Minn.,  pink 

Little  Falls,  N.  Y.,  brown 
Marquette,  Mich.,  purple 
Masillon,  0.,  yellow  drab 

Medina,  N.  Y.,  pink 

Middletown,  Ct.,  brown. 
Seneca,  Ohio,  red  “ 
Vermillion,  Ohio,  drab. . 
Warrensburgh,  Mo 


Specific 

Gravity. 

Weight 
of  a Cuba 
Foot. 

Lbs. 

2780 

173-7 

2635 

164.7 

2580 

161.2 

3°3° 

189.3 

2656 

166 

2635 

164.7 

2706 

169. 1 

2660 

166.25 

2695 

168.5 

2727 

170.5 

2630 

164.4 

2861  ; 

178.8 

2655 

165.9 

2670 

166.9 

2670 

166.9 

19°° 

xi8.8 

2685 

167.8 

2320 

i4i-3 

2640 

165 

2635 

164.7 

2700 

168.7 

2540 

158.7 

2690 

168. 1 

2750 

171.9 

2500 

156.3 

2400 

150 

2340 

146.25 

O 

00 

N 

I73-7 

2635 

164.7 

2875 

179.7 

269O 

168. 1 

2570 

171.9 

2800 

W5 

2420 

151-25 

2259 

141.2 

2110 

131-9 

224O 

140 

2260 

141.25 

2220 

138.7 

2325 

i45-3i 

214 


SPECIFIC  GRAVITY  AND  WEIGHT. 


Spec.  Grav. 

Agate 2590 

Amethyst 3920 

Carnelian 2613 

Chrysolite 2782 

Diamond,  Oriental. . . 3521 
“ Brazilian..  3444 

“ pure 3520 

Emerald 395° 


Precious  Stones. 

Spec.  Grav. 

Emerald,  aqua 

ma- 

rine 

Garnet 

“ black... 

3750 

Jasper 

Jet 

Lapis  lazuli. . . . 

Malachite 

Spec.  Gray. 

Onyx 2700 

Opal 2090 

Pearl,  Oriental 2650 

Ruby 3980 

Sapphire 3994 

Topa^ 3 s00 

Tourmaline 3070 

Turquoise 2750 


Substances.  ( 

Specific  L 
Gravity. 

Weight 
if  a Cube 
Foot. 

IVI  is  cell  an  eons. 

1090 
,001292  . 

965 

1900 

942 

988 

Lbs. 

68.125 

Atmospheric  Air 

,080728 

60.312 

Beeswax 

Bone 

118.75 

Butter 

58-875 

Camphor 

61.75 

Caoutchouc 

930 

95° 

1650 

58.125 

Cotton  

59-375 

Dynamite 

103. 125 

]£gg 

1090 

923 

936 

; 923 

1790 
1 1222 

— 

Fat  of  Beef | 

“ Hogs 

“ Mutton 

jrjax 

57-687 

58.5 

57.687 

111.875 

Gamboge 

Gtycerin*1  

1261 

78.752 

Grain  Barley 

59° 

750 

500 

36.875 

“ ’ Wheat 

46.875 

“ Oats 

31-25 

Gum  Arabic.,  _ . 

1452 

900 

1000 

i55o 

1800 

980 

90.75 

Gunpowder  loosft  ...... 

56.25 

4 ‘ shaken  . . . . 

“ solid | 

Gutta-pfM'uhn, 

62.5 

96.875 

112.5 

61.25 

Hay  old  compact 

128.8 

8.05 

Horn 

1689 

105.562 

Substances. 


Human  body 

Ice,  at  320 

Indigo 

Isinglass 

Ivory 

Lard 

Leather 

Mastic 

Myrrh 

Nitro-Glycerine. 

Opium 

Potash 

Resin 

Snow 

Soap,  Castile 

Spermaceti 

Starch 

Sugar 

“ .66 

Tallow 

Wax 


1070 

922 

1009 

mi 

1825 

947 

960 

1074 

1360 

1600 

1336 

2100 

1089 

•0833 

1071 

943 

950 

1606 

972 

1326 

941 

964 

970 


57-5 

63.062 

69.437 

114.062 

59*l87 

60 

67.125 

85 

100 

83-5 

131.25 

68.062 
5.2 

56.937 

58.937 
59-375 

100.375 

60.25 

82.875 

58.812 

60.25 

60.625 


Liquids. 

Acid,  Acetic 

“ Benzoic 

“ Citric 

“ Concentrated.. 

“ Fluoric 

“ Muriatic 

“ Nitric 

“ Nitrous 

“ Phosphoric 

“ “ solid.. 

“ Sulphuric 

Alcohol,  pure,  6o° 

“ 95  per  cent. . . . 

“ 80  “ 

“ 50  “ 

“ 40  “ ...■ 

“ 25  u 

“ 10  .... 

“ 5 “ 

“ proof  spirit,*  5c 
per  cent.,  6oa 
“ proof  spirit,  50 
per  cent.,  8o° 
Ammonia,  27.9  per  cent. 

Aquafortis,  double 

“ single 

Beer 

Benzine 

Bitumen,  liquid 

Blood  (human) 

Brandy,  .83  or  .5  of  spirit 

Bromine 

Cider 

Ether,  Acetic 

“ Muriatic..... 

“ Nitric 

“ Sulphuric.... 

Honey 

Milk 

Oil,  Anise-seed 

“ Codfish 

“ Whale 

“ Linseed., 

“ Naphtha 

“ Olive 

“ Palm 

“ Petroleum 

Rape 

Sunflower 

Turpentine  . . . 


1062 

667 

1034 

1521 

1500 

1200 

1217 

1550 

1558 

2800 

1849 

794 

816 

863 

934 

95i 

970 

986 

992 

} 934 

' 875 


1200 

1034 

850 


io54 

924 

2966 

1018 

866 

845 

mo 

7i5 

1450 

1032 

986 

923 

923 

940 

850 

9a5 

969 

880 

9*4 

926 

870 


66.375 

41.687 

64.625 

95.062 
93-75 
75  „ 

76.062 
96.875 
97-375 

175  a 
115.562 

49.622 

5i 

53-937 

58- 375 

59- 437 

60.625 

61.625 
62 

58-375 

54.687 

55-687 
81.25 
75  , 

64.625 

53- 125 

53  „ 
65-875 

57-75 

85-375 

63.625 

54-  125 
52.812 
69-375 

44.687 

90.625 

64-5 

61.625 

57.687 

57- 687 

58- 75 

53- I25 
57-l87 
60. 562 
55 

57-125 

57-875 

54- 375 


* Specific  gravity 


y of  proof  spirit  according  to  Ure’s  Table  for  Sykes's  Hydrometer,  < 


SPECIFIC  GRAVITY  AND  WEIGHT. 


215 


Substances. 

Liquids. 

Spirit,  rectified 

Steam,  at  2120 

Tar 

Vinegar  

Water,  at  320 

“ “ 39- l0 

“ “ 62°t 

“ “ 212° 

“ distilled,  at  390 

* .038 18. 


Specific 

Gravity. 

Weight 
of  a Cube 
Foot. 

Substances. 

Specific 

Gravity. 

Weight 
of  a Cube 
Foot. 

Lbs. 

Liquids. 

Lbs. 

824 

51-5 

Water,  Dead  Sea 

1240 

77-5 

.00061 

.038* 

“ Mediterranean... 

1029 

64.312 

1015 

63-437 

‘ ‘ sea 

1029 

64.312 

1080 

67-5 

“ Black  Sea 

1016 

63-5 

998.7 

62.418 

“ rain 

1000 

62.5 

998.8 

62.425 

Wine,  Burgundy 

992 

62 

997-7 

62.355 

“ Champagne 

997 

64-375 

956.4 

59-64 

“ Madeira 

1038 

62.312 

998 

62.379 

“ Port 

997 

62.312 

t x cube  inch  at  standard  temperature  = 252.5954  grains. 


Compression  of  following  fluids  under  a pressure  of  15  lbs.  per  square  inch: 
Alcohol..  .0000216  | Mercury..  .00000265  I Water..  .00004663  | Ether..  .00006158 

Elastic  Eliaids. 

1 Cube  Foot  of  Atmospheric  Air  at  320  weighs  .080728  lbs.  Avoirdupois  = 565.096 
grains , and  at  62°  532. 679  grains. 

Its  assumed  Gravity  of  1 is  Unit  for  Elastic  Fluids. 

Sp 


Spec.  Gray. 

Acetic  Ether 3.04 

Ammonia 589 

Atmos,  air,  at  320. . 1 

Azote 976 

Carbonic  acid 1.53 

“ oxide 972 

Carburet’d  Hydrog.  .559 

Chlorine 2.421 

Chloro-carbonic ...  3. 389 

Chloroform 5.3 

Cyanogen 1.815 

Gas,  coal ( '43s 

( -752 

Hydrochloric  acid . 1. 278 
Hydrocyanic  “ . .942 

Hydrogen 0692 

Muriatic  acid 1-247 

t Weight  of  a cube  foot  267.26 


Nitric  acid. 

“ oxide 

Nitrogen 

Nitrous  acid 

Nitrous  oxide .... 

Olefiant  gas 

Oxygen ...... 

Phosphurett’d  Hy- 
drogen  

Sulphuretted  Hy- 
drogen  1 

Sulphurous  acid. . 2 
Steam,  J at  2120. . . 
Smoke. 

Bitum.  Coal. . . . 

Coke 

Wood 

grains,  and  compared  with 


5.  Grav. 
.217 
1.094 

•974 

2.638 

1-527 

.9672 
1. 106 

1.77 


•47295 

.102 

.105 

•°9 

water  at 


Spec.  Grav. 

Vapor. 

Alcohol 1.613 

Bisulphuret  of 

Carbon 2.64 

Bromine 5.4 

Chloric  Ether 3.44 

Chloroform 4.2 

Ether 2.586 

Hydrochlor.  Ether  2.255 

Iodine 8.716 

Nitric  acid 3.75 

Spirits  of  Turpen- 
tine  5-013 

Sulphuric  acid  .. . 2.7 
“ Ether..  2.586 

Sulphur 2.214 

Water 623 

62°  specific  gravity  = .000  612  3. 


Weight  of  a Cube  Foot  of  Gases  at  320  F.,  and  under  Pressure  of  one  Atmos- 
phere, or  2116.4  lbs.  per  Square  Foot. 


Lbs. 

Air,  at  320 080728 

u “ 62°  076  097 

Alcohol 1302 

Carbonic  acid 12344 

Carburet.  Hydrog.  . 044  62 


Lbs. 

Chlorine 197 

Chloroform 428 

Coal  gas 03536 

Ether,  Sulphuric. . .2093 
Gaseous  steam 05022 


Sulphurous  acid 1814  lbs. 


, Lbs. 

Hydrogen 005  594 

Nitrogen 078  596 

Olefiant  gas 0795 

Oxygen 089256 

Steam 05022 


To  Compute  Weight  of  a Body  or  Substance  when 
Specific  Gravity  is  given. 

Rule.— Multiply  specific  gravity  by  unit  or  standard  of  body  or  sub- 
stance, and  product  is  the  weight. 

Or,  Divide  specific  gravity  of  body  or  substance  by  16,  and  quotient  will 
give  weight  of  a cube  foot  of  it  in  lbs. 

Example.— Specific  gravity  is  2250;  what  is  weight  of  a cube  foot  of  it? 

2250  X 62.5  = 140.625  lbs. 


WEIGHTS  OF  VARIOUS  SUBSTANCES. 


2l6 

Weights  and.  Volumes  of  various  Substances 
v “ Ordinary  Use. 


Substances. 

IMetals. 

„ ( copper  67  \ 

Brass  . . j ziuc  33  J 

“ gun  metal 

“ sheets..., 

“ wire 

Copper,  cast 

“ plates  

Iron,  cast 

“ gun  metal 

“ heavy  forging. 

“ plates 

“ wrought  bars. . . 
Lead,  cast. . . 

“ rolled. 
Mercury,  6o° 

Steel,  plates. 

“ soft... 

Tin 

Zinc,  cast . . . 

“ rolled.. 


Cube  Foot.  Cube  Inch. 


Lbs. 

488.75 

543-75 

5i3-6 

524.16 

547-25 

543-625 

450-437 

466.5 
479-5 

481.5 

486.75 
7°9-5 
7IX-75 
848.7487 
487-75 
489. 562 
455-687 
428.812 
449-437 


"Woods. 

Ash 

Bay 

Blue  Gum 

Cork 

Cedar 

Chestnut 

Hickory,  pig  nut. 

“ shell-bark. 

Lignum-vitee 

Logwood 

Mahoga^Hondur’s 

Oak,  Canadian 

“ English 

“ live,  seasoned. . 

“ white,  dry 

“ “ upland.. 

Pine,  pitch 

“ red 

“ white 

“ well-seasoned.. 
Pine,  yellow 


52.812 

51-375 

64-3 

i5 

35.062 
38.125 
49-5 
43-x25 
83.312 

57.062 
35 

66.437 
54-5 
58-25 
66.75 
53-75 
42-937 
4x-25 
36.875 
34.625 
29. 562 
33-812 


.2829 

•3147 
.297 
•3033 
•3179 
.3167 
.2607 
.27 
•2775 
.2787 
.2816 
.4106 
.4119 
.491174 
.2823 
•2833 
.2637 
.2482 
.2601 
Cube  Feet 
in  a Ton. 
42.414 
43.601 
34-S37 
149-333 

63.886 

58-754 

45.252 

51.942 

26.886 

39-255 

64 

33-7x4 

41.101 

38.455 

33-558 

41.674 

52.169 

54-303 

60.745 

64.693 

75-773 

66.248 


_ . r-  . I Cube  Feet 
Cube  Foot.  inaTon. 


Woods. 

Spruce 

Walnut,  black,  dry... 

Willow 

dry 

IVIiscellan  eous, 

Air 

Basalt,  mean 

Brick,  fire 

mean 

Coal,  anthracite — j 
bitumin.,  mean 

Cannel 

Cumberland. . . 
Welsh,  mean.. 

Coke 

Cotton,  bale,  mean . . 

‘ “ pressed 


Earth,  clay 

common  soil. 

“ gravel 
dry,  sand. 

loose 

moist,  sand.. 

1 mold.- 

mud 

£ with  gravel. 

Granite,  Quincy 

“ Susquehanna 

Gypsum 

Hay,  bale 

“ hard  pressed. . 

Ice,  at  32° 

India  rubber 

“ u vulcanized 

Limestone 

Marble,  mean 

Mortar,  dry,  mean 
Plaster  of  Paris. . . 

Water,  rain 

“ salt 

“ at  62° 


Lbs. 

3x-25 

3x-25 

36.562 

30-375 


075291 

75  a 
37.562 

102 

•75 
102.5 
80 

94-875 

84.687 

81.25 

62.5 

14-5 

20 

25  a 

120.025 

137.  i25 
109. 312 
120 
93-75 
128.125 
128. 125 
101.875 

126.25 


71.68 

71.68 

61.265 

73-744 


16.284 

21.961 

24.958 

21.854 

28 

23.609 

26.451 

27.569 
35-84 

154.48 

114  . 
89.6 

18.569 

16.335 

20.49 

18.667 

23.893 

17.482 

17.482 

21.987 

17-742 


165-75 

i3-5x4 

169 

13-254 

135-5 

i6.53x 

12 

186.66  * 

25 

89.6 

57-5 

38-95 

56-437 

39-69 
— ; 

I97-25 

1I-355  r 

167.875 

x3-343 

97.98 

22.862 

73-5 

3°-476 

62.5 

35-84 

64.312 

34-83 

62.355 

35-955 

To  Compute  Proportions  of  Two  Ingredients  in  a Com-: 
po“»d,  or  to ^Discover  Adulteration  m Metals.  • 

Rui.e. — Take  differences  of  each  specific  gravity  of  ingri 

gravity  of  body  to  proportions  of  the  ingredients. 

Examfle.-A  compound  of  gold  (spec.  grav.  = .8.888)  and  silver  (spec.  0>™.= 
xo  535)  has  a specific  gravity  of  14 ; what  is  proportion  of  each  metal  ? 

18.888 — 14=4. 888X  10.535=51.495*  14— i»S3S=3-465Xi8.M8_6|^^ 

6s-447+5i-49S:6s-447:--*4:7-835fl»«.  65-447+SM95- 5i-495..«4-  0 


WEIGHTS  OF  VARIOUS  SUBSTANCES  IN  BULK.  21/ 


Weiglits  of  Vario-as  Safbstances  per  Cube  Foot  in  13  vxlL; 


Lbs. 

Lead,  in  pigs 567 

Iron,  “ 360 

Marble,  in  blocks) 
Limestone,  “ ) ' * 17 

Trap 170 

Granite,  in  blocks 164 

Sandstone 141 


Lbs. 


Potters’  clay 

Loam 

Gravel 

Sand 

Bricks,  common. . 

• • 93 

Ice,  at  320. . . . -. 

«•  57-5 

Oak,  seasoned 

..  52 

Lbs. 


Coal,  caking 50 

Wheat 48 

Barley 38 

Fruit  and  vegetables . . 22 

Cotton  seeds 12 

Cotton 10 

Hay,  old 8 


Ash,  dry,  100  feet  BM 175  ton. 

“ white,  “ “ 141  “ 

Cement,  struck  bushel  and 

packed* 100  lbs. 

Cement,  Portland,  bushel. no  lbs. 

Cherry,  dry,  100  BM 156  ton. 

Chestnut,  dry,  100  BM  .. . .153  “ 

Coal,  anthracite,  1 cub.  yd. 

broken  and  loose  ...  1.75  yds. 

“ “ “ 1 ton..  41.5  cub.  feet. 

Coke,  ton  = 80  to  97  cub.  feet. 

Earth,  common  soil 137-125  lbs. 


Earth, loose 93-75  lbs. 

Elm,  dry,  100  feet  BM 13  ton. 

Gypsum,  ground,  str.  bush.  70  lbs. 

“ “ well  shaken  80  “ 

Hemlock,  dry,  100  feet  BM.  .093  ton. 

Hickory,  “ ‘‘  “ . .197  “ 

Masonry,  Granite, dressed. . 165  lbs. 

“ “ rough. . . 126  “ 

“ Limestone,  dres’d  165  “ 

“ Sandstone 135  “ 

“ Brick,  pressed  .. . 140  “ 

“ “ com’n,  rough.  100  u 


* One  packed  bushel  = 1.43  loose. 


Comparative  W eiglit  of  Green  and.  Seasoned  Timber. 


Timber. 

Weight  of  a Cube  Foot. 
Green.  1 Seasoned. 

Timber. 

Weight  of  a 
Green. 

, Cube  Foot. 
Seasoned. 

American  Pine 

Ash 

Beech 

Lbs. 

44-75 

58.18 

60 

Lbs. 

30-7 

50 

53-37 

Cedar  

English  Oak 

Riga  Fir 

Lbs. 

32 

71.6 

4^-75 

Lbs. 

28.25 

43-5 

35-5 

^Application  of  tlie  Ta"bles. 

When  Weight  of  a Solid  or  Liquid  Substance  is  required.  Rule. — Ascer- 
tain volume  of  substance  in  cube  feet ; multiply  it  by  unit  in  second  column 
of  tables  (its  specific  gravity),  and  divide  product  by  16 ; quotient  will  give 
weight  in  lbs. 

When  Volume  is  given  or  ascertained  in  Inches.  Rule. — Multiply  it  by 
unit  in  third  column  of  tables  (weight  of  a cube  inch),  and  product  will  give 
weight  in  lbs. 

Example.— What  is  weight  of  a cube  of  Italian  marble,  sides  being  3 feet? 

33  x 2708  = 73  1 16  02.,  which -4- 16  = 4569.75  lbs. 

Or  of  a sphere  of  cast  iron  2 inches  in  diameter? 

23  x .5236  X .2607  weight  of  a cube  inch=  1.092  lbs. 

When  Weight  of  an  Elastic  Fluid  is  required.  Rule. — Multiply  specific 
gravity  of  fluid  by  532.679  (weight  of  a cube  foot  of  air  at  62°  in  grains), 
divide  product  by  7000  (grains  in  a lb.  Avoirdupois),  and  quotient  will  give 
weight  of  a cube  foot  in  lbs. 

Example. — What  is  weight  of  a cube  foot  of  hydrogen  ? 

Specific  gravity  of  hydrogen  .0692. 

532.679  X .0692  -4-  7000  :=  .005  265  9 lbs. 

To  Compute  W eiglit  of  Cast  iVCetal  by  W eiglit  of  Pattern. 

When  Pattern  is  of  White  Pine.  Rule. — Multiply  weight  of  pattern  in 
lbs.  by  following  multipliers,  and  product  will  give  weight  of  casting : 

Iron,  14 ; Brass,  15 ; Lead,  22 ; Tin,  14 ; Zinc,  13.5. 

When  there  are  Circular  Cores  or  Prints.  Multiply  square  of  diameter  of 
core  or  print  by  its  length  in  inches,  the  product  by  .0175,  and  result  is 
weight  of  pattern  of  core  or  print  to  be  deducted  from  weight  of  pattern. 

T 


218  balloons,  shrinkage  of  castings,  etc. 


To  Compute  Weights  of  Ingredients,  tLat  of  Compound 
"being  given. 


Rule. — As  specific  gravity  of  compound  is  to  weight  of  compound,  so  are 
each  of  the  proportions  to  weight  of  its  material. 

Example.— Weight,  as  above,  being  28  lbs.,  what  are  weights  (Of  the  ingredients? 

TA  " 28  * * I7'835  ! I5-67  W1*' 

14  . 2«  ..  j65  . I2  33  Sliver, 

Note. —Specific  gravity  of  alloys  does  not  usually  follow  ratio  of  their  compo- 
nents, it  being  sometimes  greater  and  sometimes  less  than  their  mean. 

To  Compute  Capacity  of  a Balloon. 

Rule. — From  specific  gravity  of  air  in  grains  per  cube  foot,  subtract  that 
of  the  gas  with  which  it  is  inflated  ; multiply  remainder  by  volume  of  bal- 
loon in  cube  feet ; divide  product  by  7000,  and  from  quotient  subtract  weight 
of  balloon  and  its  attachments 


Example.— Diameter  of  a balloon  is  26.6  feet,  its  weight  is  100  lbs.,  and  specific 
gravity  of  the  gas  with  wThich  it  is  inflated  is  .07  (air  being  assumed  at  1);  what  is 
its  capacity,  specific  gravity  of  air  assumed  at  527.04  grains. 


- 100  = 590.04  lbs. 


527.04  — (527-04  X .07)  36-s9  X z6-63  X -5236  . 

7000 

To  Compute  Diameter  of  a Balloon. 

Weight  to  be  raised  being  given.—  By  inversion  of  preceding  rule. 

3/VV  X 7000  ~t~  0 ~~~  * ' s an(j  s'  representing  weight  of  air  and  gas 

.^5236 

in  grains  per  cube  foot,  Wr  weight  to  be  raised  in  lbs .,  and  d diameter  of  bal- 
loon in  feet. 

Illustration.—  Given  elements  in  preceding  case. 


Then 


'590.04 -b  100  X 7000^-  527- °4  ~ 36-89 


— 3/2 

.5236  V -5236 

Proof  of  Spirit  moms  Liquors. 


/Q854.69  _ 


; 26 .6  feet. 


A cube  inch  of  Proof  Spirits  weighs  234  grains ; then,  if  an  immersed 
cube  inch  of  any  heavy  body  weighs  234  grains  less  in  spirits  than  air,  it 
shows  that  the  spirit  in  "which  it  was  weighed  is  Proof. 

If  it  lose  less  of  its  weight,  the  spirit  is  above  proof ; and  if  it  lose  more, 
it  is  below  proof. 


Illustration.— A cube  inch  of  glass  weighing  700  grains  weighs  500  grains  when 
weighed  in  a certain  spirit;  what  is  the  proof  of  it? 


700  — 500  ==  200  ==  grains  = weight  lost  in  spiidt. 

: 234  ; : 1 : 1. 17  = ratio  of  proof  of  spirits  compared  to  proof  spirits , or 


Then  200  : 

1 = .17  above  proof 

Note.— For  Hydrometers  and  Rules  for  ascertaining  Proof  of  Spirits,  see  page 
67;  and  for  a very  full  treatise  on  Specific  Gravities  and  on  Floatation,  see  Jamie- 
son’s Mechanics  of  Fluids.  Lond.,  1837. 


SLriifkage  of  Castings.  i 

It  is  customary,  in  making  of  patterns  for  castings,  to  allow  for  shrinkage  . 
per  lineal  foot  of  pattern  as  follows : 

Iron,  small  cylinders  in.  per  ft. 

u Pipes =K. 

“ Girders,  beams,  etc.  = % in  15  ins. 

“ Large  cylinders, 

the  contraction  > —X<s  Per  f°ot- 


of  diam.at  top.J 
Ditto  at  bottom  . . = yg- 


Ditto  in  length = % in  16  ins.  , 

Brass,  thin =3^  in  9 ins.  1 

“ thick = % in  10  ms. 

Zinc in  a foot. 

Lead  . . “ 

Copper — %,  1 

Bismuth = A W 


GEOMETRY. 


219 


GEOMETRY. 

33  eflnitioias. 

Point  has  position,  but  not  magnitude. 

Line  is  length  without  breadth,  and  is  either  Right , Curved , or  Mixed. 
Right  Line  is  shortest  distance  between  two  points. 

Curved  Line-  is  one  that  continually  changes  its  direction. 

Mixed  Line  is  composed  of  a right  and  a curved  line. 

Superficies  has  length  and  breadth  only,  and  is  plane  or  curved. 

Solid  has  length,  breadth,  and  thickness,  or  depth. 

Angle  is  opening  of  two  lines  having  different  directions,  and  is  either 
Right , A cute , or  Obtuse. 

Right  A ngle  is  made  by  a line  perpendicular  to  another  falling  upon  it. 
Acute  Angle  is  less  than  a right  angle. 

Obtuse  A ngle  is  greater  than  a right  angle. 

Triangle  is  a figure  of  three  sides. 

Equilateral  Triangle  has  all  its  sides  equal. 

Isosceles  Triangle  has  two  of  its  sides  equal. 

Scalene  Triangle  has  all  its  sides  unequal. 

Right-angled  Triangle  has  one  right  angle. 

Obtuse-angled  Triangle  has  one  obtuse  angle. 

Acute-angled  Triangle  has  all  its  angles  acute. 

Oblique-angled  Triangle  has  no  right  angle. 

Quadrangle  or  Quadrilateral  is  a figure  of  four  sides,  and  has  following 
particular  designations— viz., 

Parallelogram , having  its  opposite  sides  parallel. 

Square , having  length  and  breadth  equal. 

Rectangle , a parallelogram  having  a right  angle. 

Rhombus  or  Lozenge , having  equal  sides,  but  its  angles  not  right  angles. 
Rhomboid , a parallelogram,  its  angles  not  being  right  angles. 

Trapezium , having  unequal  sides. 

Trapezoid , having  only  one  pair  of  opposite  sides  parallel. 

Note. — Triangle  is  sometimes  termed  a Trigon , and  a Square  a Tetragon. 

Gnomon  is  space  included  between  the  lines  forming  two  similar  parallelo- 
grams, of  which  smaller  is  inscribed  within  larger,  so  as  to  have  one  angle 
in  each  common  to  both. 


Polygons  are  plane  figures  having  more  than  four  sides,  and  are  either 
Regular  or  Irregular , according  as  their  sides  and  angles  are  equal  or  un- 
equal, and  they  are  named  from  number  of  their  sides  or  angles.  Thus : 


Pentagon  has  five  sides. 
Hexagon  “ six  “ 
Heptagon  “ seven  “ 
Octagon  u eight  “ 


Nonagon  has  nine  sides. 
Decagon  “ ten  “ 
Undecagon  “ eleven  “ 
Dodecagon  “ twelve  “ 


Circle  is  a plane  figure  bounded  by  a curved  line,  termed  Circumference 
or  Periphery. 

Diameter  is  a right  line  passing  through  centre  of  a circle  or  sphere,  and 
terminated  at  each  end  by  periphery  or  surface. 

A rc  is  any  part  of  circumference  of  a circle. 

Chord  is  a right  line  joining  extremities  of  an  arc. 

Segment  of  a circle  is  any  part  bounded  by  an  arc  and  its  chord. 

Radius  of  a circle  is  a line  drawn  from  centre  to  circumference. 

Sector  is  any  part  of  a circle  bounded  by  an  arc  and  its  two  radii. 
Semicircle  is  half  a circle. 


Quadrant  is  a quarter  of  a circle. 

Zone  is  a part  of  a circle  included  between  two  parallel  cords. 

Lune  is  space  between  the  intersecting  arcs  of  two  eccentric  circles. 


220 


GEOMETRY. 


C^Lt0of“iTs  ^rSnffrom  one  extremity  of  an  arc  perpendieu- 
lar  to  a diameter  passing  through  other  extremity,  and  sine  of  an  angle  os 

"^Lt^'^ofrmn/afglTrpart  of  diameter  intercepted  between  sine 

aUCoZ'e  of  an  are  or  angle  is  part  of  diameter  intercepted  between  sine  and 

Cmclversed  Sine  of  an  arc  or  angle  is  part  of  secondary  radius  intercepted 

hpfween  cosine  and  circumference.  . , , ...  „ 

b Tangent  is  a right  line  that  touches  a circle  without  cutting  it. 

Cotangent  is  tangent  of  complement  of  arc. 

“CLSTa;  angle  is  what  remains  after  subtracting  angle  from  90 

^Supplement  of  an  angle  is  what  remains  after  subtracting  angle  from  180 
degrees. 

To  exemplify  these  definitions,  let  Ac  6,  in  following  Figure,  be  an  assumed 
arc  of  a circle  described  with  radius  BA: 

J tr  Ac6,an  Arc  of  circle  A C E D. 

A b,  Chord  of  that  arc. 

B A an  Initial  radius. 

B C,  a Secondary  radius, 
e D d,  a Segment  of  the  circle., 

A B 6,  a Sector. 

A D E,  a Semicircle. 

C B E,  a Quadrant. 

A e d E,  a Zone. 

j a noh,  a Lune.  . 

Bo  Secant  of  arc  A cb ; written  Sec. 
b A;,’ Sine  of  arc  Ac  6;  written  Sin. 

A fc,  Versed  Sine  of  arc  A c b ; written  Yersm. 

B k or  m b,  Cosine  of  arc  A cb. 

A a.  Tangent  of  arc  A c b. 

CB  b.  Complement,  and  b B E,  Supplement  of 
arc  A cb.  _ 

. 

Vertex  of  a figure  is  its  top  or  upper  point.  In  Conic  Sections  it  is  point 
SaX  iC'perpendChr  fetTu  Cm  its  vertex 

Segment  is  a part  cut  off  by  a plane,  parallel  to  bas  . 

Frustum  is  the  part  remaining  after  segment  is  cut  o . 

Perimeter  of  a figure  is  the  sum  of  all  its  sides. 

Problem  is  something  proposed  to  be  done. 

Postulate  is  something  required.  , 

Theorem  is  something  proposed  to  be  e easv# 

Lemma  is  something  premised,  to  render 

Corollary  is  a truth  consequent  upon  a preceding  demonstration. 

Scholium  is  a remark  upon  something  going  before  . 

For  other  definitions  see  Mensuration  of  Surfaces  and  Solids,  and  Conic  Section,. 


GEOMETRY. 


221 


Lengths  of  following  Elements,  Ladins  = 1. 


|.  Angle  45° 

Angle  6o°.  [ 

Angle  450. 

Angle  6o°. 

Sine 

.707  107 

.707  107 

.866  025 

. cr 

Cosecant 

1.414214 

i-i54  7 

Cosine 

Tangent 

Cotangent  . . . 
Chord 

Versed  Sine. . 
Coversed  “ . . 

.292893 
.292  893 
I.4I42I4 

O 

•5 

•133  975 
2 

1 

.765  366 
•785  398 

J-732  °5 
^•577  349 

Secant 

Arc 

1.047  2 

Scales. 


To  Divide  a Line,  as  A B,  with  any  required  Number 
of  Equal  Darts.— Dig.  1. 


o From  A and  B draw  two  parallel  lines, 
A o,  B ?*,  to  an  indefinite  length,  and  upon 
them  point  off  required  number  of  equal 
parts,  as  A i,  2,  3,  4,  and  Bi,  2,  3,4;  join 
oB,4  i,  etc. 

Or,  point  off  on  A 0,  join  0 B,  and  draw 
the  other  lines  parallel  thereto. 


2.  A a 


rEo  Construct  a Diagonal  Scale,  as  A B.— Fig.  3. 


123 

lines  A 


Divide  a line  into  as  many  di- 
]<-  visions  as  there  are  hundreds  of 
1 feet,  spaces  of  ten  feet,  feet,  or 
inches  required. 

Draw  perpendiculars  from  each 
division  to  a parallel  line,  C D. 
Divide  them  and  one  of  divisions, 
A E,  C F,  into  spaces  of  ten  if  for 
y feet  and  hundredths,  and  twelve 
. - , . . „ 1 if  for  feet  and  inches:  draw  the 

^ T’ a.  2’  ^ 3?  et,C-*  anc*  w>h  complete  scale. 
w/mf ’’ B rePreseilti»g  ten  feet;  A to  E,  E to  G,  etc.,  will  measure  one 
1001.  a to  a,  c to  1,  1 to  2,  etc.,  will  measure  x-ioth  of  a foot.  The  several  lines 

4easUre’„nnn;T'/‘  ?ea^r?  .UP°“  linos- k,  l,  etc.,  wooth  of  a foot;  and  op  will 
measure  upon  7c,  7,  etc.,  divisions  of  i-ioth  of  a foot. 


Lines. 

To  Draw  a Perpendicular  to  a Light  Line, 

3.  as  or>  3,  c A,  Dig.  4,  or 

from  a Point  external  to 
it,  as  A,  Dig.  £>,  and.  from 
any  two  Points,  as  c d, 
Dig.  0. 

With  any  radius  as  r A,  r B,  cut  line 
at  A and  B ; then  with  a longer  radius, 
as  A 0,  B 0 , describe  arcs  cutting  each 
~l  other  at  0 , and  connect  0 r.  ( Fig.  3. ) 

B Or,  from  A,  set  off  A B equal  to  3 B 
parts  by  scale;  from  A B,  with  radii  c 
of  4 and  5 parts,  describe  arcs  cut- 
ting  at  c,  and  connect  c A.  (Fig.  4. ) 

Note. — This  method  is  useful 
where  straight  edges  are  inappli- 
cable. Any  multiples  of  numbers 
3,  4,  5 may  be  taken  with  same  ef- 

foct,  as  6,  8,  10,  or  g,  12, 15.  ^ 

From  A,  with  a sufficient  radius,  c 
cut  linq  at  o c,  and  from  them  de- 
scribe arcs  cuttiug  at  r, and  connect 
Ar.  (Fig.  5.) 

From  any  two  points,  as  c d,  at  a proper 
distance  anart  dcsrriho  tn-ro  » t. 


222 


GEOMETRY. 


To  Bisect  a Right  Line  or  an  Arc  of  a 
Circle,  and.  to  Draw  a Perpendicu- 
lar to  a Circular  or  Right  Line,  or  a 
Radial  Arc.-F ig*  *7 . 

From  A B as  centres  describe  arcs  cutting  each  other 
at  c and  d,  connect  c d,  and  line  and  arc  are  bisected 
at  e and  o. 

Line  c d is  also  perpendicular  to  a right  line  as  A B, 
and  radial  to  a circular  arc  as  A o B. 

To  Draw  a Line  Bar  all  el  to  a Given 
Right  Line,  as  c d,  Rig.  S. 

From  A B describe  arcs  Ac,  A d,  and  draw  a line  par- 
allel thereto,  touching  arcs  c and  d. 


Angles. 

To  Describe  Angles  of  ^30°  60°,  Rig 


9,  and  45°, 


From  A,  with  any  radius,  A o,  de- 
scribe o r,  and  from  o with  a like  ra- 
dius cut  it  at  r,  let  fall  perpendicular 
rs;  then  o Ar  = 6o°,  and  Ars  = 3o°. 

(Fig.  9 ) 

Set  off  any  distance,  as  A B,  erect 
perpendicular  Ao  = AB,  and  connect 
o B.  (Fig.  10.) 


To  Bisect  Inclination  of  Two  Lines, 
when  Point  of  Intersection  is  Inac- 
cessible.—Rig-  D« 

Upon  given  lines,  A B,  C D,  at  any  points  draw  perpen- 
diculars e o,  s r,  of  equal  lengths,  and  from  ojnd  sdraw 
parallels  to  their  respective  lines,  cutting  at  n,  bisect 
angle  o n s,  connect  n m,  and  line  will  bisect  lines  as  re- 
quired. 


Rectilineal  Figures. 

To  Describe  an  Octagon  upon  a Line,  as  A B.-Ri 


. is. 


JClUgOXi  -V  — 3 — - - 

From  points  A B erect  indefinite  perpendiculars  A/, Be; 
produce  A B to  m and  n,  and  bisect  angles  mAo  and  nap 
with  A u and  B r. 

Make  A u and  B r equal  to  A B,  and  draw  u z,  r v parallel 
to  A/,  and  equal  to  A B. 

From  z and  v,  as  centres,  with  a radius  equal  to  AB,  de- 
. scribe  arcs  cutting  A/,  B e,  in  / and  6.  Connect  zffe, 
and  e v. 

To  Inscribe  any  Regular  Polygon  in  a 

Circle,  or  to  Divide  Circumference  into 

a given  Number  of  Eqtial  Parts.— b i„-  13. 
If  Circle  is  to  contain  a Heptagon.  - Draw  angle  A . B at 
centre  o for  360° -7  = 5.°  4='  5i"+..or  5'f-,thef  “*  ofl  upon 
circumference  distance  A B or  remaining  angles  Ao  B. 


GEOMETRY. 


223 


To  Inscribe  a Hexagon  in. 
a Circie.— Fig-.  14. 


Draw  a diam- 
eter, A oB.  From 
A and  B as  cen- 
tres, with  A 0 and 
B o,  cut  circle  at 
cm  and  eny  and 
connect. 


m>' --'n 


To  Describe  a Hexagon 
about  a Circle.—  Fig.  IS. 


Draw  a diam- 
eter as  a 0 b ; and 
with  ao  cut  circle 
ate;  join  ac,and 
bisect  it  with  ra- 
dius o r,  through 
r draw  e r paral- 
lel to  c a,  cutting 
diameter  at  m ; 
then  with  radius 


0 m describe  circle,  within  which  describe 
a hexagon  as  above. 


Cr— 


To  Inscribe  a Pentagon  in 

a Circle.— Fig.  10. 

Draw  diameters 
A c and  m n,  at 
right  angles  to 
each  other;  bisect 
0 n in  r,  and  with 
r A describe  As; 
from  A with  A 5 
describe  5 B. 

Connect  AB,  and 
distance  is  equal  to 

one  side  of  a pentagon. 

To  Describe  a Pentagon 
upon  a Dine,  as  A B.— Fig. 


Draw  B m per- 
pendicular to  A B, 
and  equal  to  one 
half  of  it;  extend 
A m until  m n is 
equal  to  B m. 

From  A and  B, 
with  radius  Bn,  de- 

jp-w  scribe  arcs  cutting 

" 7 each  other  in  o; 

then  from  0,  with  radius  o B,  describe 
circle  A C B,  and  line  A B is  equal  to  one 
side  of  a pentagon  upon  circle  described. 
To  Describe  a Regular  Polygon  of  any  required  Number 
of  Sides.— Fig.  18. 

From  point  o,  with  distance  0 B,  describe  semicircle 
B b A,  which  divide  into  as  many  equal  parts,  Aa,ab,bc, 
etc.,  as  the  polygon  is  to  have  sides. 

Thus,  let  a Hexagon  be  required: 

From  0 to  second  point  b of  six  divisions  draw  0 b , and 
through  other  points,  c,  d,  and  e,  draw  oC,oD,  etc. 

L VS  Apply  distance  0 B,  from  B to  E,  from  E to  D,  from  D to 

A o "" -'B  C,  etc.  Join  these  points,  as  b C,  C D,  etc. 

To  Construct  a Hexagon 


To  Construct  a Square  or 
a Rectangle  on  a given 
Dine.— Fig.  19. 


19.  m;< 


Xn 


A 

B n,  and  join  0 r. 


On  A B as  cen- 
tres, with  A B as 
radius,  describe 
arcs  cutting  at 
■'IF  c;  on  c describe 
arcs  cutting  at 
o r;  and  on  o r 
describe  others, 
cutting  at  ran; 
draw  A m and 


a given  Dine.— Fig 


From  ends  of  line, 
A B,  describe  arcs 
cutting  each  other 
at  o,  and  from  o as 
a centre,  with  radius 
o A,  describe  a cir- 
cle, and  with  same 
radius  set  off  A c, 
cd,  Bffe , and  con- 
nect them. 


Inscribe  an  Octagon  in  a Circle.— Fig.  21 
Draw  diameters,  A C,  B D,  at  right 
angles,  bisect  arcs,  A B,  B C,  etc. , at  s,  r, 
o,  e,  and  join  Ao,  0 B,  etc.  (Fig.  21.) 

To  Describe  an  Octagon, 
about  a Circle.— Fig.  22. 

Describe  a square  about  circle  A B, 
draw  diagonals  c/,  e d , draw  0 i,  etc., 
perpendicular  to  diagonals  and  touch- 
insr  circle.  fFig.  22.) 


224 


GEOMETRY. 


To  Inscribe  a Square  in  a Circle.— Fig.  S3. 

Draw  line  A B through  centre  of  circle ; c 

9 take  any  radius,  as  A e,  and  describe  the 

arcs  Aee,  Bee;  connect  ee,  continuing 
line  to  C and  D ; join  AC,  AD, etc.  (Fig. 23.) 

To  Describe  a Square  about  A - 
a Circle.— Fig.  24. 

Draw  line  A B through  centre  of  circle. 

Take  any  radius,  as  A e;  describe  arcs 
Aee,  Bee;  connect  ee,  continuing  line 
to  C D. 

Describe  B r and  D r ; draw  and  extend  B r and  D r,  and  sides  A and  C parallel  to 
them.  (Fig.  24.) 


(lig.  28.)  Tq  describe  a Circular  Segment  tliat 

A\  * ^ /E  will  Lotli  fill  tlie  angle  between  two 

diverging  lines  and.  touch  tnem. 
Fig.  29. 

Bisect  inclined  lines,  A B,  D E.  by  line  e f and 
perpendicular  thereto,  B D,  to  define  nda ry  ° f seg- 

ment to  be  described.  Bisect  angles  at  B and  D by  lines 
cutting  at  0,  and  from  0,  with  radius  0 ^describe  arc 
•gVl-  _>q)  men. 

To  Draw  a Series  of  Circles  between  Two  Incnned 
Lines,  touching  them  and  each  othei . * 

qa  Bisect  given  lines  A B,  C D,  by  line  oc. 

* - — B From  a point  r in  this  line  erect  f $ perpen- 

dicular to  A B.  and  on  r describe  circle  sm, 
cutting  centre  line  at  n;  from  u erect  u n 
perpendicular  to  centre  line,  cutting  A B at 
n and  from  n describe  an  arc  n u v.  cutting 
A B at  i\  erect  x v parallel  to  r s , making  x 
-D  centre  of  next  circle  to  be  described,  with 

roHinci  nr.  11  HBfl  SO  Oil. 


Note.— Largest  circle  may  be  described  first. 


GEOMETRY. 


225 

To  Describe  a Circle  that  shall  pass  through  any  three 
given  Points,  as  A B C.— Digs.  31  and  32. 

Upon  points  A and  B, 
with  any  opening  of  a 
dividers,  describe  arcs 
cutting  each  other  at  ee. 

On  points  B C describe 
two  more  cutting  each 
other  in  points  c c. 

Draw  lines  ee  and  cc, 
and  intersection  of  these 
lines,  o , is  centre  of  circle 

ABC.  (Fig.  31.)  _ 

£CAtre  not  attainable.  — From  A B as  centres,  describe  arcs  A <7  B ft- 
B C"  6 and  B c into  any  numker  of  equal  parts,  also 

eg  and  B h into  a like  number.  Draw  A 1,  2,  3,  etc.,  and  B 1,  2,  etc  and  intersec 
tion  of  these  lines  as  at  o are  points  in  the  circle  required.  (Fig.  32./ 

Or,  let  A B C be  given  points,  connect 
A B,  AC,  C B,  and  draw  e c parallel  to  A B. 
Divide  C A into  a number  of  equal  parts' 
as  at  1,  2,  and  3,  and  from  C describe  arcs 
through  these  points  to  meet  right  lines 
from  C to  points  1,  2,  and  3,  or  A e,  and 

before  directed.  (Fig.  33.)  these  are  points  in  a circle,  to  be  drawn  as 

■Dl:aw  a Tangent  to  a To  Draw  Tangents  to  a 
Circle  from  a given  Point  Circle  from  a Point  with- 
in Circumference.  - Fig.  ont  it.— Dig.  35. 

34.  e 

34,  £ ^ > 35. 


, , . . . . ..  , , From  A draw  A o,  and  bisect  it  at  s: 

through  point  A draw  radial  line  A o,  describe  arc  through  o,  cutting  circle  at 
and  erect  perpendicular  ef  (Fig.  34.)  m n\  join  A m or  A n. 

To  Draw  from  or  to  Circumference  of  a Circle,  Lines 
leading  to  an  Inaccessible  Centre.— Fig.  36. 

•>/»  /r  v,  ^ Divide  whole  or  any  given  portion  of 

% { T Jfr  circumference  into  desired  number  of 

! I i parts;  then,  with  any  radius  less  than 

distance  of  two  divisions,  describe  arcs 
/'N  cutting  each  other,  as  A r,  b r,  c r,  d r, 
etc.;  draw  lines  b r,  c r,  etc.,  and  they 
will  lead  to  centre. 

To  draw  end  lines , as  A r,  F r.  From  b describe  arc  o,  and  with  radius  b i from 
A or  F as  centres,  cut  arcs  A r,  etc.,  and  lines  A r,  F r,  will  lead  to  centre.  ’ 

To  Describe  an  Arc,  or  Segment  of  a Circle,  of  a large 
a7  " —■  Radius.— Fig.  3 *7 . 

Draw  chord  AcB;  also  line  h D i 
parallel  with  chord,  and  at  a distance 
equal  to  height  of  segment;  bisect 
chord  in  c,  and  erect  perpendicular 

fnm  anrmimh0  AfD’  B ?’  TCt  als°  PeTendicSai°A n,  B n ^di  vhhT A B and  h i 
? a.^  r\ut^fber  e(lual  parts;  draw  lines  i i,  2 2,  etc.,  and  divide  lines  A n,  B n 
n°^fr^er  °f,equA  Parts  in  A B;  draw  lines  to  D from  each  division  in 
lines  A w,  B n,  and  at  points  of  intersection  with  former  lines  describe  arc  or  segment. 


226 


GEOMETRY. 


Ellipse. 

To  Describe  an  Ellipse  to  any  Length  and  Breadth 
given.— Eig.  38. 

38.  E Let  longest  diameter  be  C D,  and  shortest  EF.  Take 

distance  C o or  o D,  and  with  it,  from  points  E and  F, 
describe  arcs  h and  / upon  diameter  C D. 

nl  — TT] /.Jn  Insert  pins  at  h and  at  / and  loop  a string  around 

'' ' ~ 1 * 1 them  of  such  a length  that  when  a pencil  is  introduced 

within  it  it  will  just  reach  to  E or  F.  Bear  upon 
string,  sweep  it  around  centre  o,  and  it  will  describe 
ellipse. 

Note. — It  is  a property  of  Ellipse  that  sum  of  two  lines  drawn  from  foci  to  meet  in  any  point  in 
curve  is  equal  to  transverse  diameter. 

Bisect  transverse  axis  A B at  o,  and  on  centre  o 
erect  perpendicular  C D,  making  o D and  o C each 
equal  to  half  conjugate  axis.  From  C or  D,  with 
radius  A o,  cut  transverse  axis  at  s s for  foci.  Divide 
A o into  any  number  of  equal  parts,  as  i,  2,  3,  etc. 
With  radii  A 1,  B 1,  on  s and  s as  centres,  describe 
arcs,  and  repeat  this  operation  for  all  other  divis- 
ions 1,  2,  3,  etc.,  and  these  points  of  intersection  will 
give  line  of  curve. 

and.  Two  Diameters  of  an  Ellipse. 
—Eig.  40. 

Let  A B e wbe  diameters  of  an  Ellipse. 

Draw  at  pleasure  two  lines,  q q,  0 ?n,  parallel  to 
each  other,  and  equidistant  from  A and  B;  bisect 
them  in  points  h «,  and  draw  line  ur ; bisect  it 
' in  s,  and  upon  s,  as  a centre,  describe  a circle  at 
pleasure,  as  fl  v,  cutting  figure  in  points/ v. 

Draw  right  line  fv;  bisect  it  in  »,  and  through 
points  i s draw  greatest  diameter  A B,  and  through 
centre,  s,  draw  least  diameter  c w,  parallel  to  fv. 

To  Describe  an  Ellipse  approximately  DyCireular  Arcs. 

Set  off  differences  of  axes  from  centre  0 to  a and 
c or  0 A and  0 C ; draw  a c and  bisect  it,  and  set  ofT 
its  half  to  r : draw  r s parallel  to  a c,  set  off  0 n 
equal  to  0 r,  connect  n s , and  draw  parallels  r m, 

• nm:  from  m,  with  radii  m s and  s m,  describe  arcs 
through  C and  D,  and  from  n and  r describe  arcs 
through  A and  B. 

Note  —This  method  is  not  satisfactory  when  con- 
jugate axis  is  less  than  two  thirds  of  transverse  axis. 

"Witli  Arcs  of  Three  Radii-— Eig. 


/ 

ry\  \ 

T n\ 

6 1 a /r  J 

\S/ 

42. 


1 


42. 

On  transverse  axis  A B draw  rectangle  A B c d, 
on  height  0 e\  to  diagonal  A e draw  perpendicular 
dh  0 • set  off  0 r equal  to  0 e,  describe  a semicircle 
on  A V,  and  produce  O e to  l ; set  off  0 m equal  to 
el  and  on  0 describe  an  arc  with  radius  Owi;  on 
A ’with  radius  0 1,  cut  this  arc  at  a.  Thus  the  five 
centres,  0,  a,  a',  fc,  h\  are  found,  from  which  arcs 
are  described  to  form  ellipse. 

Note  —This  process  answers  for  nearly  all  pro- 
portions of  ellipses.  It  is  used  in  striking  vaults, 
stone  bridges,  etc. 


GEOMETRY. 


227 


To  Construct  an  Ellipse  from  Two 
Circles.— Eig.  43. 

Describe  two  semicircles,  as  A B,  C D,  diameters  of 
which  are  respectively  lengths  of  major  and  minor 
axes.  The  intersection  of  the  horizontal  and  vertical 
lines  drawn  from  any  radial  line  will  give  a point  in 
the  curve  G D. 


To  Construct  an  Ellipse,  wTieii  Two 
Diameters  are  Given.-Pig.  44. 

Make  c o and  A v equal  to  each  other,  but  less 
than  half  breadth.  Draw  vo,  and  from  its  centre  i 
draw  and  extend  perpendicular  at  i to  d,  draw  d v m, 
make  B?«  = Av,  draw  du  r,  from  u and  v describe 
B r and  A m,  from  d describe  m c r,  extend  c z to  s, 
and  it  will  be  centre  for  other  half  of  figure. 


To  Construct  an 


Ellipse  by  Ordinates.— Fig.  45. 

Divide  semi-transverse  axis,  as  A 6,  into  8 or  10 
divisions,  as  may  be  convenient,  and  erect  ordi- 
nates, the  lengths  of  which  are  equal  to  semi-con- 
jugate, multiplied  by  the  units  for  each  division  as 
follows: 


1 — .484  12 

2 — .661  44 

3 — .78063 

4 — . 866  03 


5 — 

6 — 


•92 7 03 

.968  24 
.992  16 


Divisions. 

1 — .45389 

2 — .6 

3 — .714  14 

4- . 8 


Tenths. 

5 — .86602  9 — *99499 

6 — .91651  10  — 1 

7 — -993  94 

8 — .999  79 


To  Construct  an  Ellipse  when  Diameters  do  not  Inter- 
sect at  Right  Angles.- Eig.  46. 

Let  A B and  C D be  given  diameters. 

Draw  boundary  lines  parallel  to  diameters, 
divide  longest  diameter  into  any  number  of 
equal  parts,  and  divide  shortest  boundary  lines 
into  same  number  of  equal  parts. 

From  one  end  of  shortest  diameter,  D,  draw 
radial  lines  through  divisions  of  longest  diame- 
ter, and  from  opposite  end,  C,  draw  radial  lines 
to  divisions  on  shortest  boundary  lines  ; the 
intersection  of  these  lines  will  give  points  in  the 
curve. 


Arcs. 

To  Describe  a,  Grothic  Arc.- Eig.  4I7. 

Take  line  A B.  At  points  A and  B draw  arcs  B a and  A c, 
and  it  will  describe  arc  required. 


To  Describe  an  Elliptic  Arc,  Chord,  and  Height  being 


:\\\ 


given.— Eig.  48. 

Bisect  A B.  at  c ; erect  perpendicular  A q,  and 
draw  line  q D equal  and  parallel  to  A c. 

Bisect  A c and  A q in  r and  n;  make  c 1 equal  to 
c D,  and  draw  line  l r q ; draw  also  line  n s D ; bisect 
B 5 D with  a line  at  right  angles,  and  cutting  line 
c D at  0;  draw  line  o q\  make  cp  equal  to  c lc,  and 
draw  line  op  i. 

Then,  from  0 as  a centre,  with  radius  0 D,  describe 
arc  sDi;  and  from  lc  and  p as  centres,  with  radius 
A lc,  describe  arcs  A s and  B i. 


228 


GEOMETRY. 


To  Describe  a,  Grotliic  Arc.— Figs.  4=0  and.  SO. 

Divide  line  A B into  three  equal  parts,  e c ; from  points 
A and  B let  fall  perpendiculars  A o and  B r,  equal  in  length 
to  two  of  divisions  of  line  A B ; s 50. 

draw  lines  o h and  r g from  points 
. e,  c;  with  length  of  cB,  describe  arcs  /C  *\ 

Ag  and  B h,  and  from  points  o and  r f \ / \ 

describe  arcs  g i and  i h.  (Fig.  49.)  j^L -}r ic 13 

Or,  divide  line  A B into  three  a \ / 

equal  parts  at  a and  &,  and  on  points  V 

A,  a,  &,  and  B,  with  distance  of  two  / \ 

divisions,  make  four  arcs  intersect-  ^ V 

6 r ing  at  c and  0.  c> 

Through  points  c,  0,  and  divisions  a,  6,  draw  lines  cf  and  0 e , on  points  a and  b 
describe  arcs  A e and  B^,  and  on  points  c 0 arcs  f s and  e s.  (I  ig.  5°-) 

Cycloid,  and  Epicycloid. 

To  Describe  a Cycloid..— Fig.  SI. 

When  a circle,  as  a wheel,  rolls  over  a 
straight  right  line,  beginning  as  at  A and 
ending  at  B,  it  completes  one  revolution, 
and  measures  a straight  line,  A B,  exactly 
equal  to  circumference  of  circle  c e r,  which 
is  termed  the  generating  circle , and  a point 
or  pencil  fixed  at  point  r in  circumference 
traces  out  a curvilinear  path,  ArB,  termed  a 
cycloid.  A B is  its  base  and  c r its  axis. 
Place  generating  circle  in  middle  of  Cy- 
■p  cloid,  as  in  figure;  draw  a line,  m n,  paral- 
lei  to  base,  cutting  circle  at  e;  and  tangent 
The  following  are  some  of  properties  of  Cycloid : 

Or,  whole  arc  of  Cycloid  ArB  = four 
times  axis  c r. 

Area  of  Cycloid  A r B A = three  times 
area  of  generating  circle  r c. 

Tangent  n i is  parallel  to  chord  e r. 


n i to  curve  at  point  n. 

Horizontal  line  e n = arc  of  circle  e r. 
Half-base  A c=half-circumference  cer. 
Arc  of  Cycloid  rn  = twice  chord  r e. 
Half  arc  of  cycloid  Ar=twice  diameter 
of  circle  r c. 


To  Describe  Curve  of  a Cycloid.—  IFi 


52. 


On  an  indefinite  line,  A B,  set  olf  co== 
circumference  of  generating  circle,  di- 
vide this  line  into  any  number  of  equal 
parts  (8  in  figure),  and  at  points  of  divis- 
ion erect  perpendiculars  thereto.  Upon 
p each  of  these  lines  describe  a circle  — 

— - x ~ ~ generating  circle.  On  c 1 take  1 x — 

.2<  0 1,  and  with  a;  as  a centre,  with  radius  x c = .75  c x,  describe  an  arc  cutting  circle 
at  1';  from  2 on  next  circle,  with  two  distances  of  1 1', ^measured  as  chords  cut 
circle  at  2'  • from  3 on  next  circle,  with  three  distances  of  1 1 , cut  circle  at  3 , and 
proceed  in  like  manner  from  each  side  until  figure  is  complete. 

To  Describe  an  Interior  Epicycloid  or  Hypocycloid.— 
Eig.  S3. 

If  generating  circle  is  rolled  on  inside  of  fundamental 
circle  *as  in  Fig.  53,  it  forms  an  interior  epicycloid , or  % 
1 hypocycloid , A c B,  which  becomes  in  this  case  nearly  a < 
J*  straight  line.  Other  points  of  reference  in  figure  cor- 
respond  to  those  in  Fig.  51.  When  diameter  of  generat- 
ing circle  is  equal  to  half  that  of  fundamental  circle^ 
epicycloid  becomes  a straight  line,  being  diameter  of 
the  larger  circle. 


* See  explanation,  Fig.  54. 


GEOMETRY. 


229 


To  Describe  an  Exterior  Epicycloid.— 
Eig.  54. 

An  Epicycloid  differs  from  a Cycloid  in  this,  that  it  is 
generated  by  a point,  o'",  in  one  circle,  o r,  rolling  upon 
circumference  of  another,  A r s,  instead  of  upon  a right 
line  or  horizontal  surface,  former  being  generating  circle 
and  latter  fundamental  circle. 

Generating  circle  is  shown  in  four  positions,  in  which 
its  generating  point  is  indicated  by  o o'  0"  o'".  A o'"  s 
is  an  Epicycloid. 


Involute. 

To  Describe  an  Involute.-Fig.  55. 

Assume  A as  centre  of  a circle,  b c 0;  a cord  laid  partly 
upon  its  circumference,  as  be-,  then  the  curve  eimn, 
described  by  a tracer  at  end  of  cord,  when  unwound  from 
a circle,  is  an  involute. 

This  curve  can  also  be  defined  by  a batten,  x,  rolling  on 
a circle,  as  s u. 


3?araT>ola. 


geometry. 


To  Describe  Curve  of*  a Parabola,  Base  and. 
Height  "being  given.— Big.  60. 

Draw  an  isosceles  triangle,  as  a b d,  base  of  which  shall  be  equal 
to  and  its  height,  c b , twice  that  of  proposed  parabola.  Divide 
each  side,  a 6,  d b,  into  any  number  of  equal  parts ; then  draw  lines 
i i,  2 2,  3 3,  etc.,  and  their  intersection  will  define  curve.  (Fig.  6o.) 

To  Describe  a Parabola,  any  Ordinate  to  -A.xis 
and  its  Abscissa  being  given.— Big.  61- 

Bisect  ordinate,  as  A o in  r;  join  m c n 

B r,  and  draw  r s perpendicular  to  it, 
meeting  axis  continued  to  s.  Set  off 
- ’ Bc,Be,  each  equal  to  os;  draw  m 

c u perpendicular  to  B s,  then  m u is  directrix  and 
B e focus;  through  e and  any  number  of  points,  i,  i, 
i etc.,  in  axis,  draw  double  ordinates  v i v,  and  on 
centre  e,  with  radii  e c,  i c,  etc.,  cut  respective  or- 
dinates at  v v , etc.,  and  trace  curve  through  these 
points.  ► 

Note.— Line  vev  passing  through  focus  is  parameter.  j's 

Spiral.  * 

To  Draw  a Spiral  about  a given  Point.— 
Big.  62. 


Assume  c the  centre.  Draw  A Ji,  divide  it  into  twice  number 
of  parts  that  there  are  to  be  revolutions  of  line.  Upon  c de- 
scribe re,  os,  A h,  and  upon  e describe  rs,os,  etc. 


Hyperbola. 

To  Describe  a Hyperbola,  Transverse  and  Conjugate 
Diameters  being  given.  Big.  63. 

Let  A B represent  transverse  diameter,  and  C D 

C°DrawC  e parallel  to  A B,  and  e r parallel  to  C D; 
draw  o e,  and  with  radius  o e,  with  o as  a centre, 
describe  circle  F e r,  cutting  transverse  axis  pro- 
duced in  F and  /;  then  will  F and  ./be  foci  of  fig- 

In  o B produced  take  any  number  of  points,  n,  n, 
etc.,  and  from  F and /as  centres,  with  A n and  B n 
as  radii,  describe  arcs  cutting  each  other  in  s,  s, 
etc.  Through  s,  s,  etc. , draw  curve  ssssBssss. 


Note.— If  straight  lines,  as  o e y and  o r y,  are  drawn  from  ^ 

tremities  e r,  they  will  be  asymptotes  of  hyperbola,  property  of  which  is  to  ap- 
proach continually  to  curve,  and  yet  never  to  touch  it. 

When  Foci  and  Conjugate  Axis  are  given.—  Let  F and /be  foci,  and  C D conjugate 

a?hroVghPcTaw|  ^parallel  to  F and  /;  then,  Was  a centre  and  oF  as  a 
radius,  describe  an  arc  cutting  g C e at  g and  e;  from  these ^points ^et  faU  Perpen- 
diculars upon  line  connecting  F and  / and  part  intercepted  between  them,  as  A B, 
will  be  transverse  axis. 

Catenary. 

To  Delineate  a Catenary,  Span  and.  Versed  Sine  being 
given. — Big.  64.  (IF.  Hildenbrand.) 

Divide  half  span,  as  A B,  into  any  required 
number  of  equal  parts,  as  i,  2,  3,  and  let  fall  BG 
and  A 0,  each  equal  to  versed  sine  of  curve ; divide 
A o into  like  number  of  parts,  1',  2',  3 as  A B. 
Connect  Ci',C  2',  and  0 3',  and  points  of  intersec- 
tion of  perpendiculars  let  fall  from  A B will  give 
points  through  which  curve  is  to  be  drawn. 

A Or,  suspend  a finely  linked  chain  against  a ver- 
tical plane,  trace  curve  from  it  on  the  plane  in  accordance  with  conditions  of  given 
length  and  height,  or  of  given  width  or  length  of  arc. 


UUgLH  ilUU  UCIgui,  v.  — - v 

Note.— For  other  methods  see  D.  R.  Clark’s  Manual,  pp.  18, 19. 


AREAS  OF  CIRCLES.  23  I 


Areas  of  Circles,  from  St  to  150. 


Diam. 

| Area. 

Diam. 

| Area. 

Diam. 

Area. 

1 1 Diam. 

Area. 

St 

.OOO  192 

3 

7.0686 

7 

38.4846 

14 

I53-938 

v 

/16 

| 7.3662 

X 

39-87I3 

Y 

156.7 

/82 

7.6699 

X 

41.2826 

Y± 

I59-485 

^L6 

.003  068 

% 

7.9798 

X 

42.7184 

Ys 

162.296 

% 

.012  272 

X 

8.2958 

Y 

44.1787 

Y 

165.13 

* 

.027612 

Yg 

X 

8.618 

8.9462 

45.6636 

47-1731 

s 

167.99 

I7O.874 

¥ 

.049087 

X 

9.2807 

% 

48.7071 

X 

173.782 

Y 

9.62H 

8 

50.2656 

15 

176.715 

/16 

■% 

9.968 

M 

51.8487 

X 

179.673 

K 

.IIO447 

% 

IO.3206 

3^ 

53-4563 

X 

182.655 

% 

.15033 

% 

IO.679 

X 

55.0884 

X 

185.661 

Si 

II.0447 

Y 

56.7451 

X 

188.692 

X 

•i9635 

% 

II.416 

% 

58.4264 

X 

I9I.748 

% 

.248505 

Ys 

n-7933 

% 

60.1322 

X 

194.828 

% 

12.177 

Ys 

61.8625 

X 

197-933 

78 

4 

12.5664 

9 

63.6174 

16 

201.062 

^6 

.371  224 

X 

12.962 

X 

65.3968 

X 

204.216 

.441  787 

13-3641 

Y 

67.2008 

X 

207.395 

13.772 

% 

69.0293 

X 

210.598 

/16 

.518487 

X 

14.1863 

Y 

70.8823 

X 

213.825 

X 

.601 322 

Xg 

14.606 

H 

72.7599 

X 

217.077 

% 

.690292 

% 

Xg 

15-033 

15-465 

K 

X 

74.6621 

76.5888 

X 

X 

220.354 

223.655 

I 

•7854 

Si 

I5-9043 

10 

78.54 

*7 

226.981 

.8866 

V 

m 

16.349 

X 

80.5158 

X 

230.331 

.99402 

Ys 

16.8002 

X 

82.5161 

X 

233.706 

X& 

1. 1075 

Xg 

17-257 

X 

84.5409 

X 

237.105 

✓£ 

1.2272 

Si 

17.7206 

x: 

86.5903 

X 

240.529 

^6 

I*353 

% 

18.19 

x 

88.6643 

X 

243-977 

X 

1.4849 

% 

18.6655 

x 

90.7628 

X 

247-45 

A& 

1.6229 

% 

19.147 

% 

92.8858 

X 

250.948 

Y 

1.767 1 

5 

I9-635 

11 

95-0334 

18 

254-47 

%& 

I-9I75 

x& 

20.129 

X 

97-2055 

X 

258.016 

M 

2.073*5 

Si 

20.629 

X 

99.4022 

X 

261.587 

x 

2.2365 

Xs 

2i-i35 

X 

101.6234 

X 

265.183 

% 

2.4053 

Si 

21.6476 

X 

103.8691 

X 

268.803 

% 

2.58 

Xg 

22.166 

X 

106.1394 

X 

272.448 

% 

2.761  2 

Ys 

22.6907 

M 

1 08.4343 

X 

276.117 

% 

2.9483 

Xg 

23.221 

X 

1 10.7537 

X 

279.811 

2 

3.1416 

Si 

23-7583 

12 

113.098 

19 

283.529 

/5l6 

3-338 

Xg 

24.301 

X 

115.466 

X 

287.272 

Si 

3-5466 

% 

24.8505 

X 

117.859 

X 

291.04 

X& 

37584 

Xg 

25.406 

X 

120.277 

X 

294.832 

Si 

3.9761 

% 

25-9673 

X 

122. 719 

X 

298.648 

%, 

4.2 

% 

26.535 

X 

125.185 

X 

302.489 

Ys 

4.4301 

% 

27.1086 

X 

127.677 

X 

306.355 

4.7066 

Xg 

27.688 

% 

130. 192 

X 

3IO-245 

1/ 

Si 

4.9087 

6 

28.2744 

13 

132.733 

20 

314.16 

X& 

5-1573 

Ys 

29.4648 

X 

135-297 

X 

318.099 

§ 

5-4ii9 

Y 

30.6797 

X 

137.887 

X 

322.063 

Xg 

£•6723 

Ys 

31.9191 

X 

140.501 

X 

326.051 

Si 

5-9396 

Y 

33-1831 

X 

I43-I39 

X 

330.064 

% 

6.2126 

Ys 

34-4717 

X 

145.802 

X 

334- 102 

% 

6.491 8 

% 

35-7848 

X 

148.49 

X 

338.164 

i % 

6.7772 

1 Ys 

37.1224 

% 

1 5 1. 202 

X 

342.25 

232 


AREAS  OF  CIRCLES. 


Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

21 

346.361 

28 

6I5-754 

35 

962.115 

% 

350.497 

X 

621.264 

X 

969 

% 

354-657 

X 

626.798 

X 

975-909 

% 

358.842 

% 

632.357 

% 

982.842 

X 

363-05I 

X 

637.941 

X 

989.8 

% 

367.285 

% 

643-549 

% 

996.783 

% 

371-543 

% 

649.182 

% 

IOO379 

X 

375.826 

X 

654.84 

% 

1010.822 

22 

380.134 

29 

660.521 

36 

IOI7.878 

X 

384.466 

X 

666.228 

X 

IO24.96 

X 

388.822 

X 

671.959 

X 

IO32.065 

X 

393-203 

X 

677.714 

% 

I°39-I95 

X 

397.609 

X 

683.494 

X 

1046.349 

% 

402.038 

% 

689.299 

% 

1053.528 

% 

406.494 

% 

695.128 

% 

1060.732 

% 

410.973 

X 

700.982 

X 

1067.96 

23 

4I5-477 

30 

706.86 

37 

1075.213 

X 

420.004 

X 

7i2.763 

X 

1082.49 

X 

424-558 

X 

718.69 

X 

1089.792 

X 

429-I35 

% 

724.642 

% 

1097.118 

X 

433-737 

X 

730.618 

X 

1104.469 

% 

438.364 

% 

736.619 

% 

1111.844 

% 

443-015 

% 

742.645 

% 

1 1 19.244 

X 

447.69 

X 

748.695 

X 

1126.669 

24 

452.39 

31 

754-769 

38  . 

1134.118 

X 

457*115 

X 

760.869 

X 

1141.591 

X 

461.864 

X 

766.992 

X 

1149.089 

% 

466.638 

X 

773-14 

X 

1156.612 

X 

471.436 

X 

779-3x3 

X 

1164.159 

% 

476.259 

X 

785-51 

X 

11 7I-73I 

% 

481.107 

% 

79I*732 

% 

1 1 79.327 

% 

485-979 

X 

797-979 

X 

1186.948 

2 5 

490.875 

32 

804.25 

39 

1 194.593 

X 

495.796 

X 

810.545 

X 

1202.263 

X 

500.742 

X 

816.865 

X 

1209.958 

% 

505-712 

% 

823.21 

X 

1217.677 

X 

510.706 

X 

829.579 

X 

1225.42 

% 

5I5-726 

X 

835-972 

X 

1233.188 

% 

520.769 

% 

842.391 

X 

1240.981. 

% 

525-838 

X 

848.833 

X 

1248.798 

26 

530.93 

33 

855.301 

40 

1256.64 

X 

536.048 

861.792 

X 

1264.506 

X 

54i-i9 

X 

868.309 

X 

1272.397 

% 

546.356 

X 

874.85 

% 

1280.312 

X 

551.547 

X 

881.415 

X 

1288.252 

% 

556.763 

X 

888.005 

% 

1296.217 

% 

562.003 

X 

894.62 

% 

1304.206 

% 

567.267 

% 

901.259 

X 

1312. 219 

27 

572.557 

34 

907.922 

41 

1320.257 

X 

577-87 

X 

914.611 

X 

1328.32 

X 

583.209 

X 

921.323 

X 

1336-407 

% 

588.571 

X 

928.061 

X 

I344-5I9 

X 

593-959 

X 

934.822 

X 

1352655 

% 

599-371 

X 

941.609 

X 

1360.816 

% 

604.807 

X 

948.42 

X 

1369.001 

X 

610.268 

X 

955*255 

X 

I377-2II 

42 

X 

X 

% 

X 

X 

% 

X 

43 

X 

X 

X 

X 


% 

44 

X 


X 

X 

4S 

X 

% 


% 

X 


46 


X 


47 


48 


X 


138545 

1393-7 

1401.99 
1410.3 

1418.63 

1426.99 
1435-37 
1443-77 
1452.2 
1460.66 
1469.14 

1477.64 
1486.17 
1494-73 
15033 
I5II-9I 
1520.53 

I529-I9 

1537.86 

1546.56 

1555-29 

1564.04 
1572.81 
1581.61 

I590-43 

1599.28 

1608.16 

1617.05 
1625.97 
1634.92 

1643.89 

1652.89 
1661.91 
1670.95 
1680.02 
1689. 1 1 
1698.23 
1707.37 
i7l6-54 
I725-73 
1734-95 
1744.19 
1753-45 
1762.74 

1772.06 
1781.4 
1790.76 
1800.15 
1809.56 
1819 

1828.46 

I837-95 

1847.46 
1856.99 
1866.55 
1876.14 


AREAS  OF  CIRCLES. 


233 


Diam. 

Area. 

| Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

49 

1885.75 

56 

2463.OI 

63 

3II7.25 

7° 

3848.46 

A 

1895.38 

y8 

2474.02 

A 

3129.64 

A 

3862.22 

A 

1905.04 

A 

2485.05 

A 

3142.04 

A 

3876 

A 

1914.72 

% 

2496. 1 1 

A 

3154.47 

a 

3889.8 

A 

1924.43 

A* 

2507.19 

A 

3166.93 

A 

3903.63 

% 

1934.16 

A 

2518.3 

& 

3I79-4I 

A. 

39I7-49 

A 

1943.91 

% 

2529-43 

% 

3191.91 

A 

3931-37 

% 

I953-69 

A 

2540.58 

A 

3204.44 

A 

3945-27 

50 

I963-5 

57 

255I-76 

64 

3217 

71 

3959-2 

1973-33 

2562.97 

A 

3229.58 

A 

3973-15 

A 

1983.18 

A 

2574.2 

A 

3242.18 

A 

3987.13 

% 

1993.06 

Vs 

2585-45 

A 

3254.81 

4, 

4001.13 

A 

2002.97 

a 

2596.73 

A 

3267.46 

A 

4015.16 

% 

2012.89 

A 

2608.03 

A 

3280.14 

4 

4029.21 

A 

2022.85 

A 

2619.36 

A 

3292.84 

A 

4043.29 

% 

2032.82 

% 

2630.71 

A 

3305-56 

A 

4057-39 

51 

2042.83 

58 

2642.09 

65 

3318.31 

72 

4071.51 

% 

2052.85 

H 

2653-49 

A 

3331.09 

A 

4085.66 

X 

2062.9 

A 

2664.91 

A 

3343-89 

A 

4099.84 

A 

2072.98 

% 

2676.36 

A 

3356.71 

A 

4114.04 

A 

2083.08 

A 

2687.84 

A 

3369-56 

A 

4128.26 

% 

2093.2 

4 

2699.33 

A 

3382.44 

A 

4142.51 

A 

2103.35 

% 

2710.86 

% 

3395-33 

A 

4156.78 

% 

2113.52 

% 

2722.41 

A 

3408.26 

A 

4171.08 

52 

2123.72 

59 

2733.98 

66 

3421.2 

73 

4185.4 

K 

2133-94 

A- 

2745-57 

X 

3434-17 

A 

4199.74 

A 

2144.19 

A 

2757.2 

A 

3447.17 

A 

4214.11 

% 

2154.46 

A 

2768.84 

A 

3460.19 

A 

4228.51 

A 

2164.76 

A 

2780.51 

A 

3473-24 

A 

4242.93 

% 

2175.08 

A 

2792.21 

A 

3486.3 

A 

4257-37 

% 

2185.42 

% 

2803.93 

A 

3499-4 

A 

4271.84 

% 

2195.79 

A 

2815.67 

A 

3512.52 

A 

4286.33 

53 

2206. 19 

60 

2827.44 

67 

3525-66 

74 

4300.85 

A 

2216.61 

A 

2839.23 

A 

3538.83 

X 

4315-39 

A 

2227.05 

A 

2851.05 

A 

3552.02 

A 

4329.96 

% 

2237.52 

% 

2862.89 

A 

3565-24 

A 

4344-55 

A 

2248.01 

A 

2874.76 

A 

3578.48 

A 

4359-17 

% 

2258.53 

A 

2886.65 

A 

3591.74 

A 

4373-8i 

% 

2269.07 

% 

2898.57 

A 

3605.04 

A 

4388.47 

% 

2279.64 

A 

2910.51 

A 

3618.35 

A 

4403.16 

54 

2290.23 

61 

2922.47 

68 

3631.69 

75 

4417.87 

A 

2300.84 

A 

2934.46 

A 

3645.05 

A 

4432.6i 

A 

2311.48 

A 

2946.48 

A 

3658.44 

A 

4447-38 

% 

2322.15 

A 

2958.52 

A 

3671.86 

A 

4462.16 

A 

2332.83 

A 

2970.58 

A 

3685.29 

A 

4476.98 

A 

2343-55 

A 

2982.67 

A 

3698.76 

A 

4491.81 

A 

2354-29 

% 

2994.78 

A 

3712.24 

A 

4506.67 

A 

2365.05 

A 

3006.92 

A 

3725.75 

A 

4521.56 

55 

2375-83 

62 

3019.08 

69 

3739-29 

76 

4536.47 

A 

2386.65 

A 

3031.26 

A 

3752.85 

X 

455i-4i 

A 

2397.48 

A 

3043-47 

A 

3766.43 

X 

4566.36 

A 

2408.34 

A 

3055.71 

A 

3780.04 

% 

4581.35 

A 

2419.23 

M 

3067.97 

A 

3793.68 

. X 

4596.36 

A 

243°-I4 

% 

3080.25 

A 

3807.34 

A 

4611.39 

X 

2441.07 

% 

3092.56 

A 

3821.02 

A 

4626.45 

A 

2452.03 

A 

3104.89 

A 

3834.73 

A 

464I-53 

234 


AREAS  OF  CIRCLES. 


Diam. 

Area. 

Diam. 

Area. 

77 

4656.64 

84 

554I-78 

% 

4671.77 

H 

5558.29 

X 

4686.92 

x 

5574.82 

% 

4702.I 

% 

5591-37 

A 

47I7-3I 

x 

5607.95 

% 

4732.54 

% 

5624.56 

% 

4747-79 

x 

5641.18 

% 

4763.07 

% 

5657-84 

78 

4778.37 

5674-5I 

ya 

4793-7 

Ya 

5691.22 

x 

4809.05 

X 

5707-94 

% 

4824.43 

Ya 

5724.69 

X 

4839-83 

x 

5741-47 

% 

4855.26 

% 

5758.27 

X 

4870.71 

x 

5775-1 

% 

4886.18 

% 

579!-94 

79 

4901.68 

86 

5808.82 

A 

4917.21 

5825.72 

x 

4932.75 

X 

5842.64 

% 

4948.33 

X 

5859-59 

A 

4963.92 

x 

5876.56 

% 

4979-55 

% 

5893-55 

% 

4995-19 

x 

59io-58 

% 

5010.86 

% 

5927-62 

So 

5026.56 

87 

5944.69 

% 

5042.28 

% 

5961.79 

X 

5058.03 

X 

5978.91 

% 

5073-79 

% 

5996-05 

x 

5089.59 

V* 

6013.22 

Ya 

5105.41 

% 

6030.41 

X 

5121.25 

x 

6047.63 

% 

5I37-I2 

% 

6064.87 

Si 

5i53-oi 

88 

6082.14 

% 

5168.93 

% 

6099.43 

X 

5184.87 

x 

6116.74 

% 

5200.83 

% 

6134.08 

u 

5216.82 

X 

615145 

Ya 

5232.84 

Ya 

6168.84 

X 

5248.88 

X 

6186.25 

% 

5264.94 

! % 

6203.69 

82 

5281.03 

' 89 

6221.15 

Vs 

5297.14 

X 

6238.64 

X 

53I3-28 

x 

6256.15 

% 

532944 

% 

6273.69 

x 

5345-63 

x 

6291.25 

% 

5361.84 

% 

6308  84 

x 

5378.o8 

% 

6326  45 

% 

5394-34 

i % 

6344.08 

83 

5410.62 

90 

6361.74 

% 

5426.93 

X 

6379.42 

x 

5443.26 

X 

6397-I3 

% 

5459-62 

Ya 

6414.86 

x 

5476.01 

X 

6432.62 

% 

5492.41 

4 

64504 

% 

5508.84 

% 

6468.21 

% 

5525-3 

% 

6486.04 

Diam. 

Area. 

Diam. 

Area. 

91 

6503-9 

98 

7542.98 

Ya 

6521.78 

% 

7562.24 

X 

6539.68 

X 

7581.52 

Ya 

6557-6I 

Ya 

7600.82 

X 

6575-56 

X 

7620.15 

% 

6593-54 

4 

7639-5 

% 

6611.55 

X 

7658.88 

Ya 

6629.57 

Ya 

7678.28 

92 

6647.63 

99 

7697.71 

Ys 

6665.7 

Ya 

7717.16 

X 

6683.8 

X 

7736.63 

% 

67OI.93 

% 

7756.13 

X 

6720.08 

X 

7775.66 

% 

6738.25 

Ya 

7795.21 

X, 

675645 

X 

7814.78 

Ya 

6774.68 

Ya 

7834.38 

93 

6792.92 

100 

7854 

Ya 

6811.2 

X 

7893-32 

X 

6829.49 

Ya 

7932.74 

Ya 

6847.82 

X 

7972.25 

X 

6866.16 

IOI 

80II.87 

Ya 

6884.53 

X 

8051.58 

X 

6902.93 

X 

8091.39 

Ya 

6921.35 

X 

8131.3 

94 

6939.79 

102 

8171.3 

Ya 

6958.26 

X 

82H.4I 

X 

6976.76 

X 

8251.61 

Ya 

6995.28 

X 

8291.91 

X 

7013.82 

103 

8332.31 

Ya 

7032-39 

¥ 

8372.81 

X. 

7050.98 

X 

8413  4 

Ya 

7069.59 

X 

8454.09 

95 

7088.23 

104 

8494.89 

X 

7106.9 

X 

8535.78 

X 

7I25-59 

X 

8576.76 

% 

7I44-3I 

X 

8617.85 

X 

7i63-°4 

105 

8659.03 

Ya 

7181.81 

X 

87OO  32 

X 

7200.6 

X 

8741.7 

Ya 

7219.41 

X 

8783  l8 

96 

7238.25 

106 

8824.75 

Ya 

7257.11 

A 

8866.43 

X 

7275-99 

A 

8908.2 

% 

7294.91 

% 

8950.07 

X 

73*3-84 

107 

8992.04 

Ys 

7332.8 

A 

9034.II 

X 

7351-79 

H 

9076.28 

Ya 

7370.79 

% 

9118.54 

97 

7389-83 

108 

9160.91 

Ya 

7408.89 

A 

9203.37 

X 

7427.97 

A 

9245  93 

Ya 

7447.08 

% 

9288.58 

X 

7466.21 

109 

9331-34 

Ya 

7485-37 

i ^ 

9374-I9 

X 

7504-55 

A 

94I7-I4 

i Ya 

7523-75 

1 X 

9460.19 

AREAS  OF  CIRCLES. 


235 


Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area.  | 

I Diam. 

Area. 

iio 

9503-34 

120 

II309.76 

130 

13273.26 

140 

I5  393-84 

H 

9546.59 

X 

11356.9  3 

A 

I3324-36 

A 

15448.87 

A 

9589-93 

3< 

1 1 404.2 

A 

I3375-56 

A 

I5  503-99 

% 

9633-37 

% 

H45I-57 

% 

13426.85 

A 

15  559.22 

in 

0676.QI 

121 

II499.O4 

131 

13478.25 

141 

15614.54 

H 

9720-55 

H 

II  546.61 

A 

13  529- 74 

3^ 

15669.96 

A 

9764.29 

A 

II594.27 

• A 

I3  58i-33 

A 

I5  725-48 

% 

9808.12 

% 

II  642.03 

A 

13633-02 

A 

15781.09 

112 

9852.06 

122 

1 1 689.89 

132 

13684.81 

142 

15836.81 

X 

Q 896.09 

3^ 

II  737-85 

A 

13  736.69 

A 

15  892.62 

A 

9940.22 

A 

II785.91 

A 

13  788.68 

A 

15948.53 

% 

9984.45 

% 

II  834  06 

A 

13840.76 

A 

16004.54 

113 

IOO28.77 

123 

II  882.32 

133 

13892.94 

143 

16060.64 

a 

IOO73.2 

a 

II930.67 

A 

13945.22 

A 

16  116.85 

A 

IO  II7.72 

A 

II  979.12 

A 

13997.6 

a 

16173.15 

% 

IO  162.34 

% 

12  027.66 

A 

14050.07 

A 

16229.5  5 

114 

10  207.06 

124 

12076.31 

134 

14 102.64 

144 

16286.05 

3^ 

10251.88 

A 

12  125.05 

3^ 

I4I55-3I 

3^ 

16342.65 

'A 

IO296.79 

A 

12173-9 

14  208.08 

A 

16399  35 

A 

IO34I.8 

A 

12  222.84 

% 

14253.09 

A 

16456.14 

115 

IO386.9I 

125 

12  271.87 

135 

I43I3-9I 

145 

16513.03 

X 

10432.12 

X 

12  321.01 

X- 

14366.98 

3€ 

16570.02 

A 

IO477.43 

A 

12370.25 

A 

14420.14 

X 

16627. 11 

A 

IO522.84 

% 

12419.58 

A 

14473-4 

A 

16684.3 

1 16 

IO568.34 

126 

12469.01 

136 

14526.76 

146 

16741.59 

A 

IO613.94 

%■ 

12518.54 

X 

14580.21 

A 

16798.97 

A 

I0659  65 

y* 

12568.17 

. A 

I4633-77 

, A, 

16856.45 

% 

IO705.44 

% 

12  618.09 

A 

14687.42 

A 

16914.03 

117 

10  751.34 

127 

12667.72 

137 

14  741.17 

147 

16971.71 

A 

10  797.34 

A 

12  717.64 

A 

14795.02 

17029.48 

A 

I0843.43 

A 

12  767.66 

A 

14848.97 

X 

17087.36 

% 

10889.62 

A 

12817.78 

A 

14903.01 

A 

I7I45-33 

n8 

IO935.9I 

128 

12867.99 

138 

14957.16 

148 

17203.4 

A 

IO982.3 

A 

12918.31 

A 

15011.4 

A 

17261.57 

A 

II  028.78 

A 

12968.72 

A 

15065.74 

A 

17319.84 

% 

HO75.37 

% 

13019.23 

A 

15  120.18 . 

, A 

17378.2 

1 19 

II  I22.05 

129 

13069.84 

139 

I5I74-7I 

149 

17436.67 

A 

II  168.83 

A 

13  120.55 

A 

15229.35 

A 

17495.23 

A 

11215.71 

A 

I3I7I-35 

A 

15  284  08 

A 

I7  553-89 

A 

II  262.69 

% 

13  222.26 

A 

15338.91 

150 

17671.5 

To  Compute  Area  of  a Circle  greater  than,  any  in  Table. 

Rule.— Divide  dimension  by  two,  three,  four,  etc.,  if  practicable  to  do  so, 
until  it  is  reduced  to  a diameter  to  be  found  in  table. 

Take  tabular  area  for  this  diameter,  multiply  it  by  square  of  divisor,  and 
product  will  give  area  required. 

Example. — What  is  area  for  a diameter  of  1.050? 

1050-^-7  = 150;  tab.  area , 150=  17  671.5,  which  x 72  = 86.5 903. 5,  area. 

To  Compute  Area  of  a Circle  in  3Teet  and.  Inches,  etc., 
by  preceding  Table. 

Rule.— -Reduce  dimension  to  inches  or  eighths,  as  the  case  may  be,  and 
take  area  in  that  term  from  table  for  that  number. 


AREAS  OE  CIRCLES. 


236 

Divide  this  number  by  64  (square  of  8)  if  it  is  in  eighths,  and  quotient  will 
give  area  in  inches,  and  divide  again  by  144  (square  of  12)  if  it  is  in  inches, 
and  quotient  will  give  area  in  feet. 

Example.— What  is  area  of  1 foot  6.375  ins.? 

1 foot  6.375  ins.  = 18.375  ins.  = 147  eighths.  Area  of  147  = 16971.71,  which  — 64 
= 265.181  25  ins.;  and  by  144  = 1.84 125  feet. 

To  Compute  Area,  of*  a Circle  Composed  of*  air  Integer 
and.  a Fraction. 

Rule. — Double,  treble,  or  quadruple  dimension  given,  until  fraction  is  in- 
creased to  a whole  number,  or  to  one  of  those  in  the  table,  as  3^,  etc., 
provided  it  is  practicable  to  do  so. 

Take  area  for  this  diameter ; and  if  it  is  double  of  that  for  which  area  is 
required,  take  one  fourth  of  it ; if  treble,  take  one  sixteenth  of  it,  etc. 

Example.— Required  area  for  a circle  of  2.1875  ins. 

2.1875  X 2 = 4.375,  area  for  which  = 15.0331,  which -4- 4 = 3. 758  ins. 

When  Diameter  is  composed  of  Integers  and  Fractions  contained  in  Table. 

Rule. — Point  off  a decimal  to  a diameter  from  table,  and  add  twice  as 
many  figures  or  ciphers  to  the  right  of  the  area  as  there  are  figures  cut  off 
from  the  diameter. 

Example  i. — What  is  area  of  9675  feet  diameter? 

Area  of  96.75  = 7351.79;  hence,  area  = 73  5i7  9oo/ee£ 

2.— What  is  area  of  24 375  feet  diameter? 

Area  of  2. 437  5 = 4. 6664 ; hence,  area  = 4 66  640  000  feet. 

To  Ascertain  Area  of*  a Circle  as  300,  3000,  etc.,  not 
contained  in  Table. 

Rule. — Take  area  of  3 or  30,  and  add  twice  the  excess  of  ciphers  to  the 
result. 

Example. — What  is  area  of  a circle  3000  feet  in  diameter? 

Area  of  30  = 706. 86,  hence  area  of  3000  = 7 068  600  feet. 

To  Compute  Area  of*  a Circle  "by*  Logarithms. 

Rule. — To  twice  log.  of  diameter  add  1.895091  (log.  of  .7854),  and  sum 
is  log.  of  area,  for  which  take  number. 

Example. — What  is  area  of  a circle  1200  feet  in  diameter? 

Log.  1200  X 2 -j-  1.895091  = 6.158362  1.895091  = 6.053453,  and  number  for 

which  = 1 130976  feet. 


Areas  of*  Birmingham  Wire  Grange. 


I>iam. 

| Area.  1 

Diam. 

Area. 

Diam. 

Area. 

Diam. 

Area. 

No. 

Sq. Inch. 

No. 

Sq. Inch. 

No. 

Sq. Inch. 

No. 

Sq. Inch. 

I 

.070  686 

IO 

.014  IO3 

19 

.OOI  385 

28 

.000  154 

2 

•063347 

II 

.OII309 

20 

.OOO  962 

29 

.OOOI33 

3 

.052  685 

12 

.009  331 

21 

.OOO  804 

30 

.OOO  1 13 

4 

.044  488 

13 

.007  088 

22 

.OO0616 

31 

.OOOO78 

5 

.038  013 

14 

.OO54II 

23 

.OOO  491 

32 

.OOO064 

6 

•032  365 

15 

.004071 

24 

.OOO  38 

33 

.000  05 

7 

.025447 

16 

.003318 

25 

.OOO314 

34 

.OOOO38 

8 

.021  382 

17 

.002642 

26 

.OOO  254 

35 

.000  02 

9 

.OI 7 203 

18 

.001  886 

27 

.000  201 

36 

.000  013 

CIRCUMFERENCES  OF  CIRCLES. 


237 


Circumferences  of  Circles,  from  to  150. 


Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

1 Diam. 

ClRCUM. 

| Diam. 

ClRCUM. 

tr 

.04909 

3 

9.4248 

8 

25.1328 

15 

47.124 

1/ 

.Ye 

9.6211 

X 

25.5255 

Xs 

47.5167 

/32 

X 

9-8I75 

X 

25.9182 

X 

47.9094 

Kg 

% 

10.014 

% 

26.3109 

% 

48.3021 

H 

.392  7 

10.2102 

X 

26.7036 

X 

48.6948 

/16 

IO.406 

. % 

27.0963 

X 

49.0875 

Kg 

•589 

% 

IO.6029 

% 

27.489 

% 

49.4802 

K 

•785  4 

X. 

IO.799 

% 

27.8817 

Vs 

49.8729 

ft/ 

X 

IO.9956 

9 

28.2744 

16 

50.2656 

VlG 

.981  75 

% 

II.I9I 

X 

28.6671 

Vs 

50.6583 

% 

1.1781 

% 

n.3883 

X 

29.0598 

% 

51.051 

Kg 

1.3744c; 

% 

II.584 

% 

29.4525 

% 

51.4437 

% 

II.781 

X 

29.8452 

X 

51.8364 

% 

1.5708 

% 

II.977 

% 

30.2379 

% 

52.2291 

V 

/16 

1.76715 

Vs 

I2.I737 

% 

30.6306 

% 

52.6218 

I-963  5 

X 

12.369 

% 

3 1 0233 

% 

53.0145 

/8 

4 

I2.5664 

10 

3i<416 

17 

53.4072 

Kg 

2.15985 

Y 

12.762 

K 

31.8087 

K 

53-7999 

% 

2.3562 

% 

I2.959I 

K 

32.2014 

% 

54.1926 

Kg 

13.155 

% 

32.5941 

% 

54-5853 

% 

2.552  55 

X 

13.3518 

X 

32.9868 

X- 

54.978 

% 

2.7489 

Kg 

13-547 

% 

33-3795 

% 

55.3707 

% 

13.7445 

% 

33.7722 

% 

55.7634 

Xg 

2-945  25 

X 

13  94 

% 

34.1649 

% 

56.1561 

1 

3.1416 

X 

14.1372 

11 

34.5576 

18 

56.5488 

Kg 

3-337  9 

X 

14-333 

X 

34-9503 

X 

56.9415 

Si 

3-534  3 

X 

14.5299 

X 

35-343 

X 

57-3342 

K 

3.7306 

% 

14-725 

% 

35.7357 

% 

57.7269 

K 

3-927 

% 

14.9226 

X 

36.1284 

X 

58.1196 

Kg 

4-1233 

X 

15.119 

% 

36.5211 

% 

58.5123 

% 

4.319  7 

Vs 

I5.3I53 

X 

36.91 38 

% 

58.905 

Kg 

4.516 

% 

15-511 

% 

37.3065 

Ks 

59-2977 

Si 

4.7124 

5 

15.708 

12 

37.6992 

J9 

59.6904 

Kg 

4.9087 

Vs 

16.1007 

X 

38.0919 

X 

60.0831 

% 

5.105  1 

X 

16.4934 

X 

38.4846 

X 

60.4758 

Kg 

5-301  4 

X 

16.8861 

% 

38.8773 

Ks 

60.8685 

% 

5497  8 

X 

17.2788 

X 

39.27 

X 

61.2612 

% 

5.6941 

% 

17.6715 

% 

39.6627 

Ks 

61.6539 

% 

5.8905 

% 

18.0642 

% 

400554 

% 

62.0466 

% 

6.086  8 

% 

18.4569 

X 

40.4481 

% 

62.4393 

2 

6.283  2 

6 

18.8496 

13 

40.8408 

20 

62.832 

Kg 

6.4795 

Vs 

19.2423 

Vs 

4I.2335 

Vs 

63.2247 

% 

6.675  9 

X 

19.635 

X 

41.6262 

X 

63.6174 

Kg 

6.872  2 

% 

20.0277 

% 

42.0189 

% 

64.0101 

Si 

7.0686 

X 

20.4204 

X 

42.4116 

X 

64.4028 

Kg 

7.2649 

X 

20.8131 

% 

42.8043 

X 

64-7955 

% 

746i  3 

% 

21.2058 

% 

43-197 

% 

65.1882 

K& 

7.6576 

% 

21.5985 

% 

43-5897 

% 

65.5809 

X 

7.854 

7 

21.9912 

14 

43.9824 

21 

65-9736 

Kg 

8.0503 

Vs 

22.3839 

X 

44-3751 

K 

66.3663 

% 

8.246  7 

X 

22.7766 

X 

44.7678 

X 

66.759 

Kg 

8-443 

% 

23.1693 

X 

45.1605 

X 

67.1517 

% 

8.6394 

X 

23.562 

X 

45.5532 

X 

67-5444 

% 

8-835  7 

% 

23-9547 

X 

45-9459 

X 

67.9371 

% 

9.032  1 

% 

24.3474 

% 

46.3386 

% 

68.3298 

X 

9.2284 

% 

24.7401 

X 

46.7313 

% 

68.7225 

238  CIRCUMFERENCES  OF  CIRCLES. 


Diam. 

ClBCUM. 

Diam. 

ClBCUM. 

Diam. 

ClBCUM. 

Diam. 

ClBCUM. 

22 

69.II52 

29 

91.1064 

36 

113.098 

43 

135.089 

A 

69.5079 

91.4991 

X 

113-49 

X 

1 35 -48 1 

A 

69.9O06 

A 

91.8918 

X 

113.883 

X 

*35-874 

X 

70.2933 

% 

92.2845 

X 

114.276 

X 

136.267 

A 

70.686 

A 

92.6772 

X 

114.668 

X 

136.66 

A 

7I.O787 

% 

93.0699 

X 

115.061 

X 

137.052 

% 

71.4714 

% 

93.4626 

X 

1 15-454 

X 

137-445 

x 

7I.864I 

% 

93-8553 

X 

115.846 

X 

137.838 

23-, 

72.2568 

30 

94.248 

37 

116.239 

44 

138.23 

X 

72.6495 

x 

94.6407 

X 

116.632 

X 

138.623 

A 

73.0422 

A 

95-0334 

X 

117.025 

X 

139.016 

x 

73-4349 

% 

-95.4261 

X 

1 1 7.41 7 

X 

139.408 

x 

73.8276 

A 

95.8188 

X 

117.81 

X 

139.801 

x 

74.2203 

x 

96.2115 

X 

118.203 

X 

140. 194 

X 

74-6i3 

% 

96.6042 

X 

118.595 

X 

140.587 

% 

75-0057 

x 

96.9969 

X 

118.988 

X 

140.979 

24 

75-3984 

3* 

97.3896 

38 

119.381 

45 

141.372 

A 

75-79H 

>8 

97.7823 

X 

119.773 

X 

141.765 

A 

76.1838 

X 

98-175 

X 

120.166 

X 

I42*i57 

% 

76.5765 

% 

98.5677 

X 

120.559 

X 

142.55 

A 

76.9692 

X 

98.9604 

X 

120.952 

X 

142.943 

% 

77.3619 

X 

99-3531 

X 

121.344 

X 

1 43 -335 

% 

77-7546 

% 

99-7458 

X 

121.737 

X 

143.728 

% 

78.1473 

% 

100.1385 

X 

122.13 

x 

144.121 

25 

78.54 

32 

100.5312 

39 

122.522 

46 

144.514 

A 

78.9327 

X 

100.9239 

X 

122.915 

X 

144.906 

A 

79-3254 

X 

101.3166 

X 

123.308 

X 

145.299 

% 

79.7181 

% 

101.7093 

X 

123.7 

X 

145.692 

A 

80.1108 

% 

102. 102 

X 

124.093 

X 

146.084 

% 

80.5035  | 

X 

102.4947 

X 

124.486 

X 

146.477 

% 

80.8962 

X 

102.8874 

X 

124.879 

X 

146.87 

% 

81.2889 

% 

103.2801 

X 

125.271 

X 

147.262 

26 

81.6816 

33 

103.673 

40 

125.664 

47 

147-635 

A 

82.0743 

X 

104.065 

X 

126.057 

X 

147.048 

A 

82.467 

X 

104.458 

X 

126.449 

X 

148.441 

% 

82.8597 

X 

104.851 

X 

126.842 

X 

148.833 

A, 

83.2524 

X 

105.244 

X. 

127.235 

X 

149.226 

% 

83.6451 

% 

105.636 

X 

127.627 

X 

149.619 

% 

84.0378 

X 

106.029 

X 

128.02 

X 

150.01 1 

% 

84-4305 

X 

106.422 

X 

128.413 

X 

150.404 

27 

84.8232 

34 

106.814 

41 

128.806 

48 

150.797 

A 

85.2159 

X 

107.207 

129.198 

X 

151.189 

A 

85.6086 

X 

107.6 

X 

129.591 

X 

151.582 

% 

86.0013 

X 

107.992 

X 

129.984 

X 

I5I-975 

A 

86.394 

X 

108.385 

X 

130.376 

X 

152.368 

% 

86.7867 

X 

108.778 

X 

130.769 

X 

152.76 

% 

87.1794 

X 

109.171 

X 

131.162 

X 

I53-I53 

A 

87.5721 

X 

109.563 

X 

I3I-554 

X 

I53-546 

28 

87.9648 

35 

109.956 

42 

I3I-947 

49 

I53-938 

A 

88.3575 

X 

1 10.349 

X 

132.34 

X 

I54-33I 

A 

88.7502 

X 

no.  741 

X 

132.733 

X 

154.724 

% 

89.1429 

X 

in-134 

X 

133-125 

X 

I55-ii6 

A 

89-5356 

X 

in. 527 

X 

i33-5i8 

X 

I55-509 

%. 

89.9283 

X 

111.919 

X 

I33-911 

If 

155.902 

% 

90.32 1 

X 

112.312 

X 

I34-303 

X 

156.295 

A 

90.7137 

X 

112.705 

X 

134.696 

X 

156.687 

CIRCUMFERENCES  OF  CIRCLES. 


239 


Diam. 

r ClRCUM. 

II  Diam. 

I ClRCUM. 

Diam. 

ClRCUM. 

Diam. 

ClRCUM. 

50 

157.08 

' 57 

1 79.071 

64 

201.062 

71 

223.054 

X 

157*473 

X 

179.464 

X 

201.455 

X 

223.446 

X 

157*865 

K 

179.857 

X 

201.848 

X 

223.839 

X 

158.258 

% 

180.249 

X 

202.24 

X 

224.232 

X 

158.651 

X 

180.642 

X 

202.633 

X 

224.624 

H 

i59*°43 

% 

181.035 

X 

203.026 

X 

225.017 

X 

I59-436 

% 

181.427 

% 

203.419 

% 

225.41 

% 

159.829 

% 

181.82 

X 

203.811 

X 

225.802 

51 

160.222 

58 

182.213 

65 

204.204 

72 

226.195 

% 

160.614 

X 

182.605 

X 

204.597 

X 

226.588 

U 

161.007 

X 

182.998 

X 

204.989 

X 

226.981 

% 

161.4 

% 

183.391 

X 

205.382 

X 

227.373 

X 

161.792 

X 

183.784 

X 

205.775 

X 

227.766 

% 

162.185 

% 

184.176 

X 

206.167 

X 

228.159 

X 

162.578 

% 

184.569 

X 

206.56 

% 

228.551 

% 

162.97 

% 

184.962 

X 

206.953 

X 

228.944 

52 

163.363 

59 

185.354 

66 

207.346 

73 

229.337 

H 

163.756 

X 

185.747 

X 

207.738 

229.729 

X 

164.149 

X 

186.14 

X 

208.131 

X 

230.122 

% 

164.541 

% 

186.532 

X 

208.524 

X 

230.515 

X 

164.934 

X 

186.925 

X 

208.916 

X 

230.908 

% 

165.327 

% 

187.318 

X 

209.309 

X 

231.3 

X 

165.719 

% 

187.711 

M 

209.702 

% 

231.693 

X 

166.112 

X 

188.103 

X 

210.094 

X 

232.086 

53 

166.505 

60 

188.496 

67 

210.487 

74 

232.478 

X 

166.897 

188.889 

X 

210.88 

232.871 

X 

167.29 

X 

189.281 

X 

211.273 

233.264 

X 

167.683 

% 

189.674 

X 

211.665 

X 

233*656 

X 

168.076 

X 

190.067 

X 

212.058 

X 

234.049 

X 

168.468 

% 

190.459 

X 

212.451 

X 

234.442 

X 

168.861 

X 

190.852 

M 

212.843 

X 

234*835 

% 

169.254 

% 

191-245 

X 

213.236 

X 

235.227 

54 

169.646 

6t 

191.638 

68 

213.629 

75 

235.62 

X 

170.039 

X 

192.03 

X 

2 14.02 1 

X' 

236.013 

x 

170.432 

X 

192.423 

X 

214.414 

X 

236.405 

x 

170.824 

% 

192.816 

X 

214.807 

X 

236.798 

X 

171.217 

X 

193.208 

X 

215.2 

X 

237*I9I 

% 

171.61 

% 

193.601 

X 

215.592 

X 

237.583 

x 

172.003 

% 

I93*994 

X 

215.985 

X 

237*976 

% 

172.395 

X 

194.386 

X 

216.378 

X 

238.369 

55 

172.788 

62 

194.779 

69 

216.77 

76 

238.762 

X 

173.181 

X 

195*172 

217.163 

X 

239*154 

X 

173*573 

X 

195*565 

X 

217*556 

X 

239*547 

% 

173.966 

X 

195*957 

X 

217.948 

X 

239.94 

X 

1 74*359 

X 

196.35 

X 

218.341 

X 

240.332 

% 

174*751 

X 

196.743 

X 

218.734 

X 

240.725 

% 

I75-I44 

% 

197*135 

% 

219. 127 

X 

241.118 

% 

175*537 

: X 

197.528 

X 

219.519 

X 

241.51 

56 

175*93 

63 

197. 921 

70 

219.912 

77 

241.903 

X 

176.322 

X 

198.313 

X 

220.305 

'X- 

242.296 

X 

176.715 

X 

198.706 

X 

220.697 

X 

242.689 

% 

177.108 

X 

*99-099 

X 

221.09 

X 

243.081 

X 

177*5 

X 

199.492 

X 

221.483 

X 

243.474 

$ 

177.893 

X 

199.884 

X 

221.875 

X 

243.867 

% 

178.286 

% 

200.277 

% 

222.268 

X 

244.259 

% 

178.678  1 

X 

200.67 

X 

222.661 

X 

244.652 

240 


CIRCUMFERENCES  OF  CIRCLES. 


Diam. 

Circum.  ' ! 

Diam. 

Circum. 

Diam.  | 

78 

245-045 

^85 

267.036 

92t/ 

X 

245-437 

% 

267.429 

X 

X 

245-83  , 

M 

267.821 

X 

X 

246.223 

% 

268.214 

X 

X 

246.616 

X 

268.607 

X 

X 

247.008 

% 

268.999 

X 

X 

247.401 

% 

269.392 

X 

% 

247.794 

% 

269.785 

X 

79 

248.186 

86 

27O.I78 

931/ 

X 

248.579 

X 

27O.57 

X 

248.972 

X 

270.963 

X 

X 

249.364 

% 

271.356 

X 

X 

249-757 

X, 

271.748 

X. 

X 

250.15 

% 

272. 141 

X 

X 

250.543 

% 

272.534 

X 

X 

250.935 

% 

272.926 

X 

80 

251.328 

8 ‘7 

273-3I9 

94 

Vs 

251. 721 

ft 

273-712 

X 

K 

252.113 

274.105 

X 

X 

252.506 

X 

274.497 

X 

X 

252.899 

X 

274.89 

X 

X 

253.291 

ft 

275.283 

X 

X 

253.684 

275-675 

X 

% 

254.077 

% 

276.068 

X 

81 

254-47 

88 

276.461 

95 

X 

254.862 

X 

276.853 

X 

X 

255-255 

X 

277.246 

X 

X 

255.648 

% 

277.629 

X 

X 

256.04 

X 

278.032 

X 

X 

256.433 

% 

278.424 

X 

X 

256.826 

% 

278.817 

X 

X 

257.218 

% 

279.21 

X 

82 

257.611 

89 

279.602 

96 

X 

258.004 

X 

279-995 

X 

X 

258.397 

X 

280.388 

X 

X 

258.789 

X 

280.78 

X 

X 

259.182 

X 

281.173 

X 

X 

259-575 

ft 

281.566 

$ 

X 

259.967 

281.959 

X 

X 

260.36 

% 

282.351 

X 

83 

260.753 

90 

282.744 

97 

X 

261.145 

X 

283.137 

X 

X 

261.538 

X 

283.529 

X 

X 

261.931 

X 

283.922 

X 

X 

262.324 

X 

284.315 

x 

X 

262.716 

X 

284.707 

g 

X 

263.109 

X 

285.1 

X 

X 

263.502 

X 

285.493 

X 

84 

263.894 

91 

285.886 

98 

X 

264.287 

X 

286.278 

x 

X 

264.68 

X 

286.671 

X 

X 

265.072 

X 

287.064 

X 

. X 

265.465 

X 

287.456 

X 

X 

265.858 

% 

287.849 

X 

266.251 

% 

288.242 

X 

X 

266.643 

\ % 

288.634 

X 

289.027 

289.42 

289.813 

290.205 

290.598 

290.991 

291.383 

291.776 

292.169 

292.562 

292.954 

293-347 

293- 74 

294-  132 

294- 525 

294.918 

295- 31 
295-703 
296.096 
296.488 
296.881 
297.274 
297.667 
298.059 
298.452 
298.845 
299.237 
299.63 
300.023 
300.415 
300.808 
301. 201 

301-594 

301.986 

302.379 

302.772 

3°3-i64 

303- 557 
3°3-95 

304- 342 


305-521 


308.27 


309.84 


Diam. 

Circum. 

99 

311.018 

ft 

311.411 

X 

311.804 

% 

312.196 

ft 

312.589 

ft 

312.982 

% 

313-375 

% 

3x3-767 

100 

314.16 

X 

3 1 4-945 

X 

315-731 

X 

316.516 

IOI 

317.302 

X 

318.087 

X 

318.872 

X 

3j9-658 

102 

320.443 

X 

321.229 

X 

322.014 

X 

322.799 

io31/ 

323-585 

X 

324.37 

X 

325-156 

X 

325-941 

104 

326.726 

X 

327.512 

X 

328.297 

X 

329.083 

105 

329.868 

X 

330-653 

X 

33 1 -439 

X 

i 

332.224 

106 

333-01 

X 

1 

333-795 

X 

334-58 

X 

335-366 

107 

1 

336.151 

X 

336.937 

34 

337.722 

338.507 

108 

. 

339-293 

: X 

340.078 

K 

340.864 

l % 

341.649 

» io9  . 

342.434 

) X 

1 

343-22 

: X 

344-005 

i-  X 

344-791 

7 IIO 

345.576 

X 

346.361 

2 X 

347-147 

5 x 

347.932 

3 III 

348.718 

34 

349-503 

3 ft 

350.288 

5 % 

351-074 

CIRCUMFERENCES  OF  CIRCLES.  24 1 


Diam. 

ClRCUM. 

| Diam. 

Circum. 

Diam. 

1 Circum. 

Diam. 

J Circum. 

1 12 

351-859 

121 

380.134 

130 

408.408 

*39 

436.682 

X 

352.645 

X 

380.919 

X 

409.192 

X 

437-467 

X 

353-43 

X 

381.704 

X 

409.979 

X 

438.253 

X 

354-215 

X 

382.49 

X 

410.763 

X 

439-037 

1 13 

355- 001 

122 

383.275 

131 

411-55 

140 

439.824 

X 

355-786 

X 

384.061 

X 

412.334 

X 

440.608 

X 

356.572 

X 

384.846 

X 

413.12 

X 

441-395 

x 

357-357 

X 

385.631 

X 

413-905 

X 

442.179 

1 14 

358.142 

123 

386.417 

132 

414.691 

141 

442.966 

X 

358.928 

X 

387.202 

X 

415.476 

X 

443-75 

X 

359-7I3 

X 

387.988 

X 

416.262 

X 

444-536 

X 

360.499 

% 

388.773 

X 

417.046 

X 

445-321 

1 15 

361.284 

124 

389-558 

133 

417-833 

142 

446.107 

X 

362.069 

¥ 

390.344 

X 

418.617 

X 

446.891 

X 

362.855 

X 

39I-I29 

X 

419.404 

X 

447.678 

X 

363-64 

X 

39I-9I5 

X 

420.188 

X 

448.462 

116 

364.426 

125 

392.7 

134 

420.974 

*43 

449.249 

X 

365.211 

X 

393-484 

X 

421.759 

X 

450.033 

X 

365-996 

X 

394.271 

X 

422.545 

X 

450.82 

X 

366.782 

X 

395-055 

X 

423-33 

X 

451-604 

117 

367.567 

126 

395-842 

135 

424.116 

144 

452.39 

X 

368.353 

X 

396.626 

X 

424.9 

X 

453-175 

X 

369.138 

X 

397.412 

X 

425.687 

X 

453-96i 

X 

369.923 

X 

398.197 

X 

426.471 

X 

454-745 

118 

370.709 

127 

398.983 

136 

427.258 

I45 

455-532 

X 

371.494 

X 

399.768 

X 

428.042 

X 

456.316 

X 

372.28 

X 

400.554 

X 

428.828 

X 

457-103 

X 

373.065 

X 

401.338 

X 

429.613 

146 

458.674 

119 

373-85 

128 

402.125 

137 

430.399 

X 

460.244 

X 

374-636 

X 

402.909 

X 

431-183 

147 

461.815 

X 

375-421 

X 

403.696 

X 

431-97 

X 

463.386 

X 

376.207 

X 

404.48 

X 

432.754 

148 

464-957 

120 

376.992 

129 

405.266 

138 

433-541 

X 

466.528 

X 

377-777 

X 

406.051 

X 

434-325 

149 

468.098 

X 

378.563 

X 

406.837 

X 

435- I 12 

X 

469.669 

X 

379-348 

X 

407.622 

X 

435-896  1 

150 

471.24 

To  Compute  Circumference  of  a Diameter  greater  tban 
any-  in  preceding  Table. 

Rule. — Divide  dimension  by  two,  three,  four,  etc.,  if  practicable  to  do  so, 
until  it  is  reduced  to  a diameter  in  table. 

Take  tabular  circumference  for  this  dimension,  multiply  it  by  divisor, 
according  as  it  was  divided,  and  product  will  give  circumference  required. 

Example. — What  is  circumference  for  a diameter  of  1050? 

1050-^-7  = 150;  tab.  circum 150  = 471.24,  which  X 7 = 3298.68,  circumference. 

To  Compute  Circumference  of  a Diameter  in  Feet  and 
Indies,  etc.,  "by"  preceding  Table. 

Rule. — Reduce  dimension  to  inches  or  eighths,  as  the  case  may  be,  and 
take  circumference  in  that  term  from  table  for  that  number. 

Divide  this  number  by  8 if  it  is  in  eighths,  and  by  12  if  in  inches,  and 
quotient  will  give  circumference  in  feet. 

X 


CIRCUMFERENCES  OF  CIRCLES. 


242 

Example.— Required  circumference  of  a circle  of  1 foot  6.375  ins. 

1 foot  6.375  ins.  = 18.375  ins.  = 147  eighths.  Circum.  of  147  = 461.815.  which -5-  8 
= 57-727  ins.;  and  by  12  = 4.810 6 feet.  • 

Compute  Circumference  for  a Diameter  composed  of 
an  integer  and  a Fraction. 

Rule. — Double,  treble,  or  quadruple  dimension  given,  until  fraction  is  in- 
creased to  a whole  number  or  to  one  of  those  in  the  table,  as  etc.,  pro- 

vided it  is  practicable  to  do  so. 

Take  circumference  for  this  diameter ; and  if  it  is  double  of  that  for  which 
circumference  is  required,  take  one  half  of  it ; if  treble,  take  one  third  of  it ; 
and  if  quadruple,  one  fourth  of  it. 

Example.— Required  circumference  of  2.218  75  ins. 

2.21875  X 2 = 4.4375,  which  x 2 = 8.875;  circum.  forf\diich  = 27.8817,  which  -i-  4 
= 6.9704  ins. 

When  Diameter  consists  of  Integers  and  Fractions  contained  in  Table. 

Rule.— Point  a decimal  to  a diameter  in  table,  take  circumference  from 
table,  and  add  as  many  figures  to  the  right  as  there  are  figures  cut  off. 

Example.— What  is  circumference  of  a circle  9675  feet  in  diameter? 

Circumference  of  96. 75  — 303. 95 ; hence,  circumference  of  9675  = 30  395 

To  Ascertain  Circumference  for  a Diameter,  as  (500, 
£5000,  etc.,  not  contained  in  Talxle. 

Rule. — Take  circumference  of  5 or  50  from  table,  and  add  the  excess  of 
ciphers  to  the  result. 

Example.— What  is  circumference  of  a circle  8000  feet  in  diameter? 

Circumference  of  80  ==  251. 38;  hence,  circumference  of  8000  = 25 138  feet. 


To  Compute  Circumference  of  a Oircle  by  Logarithms. 

Rule.— To  log.  of  diameter  add  .497  01  G°»-  3-I4I6),  and  sum  is  log.  *j 

of  circumference,  from  which  take  number. 

Example. — What  is  circumference  of  a circle  1200  feet  in  diameter? 

Log.  1200=  3-079 18 -f-. 497 01  = 3.576 19,  and  number  for  which  = 3769.91 /erf. 


Circumferences  of  Birmingham  Wire  Gauge. 


Circum. 

I Diam.  1 

Circum. 

Diam. 

Circum. 

Diam. 

Circum.  j 

Ins. 

.942  48 
.892  21 
.81367 
•747  7 
.691  15 
.637  74 
.565  49 
.51836 
.46495 

| No.  ' 
IO 
11 
12 
13 
14 
. 15 

I 

I 

1 18 

Ins. 

.42097 
.37699 
.342  43 
.298  45 
.260  75 
.226  19 
.2042 
.18221 
.15394 

No. 

19 

20 
21 
22 

23 

24 

25 

26 

27 

Ins. 

.13195 
.IO995 
.100  53 
.087  96 
.078  54 
.069 II 
.062  83 
.05655 
.050  26 

No. 

28 

29 

30 

31 

32 

33 

34 

35 

36 

Ins.  | 

.04398  1 

.040  84  j 

•037  7 ; 

.031 41  < 

.028  27 
.025  13 
.021  99 

.015  71 
.012  57 

o\oi  -£*  oj  io  h vb  oo-<i  bvoi  £.  d)  to  h vo  bcCj  b'Oi  4^  cb 


AREAS  AND  CIRCUMFERENCES  OF  CIRCLES. 


243 


Areas  and  Circumferences.  {Advancing  by  Tenths.) 


Diam. 


Area. 


Circum. 


*7 

.8 

•9 

3 


.007  854 

-3H 16 

.031416 

.628  32 

.070  686 

.942  48 

.125664 

1.2566 

•19635 

1.5708 

.282  744 

1.885 

.384846 

2.199 1 

.502656 

25133 

.636 174 

2.8274 

•7854 

3.1416 

•950  3 

34558 

1-131 

3.7699 

I-3273 

4.084 1 

1-5394 

4 398  2 

1.767  1 

4.7124 

2.010  6 

5.0266 

2.2698 

5-340  7 

2.5447 

5-6549 

2.8353 

5-969 

3.1416 

6.2832 

3-4636 

6-5974 

3-8oi  3 

6.9115 

4.1548 

72257 

4-5239 

7-5398 

4.908  7 

7-854 

5-3093 

8.1682 

5-7256 

8.4823 

6.157  5 

8.7965 

6.605  2 

9.1106 

7.068  6 

9.4248 

.2 

•3 

•4 

•5 

.6 

•7 

.8 

•9 

4 

.1 

.2 

•3 

•4 

•5 

.6 

•7 

.8 

•9 

5 


.2 

•3 

•4 

•5 


7- 547  7 
8.042  5 

8- 553 
9.0792 
9.621 1 

10.1788 

10.7521 

11.3412 

n-9459 

12.5664 

13.2026 

!3-8545 

14.522 

15-2053 

159043 

16.619 1 

1 7-349  5 

18.0956 

18.8575 

I9-635 

20.428  3 

21.2372 

22.061 9 

22.9023 

23-7583 


9-739 
10.053 1 
10.3673 
10.681 4 
10.9956 
11.3098 
11.6239 
11.9381 
12.252  2 
12  5664 
12.8806 
I3-I94  7 
135089 
13-823 
14.1372 
14.4514 
14765  5 
15.0797 
I5-3938 
15.708 
16.022  2 
16.3363 
16.6505 
16.9646 
17.2788 


Diam. 

| Area. 

Circum. 

.6 

24.6301 

17-593 

•7 

25.5176 

17.9071 

.8 

26  4209 

18.2213 

•9 

27-3398 

18.5354 

6 

28.2744 

18.8496 

.1 

29.2247 

19.1638 

.2 

30.1908 

19.4779 

-3 

31.1725 

19.7921 

-4 

32.17 

20.1062 

-5 

33-I83I 

20.4204 

.6 

34.212 

20.7346 

-7 

35-2566 

21.0487 

.8 

36.3169 

21.3629 

-9 

37-3929 

21.677 

7 

38.4846 

21.9912 

.1 

39-592 

22.3054 

.2 

40.7151 

22.6195 

-3 

41.854 

22-9337 

•4 

43.0085 

23.2478 

•5 

44.1787 

23.562 

.6 

45-3647 

23.8762 

-7 

46.5664 

24.1903 

.8 

47.7837 

24-5045 

-9 

49.0168 

24.8186 

8 

50.2656 

25.1328 

.1 

51.5301 

25-447 

.2 

52.8103 

25.7611 

-3 

54.1062 

26.0753 

•4 

55.4178 

26.3894 

-5 

56.7451 

26.7036 

.6 

58.0882 

27.OI78 

-7 

59.4469 

27-33I9 

.8 

60.8214 

27  6461 

-9 

62.2115 

27  9602 

9 

63.6174 

28.2744 

.1 

65.039 

28.5886 

.2 

66.4763 

28.9027 

•3 

67.9292 

29.2169 

•4 

693979 

29-53I 

•5 

70.8823 

29.8452 

.6 

72.3825 

30.1594 

-7 

73.8983 

30.4735 

.8 

75.4298 

30.7877 

-9 

76.9771 

31.1018 

10 

78.54 

31.416 

.1 

80.1187 

3I-73°2 

.2 

81.713 

32.0443 

-3 

83.3231 

32.3585 

-4 

84.9489 

32.6726 

-5 

86.5903 

32.9868 

.6 

88.2475 

33-301 

-7 

89.9204 

336151 

.8 

91.6091 

33-9293 

•9 

93-3I34 

34-2434 

244  AREAS  and  circumferences  of  circles. 


Diam. 

Area. 

ClRCUM. 

Diam.  I 

Area. 

ClRCUM. 

II 

95-0334 

34-5576 

•5 

213.8251 

5I.8364 

.1 

96. 7691 

34.8718 

.6 

216.4248 

52.1505 

.2 

98.5206 

35-I859 

•7 

219.0402 

52.4647 

.3 

100.2877 

35-5ooi 

.8 

221.6713 

52.7789 

•4 

102.0706 

35.8142 

•9 

224.3181 

53-093 

.5 

103.8691 

36.1284 

17 

226.9806 

53.4072 

.6 

105.6834 

36.4426 

.1 

229.6588 

53-7214 

*7 

107.5134 

36.7567 

.2 

232.3527 

54-0355 

.8 

109.3591 

37.0709 

•3 

235.0624 

54-3497 

•9 

1 1 1.2205 

37-385 

•4 

237-7877 

54.6638 

12 

113.0976 

37.6992 

•5 

240.5287 

54-978 

.1 

1 14.9904 

38.0134 

.6 

243.2855 

55-2922 

.2 

116.8989 

38-3275 

•7 

246.058 

55.6063 

.3 

118.8232 

38.6417 

.8 

248.8461 

55.9205 

*4 

120.7631 

38.9558 

•9 

251.65 

56.2346 

•5 

122.7187 

39-27 

18 

254.4696 

56.5488 

.6 

124.6901 

39-5842 

.1 

257.3049 

56.863 

.7 

126.6772 

39-8983 

.2 

260.1559 

57-1771 

.8 

128.6799 

40.2125 

•3 

263.0226 

57-4913 

•9 

130.6984 

40.5266 

•4 

265.905 

57-8054 

13 

132.7326 

40.8408 

•5 

268.8031 

58.1196 

.1 

134.7825 

4I-I55 

.6 

271.717 

58.4338 

.2 

136.8481 

41.4691 

•7 

274.6465 

58.7479 

.3 

138.9294 

4I-7833 

.8 

277-5918 

59.0621 

•4 

141.0264 

42.0974 

•9 

280.5527 

59-3762 

.5 

i43-I39I 

42.4116 

19 

283.5294 

59.6904 

.6 

145.2676 

42.7258 

.1 

286.5218 

60.0046 

.7 

147.4H7 

43-0399 

.2 

289.5299 

60.3187 

.8 

149.5716 

43-3541 

•3 

292.5536 

60.6329 

•9 

I5I-747I 

43.6682 

-4 

295  593i 

60.947 

14 

I53-9384 

43.9824 

-5 

298.6483 

61.2612 

.1 

156.1454 

44.2966 

.6 

301. 7 193 

6i.5754 

.2 

158.3681 

44.6107 

-7 

304.806 

61.8895 

.3 

160.6064 

44.9249 

.8 

307.9082 

62.2037 

•4 

162.8605 

45-239 

•9 

311.0263 

62.5178 

•5 

165.1303 

45.5532 

20 

314.16 

62.832 

.6 

167.4159 

45.8674 

.1 

317.3094 

63.1462 

.7 

169.7171 

46.1815 

.2 

320.4746 

63.4603 

.8 

172.034 

46.4957 

•3 

323-6555 

63-7745 

•9 

174.3667 

46.8098 

-4 

326.8521 

64.0886 

15  ' 

i76-7T5 

47.124 

•5 

330.0643 

64.4028 

.1 

179.0791 

47.4382 

.6 

333-2923 

64-7*7 

.2 

181.4588 

47-7523 

-7 

336.536 

65.0311 

.3 

183.8543 

48.0665 

.8 

339-7955 

65-3453 

.4 

186.2655 

48.3806 

•9 

343.0706 

65.6594 

.5 

188.6924 

48.6948 

21 

346.3614 

65.9736 

.6 

i9I-I349 

49.009 

.1 

349.6679 

66.2878 

.7 

I93-5932 

49-3231 

.2 

352.9902 

66.6019 

.8 

196.0673 

49.6373 

•3 

356.3281 

66.9161 

•9 

198.557 

49-9514 

•4 

359.6818 

67.2302 

16 

201.0624 

50.2656 

•5 

3630511 

67-5444 

.1 

203.5835 

50-5797 

.6 

1 366.4362 

67.8586 

.2 

206.1204 

50.8939 

•7 

i 369-837 

68.1727 

208.6729 

51.2081 

.8 

1 373-2535 

68.4869 

•j 

•4 

211.2412 

51.5222 

! -9 

.!  376.6857 

68.801 

AREAS  AND  CIRCUMFERENCES  OF  CIRCLES. 


245 


Diam. 

Area. 

ClRCUM. 

22 

380.1336 

69.II52 

.1 

383.5972 

69.4294 

.2 

387-0765 

69-7435 

•3 

! 390-57I6 

70.0577 

-4 

394.0823 

70.3718 

•5 

397.6087 

70.686 

.6 

401. 1509 

71.0002 

-7 

404.7088 

7I-3I43 

.8 

408.2823 

71.6285 

•9 

411.8716 

71.9426 

23 

415.4766 

72.2568 

.1 

419.0973 

72.571 

.2 

422.7337 

72.8851 

•3 

426.3858 

73-1993 

•4 

430.0536 

735134 

•5 

433-7371 

73.8276 

.6 

437.4364 

74.1418 

•7 

44i-i5i3 

74-4559 

.8 

444.882 

74.7701 

•9 

448.6283 

75.0842 

24 

452.3904 

75-3984 

.1 

456.1682 

75-7I26 

.2 

459.9617 

76.0267 

•3 

463.7708 

76.3409 

•4 

467-5957 

76-655 

•5 

47I-4363 

76.9692 

.6 

475.2927 

77.2834 

•7 

479.1647 

77-5975 

.8 

483.0524 

77.9117 

•9 

486.9559 

78.2258 

25 

490.875 

78.54 

.1 

494.8099 

78.8542 

.2 

498.7604 

79.1683 

•3 

502.7267 

79.4825 

•4 

506.7087 

79.7966 

•5 

510.7063 

80.1108 

.6 

514.7196 

80.425 

•7 

518.7488 

80.7391 

.8 

522.7937 

81.0533 

•9 

526.8542 

81.3674 

26 

530.9304 

81.6816 

.1 

535.0223 

81.9958 

.2 

539-13 

82.3099 

•3 

543-2533 

82.6241 

•4 

547.3924 

82.9382 

•5 

55I-547I 

83.2524 

.6 

555-7I76 

83.5666 

•7 

559.9038 

83.8807 

.8 

564.1057 

84.1949 

•9 

568.3233 

84.509 

27 

572.5566 

84.8232 

.1 

576.8056 

85-I374 

.2 

581.0703 

85-45I5 

•3 

585-3508 

85-7657 

•4 

589.6469 

86.0798 

Diam. 

Area. 

ClRCUM. 

•5 

593-9587 

86.394 

.6 

598.286  3 

86.7082 

•7 

602.6296 

87.0223 

.8 

606.9885 

87-3365 

•9 

611.3632 

87.6506 

28 

6I5-7536 

87.9648 

.1 

620.1597 

88.279 

.2 

624.5815 

88.5931 

•3 

629.OI9 

88.9O73 

•4 

633.4722 

89.2214 

•5 

637.9411 

89-5356 

.6 

642.4258 

89.8498 

•7 

646.9261 

9°-i639 

.8 

651.4422 

90.4781 

•9 

655-9739 

90.7922 

29 

660.5214 

91.1064 

.1 

665.0846 

91.4206 

.2 

669.6635 

9x-7347 

•3 

674.258 

92.0489 

•4 

678.8683 

92.363 

•5 

683-4943 

92.6772 

.6 

688.1361 

92.9914 

•7 

692.7935 

93-3055 

.8 

697.4666 

93.6197 

•9 

702.1555 

93-9338 

30 

706.86 

94.248 

.1 

711.5803 

94.5622 

.2 

716.3162 

94.8763 

•3 

721.0679 

95-I905 

-4 

725-8353 

95-5046 

•5 

730.6183 

95.8188 

.6 

735-4I7I 

96.133 

•7 

740.2316 

96.4471 

.8 

745.0619 

96.7613 

•9 

7499078 

97-0754 

3i 

754.7694 

97.3896 

.1 

759.6467 

97.7038 

.2 

764.5398 

98.0179 

•3 

769.4485 

98.3321 

•4 

774-373 

98.6462 

•5 

779-313I 

98.9604 

.6 

784.269 

99.2746 

•7 

789.2406 

99.5887 

.8 

794.2279 

99.9029 

•9 

799.2309 

100.217 

32 

804.2496 

100.5312 

.1 

809.284 

100.8454 

.2 

8i4-334i 

101.1595 

•3 

819.4 

IOI-4737 

•4 

824.4815 

101.7878 

•5 

829.5787 

102. 102 

.6 

834.6917 

102.4162: 

•7 

839.8204 

1 02. 7303; 

.8 

844.9647 

103.0445 

•9 

850.1248 

103.3586^ 

X* 


246  AREAS  AND  CIRCUMFERENCES  OF  CIRCLES. 


Diam. 

Area. 

ClRCUM. 

Diam. 

33 

855*3006 

IO3.6728 

•5 

.1 

860.4921 

IO3.987 

.6 

.2 

865.6993 

I O4. 301 1 

•7 

•3 

870.9222 

104.6153 

.8 

•4 

876.1608 

IO4.9294 

•9 

.5 

881.4151 

IO5.2436 

39 

.6 

886.6852 

I05-5578 

.1 

.7 

891.9709 

IO5.87I9 

.2 

.8 

897*2724 

106.  l86l 

•3 

•9 

902.5895 

106.5002 

•4 

34 

QO7.Q224 

106.8144 

•5 

.1 

9I3*27I 

107.1286 

.6 

.2 

9 1 8*6353 

IO7.4427 

•7 

*3 

924.0152 

IO7.7569 

.8 

*4 

929.4109 

108,071 

♦9 

.5 

934.8223 

IO8.3852 

40 

.6 

940.2495 

IO8.6994 

.1 

*7 

945.6923 

109.0135 

.2 

.8 

95i*i5°8 

IO9.3277 

*3 

•9 

956.6251 

IO9.6418 

*4 

35 

962.115 

IO9.956 

*5 

.1 

967.6207 

1 10.2702 

.6 

.2 

973.142 

IIO.5843 

•7 

.3 

978.6791 

I IO.8985 

.8 

•4 

984.2319 

III.2I26 

•9 

*5 

989.8003 

III.5268 

4i 

.6 

995*3845 

III.84I 

.1 

.7 

1000.9844 

II2.I55I 

.2 

.8 

1006.6001 

II2.4693 

*3 

*9 

1012. 2314 

II2.7834 

•4 

36 

1017.8784 

H3.0976 

•5 

.1 

1023.5411 

II3.4II8 

.6 

.2 

1029.2196 

II3-7259 

*7 

• -.3 

I°34*9I37 

I I4.O4OI 

.8 

.4 

1040.6236 

114*3542 

•9 

•5 

1046.3491 

II4.6684 

42 

.6 

1052.0904 

II4.9826 

.1 

.7 

1057.8474 

II5.2967 

.2 

.8 

1063.6201 

II5.61O9 

•3 

'9 

1069.4085 

II5-925 

*4 

31 

1075.2126 

Il6.2392 

•5 

.1 

1081.0324 

116.5534 

.6 

.2 

1086.8679 

II6.8675 

•7 

*3 

1092.7192 

II7.1817 

.8 

•4 

1098.5861 

117.4958 

•9 

*5 

1104.4687 

II7.81 

43 

.6 

1110.3671 

Il8.I242 

.1 

.7 

1116.2812 

H8.4383 

.2 

.8 

1122.2109 

118.7525 

•3 

•9 

1128.1564 

II9.0666 

•4 

38 

1134.1176 

II9.3808 

•5 

.1 

1140.0945 

II9-695 

.6 

.2 

j:  146.0871 

120.0091 

•7 

•3 

1152.0954 

120.3233 

.8 

•4 

1158.1194 

120.6374 

•9 

1164.1591 

1170.2146 

1176.2857 

1182.3726 

1188.4751 

1194*5934 

1200.7274 

1206.8771 

1213.0424 

1219.2235 

1225.4203 

1231.6329 

1237.8611 

1244.105 

1250.3647 

1256.64 

1262.9311 

1269.2378 

1275.5603 

1281.8985 

1288.2523 

1294.6219 

1301.0072 

1307.4083 

i3J3  825 

1320.2574 

1326.7055 

i333*l694 

1 339*6489 

1346.1442 

1352*6551 

1359*  I8i8 

1365*7242 

1372.2823 

1378.8561 

I385-4456 

1392.0508 

1398.6717 

1405.3084 

1411.9607 

1418.6287 

i425*3I25 

1432.012 

1438.7271 

1445.458 

1452.2046 

1458.9669 

I465-7449 

I472-5386 

I479*348 

1486.1731 

1493.014 

1499.8705 

1506.7428 

I5i3*63°7 


Circum. 


120.9516 

121.2658 

121.5799 

121.8941 

122.2082 

122.5224 

122.8366 

123.1507 

123.4649 

123.779 

124.0932 

124.4074 

124.7215 

125.0357 

125.3498 

125.664 

125.9782 

126.2923 

126.6065 

126.9206 

127.2348 

127.549 

127.8631 

128.1773 

128.4914 

128.8056 

129.1198 

129*4339 

129.7481 

130.0622 

i3°*3764 

130.6906 

131.0047 

i3I*3l89 

i3I*633 

i3I*9472 

132.2614 

132.5755 

132.8897 

133*2038 

133.518 

133.8322 

134.1463 

134.4605 

I34-7746 

135.0888 

135*403 

I35*7i7i 

1360313 

136.3454 

136.6596 

136.9738 

137.2879 

137.6021 

i37*9i62 


AREAS  AND  CIRCUMFERENCES  OF  CIRCLES.  247 


Diam. 

Area. 

ClRCUM. 

44 

1520.5344 

I38.23O4 

.1 

1527.4538 

I38.5446 

.2 

1534.3889 

I38.8587 

•3 

1541  *3396 

139. 1 729 

•4 

1548.3061 

I39.487 

•5 

1555.2883 

139.8012 

.6 

1562.2863 

140. 1 154 

•7 

1569.2999 

140.4295 

.8 

1576.3292 

I4O.7437 

•9 

I583-3743 

I4I.O578 

45 

1590-435 

I4I-372 

.1 

I597-5II5 

141.6862 

.2 

1604.6036 

I42.OOO3 

•3 

1611,7115 

I42-3I45 

•4 

1618.8351 

142.6286 

•5 

1625.9743 

I42.9428 

.6 

1633.1293 

143-257 

•7 

1640.3 

I43-57H 

.8 

1647.4865 

I43-8853 

•9 

1654.6886 

I44-I994 

46 

1661.9064 

I44.5I36 

.1 

1669.1399 

I44.8278 

.2 

1676.3892 

145. 1419 

•3 

1683.6541 

I45-456I 

•4 

1690.9348 

145.7702 

•5 

1698.2311 

I46.0844 

.6 

I705-5432 

I46.3986 

•7 

1712.871 

I46.7I27 

.8 

1720.2145 

I47.O269 

•9 

I727-5737 

I47-34I 

47 

1734.9486 

147.6552 

.1 

1742.3392 

I47.9694 

.2 

1749-74 55 

I48.2835 

•3 

1757.1676 

I48.5977 

•4 

1764.6053 

I48.9U8 

•5 

1772.0587 

149.226 

.6 

I779-5279 

149.5402 

•7 

1787.0128 

I49-8543 

.8 

I794-5I33 

150.1685 

•9 

1802.0296 

I5O.4826 

48 

1809.5616 

I5O.7968 

.1 

1817.1093 

i5i.HI 

.2 

1824.6727 

I5I-425I 

•3 

1832.2518 

151.7393 

•4 

1839.8466 

152.0534 

•5 

1847.4571 

I52.3676 

.6 

1855.0834 

I52.68l8 

•7 

1862.7253 

I52.9959 

.8 

1870.383 

I53-3IOI 

•9 

1878.0563 

153.6242 

49 

1885.7454 

153-9384 

.1 

1 893 -4502 

154.2526 

.2 

1901. 1 707 

154-5667 

•3 

1908.9068 

154.8809 

•4 

1916.6587 

I55-I95 

Diam. 

Area. 

ClRCUM. 

•5 

1924.4263 

I55-5092 

.6 

1932.2097 

155.8234 

•7 

1940.0087 

156.1375 

.8 

1947.8234 

156.4517 

•9 

I955-6539 

I56.7658 

50 

I963-5 

157.08 

.1 

1971.3619 

157.3942 

.2 

1979.2394 

I57-7083 

•3 

1987.1327 

158.0225 

•4 

I995.O417 

I58.3366 

•5 

2002.9663 

I58.6509 

.6 

2010.9067 

I58.965 

•7 

2018.8628 

159.2791 

.8 

2026.8347 

1 59-5933 

•9 

203^.8222 

I59-9074 

5i 

2042.8254 

160.2216 

.1 

2050.8443 

160.5358 

.2 

2058.879 

160.8499 

•3 

2066.9293 

161.1641 

•4 

2074.9954 

161.4782 

•5 

2083.0771 

161.7924 

.6 

2091.1746 

162.1066 

•7 

2099.2878 

162.4207 

.8 

2107.4167 

162.7349 

•9 

2115.5613 

163.049 

52 

2123.7216 

163.3632 

.1 

2i3i-8976 

163.6774 

.2 

2140.0893 

i63.99i5 

•3 

2148.2968 

164.3057 

•4 

2156.5199 

164.6198 

•5 

2164.7587 

164.934 

.6 

2173-0133 

165.2482 

•7 

2181.2836 

165.5623 

.8 

2189.5695 

165.8765 

•9 

2197.8712 

166.1906 

53 

2206.1886 

166.5048 

.1 

2214.5217 

166.819 

.2 

2222.8705 

167.1331 

•3 

2231.235 

167.4473 

•4 

2239.6152 

167.7614 

•5 

2248.0111 

168.0756 

.6 

2256.4228 

168.3898 

•7 

2264  8501 

168.7039 

.8 

2273.2932 

169.0181 

•9 

2281.7519 

169.3322 

54 

2290.2264 

169.6464 

.1 

2298.7166 

169.9606 

.2 

2307.2225 

170.2747 

•3 

23I5-744 

170.5889 

•4 

2324.2813 

170.903 

•5 

2332.8343 

171.2172 

.6 

2341.4031 

I7I-53I4 

•7 

2349.9875 

171-8455 

.8 

2358.5876 

172.1597 

•9 

2367.2035 

172.4738 

248  AREAS  AND  CIRCUMFERENCES  OF  CIRCLES. 


Diam. 

Area. 

ClRCUM. 

Diam. 

Area. 

55 

2375-835 

I72.788 

-5 

2874.7603 

.1 

2384.4823 

1 73. 1022 

.6 

2884.2715 

.2 

2393 •I452 

I73-4I63 

•7 

2893.7984 

.3 

2401.8239 

I73-7305 

.8 

2903-3411 

•4 

2410.5183 

I74.O446 

•9 

2912.8994 

-5 

2419.2283 

174.3588 

61 

2922.4734 

.6 

2427.9541 

I74-673 

.1 

2932.0631 

•7 

2436.6957 

174.9871 

.2 

2941.6686 

.8 

2445.4529 

I75-30I3 

•3 

2951.2897 

•9 

2454.2258 

I75-6I54 

•4 

2960.9266 

56 

2463.0144 

175.9296 

•5 

2970-5791 

.1 

2471.8187 

176.2438 

.6 

2980.2474 

.2 

2480.6388 

I76-5579 

•7 

2989 -93 1 4 

.3 

2489.4745 

I76.872I 

.8 

2999.6311 

•4 

2498.326  „ 

177.1862 

•9 

3009.3465 

•5 

2507-1931 

I77.5OO4 

62 

3019.0776 

.6 

2516.076 

177.8146 

.1 

3028.8244 

•7 

2524-9736  1 

178.1287 

.2 

3038.5869 

.8 

2533-8889 

178.4429 

•3 

3048.3652 

•9 

2542.8189 

178.757 

•4 

3058.1591 

57 

2551-7646 

I79.O712 

•5 

3067.9687 

.1 

2560.726 

I79-3854 

.6 

3077.7941 

.2 

2569-7031 

I79.6995 

•7 

3087.6341 

.3 

2578.696 

180.0137 

.8 

3097 •49I9 

•4 

2587.7045 

180.3278 

•9 

3107.3644 

•5 

2596.7287 

180.642 

63 

3117.2526 

.6 

2605.7687 

180.9562 

.1 

3127.1565 

.7 

2614.8244 

181.2703 

.2 

3137.0761 

.8 

2623.8957 

181.5845 

-3 

3147.0114 

•9 

2632.9828 

181.8986 

-4 

3156.9624 

58 

2642.0856 

182.2128 

•5 

3166.9291 

.1 

2651.2041 

182.527 

.6 

3176.9116 

.2 

2660.3383 

182.84II 

•7 

3186.9097 

.3 

2669.4882 

183.1553 

.8 

3196.9236 

•4 

2678.6538 

183.4694 

•9 

3206.9531 

•5 

2687.8351 

183.7836 

64 

3216.9984 

.6 

2697.0322 

184.0978 

.1 

3227.0594 

.7 

2706.2449 

184.4119 

.2 

3237-i36i 

.8 

-27I5-4734 

184.7261 

•3 

3247.2284 

•9 

2724-7i75 

185.0402 

•4 

3257-3365 

59 

2733-9774 

185.3544 

•5 

3267.460  3 

.1 

2743-253 

185.6686 

.6 

3277-5999 

.2 

2752-5443 

185.9827 

•7 

3287.7551 

.3 

2761.8512 

186.2969 

.8 

3297.9261 

•4 

277I-I739 

l86.6l  I 

-9 

3308.1127 

.5 

2780.5123 

186.9252 

65 

33i8-3i5 

.6 

2789.866  5 

187.2394 

.1 

3328.5331 

.7 

2799.2363 

I87-5535 

.2 

3338.7668 

.8 

2808  6218 

187.8677 

•3 

3349-oi63 

•9 

2818.0231 

I88.l8l8 

•4 

3359-28I5 

60 

2827.44 

188.496 

•5 

3369.5623 

.1 

2836.8727 

188.8102 

.6 

3379.8589 

.2 

2846.321 

189.1243 

•7 

3390-I712 

.3 

2855.7851 

189.4385 

.8 

3400.4993 

•4 

2865.2649 

189.7526 

•9 

3410.843 

ClRCUM. 

190.0668 
I9O.38I 
I9O.695I 
1 9 1. 0093 

191.3234 

I9I.6376 

191.9518 

192.2659 

192.5801 

192.8942 

193.2084 

193.5226 

I93-8367 

I94. 1509 
I94.465 


194.7792 

195-0934 

I95-4075 

I95-72I7 

I96.O358 

196*35 

I96.6642 

I96.9783 

197.2925 

I97.6066 

I97.9208 

I98.235 

198.5491 

I98.8633 

I99-I774 

199.4916 

199:8058 

200.1199 

200.4341 

200.7A82 


201.0624 
201.3766 
201.6907 
202.0049 
202.319 
202.6332 
202-9474 
203.2615 
203.5757 
203  8898 


204.204 

204.5182 

204.8323 

2O5.I465 

205.4606 

205.7748 

206.089 

206.4O3I 

206.7173 

207.0314 


AREAS  AND  CIRCUMFERENCES  OF  CIRCLES.  249 


Diam. 

Area. 

ClKCUM. 

66 

3421.2024 

207.3456 

.1 

3431-5775 

2O7.6598 

.2 

3441.9684 

207.9739 

•3 

3452.3749 

208.2881 

•4 

3462.7972 

208.6022 

•5 

3473-2351 

208.9164 

.6 

3483.6888 

209.2306 

•7 

3494.1582 

209.5447 

.8 

3504.6433 

2O9.8589 

•9 

3515.1441 

2IO.I73 

67 

3525.6606 

2IO.4872 

.1 

3536.1928 

210.8014 

.2 

3546.7407 

2II.II55 

•3 

3557-3044 

21 1.4297 

•4 

3567.8837 

2II.7438 

•5 

3578.4787 

212.058 

.6 

3589.0895 

212.3722 

•7 

3599.716 

212.6863 

.8 

3610.3581 

213.OOO5 

•9 

3621.016 

213.3146 

68 

3631.6896 

213.6288 

.1 

3642.3789 

213-943 

.2 

3653.0839 

214.257I 

•3 

3663.805 

2I4-57I3 

•4 

3674.541 

214-8354 

•5 

3685.2931 

215.1996 

.6 

3696.061 

215-5138 

•7 

3706.8445 

215.8279 

.8 

3717.6438 

216.1421 

•9 

3728.4587 

216.4562 

69 

3739.2894 

216.7704 

.1 

3750.1358 

217.0846 

.2 

3760.9979 

217.3987 

•3 

3771.8756 

217. 7129 

•4 

3782.7691 

218.027 

•5 

3793-6783 

218.3412 

.6 

3804.6033 

218.6554 

•7 

3815.5439 

218.9695 

.8 

3826.5002 

219.2837 

•9 

3837.4722 

219.5978 

70 

3848.46 

219.912 

.1 

3859.4635 

220.2262 

.2 

3870.4826 

220.5403 

•3 

3881.5175 

220.8545 

•4 

3892.5681 

221.1686 

•5 

3903.6343 

221.4828 

.6 

3914.7163 

221.797 

•7 

3925.814 

222. IIII 

.8 

3936.9275 

222.4253 

•9 

3948.9566 

222.7394 

7i 

3959.2014 

223.0536 

.1 

3970.3619 

223.3678 

.2 

3981.5382 

223.6819 

•3 

3992.7301 

223.9961 

•4 

4003.9378  | 

224.3102 

Diam. 

Area. 

ClRCUM. 

•5 

4015.1611 

224.6244 

.6 

4026.4002 

224.9386 

•7 

4037-655 

225.2527 

.8 

4048.9255 

225.5669 

•9 

4060.2117 

225.881 

72 

4071.5136 

226.1952 

.1 

4082.8312 

226.5094 

.2 

4094.1645 

226.8235 

•3 

4105.5136 

227.I377 

•4 

4116.8783 

227.4518 

' -5 

4128.2587 

227.766 

.6 

4I39-655 

228.0802 

•7 

4151.0668 

228.3943 

.8 

4162.4943 

228.7085 

•9 

4I73-9376 

229.0226 

73 

4185.3966 

229.3368 

.1 

4196.8713 

229.651 

.2 

4208.3617 

229.965I 

•3 

4219.8678 

230-2793 

•4 

4231.3896 

230.5934 

•5 

4242.9271 

23O.9O76 

.6 

4254.4804 

231.2218 

•7 

4266.0493 

231-5359 

.8 

4277.634 

231.8501 

•9 

4289.2343 

232.1642 

74 

4300.8504 

232.4784 

.1 

4312.4822 

232.7926 

.2 

4324.1297 

233.I067 

•3 

4335-7928 

2334209 

•4 

4347-47I7 

233-735 

•5 

4359.1663 

234  0492 

.6 

4370.8767 

234.3634 

•7 

4382.6027 

234-6775 

.8 

4394-3444 

234.9917 

•9 

4406.1019 

235-3058 

75 

4417.875 

235.62 

.1 

4429.6639 

235-9342 

.2 

4441.4684 

236.2483 

•3 

4453.2887 

236.5625 

•4 

4465.1247 

236.8766 

•5 

4476.9763 

237.1908 

.6 

4488.8437 

237-505 

•7 

4500.7268 

237.8191 

.8 

4512.6257 

238.1333 

•9 

4524.5402 

238.4474 

76 

4536.4704 

238.7616 

.1 

4548.4163 

239-0758 

.2 

4560.378 

239.3899 

•3 

4572.3553 

239.7041 

•4 

4584.3484 

240.0182 

•5 

4596.3571 

240.3324 

.6 

4608.3816 

240.6466 

•7 

4620.4218 

240.9607 

.8 

4632.4777 

241.2749 

•9 

4644.5493 

241.589 

250 


AREAS  AND  CIRCUMFERENCES  OF  CIRCLES. 


Diam.  I 

Area. 

ClRCL’M. 

Diam.  | 

77 

4656.6366 

241.9032 

*5  | 

.1 

4668.7396 

242.2174 

.6  1 

4680.8583 

242-53i5 

•7 

.3 

4602.9928 

242.8457 

.8 

•4 

47°5-I429 

243>i598 

•9 

•5 

47i7-3o87 

243-474 

83 

.6 

4729.4903 

243.7882 

.1 

•7 

4741.6876 

244.1023 

.2 

.8 

4753-9005 

244.4165 

•3 

•9 

4766.1292 

244.7306 

•4 

78 

4778.3736 

245.0448  N 

•5 

.1 

4790-6337 

245-359 

.6 

.2 

4802.9095 

245-6731 

•7 

.3 

4815.201 

245-9873 

.8 

.4 

4827.5082 

246.3014 

•9 

.5 

4839.8311 

246.6156 

84 

.6 

4852.1698 

246.9298 

.1 

.7 

4864.5241 

247.2439 

.2 

.8 

4876.8942 

247-558i 

•3 

•9 

4889.2799 

247.8722 

•4 

79 

4901.6814 

248.1864 

•5 

.1 

4914.0986 

248.5006 

.6 

.2 

4926.5315 

248.8147 

•7 

•3 

4938-98 

249.1289 

.8 

•4 

4951.4443 

249.443  ! 

•9 

•5 

4963.9243 

249-7572  i 

85 

.6 

4976.4201 

250.0714 

.1 

.7 

4988.9315 

250.3855 

.2 

.8 

5001.4586 

250.6997 

•3 

•9 

5014.0015 

251.0138 

•4 

80 

5026.56 

251.328 

! -5 

.1 

5039 -1 343 

251.6422 

•6 

.2 

5051.7242 

251-9563 

•7 

.3 

5064.3299 

252.2705 

.8 

•4 

5076.9513 

252.5846 

■9 

•5 

5089.5883 

252.8988 

86 

.6 

5102.2411 

253213 

.1 

•7 

5114.9096 

253-527! 

.2 

.8 

5127.5939 

253-8413 

•3 

•9 

5140.2938 

254-I554 

•4 

81 

5153.0094 

254.4696 

•5 

.1 

5165.7407 

254.7838 

.6 

.2 

5178.4878 

255-0979 

•7 

.3 

5191.2505 

255-4121 

! .8 

•4 

5204.0289 

255.7262 

•9 

.5 

5216.8231 

256.0404 

87 

.6 

5229.633 

256.3546 

.1 

.7 

5242.4586 

256.6687 

.2 

.8 

5255-2999 

256.9829 

•3 

•9 

5268.1569 

257.297 

1 *4 

82 

5281.0296 

257.6112 

•5 

.1 

5293-9l8 

257-9254 

.6 

.2 

5306.8221 

258.2395 

•7 

.3 

5319.742 

258.5537 

.8 

•4 

5332.6775 

258  8678 

•9 

5345.6287 

5358.5957 

537I-5784 

5384-5767 

5397-5908 

5410.6206 

5423.6661 

5436.7273 

5449.8042 

5462.8968 

5476.0051 

5489.1292 

5502.2689 

5515.4244 

5528.5955 

5541.7824 

5554-985 

5568.2033 

5581.4372 

55946869 

5607.9523 

5621.2335 

5634-5303 

5647.8428 

5661.1711 

5674-5I5 

5687.8747 

57oi-25 

5714.6411 

5728.0479 

5741.4703 

5754.9085 

5768.3624 

5781.8321 

5795-3174 

5808.8184 

5822.3351 

5835.8676 

58494157 

15862.9796 

5876.5591 

5890.1544 

5903-7654 

59I7-3921 

5931.0345 

5944.6926 

5958.3644 

5972.0559 

5985.7612 

5999.4821 

6013.2187 

6026.9711 

6040.7392 

6054.5229 

6068.3224 


259.182 

2594962 

259.8103 

260.1245 

260.4386 

260.7528 

261.067 

261.3811 

261.6953 

262.0094 

262.3236 

262.6378 

262.9519 

263.2661 

263.5802 

263.8944 

264.2086 

264.5227 

264.8369 

265.151 

265.4652 

265.7794 

266.0935 

266.4077 

266.7218 

267.036 

267.3502 

267.6643 

267.9785 

268.2926 

268.6068 

268.921 

269.2351 

269.5493 

269.8634 

270.1776 

270.4918 

270.8059 

271.1201 

271.4342 

271.7484 

272.0626 

272.3767 

272.6909 

273.005 

273.3192 

273  6334 

273  9475 

274.2617 

2745758 

274.89 

275.2042 

275-5183 

275-8325 

276.1466 


AREAS  AND  CIRCUMFERENCES  OF  CIRCLES.  25 


Diam. 

Area. 

ClRCUM. 

88 

6082.1376 

276.4608 

.1 

6095.9685 

276.775 

.2 

6109  8151 

277.089I 

•3 

6123.6774 

277.4O33 

•4 

6I37-5554 

277.7174 

•5 

6151.4491 

278.O316 

.6 

6165.3586 

278.3458 

•7 

6179.2837 

278.6599 

.8 

6193.2246 

278.974I 

•9 

! 6207.1811 

279.2882 

89 

6221.1534 

279.6024 

.1 

6235.1414 

279.9166 

.2 

6249.1451 

280.2307 

•3 

6263.1644 

280.5449 

•4 

6277.1995 

28O.859 

•5 

6291.2503 

28l.I732 

.6 

6305-3169 

281.4874 

•7 

63I9-3991 

281.8015 

.8 

6333497 

282.1157 

•9 

6347.6107 

282.4298 

90 

6361.74 

282.744 

.1 

6375-885I 

283  0582 

.2 

6390.0458 

283.3723 

•3 

6404.2223 

283.6865 

•4 

! 6418.4144 

284.OOO6 

•5 

6432.622  3 

284.3I48 

.6 

6446.8459 

284.629 

•7 

6461.0852 

284.943I 

.8 

6475-3403 

285.2573 

•9 

6489.61 1 

285.5714 

9i 

6503.8974 

285.8856 

.1 

6518.1995 

286.I998 

.2 

6532-5174 

286.5139 

•3 

6546.8509 

286.828l 

•4 

6561.2002 

287.I422 

•5 

6575-5651 

287.4564 

.6 

6589-9458 

287.7706 

•7 

6604.3422 

288.0847 

.8 

6618.7543 

288.3989 

•9 

6633.1821 

288.7I3 

92 

6647.6256 

289.O272 

.1 

6662.0848 

289.34I4 

.2 

6676.5598 

289.6555 

•3 

6691.0504 

289.9697 

•4 

6705.5567 

29O.2838 

•5 

6720.0787 

29O.598 

.6 

6734.6165 

29O.9I2 1 

•7 

6749.17 

291.2263 

.8 

6763.7391 

291.5405 

•9 

6778.324 

29I.8546 

93 

6792.9246 

292.1688 

.1 

6807.5409 

292.483 

.2 

6822.1729 

292.797I 

•3 

6836.8206 

293.  II I3 

•4 

6851.484  | 

293  4254  1 

Diam. 

Area. 

ClRCUM. 

•5 

6866.1631 

293-7396 

.6 

6880.858 

294.O538 

•7 

6895.5685 

2943679 

.8 

6910.2948 

294.6821 

•9 

6925.0367 

294.9962 

94 

6939-7944 

295.3IO4 

.1 

6954.5678 

295.6246 

.2 

6969-3569 

295-9387 

•3 

6984.1616 

296.2529 

-4 

6998.9821 

296.567 

-5 

7013.8183 

296.8812 

.6 

7028.6703 

297-I954 

•7 

7043.5379 

297-5095 

.8 

7058.4212 

297.8237 

•9 

7073.3203 

298.I378 

95 

7088.235 

298.452 

.1 

7103.1655 

298.7662 

.2 

7118.II16 

299.0803 

•3 

7133-0735 

299-3945 

•4 

7148.0511 

299.7086 

•5 

7163.0443 

300.0228 

.6 

7 1 78.0533 

300.337 

•7 

7193.078 

300.6511 

.8 

7208.1185 

3OO.9653 

•9 

7223.1746 

301.2794 

96 

7238.2464 

3OI.5936 

.1 

72533339 

3OI.9O78 

.2 

7268.4372 

302.2219 

•3 

7283.5561 

302.5361 

•4 

7298.6908 

302.8502 

•5 

7313.8411 

3O3.1644 

.6 

7329  0072 

3O3.4786 

•7 

7344.189 

3O3.7927 

.8 

7359-3865 

3O4.I069 

•9  ' 

7374-5997 

304-42I 

97 

7389.8286 

304-7352 

.1 

7405.0732 

305.0494 

.2 

7420.3335 

305-3635 

•3 

7435.6096 

3O5.6777 

•4 

7450.9013 

3O5.9918 

•5 

7466.2087 

306.306 

.6 

7481.5319 

306.6202 

•7 

7496.8708 

306.9343 

.8 

7512.2253 

307.2485 

•9 

7527-5956 

3O7.5626 

98 

7542.9816 

307.876 8 

.1 

7558.3833 

308.191 

.2 

7573-8007 

308.5051 

-3 

7589-2338 

308.8193 

•4 

7604.6826 

309.1334 

-5 

7620.1471 

309.4476 

.6 

7635.6274 

309.7618 

•7 

7651.1233 

310.0759 

.8 

7666.635 

3103901 

•9 

7682.1623  | 

310.7042 

252  AREAS  AND  CIRCUMFERENCES  OF  CIRCLES. 


Diam. 

Area. 

ClRCUM. 

Diam. 

99 

7697.7054 

3II.O184 

•5 

.1 

77I3*2642 

311.3326 

.6 

.2 

7728.8337 

3II.6467 

•7 

•3 

7744.4288 

3II.9609 

.8 

•4 

7760.0347 

3I2.275 

•9 

Area. 


7775*6563 

779I,2937 

7806.9467 

7822.6154 

7838.2999 


712.5892 

312.9034 

3I3*2I75 

3i3*53i7 

3I3-8458 


To  Compute  Area  or  Circumference  of  a Diameter  greater 
than  any  in.  preceding  Table. 

See  Rules,  pages  235-6  and  241-2. 

Or,  If  Diameter  exceeds  100  and  is  less  than  1001. 

Put  a decimal  point,  and  take  out  area  or  circumference  as  for  a Whole 
Number  by  removing  decimal  point,  if  for  an  area,  two  places  to  right , and 
if  for  a circumference,  one  place. 

Example.— What  is  area  and  what  circumference  of  a circle  967  feet  in  diame- 

tGArea  of  96.7  is  7344.189;  hence,  for  967  it  is  734  418.9;  and  circumference  of  96.7 
is  303.7927,  and  for  967  it  is  3037.927. 

To  Compute  Area  and  Circumference  of  a Circle  by  Log- 
arithms. 

See  Rules,  pages  236,  242. 

Areas  and.  Circumferences  of  Circles. 

From  i to  50  Feet  {advancing  by  an  Inch). 

Or,  From  i to  50  Inches  {advancing  by  a Twelfth). 


Diam. 

Area. 

ClRCUM. 

Diam. 

Area. 

ClRCUM. 

I ft' 

1 

2 

Feet. 

.7854 

•92I7 

1.069 

Feet. 

3*I4l6 

3*4034 

3.6652 

3 A 

1 

2 

Feet. 

7.0686 

7.4668 

7.8758 

Feet. 

9.4248 

9,6866 

9.9484 

3 

1.2272 

3*927 

3 

8.2958 

10.2102 

A 

*•3963 

4.1888 

4 

8.7267 

IO.472 

T- 

5 

6 

i*5763 

1.7671 

4.4506 

4*7I24 

5 

6 

9.1685 

9.6211 

10.7338 

IO.9956 

7 

1.969 

4*9742 

7 

IO.0848 

II.2574 

8 

2.1817 

5*236 

8 

10.5593 

II.5192 

9 

2.4053 

5*4978 

9 

II.O447 

II.781 

10 

2.6398 

5*7596 

10 

11*541 

I2.O428 

11 

2.8853 

6.0214 

11 

12.0483 

12.3046 

2 ft. 

3*I4l6 

6.2832 

4 /*• 

12.5664 

12.5664 

1 

3.4088 

6-545 

1 

130955 

12.8282 

2 

3.687 

6.8068 

2 

I3-6354 

I3.O9 

3 

4 

3-9761 

4.2761 

7.0686 

7*3304 

3 

4 

14.1863 

14.7481 

I3-35I8 

13.6136 

5 

4.5869 

7.5922 

5 

15.3208 

I3-8754 

6 

4.9087 

7*854 

6 

I5-9043 

14.1372 

7 

8 

5*24i5 

5.5852 

8.1158 

8.3776 

7 

8 

16.4989 

17.1043 

I4.499 

14.6608 

9 

10 

11 

5-9396 

6.305 

6.6814 

8.6394 
8.9012 
9 i63 

9 

10 

1 11 

17.7206 

18.3478 

18.9859 

14.9226 
15.1844 
15  4462 

AREAS  AND  CIRCUMFERENCES  OF  CIRCLES.  253 


Diam 

Area. 

ClRCUM. 

Feet. 

Feet. 

5 A 

I9-635 

15.708 

1 

20.2949 

15.9698 

2 

20.9658 

16.2316 

3 

21.6476 

16.4934 

4 

22.3403 

16.7552 

5 

23  0439 

I7.OI7 

6 

23-7583 

17.2788 

7 

24.4837 

17.5406 

8 

25.22 

17.8024 

9 

25-9673 

18.0642 

10 

26.7254 

18.326 

11 

27.4944 

18.5878 

6 ft, 

28.2744 

18.8496 

I 

29.0653 

I9.III4 

2 

29.867 

I9-3732 

3 

30.6797 

I9-635 

4 

3I-5033 

19.8968 

5 

32.3378 

20.1586 

6 

33-I83I 

20.4204 

7 

34.0394 

20.6822 

8 

34.9067 

20.944 

9 

35-7848 

21.2058 

10 

36.6738 

21.4676 

11 

37-5738 

21.7294 

7/<- 

38.4846 

21.9912 

1 

39.4064 

22.253 

2 

40.339 

22.5148 

3 

41.2826 

22.7766 

4 

42.2371 

23.0384 

5 

43.2025 

23.3002 

6 

44.1787 

23.562 

7 

45-I659 

23.8238 

8 

46.164.I 

24.0856 

9 

47.I73I 

24-3474 

10 

48.193 

24.6092 

11 

49.2238 

24.871 

8 ft. 

50.2656 

25.1328 

1 

5I-3I83 

25-3946 

2 

52.3818 

25.6564 

3 

53.4563 

25.9182 

4 

54-54I7 

26.18 

5 

55.638 

26.4418 

6 

56.7451 

26.7036 

7 

57.8632 

26.9654 

8 

58.9923 

27.2272 

9 

60.1322 

27.489 

10 

61.283 

27.7508 

11 

62.4448 

28.0126 

9 ft' 

63.6174 

28.2744 

1 

64.801 

28.5362 

2 

65.9954 

28.798 

3 

67.2008 

29.0598 

4 

68.417 

29.3216 

5 

69.6442 

29-5834 

Diam. 

Area. 

ClRCUM. 

6 

Feet. 

70.8823 

Feet. 

29.8452 

7 

72-i3I4 

3O.IO7 

8 

73.3913 

30.3688 

9 

74.6621 

30.6306 

10 

75-9439 

30.8924 

11 

77-2365 

31.1542 

10  ft. 

78.54 

31.416 

1 

79-8545 

31.6778 

2 

81.1798 

3I-9396 

3 

82.5161 

32.2014 

4 

83-8633 

32.4632 

5 

85.2214 

32.725 

6 

86.5903 

32.9868 

7 

87.9703 

33.2486 

8 

89.3611 

33-5104 

9 

90.7628 

33.7722 

10 

92.1754 

34-034 

11 

93-599 

34-2958 

11ft. 

95-0334 

34-5576 

1 

96.4787 

34.8194 

2 

97-935 

35.0812 

3 

99.4022 

35-343 

4 

100.8803 

35.6048 

5 

102.3693 

35.8666 

6 

103.8691 

36.1284 

7 

105.38 

36.3902 

8 

106.9017 

36.652 

9 

108.4343 

36.9138 

10 

109.9778 

37-I756 

11 

in-5323 

37-4374 

12  ft. 

113.0976 

37.6992 

1 

114.6739 

37.961 

2 

116.261 

38.2228 

3 

117.8591 

38.4846 

4 

119.468 

38.7464 

5 

121.088 

39.0082 

6 

122.7187 

39-27 

7 

124.3605 

39-53I8 

8 

126.0131 

39-7936 

9 

127.6766 

40.0554 

10 

129.351 

4°-3I72 

11 

131.0366 

40.579 

13  A 

132.7326 

40.8408 

1 

134.4398 

41.1026 

2 

136.1578 

41.3644 

3 

137.8868 

41.6262 

4 

139.6267 

41.888 

5 

I4I-3774 

42.1498 

6 

I43-I39I 

42.4116 

7 

I44-9II7 

42.6734 

8 

146.6953 

42.9352 

9 

148.4897 

43-197 

10 

150.295 

43-4588 

11 

152.1113 

43.7206 

Y 


\ J 

>1  AM. 

I ft- 

I 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

5 ft* 

i 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

6/*. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

iT  ft. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

i8  ft. 

i 

2 

3 

4 

5 


IE  AS  AND  CIRCUMFERENCES  OF  CIRCLE! 


Feet. 

1539384 

I55-7764 

157.6254 

I59-4853 

161.3561 

163.2378 

165.1303 

167.0338 

168.9483 

170.8736 

172.8098 

I74-7569 

i76-7i5 

178.684 

180.6638 

182.6546 

184.6563 

186.6689 

188.6924 

190.7267 

192. 7721 

194.8283 

196.8954 

198.9734 

201.0624 

203.1622 

205.273 

207.3947 

209.5273 

211.6707 

213.8252 

215.9904 

218.1667 

220.3538 

222.5518 

224.7607 

226.9806 

229.2113 

23I-453 

233-7056 

235.9691 

238.2434 

240.5287 

242.8249 

245. 132 1 

247.4501 

249-779 

252.1188 

254.4696 

256.8312 

259.2038 

261.5873 

263.9817 

266.3869 


ClRCUM. 

Diam. 

Area. 

Feet. 

Feet. 

43.9824 

6 

268.8031 

44.2442 

7 

271.2302 

44.506 

8 

273.6683 

44.7678 

9 

276.II72 

45.0296 

IO 

278.577 

45.2914 

11 

281.0477 

45-5532 

19  .A 

283.5294 

45-815 

1 

286.0219 

46.0768 

2 

288.5255 

46.3386 

3 

291.0398 

46.6004 

4 

293.5651 

46.8622 

5 

296.IOI2 

47.124 

6 

298.6483 

47-3858 

7 

3OI.2064 

47-6476 

8 

303-7753 

47.9094 

9 

306.3551 

48.1712 

IO 

308.9458 

48.433 

11 

3H-5475 

48.6948 

20  j65. 

314.16 

48.9566 

1 

316.7834 

49.2184 

2 

319.4178 

49.4802 

3 

322.0631 

49.742 

4 

324-7I93 

50.0038 

5 

327.3864 

50.2656 

6 

330.0643 

50-5274 

7 

332.7532 

50.7892 

8 

335-4531 

51-051 

9 

338.1638 

51.3128 

IO 

340.8854 

5I-5746 

11 

343.618 

51.8364 

21  ft. 

346.3614 

52.0982 

1 

349-H57 

52-36 

2 

351.881 

52.6218 

3 

354-6572 

52.8836 

4 

357-4442 

53-I454 

5 

360.2422 

53-4072 

6 

363.0511 

53-669 

7 

365.8709 

53-93o8 

8 

368.7017 

54.1926 

9 

371-5433 

54-4544 

IO 

374-3958 

54.7162 

i -11 

377-2592 

54-978 

! 22  ft. 

380.1336 

55-2398 

1 

383.0188  ! 

55-5oi6 

2 

385-9I5  1 

55-7634 

3 

388.8221 

56.0252 

4 

391-74 

56.287 

5 

394.6689 

I 56.5488 

6 

397.6087 

56.8106 

7 

400.5594 

57.0724 

8 

403.521 1 

| 57-3342 

9 

406.4936 

! 57  596 

IO 

409.477 

l 57-8578 

11 

412.4713 

AREAS  AND  CIRCUMFERENCES  OF  CIRCLES.  255 


Diam. 

Area. 

ClRCUM. 

Feet. 

Feet. 

2 3 A 

415.4766 

72.2568 

I 

418.4927 

72.5186 

2 

421.5198 

72.7804 

3 

424-5578 

73.0422 

4 

427.6067 

73-304 

5 

4.IO.6664 

73-5658 

6 

433-7371 

73.8276 

7 

436.8187 

74.0894 

8 

439-91 

74-3512 

9 

443.0147 

74-6i3 

IO 

446.129 

74.8748 

ii 

449.2542 

75.1366 

24  A 

452.3904 

75-3984 

I 

455-5374 

75.6602 

2 

458.6954 

75.922 

3 

461.8643 

76.1838 

4 

465.044 

76.4456 

5 

468.2347 

76.7074 

6 

47I-4363 

76.9692 

7 

474.6488 

77.231 

8 

477-872 3 

77.4928 

9 

481.1066 

77-7546 

IO 

484.3518 

78.0164 

ii 

487.6076 

78.2782 

25  A 

490.875 

78.54 

1 

494.1529 

78.8018 

2 

497.4418 

79.0636 

3 

500.7416 

793254 

4 

504.0523 

79.5872 

5 

507-3738 

79.849 

6 

510.7063 

80.1108 

7 

514.0485 

80.3726 

8 

517.404 

80.6344 

9 

520.7693 

80.8962 

IO 

524.1454 

81.158 

ii 

527.5324 

81.4198 

26  ft. 

530.9304 

81.6816 

1 

534-3397 

81.9434 

2 

537-759 

82.2052 

3 

541.1897 

82.467 

4 

544-63I3 

82.7288 

5 

548.0837 

82.9906 

6 

55I-547I 

83.2524 

7 

555-0214 

83.5142 

8 

558.5066 

83.776 

9 

562.0028 

84.0378 

IO 

565.5098 

84.2996 

11 

569.0277 

84.5614 

27  A 

572.5566 

84.8232 

1 

576.0963 

85.085 

2 

579.6467 

85.3468 

3 

583.2086 

85.6086 

4 

586.781 

85.8704 

5 

590.3644 

86.1322 

Diam. 

Area. 

ClRCUM. 

Feet. 

Feet. 

6 

593-9587 

86.394 

7 

597-5639 

86.6558 

8 

60I.18 

86.9176 

9 

604.8071 

87.1794 

10 

608.445 

87.4412 

11 

612.0938 

87.703 

to 

00 

> 

6I5-7536 

87.9648 

1 

619.4242 

88.2266 

2 

623.IO58 

88.4884 

3 

626.7983 

88.7502 

4 

630.5016 

89.OI2 

5 

634.2159 

89.2738 

6 

637.94II 

89-5356 

7 

641.6772 

89.7974 

8 

645.4243 

90.0592 

9 

649.1822 

90.321 

10 

652.951 

90.5828 

11 

656.7307 

90.8446 

29  A 

660.5214 

91.1064 

1 

664.3229 

91.3682 

2 

668.1354 

9t-63 

3 

671.9588 

91.8918 

4 

675-793I 

92.1536 

5 

679.6382 

92.4154 

6 

683.4943 

92.6772 

7 

687.3613 

92.939 

8 

691.2393 

93.2008 

9 

695.1281 

93.4626 

10 

699.0278 

93-7244 

11 

702.9384 

93.9862 

2>o  ft- 

706.86 

94.248 

1 

710.7924 

94.5098 

2 

7I4-7358 

94.7716 

3 

718.6901 

95-0334 

4 

722.6553 

95.2952 

5 

726.6313 

95-557 

6 

730.6183 

95.8188 

7 

734.6162 

96.0806 

8 

738.6251 

96.3424 

9 

742.6448 

96.6042 

10 

746.6754 

96.866 

11 

750.7164 

97.1278 

3l/^ 

754.7694 

97.3896 

1 

758.8327 

97.6514 

2 

762.907 

97.9132 

3 

766.9922 

98.175 

4 

771.0883 

98.4368. 

5 

775-1952 

98.6986 

6 

779-3I3I 

98.9604 

7 

783.4419 

99.2222 

8 

787.5817 

99.484 

9 

79I-7323 

99-7458 

10 

795.8938 

100.0076 

11 

800.0662 

100.2694 

256  AREAS  AND  CIRCUMFERENCES  OF  CIRCLES. 


Diam. 

Area. 

ClRCUM. 

Diam. 

32  ft. 

Feet. 

804.2496 

Feet. 

IOO.5312 

6 

I 

808.4439 

IOO.793 

7 

2 

812.649 

IOI.0548 

8 

3 

816.8651 

IOI.3166 

9 

4 ’ 

821.092 

IOI.5784 

10 

5 

825.3299 

IOI.8402 

11 

6 

829.5787 

102. 102 

37  A 

7 

833-8384 

IO2.3638 

1 

8 

838.IO9I 

IO2.6256 

2 

9 

842.3906 

IO2.8874 

3 

10 

846.683 

I03-I492 

4 

11 

850.9863 

IO3.4I  I 

5 

33  ft: 

855-3006 

IO3.6728 

6 

I 

859.6257 

I03-9346 

7 

2 

863.9618 

104.1964 

8 

3 

868.3088 

104.4582 

9 

4 

872.6667 

104.72 

10 

5 

877-0354 

104.9818 

11 

6 

881.4151 

105.2436 

38  A 

7 

885.8057 

105-5054 

1 

8 

890.2073 

105.7672 

2 

9 

894.6197 

106.029 

3 

10 

899.043 

106.2908 

4 

11 

903.4772 

106.5526 

5 

34  A 

907.9224 

106.8144 

6 

1 

912.3784 

107.0762 

7 

2 

916.8454 

107.338 

8 

3 

92I-3233 

107.5998 

9 

4 

925.812 

107.8616 

10 

5 

930.31 1 7 

108.1234 

11 

6 

934.8223 

108.3852 

39  A 

7 

939-3439 

108.647 

1 

8 

943.8763 

108.9088 

2 

9 

948.4196 

109.1706 

3 

10 

952.9738 

109.4324 

4 

11 

957-5392 

109.6942 

5 

35  A 

962.115 

109.956 

6 

1 

966.7019 

110.2178 

7 

2 

971.2998 

110.4796 

8 

3 

975.9086 

1 10.7414 

9 

4 

980.5287 

hi  .0032 

10 

5 

985.1588 

111.265 

11 

6 

989.8005 

111.5268 

40  A- 

7 

994-4527 

111.7886 

1 

8 

999.116 

1 1 2.0504 

2 

9 

1003.7903 

112.3122 

3 

. 10 

1008.4754 

112.574 

4 

11 

1013.1714 

112.8358 

5 

36A 

1017.8784 

113.0976 

6 

1 

1022.5962 

1 13-3594 

7 

2 

1027.325 

113.6212 

8 

3 

1032.0647 

113.883 

9 

4 

1036.8153 

114.1448 

10 

5 

1041.5767 

114.4066 

11 

ClRCUM. 


Feet. 

IO46.3491 
IO51.1324 
IO55.9266 
1060.7318 
1065.5478 
IO7O.3747 
1075.2126 
1080.0613 
1084.921 
1089.7916 
IO94.673I 
1099.5654 
I IO4.4687 
IIO9.3829 
III4.308 
III9.244I 
II24.I9I 
II29.I489 
1 134.  H 76 
1139.0972 
II44.0878 
II49.0893 
II54.IOI7 

H59*I249 

1164.1591 

1169.2042 

1174.2603 

n 79.3272 

1184.405 

1189.4937 

H94-5934 

1199.7039 

1204.8254 

1209.9578 

1215.101 

1220.2552 

1225.4203 

1230.5963 

i235-7833 

1240.9811 

1246.1898 

1251.4094 

1256.64 

1261.8814 

1267.1338 

1272.3971 

1277.6712 

1282.9563 

1288.2523 

I293-5592 

1298.877 

1304.2058 

i3°9-5454 

i3i4-8959 


Feet. 

II4.6684 

1 14.9302 

II5.I92 

II5-4538 

II5-7I56 

H5-9774 

116.2392 

116.501 

116.7628 

117.0246 

117.2864 

117.5482 

117.81 

118.0718 

118.3336 

118.5954 

118.8572 

119.119 

119.3808 

119.6426 

1 19.9044 

120.1662 

120.428 

120.6898 

120.9516 

121.2134 

121.4758 

121.737 

121.9988 

122.2606 

122.5224 

122.7848 

123.046 

123.3078 

123.5696 

123.8314 

124.0932 

124-355 

124.6168 

124.8786 

125. 1404 

125.4022 

125.664 

125.9258 

126.1876 

126.4494 

126.7112 

126.973 

127.2348 

127.4966 

127.7584 

128.0202 

128.282 

128.5438 


A 

i 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

A 

i 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

A 

i 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

A 

i 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

A 

i 

2 

3 

4 

5 


AND  CIRCUMFERENCES  OF  CIRCLES.  257 


Area. 


Feet. 

132O.2574 

1325.6297 

I33I-OI3 

I33^*4°72 

1341.8123 

1347.2282 

1352.6551 

1358.0929 

1363-5416 

1369.0013 

i374-47i8 

1379-9532 

I385-4456 

1390.9488 

1396.463 

1401.9881 

1407.5241 

1413.0709 

1418.6287 

1424.1974 

1429.777 

1435-3676 

1440.969 

1446.5813 

1452.2046 

1457.8387 

1463-4838 

1469.1398 

1474.8066 

1480.4844 

1486.1731 

1491.8717 

1497.5833 

1503.3047 

1509037 

1514.7802 

1520.5344 

1526.2994 

1532.0754 

1537.8623 

1543.66 

1549.4687 

1555-2883 

1561.1188 

1566.9603 

1572.8126 

15786756 

1584.5499 

1590.435 

1596.3309 

1602.2378 

1608.1556 

1614.0843 

1620.0238 


ClRCUM. 

Feet. 

128.8056 

129.0674 

I29.3292 

129.591 

129.852  8 

130.1146 

130.3764 

130.6382 

130.9 

131.1618 

131.4236 

131.6854 

I3I-9472 

132.209 

132.4708 

132.7326 

132.9944 

133.2562 

i33-5i8 

133- 7798 

134- °4I6 

134- 3034 
134.5652 
134.827 

135.0888 

iV35-35o6 

135.6124 

135- 8742 

136.136 

136.3978 

136.6596 

136.9214 

137.1832 

137- 445 
137.7068 
137.9686 
138.2304 
138.4922 

138- 754 
139.0158 
139.2776 

139- 5394 
139.8012 
140.063 
140.3248 
140.5866 
140.8484 
141.1102 

I4I-372 

141.6338 

141.8956 

142.1574 

142.4192 

142.681 


Diam. 

Area. 

ClRCUM. 

Feet. 

Feet. 

6 

1625.9743 

142.9428 

7 

1631.9357 

143.2046 

8 

1637.9081 

143.4664 

9 

1643.8913 

143.7282 

10 

1649.8854 

143-99 

11 

1655.8904 

144.2518 

46  A 

1661.9064 

144.5136 

1 

1667.9332 

144-7754 

2 

1673.971 

145.0372 

3 

1680.0197 

145.299 

4 

1686.0792 

145.5608 

5 

1692.1497 

145.8226 

6 

1698.2311 

146.0844 

7 

1704.3195 

146.3462 

8 

I7IO.4267 

146.608 

9 

1716.5408 

146.8698 

10 

1722.6658 

147.1316 

11 

1728.8017 

147-3934 

47  A- 

1734.9486 

147.6552 

1 

1741.1063 

147  917 

2 

1747.275 

148.1788 

3 

I753-4546 

148.44.06 

4 

1 759.645 1 

148.7024 

5 

1765.8464 

148.9642 

6 

1772.0587 

149.226 

7 

1778.2819 

149.4878 

8 

1784.516 

149.7496 

9 

1790.7611 

1 50.01 14 

10 

I797-OI7 

150.2732 

11 

1803.2838 

150.535 

48  A- 

1809.5616 

150.7968 

1 

1815.8502 

151.0586 

2 

1822.1498 

1 5 1. 3204 

3 

1828.4603 

151.5822 

4 

1834.7817 

151.844 

5 

1841.1139 

152.1058 

- 6 

1847.4571 

152.3676 

7 

1853.8112 

152.6294 

8 

1860.1763 

152.8912 

9 

1866.5522 

I53-I53 

IO 

1872.939 

153.4148 

11 

1879.3367 

153  6766 

49  A 

1885.7454 

153-9384 

1 

1892.1649 

154.2002 

2 

1898.5954 

154.462 

3 

1905.0368 

154.7238 

4 

1911.4897 

154.9856 

5 

1917.9522 

155.2474. 

6 

1924.4263 

155-5092 

7 

I93°-9II3 

I55-77I 

8 

I937-4073 

156.0328 

9 

I943-9I42 

156.2946 

IO 

1950.4318 

156.5564 

11 

1956.9604 

156.8182 

50  A 

I963-5 

i57-o8 

258 


SIDES  OF  SQUARES  EQUAL  TO  AREAS. 


Sides  of  Squares— equal  in.  Area  to 

Diameter  from  1 to  100. 

Side  of  Sq. 


)iam. 

Side  of  Sq. 

Diam.  S 

I 

.8862 

14  ] 

X 

I.IO78 

X 3 

X 

I-3293 

X ] 

% 

1.5509 

% 

2 

I.7724 

15 

X 

I.994 

X 

X 

2.2156 

X 

% 

2.4371 

% 

3 

2.6587 

16 

2.8802 

X 

3.IO18 

X 

X 

3-3233 

% 

4 

3-5449 

17 

X 

3-7665 

X 

X 

3.988 

X 

% 

4.2096 

% 

5 

4-4311 

18 

X 

4.6527 

X 

X 

4.8742 

X 

% 

5-0958 

% 

6 

5-3T74 

X 

5.5389 

X 

X 

5-7605 

X 

% 

5.982 

% 

7 

6.2036 

20 

X 

6.4251 

X 

X 

6.6467 

X 

% 

6.8683 

% 

8 

7.0898 

21 

X 

7-3TI4 

X 

X 

7-5329 

X 

X 

7-7545 

% 

9 

7.976 

22 

X 

8.1976 

X 

X 

8.4192 

X 

% 

8.6407 

% 

10 

8.8623 

23 

X 

9.0838 

X 

X 

9-3054 

X 

% 

9.5269 

H 

II 

9-7485 

24 

X 

9-97 

X 

X 

10.1916 

X 

% 

10.4132 

X 

12 

10.6347 

25 

X 

10.8563 

X 

X 

11.0778 

X 

% 

11.2994 

% 

13 

11.5209 

26 

X 

11.7425 

X 

X 

11.9641 

X 

% 

12.1856 

% 

14.4012 

14.6227 

14.8443 

15.0659 

15.2874 

I5-509 

I5-7305 

15-9521 

16.1736 

16.3952 

16.6168 

16.8383 

17.0599 

17.2814 

I7-503 

17.7245 

17.9461 

18.1677 

18.3892 

18.6108 

18.8323 

19-0539 

19.2754 

19.497 

19.7185 

19.94OI 

20.1617 

20.3832 

20.6048 

20.8263 

21.0479 

21.2694 

21.491 

21.7126 

21.9341 

22.1557 

22.3772 

22.5988 

22.8203 

23.0419 

23.2634 

23-485 

23.7066 


27 


28 


29 


X 


30 


X 

X 

% 

I 

¥ 


32 


3^ 


33^ 


34 


X 


35 


36 


X 


37 


38 


3^ 


X 


% 

39x 

% 


23.9281 

24.1497 

24.3712 

24.5928 

24.8144 

25-0359 

25-2575 

25-479 

25.7006 

25.9221 

26.1437 

26.3653 

26.5868 

26.8084 

27.0299 

27-2515 

27-473 

27.6946 

27.9161 

28.1377 

28.3593 

28.5808 

28.8024 

29.0239 

29- 2455 
29.467 
29.6886 
29.9102 
30.1317 

30- 3533 
30.5748 
30.7964 
31.0179 

31- 2395 
31.4611 
31.6826 
31.9042 
32.1257 

32- 3473 

32.5688 

32.7904 
I 33  0112 

33- 2335 

33- 4551 
33.6766 
33.8982 

34- H97 

34- 3413 
34.5628 
34.7884 
35.006 

35- 2275 


a Circle. 


Diam. 

Side  of  Sq. 

40 

35-4491 

X 

35.6706 

X 

35.8922 

% 

36-1137 

41 

36.3353 

X 

36.5569 

X 

36.7784 

42 


3€ 


43 


% 

44 

X 

X 

% 

45 


46 

X 

% 

47 

X 


49 


X 

X 

% 


X 

X 

% 

50 

X 


51 


X 

X 

% 


52 


X 


37 

37-2215 

37-4431 

37.6646 

37.8862 

38.1078 

38.3293 

385509 

38.7724 

38.994 

39-2155 

39-4371 

39-6587 

39.8802 

40.1018 

40.3233 

40.5449 

40.7664 

40.988 

41.2096 

4I-4311 

41.6527 

41.8742 

42.0958 

42.3173 

42.539 

42.7604 

42.982 

43.2036 

43- 425I 
43.6467 
43.8682 
44.0898 

44- 3H3 
44-5329 
44-7545 
44.976 
45.1976 
45.4191 
45.6407 
45.8622 
46.0838 

46-3054 

46.5269 

46-7485 


SIDES  OF  SQUARES  EQUAL  TO  AREAS. 


Diam. 

Side  of  Sq. 

1 Diam. 

Side  of  Sq. 

Diam. 

Side  of  Sq. 

I Diam 

53w 

46.97 

65 

57.6047 

77 

68.2395 

89 

34 

A 

47.1916 

A 

57.8263 

X 

68.461 

34 

47-4I3I 

34 

58.0479 

A 

68.6826 

3^ 

54 

47-6347 

54 

58.2694 

% 

68.9041 

54 

54 

47.8562 

66 

58.491 

78 

69.1257 

90 

Va 

a 

48.0778 

X 

58.7125 

34 

69.3473 

a 

48.2994 

A 

58.9341 

A 

69.5688 

7± 

1/ 

% 

48.5209 

% 

59-1556 

54 

69.7904 

72 

55 

48.7425 

67 

59-3772 

79 

7O.OII9 

/4 

a 

48.964 

X 

59-5988 

34 

70.2335 

91 

a 

49- 1 856 

A 

59.8203 

X 

70.455 

A 

54 

49.4071 

54 

60.0419 

% 

70.6766 

Ai 

56 

49.6287 

68 

60.2634 

80 

70.8981 

54 

3€ 

49-8503 

% 

60.485 

34 

7I-II97 

92 

3? 

50.0718 

A 

60.7065 

34 

7I-34I3 

3i 

% 

50.2934 

% 

60.9281 

Si 

71.5628 

34 

57  w 

50.5H9 

69 

61.1497 

81 

7 x-  7844 

54 

A 

50.7365 

* 

61.3712 

A 

72.0059 

A 

50958 

61.5928 

A 

72.2275 

34 

54 

51.1796 

h 

61.8143 

% 

72.4491 

34 

58 

51.4012 

70 

62.0359 

82 

72.6706 

54 

A 

51.6227 

1 « 

62.2574 

34 

72.8921 

94 

3? 

5I-8443 

i x 

62.479 

X 

73-II37 

% 

52.0658 

i 54 

62.7006 

Si 

73-3353 

A 

\/ 

59 

52.2874 

71 

62.9221 

83 

73-5568 

A 

3? 

A 

52.5089 

34 

63-I437 

34 

73.7784 

3 1 

52.7305 

x 

63-3652 

X 

73-9999 

95 

54 

52.9521  , 

% 

63.5868 

Si 

74.2215 

A 

60 

53-I736 

72 

63.8083 

84 

74-4431 

A 

A 

53-3952 

A 

64.0299 

34 

74.6647 

% 

3? 

53-6167 

A 

64.2514 

A* 

74.8862 

96 

5i 

53-8383 

54 

64.4730 

% 

75.1077 

X 

61 

54-0598 

73 

64.6946 

85 

75-3293 

34 

a 

54.2814 

! 34 

64.9161 

A 

75.55o8 

54 

3? 

% 

54-503  1 
54-7245 

X 

54 

65.1377 

65.3592 

A 

% 

75.7724 

75-9934 

97 

1/ 

62 

54.9461 

74 

65.5808 

86 

76.2155 

/4r 

34 

a 

55.1676 

34 

65.8023 

A 

76.4371 

54 

a 

55-3892 

A 

66.0239 

A 

76.6586 

% 

55  6107  | 

% 

66.2455 

% 

76.8802 

98 

63 

55.8323 

75 

66.467 

87 

77.1017 

A 

1/ 

a 

a 

56.0538 

56.2754 

34 

34 

66  6886 
66.9104 

A 

A 

77-3233 

77-5449 

A 

54 

54 

56.497 

Si 

67.1317 

% 

77.7664 

99 

64 

56.7185 

76 

67-3532 

88 

77.988 

A 

A 

56.9401 

A 

67.5748 

34 

78.2095 

A 

3? 

57.1616 

A | 

67.7964 

A 

78.4316  I 

54 

54 

57-3832  i! 

Si  1 

68.0179 

% 

78.6526  j 

100 

Application  of  Table. 

To  Ascertain,  a Square  tliat  Las  same  Area  a 
Circle. 

Illus.— If  side  of  a square  that  has  same  area  as  a circle  of  7-3  2s  i 
By  Table  of  Areas,  page  233,  opposite  to  73.25  is  4214.11;  and  i 
04.9161,  which  is  side  of  a square  having  same  area  as  a circle  of  tha 


259 

Side  of  Sq. 

78.8742 

79-0957 

79-3173 

79-5389 

79.7604 

79.982 

80.2035 

80.4251 

80.6467 

80.8682 

81.0898 

81.3113 

81.5329 

81.7544 

81.976 

82.1975 

82.4191 

82.6407 

82.8622 

83.0838 

83-3053 

83.5269 

83.7484 

83-97 

84.1916 

84.4131 

84.6347 

84.8562 

85.0778 

85-2993 

85.5209 

85-7425 

85.9646 

86.185 

86.4071 

86.6289 

86.8502 

87.0718 

87-2933 

87-5449 
: 87.7364 
! 87.958 
I 88  1796 
| 88.4011 

I 88.6227 


a Griven 

i.  is  required, 
this  table  is 
diameter. 


26o 


LENGTHS  OF  CIRCULAR  ARCS. 


Lengths  of  Circular  Arcs,  up  to  a Seiuicircle. 
Diameter  of  a Circle  — i,  and  divided  into  1000  equal  Parts. 


H’ght. 

Length. 

H’ght. 

Length.  | I 

rght.; 

Length.  1 1 

H’ght. 

Length.  I 

I’ght.;  I 

.1 

I.O2645 

•15 

I.O58Q6 

.2  1 : 

[.IO348 

-25 

I- 159  12  - 

3 1 

.IOI 

I.O2698 

•151 

I.05973! 

.201  : 

1. 104  47 

.251 

I.16033  . 

3QI  1 

.102 

I.O27  52 

.152 

I.06051  I 

.202 

1.105  48 

.252 

I.16157  . 

,302  1 

.103 

I.O2806 

•153 

I.061  3 

.203 

I.I065 

-253 

I.16279  . 

•303  3 

.IO4 

1.0286 

•154 

1 .062  09 

.204 

1.107  52 

•254 

I.16402 

■304  3 

.105 

I.O29  14 

.155 

1.062  88 

.205 

I.IO855 

-255 

I.16526 

.305  3 

.106 

I.O297 

.156 

1 .063  68 

.206 

I.IO958 

.256 

I.16649 

.306  3 

.107 

1 .030  26 

.157 

1.06449 

.207 

I.II062 

-257 

I.167  74 

•307  ] 

.IO8 

1 .030  82 

.158 

1.0653 

.208 

I. II165 

.258 

I.16899 

.308  ] 

.IO9 

1.03139 

-159 

1.066 11 

.209 

I.II269 

-259 

I.17O  24 

.309  : 

.11 

I.03196 

.16 

1.06693 

.21 

I.II374 

.26 

I-I7I5 

•31 

.III 

I.O32  54 

.l6l 

1.067  75 

.211 

I.II479 

.261 

1. 17275 

•333 

.112 

I.O33  12 

.162 

1 .068  58 

.212 

I.II584 

.262 

1.17401 

•312 

•II3 

1-033  7 1 

.163 

1.06941 

-213 ! 

I.I1692 

.263 

1.17527 

•333 

.114 

1.0343 

.164 

1.07025 

.214 

I.II796 

.264 

1.17655 

•334 

.115 

1.0349 

.165 

1. 07 1 09 

.215 

I.II904 

.265 

I.17784 

•335 

.Il6 

1-03551 

.166 

1. 07 1 94 

.216 

1 .120  1 1 

.266 

I.I79I2 

•3l6 

-II7 

1.03611 

.167 

1.072  79 

.217 

I.I2I  l8 

.267 

I.1804 

•337 

.118 

1 .036  72 

.168 

1-073  65 

.218 

1 .122  25 

.268 

I.18162 

.318 

.119 

1-037  34 

.169 

1.074  51 

.219  ! 

1.123  34 

.269 

I.182  94 

•339 

.12 

1.03797 

•17 

1-075  37 

.22 

1.12445 

.27 

I.18428 

•32 

.121 

1.0386 

.171 

1.076  24 

.221 

1. 125  56 

.271 

I.18557 

.321 

.122 

1.03923 

.172 

1.077  11 

.222 

1.12663 

.272 

1.186  88 

.322 

.123 

1.03987 

•173 

1.07799 

.223 

1. 12774 

•273 

1. 188 19 

•323 

.124 

1.040  51 

.174 

1.07888 

.224 

1.12885 

•274 

1.18969 

•324 

.125 

1. 041 16 

• 175 

1.07977 

.225 

1. 129  97 

.275 

1.19082 

.325 

.126 

1.04 1 81 

.176 

1.08066 

.226 

1.13108 

.276 

1. 192 14 

.326 

.127 

1.04247 

.177 

1. 08 1 56 

.227 

1. 132 19 

•277 

I-I9345 

•327 

.128 

1-04313 

.178 

1.082  46 

.228 

II333I 

.278 

1.19477 

.328 

.129 

1.0438 

.179 

1-08337 

.229 

1.13444 

.279 

1.196 1 

.329 

.13 

1.04447 

.18 

1 .084  28 

•23 

1. 13557 

.28 

1. 197  43 

•33 

•I31 

1-045  15 

.l8l 

1.085  19 

.231 

1.13671 

.281 

1.19887 

•331 

.132 

1.045  84 

.182 

1.086 11 

.232 

1.13786 

.282 

1.200 11 

.332 

•1.33 

1.04652 

.183 

1.08704 

•233 

I-I3903 

.283 

1. 201 46 

•333 

• I34 

1.04722 

.184 

1.08797 

•234 

1. 140  2 

.284 

1.202  82 

•334 

•I35 

1 .047  92 

.185 

1 .088  9 

•235 

1.141 36 

.285 

1.204 19 

-335 

.136 

1.048  62 

.186 

1.089  84 

.236 

1. 142  47 

.286 

1.205  58 

•336 

.137 

1.04932 

.187 

1.090  79 

•237 

i-i4363 

.287 

1.20696 

•337 

.138 

1.05003 

.188 

1. 091  74 

.238 

1.1448 

.288 

1.208  28 

•338 

.139 

1-05075 

.189 

1.09269 

•239 

I-I4597 

.289 

1.20967 

•339 

.14 

1. 05 1 47 

.19 

1.09365 

.24 

I-I47I4 

.29 

1. 21202 

•34 

.141 

1 .052  2 

.191 

1.09461 

.241 

1.14831 

.291 

1.21239 

•341 

.142 

1.05293 

, .I92 

1-095  57 

.242 

1. 149  49 

► .292 

1.21381 

■342 

• i43 

1.05367 

•193 

1.09654 

•243 

1.15067 

’ -293 

i 1.2152 

•343 

.144 

1.054  41 

.194 

1-09752 

.244 

1.15186 

> .294 

. 1.21658 

•344 

.145 

1.05516 

* -195 

1.0985 

-245 

I-I53 

> .295 

: 1.21794 

*345 

.146 

• 1-05591 

.196 

1 1.09949 

1 .246 

• I-I542S 

) .296 

) 1.21926 

•346 

.147 

' 1.05667 

' I -197 

1.10048 

1 -247 

I-I554S 

) .297  1.220  01 

•347 

.148 

1 1-05745 

! -198 

; 1.10147 

.248 

■ 1.1567 

.298  1.22203 

•348 

.145 

) 1.0581c 

> .195 

) 1. 102  47 

- .245 

► 1-I579] 

[ .29911.22347 

•349 

1.24946 
1.25095 
1.25243 
1-25391 
1-25539 
1.256  86 
1.25836 
1.25987 
1.26137 
1.262  86 
1.26437 
1.265  88 
1.2674 
1.268  92 
1.27044 
1. 271 96 
1.27349 
1.27502 
| 1.27656 
1.2781 
! 1.27964 
1.281 18 
1.282  73 

1.284  28 

1.285  83 
1.28739 
1.288  95 
1.29052 
1.29209 
1.29366 

1.295  23 

1.29681 

1.29839 


LENGTHS  OF  CIRCULAR  ARCS. 


26l 


H’ght. 

| Length. 

H’ght 

Length. 

H’ght 

Length. 

H’ght. 

Length. 

H’ght. 

Length. 

•35 

I.29997 

.38 

1.34899 

•4i 

I.4OO77 

•44 

1-455  12 

•47 

I.5H85 

•35i 

I.301  56 

.381 

1.35068 

.411 

I.402  54 

.441 

1.45697 

.471 

I.5I3  78 

•352 

I-303I5 

.382 

1-35237 

.412 

I.40432 

•442 

1.45883 

•472 

i.5x5  7i 

•353 

I.30474 

.383 

1.35406 

•4i3 

I.406  I 

•443 

1.46069 

•473 

I-5I7  64 

•354 

I.30634 

•384 

i-355  75 

•4r4 

I.40788 

•444 

1.462  55 

•474 

1.51958 

•355 

I.30794 

•385 

1-35744 

•4i5 

I.40966 

•445 

1.46441 

•475 

1.521  52 

•356 

1-30954 

•386 

I-359I4 

.416 

I.41145 

•446 

r. 466  28 

•476 

1.52346 

•357 

I-3III5 

•387 

1.36084 

.417 

I.41324 

•447 

1.46815 

•477 

1 .525  41 

•358 

I.31276 

.388 

1.36254 

.418 

I-4I503 

•448 

1.47002 

.478 

1.52736 

•359 

I-3I437 

•389 

1-36425 

.419 

I.41682 

•449 

1.47189 

•479 

1.52931 

•36 

I-3I599 

•39 

1.36596 

.42 

I.41861 

•45 

1 -473  77 

.48 

I-53I  26 

.361 

i-3i76i 

•39i 

1.36767 

.421 

I.42041 

•45i 

1 -475  65 

.481 

I-533  22 

.362 

1-31923 

•392 

I-36939 

.422 

1.422  22 

•452 

1 -477  53 

.482 

i.535  i8 

•363 

1.32086 

•393 

1.37111! 

•423 

I.424O2 

•453 

1.47942 

•483 

I-537  14 

•364 

1.32249 

•394 

1.37283 

•424 

I.425  83 

•454 

1.48131 

•484 

1-539  1 

•365 

1-32413 

•395 

1-37455; 

•425 

I.42764 

•455 

1.4832 

•485 

1. 541  06 

.366 

I-325  77 

•396 

1.37628: 

.426 

I.42945 

•456 

1.48509 

.486 

1.54302 

•367 

1-32741 

•397 

1.37801 

.427 

1. 431 27 

•457 

1.48699 

.487 

1-544  99 

.368 

1.32905 

•398 

1-379  74 

.428 

I-43309 

•458 

1.488  89 

.488 

1.54696 

•369 

1.33069 

•399 

1.38148 

•429 

I.434  91 

•459 

1-490  79 

.489 

•49 

1.54893 

I-55°9 

•37 

I-332  34 

•4 

1.38322 

•43 

i-436  73 

.46 

1.49269 

•49 1 

1.552  88 

•37i 

1-33399 

.401 

1.38496 

•43i 

1.43856 

.461 

1.4946 

•492 

1.554  86 

•372 

I-33564 

.402 

1.38671 

•432 

1.44039 

.462 

1.49651 

•493 

i-556  85 

•373 

1-3373 

•403 

1.388  46 

•433 

1.442  22 

•463 

1 .498  42 

•494 

I.558  54 

•374 

1-33896  ; 

•404 

1.39021 

•434 

1.44405 

•464 

1 -5oo  33 

•495 

1.560  83 

•375 

1-34063 

•405 

I-391 96 

•435 

1.445  89 

•465 

1.502  24 

•496 

1.562  82 

•376 

1.34229 

.406 

1-393  72 

•436 

1-447  73 

.466 

1.50416 

•497 

1.56481 

•377 

1 -343  96  ! 

•407 

1-39548 

•437  : 

1-44957 

•467 

1 .506  08 

•498 

1.5668 

•378 

1-34563  ! 

.408 

1.39724 

•438 

I-45I42 

.468 

1.508 

•499 

1.568  79 

•379 

I*347  31  ; 

•409 

!-399 

•439  i 

i-45327ll 

•469 

1.50992 

•5  1 

1-570  79 

To  ^Ascertain  Length,  of  an  TAro  of1  a Circle  by  pre- 
ceding Table. 

Rule.— Divide  height  by  base,  find  quotient  in  column  of  heights,  take 
length  for  that  height  opposite  to  it  in  next  column  on  the  right  hand. 
Multiply  length  thus  obtained  by  base  of  arc,  and  product  will  give  length. 

Example.— What  is  length  of  an  arc  of  a circle,  base  or  span  of  it  being  100  feet 
and  height  25?  ’ 

25 100  = .25;  and  .25,  per  table,  = 1. 159 12,  length  of  base,  which , multiplied  by 
100  = 115.912  feet.  ’ * J 

When , in  division  of  a height  by  base , the  quotient  has  a remainder  after 
third  place  of  decimals , and  great  accuracy  is  required . 

Rule.— Take  length  for  first  three  figures,  subtract  it  from  next  following 
length ; multiply  remainder  by  this  fractional  remainder,  add  product  to 
first  length,  and  sum  will  give  length  for  whole  quotient. 

Example.— What  is  length  of  an  arc  of  a circle,  base  of  which  is  « feet  and 
height  or  versed  sine  8 feet?  * ’ 


,Jtj5="‘22*5714'’  ^butor  length  for  .228  = 1.13331,  and  for  .229  = !.  13444 
the  difference  between  which  is  .001 13.  Then  .5714  x .001 13  = .000645682. 

Hence  .228  = 1. 13331",  ■ 

and  .0005714=  .000645682 

„„„  ^ 7.  , , 1-133955682,  the  sum  by  which  base  of 

arc  is  to  be  multiplied  ; and  1. 133  955  682  X 35  = 39.688  45  feet . J 


frees.  | 

I 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

12 

13 

14 

i5 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

3° 

3i 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

rr« 

Ru 

)ly  it 

Exj 


OF  CIRCULAR  ARCS. 


►F  Circular  .Arcs  from  1°  to  1£ 
(Radius  = 1.) 


sgrees. 

Length.  | j 

Degrees. 

Length. 

Degrees.  \ 

46 

.8028  i 

91 

I.5882 

136 

47 

.8203 

92 

I.6057 

137 

48 

•8377 

93 

I.623I 

138 

49 

•8552 

94 

I.6406 

139 

50 

51 

52 

53 

54 

.8727 
.8901  1 

.9076 
•925 

.9424  ! 

95 

96 

97 

98 

99 

I.6581 

1.6755 

I.693 

I.7IO4 

I.7279 

140 

141 

142 

143 

144 

55 

•9599  ! 

100 

1*7453 

145 

56 

•9774  :: 

IOI 

I.7628 

146 

57 

.9948  j 

102 

I.7802 

147 

58 

1. 0123  j 

103 

1.7977 

148 

59 

1.0297  ! 

104 

I.8151 

149 

60 

1.0472  j 

105 

106 

I.8326 

I.85 

150 

61 

62 

1 .0646 
1.0821 

107 

108 

I.8675 

I.8849 

151 

152 

63 

64 

1.0995 

1.117 

109 

I.9024 

153 

154 

65 

1 -1345 

j no 

I-9I99 

155 

66 

1. 1519 

! ill 

1-9373 

156 

67 

1.1694 

I 1 12 

1.9548 

157 

68 

1. 1868 

113 

1.9722 

158 

69 

1.2043 

114 

1.9897 

159 

1. 2217 

115 

2.0071 

160 

70 

Il6 

2.0246 

161 

71 

72 

1.2392  k 
1.2566 

117 

Il8 

2.042 

2.0595 

162 

16-2 

73 

74 

1.2741 

1.2915 

119 

2.0769 

164 

75 

1.309 

120 

2.0944 

, 165 

76 

1.3264 

121 

2.1118 

166 

77 

1-3439 

122 

2.1293 

1 167 

78 

i-36i3 

123 

2.1467 

168 

79 

1.3788 

124 

2.1642 

169 

80 

81 

82 

83 

84 

i-3963 
I-4I37 
I-43I2 
1.4486 
1 .4661 

125 

126 
127 
128 
129 

2.1817 
2. 1991 
2.2166 
2.2304 
2.2515 

j 170 
171 
! J72 

173 

174 

85 

1-4835 

130 

2.26S9 

| 175 

86 

1.501 

131 

2.2864 

; 176 

87 

1.5184 

132 

2.3038 

; i77 

88 

1-5359 

133 

2.3213 

1 

89 

1-5533 

134 

2.3387 

179 

90 

1.5708 

135 

2.3562 

180 

.11.  Length,  of  a Circular  Arc  by 
)lumn  opposite  to  degrees  of  arc,  take  length, 
circle. 

;r  of  degrees  in  an  arc  are  450,  and  diameter  of  cir< 
igthx  54-2  = !. 9635/^. 


LENGTHS  OF  ELLIPTIC  ARCS. 


263 


Lengths  of  Elliptic  Arcs. 

Up  to  a Semi-ellipse. 

Transverse  Diameter  = 1,  and  divided  into  1000  equal  Parts. 


H’glit. 

1 Length. 

'H’ght 

Length. 

H’ght 

Length. 

H’ght 

Length. 

H’ght 

.1  Length. 

.1 

1. 04 1 62 

•15 

1-0933 

.2 

1. 150  14 

•25 

1. 21 1 36 

•3 

I.27669 

.IOI 

I.O42  62 

•151 

1.09448 

I .201 

I-I5I3I 

.251 

I.21263 

.301 

I.27803 

.102 

I.O4362 

•152 

1.09558 

.202 

I.15248 

.252 

1. 2139 

.302 

I-27937 

.103 

1 .044  62 

•153 

1.09669 

I -203 

I.15366 

•253 

I.21517 

•303 

1.280  71 

.104 

I.O45  62 

1 -154 

1.0978 

.204 

I.15484 

•254 

I.21644 

•304 

1.28205 

.105 

I.O4662 

•155 

1.09891 

.205 

I.15602 

•255 

1. 217  72 

•305 

1.28339 

.106 

1 .047  62 

.156 

1. 100  02 

.206 

1. 1572 

.256 

1. 219 

.306 

1.284  74 

.107 

I.O4862 

•157 

1. 101 13 

.207 

1-15838 

•257 

1.220  28 

•307 

1.286  09 

.108 

1.049  62 

.158 

1. 102  24 

.208 

I-I5957 

.258 

1. 221  56 

.308 

1.28744 

.IO9 

1.05063 

•159 

iio335 

.209 

1. 160  76 

•259 

I.22284 

•309 

1.288  79 

.11 

1.05164 

.16 

1. 104  47 

.21 

1.161 96 

.26 

1.224  12 

•3i 

1.290 14 

.III 

1.05265 

.161 

1.1056 

.211 

1.163  *6 

.261 

1. 2254I 

•3IX 

1. 291  49 

.112 

1.05366 

.162 

1.10672 

.212 

1.16436 

.262 

1.226  7 

.312 

1.292  85 

•113 

1.05467 

.163 

1. 107  84 

•213 

1-16557 

.263 

I.22799 

•3X3 

1.294  21 

.114 

1.05568 

.164 

1.10896 

.214 

1. 166  78 

.264 

I.22928 

•3X4 

1-295  57 

•115 

1.05669 

.165 

1.11008 

•215 

1.16799 

.265 

I.23057 

•3X5 

1.29603 

.Il6 

1-057  7 

.166 

1. hi  2 

.2l6 

1.1692 

.266 

I.23186 

•3l6 

1.298  29 

.117  I.O5872 

.167 

1.11232 

.217 

1. 1 70  41 

.267 

1 -233  15 

•3i7 

1.29965 

.118 

1.05974 

.168 

I.H344 

.2l8 

I-I71 63 

.268 

x- 234  45 

.318 

1. 301 02 

.119 

1 .060  76 

.169 

1.11456 

.219 

1.17285 

.269 

!-235  75 

•3X9 

1.30239 

.12 

! 1. 061 78 

•17 

1.11569 

.22 

1. 17407 

.27 

1 -237  05 

•32 

1 -303  76 

.121 

1.0628 

.171 

1.11682 

.221 

I-I75  29 

.271 

1 -238  35 

•321 

1 -305  13 

.122 

1.06382 

.172 

I-II795 

.222 

1.17651 

.272 

1.23966 

.322 

1-3065 

.123 

1 1.06484 

•173 

1.11908 

.223 

I-I77  74 

•273 

1.24097 

•323 

1.30787 

.124 

1.06586 

.174 

1.12021 

.224 

1.17897 

•274 

1.242  28 

•324 

1.30924 

.125 

1.06689 

•175 

1.12134 

.225 

1.1802 

•275 

1 -243  59 

•325 

1.31061 

.126 

1.067  92 

.176 

1. 12247 

.226 

1.18143 

.276 

1.2448 

.326 

1.3H98 

.127 

1.06895 

.177 

1.1236 

.227 

1.18266 

•277 

1.24612 

•327 

r-3T335 

.128  ; 

1.06998 

.178 

1. 124  73 

.228 

1.1839 

.278 

1.24744 

.328 

I-3I4  72 

.I29 

1. 070  01 

.179 

1.12586 

.229 

1.18514 

•279 

1.248  76 

•329 

1.316 1 

•X3 

1.07204 

.18 

1.12699 

•23 

1.18638 

.28 

1.250 1 

•33 

i-3i748 

•131 

1.07308 

.181 

1.128 13 

.231 

1.18762 

.281 

1.25142 

•33i 

1.31886 

.132 

1.074 12! 

.182 

1. 12927 

.232 

1. 188  86 

.282 

1.25274 

•332 

1.320  24 

•x33 

1.075 16 1 

.183 

1.13041 

•233 

1. 190 1 

.283 

1.254  06 

•333 

1.321 62 

•134 

1.07621 

.184 

I-I3I55 

•234 

I-I91 34 

.284 

1-25538 

•334 

1-32  3 

•x35 

1.07726 

.185 

1.13269 

•235 

1.19258 

.285 

1.2567 

•335 

1.32438 

.136 

1.07831 

.186 

I-I33  83 

.236 

1.19382 

.286 

1.25803 

•336 

1.325  76 

•x37 

1.07937 

.187 

I-I3497 

•237 

1.19506 

.287 

1.25936 

•337 

1-327  x5 

.138 

1 .080  43 

.188 

1.136  11 

.238 

1.1963 

.288 

1.26069 

•338 

1.328  54 

-39: 

1. 08 1 49 

.189 

1.13726 

•239 

I-I9755 

.289 

1.262  02 

•339 

1 -329  93 

.14 

1.082  55 

.19 

1.13841 

•24 

1.1988 

.29 

1-26335 

•34 

I-331 32 

.141 

1.083  62 

.191 

I-I39  56 

.241 

1.20005 

.291 

1.26468 

•34x 

1.332  72 

.142 

1.08469 

.192 

1. 140  71 

.242 

1. 201 3 

.292 

1.26601 

.342 

1 33412 

.143 

1.085  76 

•I93 

1.141 86 

•243 

1.202  55 

•293 

1.26734 

■343  1 

1 335  52 

•x44 

1.08684 

.194 

1. 143  01 

.244 

1.2038 

•294 

1.26867 

•344  | 

1 33692 

•x45 

1.08792 

•195 

1. 144 16 

•245 

1.205  °6 

•295 

1.27 

•345 

x-33833 

.146 

1.08901 

.196 

i-i453i 

.246 

1.20632 

.296 

1.27133 

•346 

x-339  74 

•x47 

1.090 1 

.197 

1.146  46 

.247 

1.20758 

.297 

1.27267 

•347 

I-341 15 

.148 

1.091 19 

.198 

1.14762 

.248 

1.208  84 

.298 

1. 27401 

•348 

1.342  56 

.149 

1.092  28; 

.199 

1.14888 

.249 

1. 210 1 

.299 

i,27535 

•349 

1-34397 

264 


LENGTHS  OF  ELLIPTIC  ARCS. 


H’ght.  Length.  H’ght 


•35 

•35i 

•352 

•353 

•354 

•355 

•356 

•357 

.358 


1-345  39 
1.34681 

1.34823 
i-34965 
1.35108 
1.352  51 
1.35394 
1-355  37 
_ 1.3568 

•359  I1  *358  23 
.36  I 1.35967 
.361  I 1.361 11 
.362  | 1.36255 
.363  | I.36399 
•364i  1.36543 
.36s  1.36688 

.366  1 1.36833 
.367  ! 1-36978 
.368  1-371  23 
.369  1-37268 

1-374  14 
1.37662 
1.37708 
1.378  54 
.^,1.38 
•375  j 1-38146 
.376  1 1.38292 
•377  | 1.38439 
.378  i 1-38585 
•379  i 1 *387  32 
.38  U.38879 


•37 

•37i 

.372 

•373 

•374 


Length. 


1 1 H’ght.  I Length.  I j H’ght. I Length.  | H’ght.j  Length. 


.381 

.382 

.383 

•384 

.385 

.386 

•387 

.388 

•389 

•39 


1.39024 
I-391 69 
I-393I4 
1-394  59 
1.39605 
1.397  5i 
1.39897 
1.40043 
1 .401  89 

w 1.40335 
.391  j 1.40481 
.392 ! 1.40627 

•393  I-4°7  73 

.394;  1.409 19 

•395  ! i.4io65 
1.412  11 
i*4I3  57 
1.41504 

1.4:1651 
1.41798 
1.41945 
1.420  92 
1.42239 
1.42386 


•396 

•397 

•398 

•399 

•4 

.401 

.402 

•403 

•404 


.405 

.406 

.407 

.408 

.409 

.41 

.411 

.412 

•4i3 

.414 

•4i5 

.416 

.417 

.418 

.419 

.42 

.421 

.422 

.423 

.424 

.425 

.426 

•427 

.428 

.429 

•43 

•431 

•432 

•433 

•434 

•435 

•436 

■437 

•438 

■439 

.44 

,441 

,442 

■443 

■444 

■445 

.446 

■447 

.448 

.449 

•45 

.451 

.452 

•453 

•454 

•455 

•456 

•457 

.458 

•459 


1.425  33  j 
1.42681  J1 
1.42829  | 
1.42977, 
I-43I  25 II 
1.432  73 

1.434  21 

1.435  69 
1.43718 
1.43867 
1.440 16 
1.44165 
1.443  H 

1.44463 
1.44613 
I.44763 
1.449  x3 
1.45064 
1.452 14 

1.45364 

1-455  J5 
1.45665 
1.45815 

1.45966 
1.461 67 
1.46268 
1 .464 19 

1-4657 

1.46721 
1.468  72 

1.47023 
1. 471  74 

1.47326 
1.474  78 

1-4763 

1.47782 
1-47934 
1 .480  86 
1.482  38 
1.48391 
1.48544 
1.486  97 
1.4885 
1.49003 
I-491  57 
1.493 1 1 

1.49465 
1.496 18 

1.497  71 

1-499  24 

1.50077 

1.5023 

1.50383 

1.50536 

1.50689 


.46 

.461 

.462 

•463 

•464 

•465 

.466 

•467 

.468 

.469 

•47 

.471 

.472 

•473 

•474 

•475 

.476 

•477 

.478 

■479 

.48 

.481 

.482 

■483 

•484 

■485 

.486 

.487 

.488 

.489 


•49 

•491 

.492 

•493 

•494 

•495 

•496 

■497 

•498 

■499 

•5 

.501 

.502 

.503 

.504 

.505 

.506 

.507 

.508 

•509 

.51 

•5” 

.512 

•5i3 

•5i4 


.515  i.594o8 

.516  1.59564 

.517  1-597  2 
.518  i.59876 
.519  1.60032 
.52  1. 601 88 
.521  1.60344 
.522  1.605 
.523  1.60656 
.524  1.60812 
.525  1.60968 
.526  1.61124 
.527  1.6128 
.528  1.61436 
.529  1.61592 
.53  1.61748 

.531  1.61904 
.532  1.6206 
.533  1.62216 
•534  1.62372 
•535  1.62528 

.536  1.62684 
.537  1.6284 
.538  1.62996 
•539  1-63152 
•54  1.63309 

.541  1.63465 

,542  1.63623 
•543  1-6378 
.544  1.63937 
.545  1.64094 
.546  1.642  51 
.547  1.64408 
.548  1.64565 
.549  1.64722 
.55  1.64879 

.551  1.65036 

.552  1.65193 
•553  1-653  5 
•554  1 -655  07 
-555  1-65665 

.556  1.65823 

•557  1.65981 

.558  | 1. 661 39 

.559  1.66297 
.56  1.66455 

.561  1.66613 
.562  1.66771 
.563  1.66929 
.564  1.67087 
.565  1-67245 
1.58784  1 .566!  1.67403 

1.5894  -567;  i-6756i( 

1.59096  .568  1.67719 
1.59252:  .569  ; i-678  77 


I.508  42  ; 
1.50996 
I-511  5 
1-51304! 
1.51458 

I.516  12  ; 
I.51766; 
I.5I92 

1.520  74 
1.52229 

1.523  84 

1.525  39 
1.52691 

1.52849 

1.530  04 
I-531  59 
1-533  *4  j 
1.53469: 
I-53625 
i.537  8i  | 
1-539371 
i.540  93> 
1.542  49 
I.54405 
i.545  6i 

1.547  18 

1.548  75 
1.55032 
1.55189 
1.55346 
1.55503' 
I.5566  I 
i.558i7i 
1-559  74 
1.561 31 

1.56289 

1.56447 

1.56605 

1.56763 

1.56921 
1.57089 
1.572  34 
1.57389 
1-57544 
1.57699 
1.57854 
1.58009 
1.58164 
1 I.583  x9 
1.584  74 
1.58629 


.57  1 1.68036 

j57 1 1 1.68195 
.572  j 1.68354 
•573  i-685  13 
•574  1-68672 
•575  1.68831 
.576  | 1.6899 
•577  1-69149 
.578  1.69308 
•579  i 1.69467 


.58 

.581 

.582 

.583 

.584 

.585 

.586 

•587 

.588 

.589 

•59 
•59i 
•592 
593 
•594 
■595 
■596 
•597 
•598 
•599 
.6 
.601 
.602 
.603 
.604 
.605 
.606 
.607 
.608 
.609 
.61 
.611 
.612 

.613 

.614 

.615 

.616 

.617 

.618 

.619 

.620 

.621 

.622 

.623 

.624 


1.69626 

1.69785 

1.69945 

I.  701 05 
1.702  64 
1.70424 
1.705  84 
1.70745 
1.70905 

I I. 71065 
1. 71225 
1. 712  86 

.71546 
i*7 1 7 °7 
1.71868 

1.720  29 

1. 721  9 

1-7235 

1.725  11 

1.72672 

1.72833 
1.72994 
1.73155 
1-733 16 

1.73638 
1-73799  : 
I-7396 

1. 741  21 

1.742  83 
1.74444 
1.74605 
1.74767  , 

1.74929  ' 

I.75091  : 
1.752  52  < 
1-754  J4  ; 
1.755  76  , 
1.75738 
1-759 
1.760  62 
1.762  24 

1.76386 

1.765  48 
1.7671 


LENGTHS  OF  ELLIPTIC  ARCS. 


H’ght. 

Length. 

| H’ght 

. Length. 

H’ght. 

.625 

I.768  72 

.68 

I.858  74 

•735 

.626 

I.77034 

.681 

I.86039 

•736 

.627 

1. 771  97 

.682 

I.862  05 

•737 

.628 

I-773  59 

.683 

I.8637 

•738 

.629 

1 1.775  2i 

.684 

1-865  35 

•739 

•63 

1.77684 

.685 

I.867 

•74 

.631 

; J-77847 

.686 

x. 868  66 

•74i 

.63  2 

1.780  09 

.687 

1.87031 

•742 

•633 

1.781  72 

.688 

1.871 96 

•743 

•634 

I.78335 

.689 

1.87362 

•744 

.635 

1.784  98 

.69 

1.87527 

•745 

.636 

1.7866 

.691 

1.87693 

.746 

•637 

1.788  23 

.692 

1.87859 

•747 

.638 

1.789  86 

•693 

1.880  24 

•748 

•639 

1. 791 49 

.694 

1. 881 9 

•749 

.64 

1.793  12 

•695 

1.88356 

•75 

.641 

1-794  75 

.696 

1.885  22 

•75* 

.642 

1.70038 

.697 

1.88688 

•752 

•643 

1.79801 

.698 

1.888  54 

•753 

.644 

1.79964 

.699 

1.8902 

•754 

•645 

1. 801  27 

•7 

1.891 86 

•755 

.646 

1.8029 

.701 

1.89352 

•756 

.647 

1.80454 

.702 

1.895  19 

•757 

.648 

1.806 17 : 

•703 

1.896  85 

•758 

.649 

1.8078 

•704 

1.89851 

•759 

.65 

1.80943! 

•705 

1.900 17 

.76 

.651 

1.811  07 

.706 

1. 901  84 

.761 

.652 

1.812  71 1 

•707 

1-9035 

.762 

•653 

i-8i4  35 

.708 

1-90517 

.763 

.654 

1.81599! 

•709 

x. 906  84 

.764 

•655 

1.81763! 

•7i 

1.90852 

•765 

.656 

1.81928 

.7x1 

1.910  19 

.766 

•657 

1.82091 

.712 

1.911  87 

.767 

.658 

1.82255 

•7i3 

I-9I3  55 

.768 

•659 

1.824  19 

.714 

I-9I5  23 

.769 

.66 

1.825  83 

.715 

1.91691 

•77 

.661 

1.82747 

.716 

1.91859 

•77* 

.662 

1.829  11 

.717 

1.92027 

•772 

.663 

1-830  75 

.718 

1-92195 

•773 

.664 

1.8324 

.719 

1.92363 

•774 

.665 

1.83404 

.72 

1-925  3* 

•775 

.666 

1.835  68 

.721 

1.927 

•776 

.667 

1-837  33 

.722 

1.92868. 

•777 

.668 

1.83897 

•723 

x. 93^36 

.778 

.669 

1.84061 

.724 

1 -933  04 

•779 

.67 

1.842  26 

•725 

1-93373 

.78 

.671 

1.84391 

.726 

*•935  4* 

.781 

.672 

1.845  56 

‘73? 

*•937  * 

.782 

•67  3 

1.847  2 

.728, 

1.93878 

•783 

.674 

1.84885 

,7291 

1.94046 

•784 

•675 

1-8505 

•73 

1.942  15 

•785 

.676 

I-853-JL5  I 

•73i 

*•94383 

.786 

.677 

1-853  79: 

•732 

I-945  52 

.787 

.678 

1-85544 

•733 

1.94721 

.788 

.679 

1.85709:! 

•734 

1.9489  | 

•789 

Length. 


H’ght. 


Length. 


H’ght. 


1 -950  59 
1.952  28 

1 -953  97 
1.95566 

*•95735 

1-95994 

1 .960  74 
x .962  44 
1.964  14 

1.96583 
I-967  53 
1.96923 
1.97093 
1.972  62 
1.97432 
1.97602 
1.97772 

1 -979  43 
1.981 13 
1.98283 
I-984  53 
1.986  23 
1.98794 
1.98964 
I-991 34 

1- 99305 
1.99476 
1.99647 
1.998  18 
1.99989 
2.001 6 
2.00331 

2.005  02 

2 .006  73 
2.008  44 
2.010x6  1 
2.01187  I 
2.0x359  ! 
2.01531 
2.01702 
2.0x874 
2.020  45 

2.022  17 

2.023  89 
2.02561 
2.02733 
2.02907 
2.0308 
2.03252 
2.03425 

2- 035  98 
2.037  71 
2.03944 
2.041 17 
2.0429 

Z 


•79 

.791 

.792 

•793 

•794 

•795 

.796 

•797 

.798 

•799 

.8 

.801 

.802 

.803 

.804 

•805 

.806 

.807 

.808 

.809 

.81 

.811 

.812 

•813 

.814 

.8x5 

.816 

.817 

.818 

.8x9 

.82 


.821 

.822 

.823 

.824 

.825 

.826 

.827 

.828 

.829 

•83 

.831 

.832 

•833 

•834 

•835 

.836 

•837 

.838 

•839 

.84 

.841 

.842 

•843 

-844 


2.04462 

2.04635 

2.04809 

2.04983 

2.05157 

2.05331 
2-05505 
2.056  79 
2.05853 
2.060  27 
2.062  02 
2.06377 
2.065  52 
2.067  27 
2.06901 
2.070  76 
2.07251 
2.07427 
2.07602 
2.07777 

2.07953 

2.081  28 

2.08304 

2.0848 

2.086  56 

2.08832 

2.09008 

2.091 98 

2.0936 

2.095  36 

2.09712 

2.098  88 

2.10065 

2.102  42 

2.104 19 

2.105  96 

2.107  73 

2.1095 

2. in  27 

2.11304 

2.11481 

2.11659 

2.11837 

2.120 15 

2.121 97 

2.123  71 

2.125  49 

2.12727 

2.12905 

2.13083 

2.13261 

2.13439 

2.13618,. 

2.13797 

2-I3976  J 


•845 

.846 

•847 

.848 

•849 

.85 

.851 

.852 

.853 

•854 

.855 

.856 

.857 

858 

•859 

,86 

,861 

,862 

,863 

.864 

.865 

.866 

.867 

.868 

.869 

.87 

.871 

.872 

.873 

•874 

•875 

.876 

•877 

.878 

•879 

.88 

.881 

.882 

.883 

.884 

.885 

.886 

.887 

.888 


.889 
.89  j 
.891 
.892 

•893 

-S94 

.895 

.896 

-89? 

.898, 
•8 99 ' 


265 

Length. 


2.14155 

2.143  34 

2.145  13 

2.146  92 
2.148  71 
2.I505 
2.15229 
2.15409 
2.15589 
2-1577 
2-1595 
2.l6l  3 
2.16309 
2.16489 
2.16668 
2.168  48 
2.17028 
2.17209 
2.17389 
2-175  7 
2.177  51 
2-I7932 
2.l8l  13 
2.18294 
2.184  75 
2.186  56 
2.18837 
2.190 18 
2.192 
2.19382 
2.19564 
2.19746 
2.19928 
2.201 1 
2.202  92 
2.204  74 
2.20656 
2.20839 
2.2x022 
2.21205 
2.21388 
221571 
2.21754 
2.21937 
2.221 2 
2.22303 
2.22486- 
2.226  7 
2.228  54 
2.23038. 
2.232  22: 
2.234  q6j 
2-2359; 
2-23774; 

2 • 239.58 


266  LENGTHS  OF  ELLIPTIC  ARCS. 


H’ght.  | 

•9  ! 

.901 1 
•9°2 
•903 
•9°4 

’9°§  I 
.906  1 
.907  1 
.908 
.909 


Length. 


2.241  42 
2.243  25 
2 24508 
2.24691 
2.248  74 
2.25057 
2.2524 
2.25423 
2.256  06 
2.25789 


H’ght.  | 
.92 


.921 

.922 

•923 

•924 

•925 

.926 

.927 

.928 

.929 


Length. 


H’ght. 1 


2.27803 
2.27987] 
2.28l  7 I 
2.283  54 
2.28537 
2.287  2 
2.289  03 
2.29086 
2.292  7 
2.294  53 


•94  1 

.941 

.942 

•943 

•944 

•945 

.946 

•947 

.948 

•949 


.91  2.25972 

.911  j 2.26155 
.912  j 2.26338 
.913  | 2.26521 
.914  2.26704 
.915  ] 2.26888 
.916  2.27071 
.917  ; 2.27254 
.918  2.27437 
.919  2.2762 


•93 

•93i 

•932 

•933 

•934 

•935 

•936 

•937 

•938 

•939 


2.29636 

•95 

2.2982 

•95 1 

2.30004 

•952 

2.30188 

•953 

2.303  73 

•954 

2.305  57 

•955 

2.30741 

•956 

2.30926 

•957 

2.311 11 

•958 

2.31295 

•959 

Length.  | 

H’ght. 

Length,  j 

H’ght.  1 

2-3T4  79  j 

•96 

2-352  41 

.98 

2.31666 

.961 

2.35431 

.981 

2.31852; 

.962 

2.35621 

.982 

2.32038 

•963 

2.358  I 

.983 

2.32224 

.964 

2.36 

.984 

2.324  1 1 

.965 

2.361  91 

.985 

2.32598 

.966 

2.36381 

.986 

2.32785 

.967 

2.36571 

.987 

2.32972 

.968 

2.367  62 

.988 

2.3316 

.969 

2.36952 

.989 

•99 

2.33348 

•97 

2.371 43 

•991 

2-335  37 

.971 

2-37334 

.992 

2.337  26 

.972 

2.37525 

•993 

2.33915 

•973 

2.377  16 

•994 

2.341  °4 

•974 

2.37908 

•995 

2.342  93 

•975 

2.381 

.996 

2.34483 

.976 

2.38291 

•997 

2346  73 

•977 

2.384  82 

•998 

2.348  62 

.978 

2.386  73 

•999 

1 2.35051 

1-979 

2.388  64 

1. 

Length. 


2.390  55 
2.39247 
2-39439 
2.39631 
2.39823 
2.40016 
2.402  08 

2.404 

2.405  92 
2.407  84 
2.40976 
2.4II  69 
2.41362 
2.41556 
2.41749 
2.41943 
2.42136 
2.42329 
2.425  22 
2.427  15 
2.42908 


To  Ascei'tsain.  Eeragtli  of*  an  Elliptic  Arc  (riglit  Semi- 
Ellipse)  by  preceding  Talole. 

Rule. — Divide  height  by  base,  find  quotient  in  column  of  heights,  and 
take  length  for  that  height  from  next  right-hand  column.  Multiply  length 
thus  obtained  by  base  of  arc,  and  product  will  give  length. 

Example.— What  is  length  of  arc  of  a semi-ellipse,  base  being  70  feet,  and  height  • 
30. 10  feet  ? 

30.  io-j-  70  — .43 ; and  43, .-per  table,  = 1.46268. 

Then  1.46268  X 70  = 102.387 6 feet. 

When  Carve  is  not  that  of  a right  Semi-Ellipse , Height  being  half  of  Trans- 
verse Diameter. 

Rule. — Divide  half  base  by  twice  height,  then  proceed  as  in  preceding 
example ; multiply  tabular  length  by  twice  height,  and  product  will  give  , 
length. 

Example. — What  is  length  of  arc  of  a semi-ellipse,  height  being  35  feet,  and  base  ] 
60  feet? 


60  -r-  2 = 30,  and  30-7-35X2  = . 428,  tabular  length  of  which  = 1.459  66. 

Then  1.45966  X 35  X 21=  102.1762  feet. 

When , in  Division  of  a Height  by  Base , Quotient  has  a Remainder  after 
third  Place  of  Decimals,  and  great  Accuracy  is  required, 

Rule. — Take  length  for  first  three  figures,  subtract  it  from  next  following  j 
length;  multiply  remainder  by  this  fractional  remainder,  add  product  to  j 
first  length,  and  sum  will  give  length  for  whole  quotient. 

Example.  — What  is  length  of  an  arc  of  a semi-ellipse,  base  being  171.3  feet  and  < 
height  125  feet? 

171.3  -4-  2 = 85.65.  and  125  X 2 = 250.  171. 3 -r-  250  = .3426  ; tabular  length  for  } 

.342  = 1.334  I2>  aD(l  -343  =■  i-335  52,  the  difference  between  which  is  .0014. 

Then  .6  X .0014  = .0084. 

Hence,  .342  = 1.334 12 
.0006=  .0084 

1.342  52,  the  sum  by  ivhicli  base  of  arc 
is  to  be  multiplied ; and  1.34252  X 171. 3 = 229.973  676/e^. 


AREAS  OF  SEGMENTS  OF  A CIRCLE. 


267 


Areas  of  Segments  of  a Circle. 

Th  e Diameter  of  a Circle  = 1,  and  divided  into  1000  equal  Parts. 


Verse* 

Sine. 

Seg.  Area. 

.OOI 

.OOOO4 

.002 

.OOO  12 

.OO3 

.000  22 

.OO4 

.OOO34 

.005 

.OOO47 

.006 

.000  62 

.007 

.OOO78 

.008 

.OOO95 

.009 

.OOI  13 

.01 

•OOI33 

.Oil 

•OOI53 

.012 

•001  75 

.013 

.OOI  97 

.014 

.002  2 

.015 

.OO244 

.016 

.002  68 

.017 

.002  94 

.018 

; -0032 

.019 

•OO347 

.02 

•OO375 

.021 

.OO403 

.022 

.OO432 

.023 

.OO462 

.024 

; .OO492 

.025 

.00523 

.026 

; -00555 

.027 

.00587 

.028 

.00619 

.029  ! 

•00653 

•03  I 

.00686 

•031  j 

.00721 

.032 

.00756 

•033 

.00791 

•034 

.008  27 

•035 

.00864 

.036 

.00901 

•037 

.00938 

•038  ; 

.009  76 

•039 

.01015 

*°4  ; 

.01054 

! -041 

.01093 

f .O42 

•01133 

! -°43  | 

.on  73 

•044 

.012  14 

•045 

•OI255 

.046 

.OI297 

•047 

•01339 

.048 

.OI382 

.049 

•OI425 

•05 

.OI468 

.051 

.015  12 

Versed 

Sine. 

Seg.  Area. 

Versed 

Sine. 

Seg.  Area. 

1 1 Versed 
J Sine. 

Seg.  Area. 

V ersed 
Sine. 

Seg.  Area. 

.052 

.015  56 

.103 

.042  69 

•154 

•076  75 

.205 

.11584 

•053 

.Ol6oi 

.IO4 

.0431 

j -x55 

•07747 

.206 

.11665 

•054 

.01646 

.105 

.04391 

1 .156 

.0782 

.207 

.11746 

•055 

.01691 

.106 

.044  52 

, •I57 

.07892 

.208 

.11827 

.056 

•01737 

.107 

•045  14 

: .158 

•07965 

.209 

.11908 

•057 

.OI783 

.108 

•045  75 

•z59 

.080  38 

.21 

.119  9 

.058 

.0183 

.109 

.04638 

j .16 

.081 11 

.211 

.120  71 

•059 

.Ol8  77 

.11 

•047 

.161 

.08 1 85 

.212 

•121  53 

.06 

.OI924 

.III 

.04763 

.162 

.082  58 

•213 

•12235 

.061 

.OI972 

.112 

.048  26 

.163 

.08332 

.214 

.123 17 

.062 

.020  2 

•113 

.048  89 

.164 

.084  06 

.215 

•12399 

.063 

.020  68 

.114 

•04953 

‘ .165 

.084  8 

.216 

.12481 

.064 

.021  17 

•115 

.050  16 

.166 

•08554 

.217 

•12563 

.065 

.021  65 

• Il6 

.0508 

.167 

.086  29 

.218 

.126  46 

.066 

.022  15 

.117 

•05145 

.168 

.08704 

.219 

.127  28 

.067 

.O2265 

.Il8 

.05209 

.169 

•087  79 

.22 

.128  II 

.068 

•023  15 

•I I9 

•052  74 

•x7 

.088  53 

.221 

.12894 

.069 

.02336 

.12 

•05338 

.171 

.089 29 

.222 

.12977 

•07 

.O24  1 7 

.121 

.05404 

.172 

.09004 

•223 

.1306 

.071 

.024  68 

.122 

.05469 

•x73 

.0908 

.224 

•13144 

.072 

.025  19 

.123 

•055  34 

•I74 

•09x55 

.225 

.13227 

•073 

.02571 

.124 

.056 

•175 

.09231 

.226 

•133 II 

.074 

.026  24 

.125 

.05666 

.176 

•09307 

.227 

•133  94 

.075 

.026  76 

.126 

•057  33 

•J77 

•09384 

.228 

•134  78 

.076 

.027  29 

.127 

•05799 

.178 

.0946 

.229 

•135  62 

•077 

.027  82 

^.128 

.05866 

•x79 

•095  37 

•23 

.136  46 

.078 

.02835 

*129 

•05933 

.18 

.09613 

.231 

•I373I 

.079 

.02889 

•13 

.06 

.181 

.0969 

.232 

•13815 

.08 

.02943 

•131 

.060  67 

.182 

.097  67 

•233 

•139 

.081 

.02997 

.132 

•06135 

.183 

.09845 

•234 

•139  84 

.082 

.03052 

•133 

.06203 

.184 

.09922 

•235 

.14069 

.083 

.031 07 

•134 

.062  71 

.185 

.1 

.236 

•141  54 

.084 

.031 62 

•x35 

•06339 

.186 

.100  77 

•237 

•142  39 

.085 

.032 18 

.136 

.064  07 

.187 

•ioi  55 

•238 

•143  24 

.086 

.032  74 

•x37 

.064  76 

.188 

.10233 

•239 

.14409 

.087 

.088 

.089 

•0333 

•03387 

.138 

•T39 

•065  45 
.066 14 

.189 

.19 

.10312 

.1039 

.24 

.24I 

•14494 
.145  8 

•034  44 

.14 

.066  83 

.191 

.104  68 

.242 

.14665 

.09 

•03501 

.141 

•06753 

.192 

•10547 

•243 

•I475I 

.091 

•035  58 

.142 

.068  22 

•T93 

.10626 

.244 

•14837 

.092 

.036 16 

•i43 

.068  92 

.194 

.10705 

•245 

.149  23 

•093 

.036  74 

.144 

.069  62 

•I95 

.107  84 

.246 

.15009 

•094 

•03732 

•i45 

•07033 

.196 

.10864 

•247 

•15095 

•095 

•0379 

.146 

.071 03 

.197 

.10943 

.248 

.151  82 

.096 

•03849 

.147 

.07174 

.198 

.11023 

.249 

.15268 

.097 

.03908 

.148 

•07245 

•I99 

.111 02 

.25 

•i  53  55 

.098 

.03968 

.149 

•073  16 

.2 

.111  82 

.251 

•I5441 

.099 

.040  27 

•T5 

•07387 

.201 

.11262 

.252 

• 15528 

.1 

.040  87 

•x5i 

•07459 

.202 

•ii343 

•253 

•15615 

.IOI 

.041 48 

.152 

•07531 

.203 

.11423 

•254 

.15702 

.102 

.04208  | 

•153 

.07603 

.204 

•11503 

•255 

•15789 

268 


AREAS  OF  SEGMENTS  OF  A CIRCLE. 


Seg.  Area. 


.256 

.257 

.258 

•259 

.26 

.261 

.262 

.263 

.264 

.265 

.266 

.267 

.268 

.269 

.27 

.27I 

.272 

•273 

.274 

•275 

.276 

.277 

.278 

.279 

.28 

.281 

.282 

.283 

.284 

.285 

.286 

.287 

.288 

.289 

.29 

.291 

.292 

•293 

.294 

•295 

.296 

.297 

.298 

•299 

•3 

•301 

.302 

•303 

•304 


.158  76 
• i5964 
.16051 
.16139 
.162  26 
.16314 


| Versed 
Sine. 

•305 

.306 

•307 

.308 

•309 

•31 


Seg.  Area. 


.16402 

.311 

.1649 

•312 

.16578 

•3i3 

.16666 

.314 

.16755 

•315 

.168  44 

•3l6 

.16931 

•3i7 

.1702 

.318 

.17109 

•3I9 

• I71 97 

.32 

.17287 

.321 

.173  76 

.322 

.17465 

.323 

.175  54 

•324 

.17643 

.325 

.17733 

.326 

.17822 

.327 

.179 12 

.328 

.18002 

•329 

.18092 

•33 

.181 82 

.33i 

.182  72 

•332 

.18361 

•333 

.18452 

•334 

.185  42 

•335 

.18633 

•336 

.18723 

•337 

.188 14 

.338 

.18905 

•339 

.18995 

•34 

.19086 

•34i 

• I9I  77 

.342 

.192  68 

•343 

.1936 

•344 

.19451 

•345 

.19542 

.346 

.19634 

•347 

.197  25 

•348 

.19817 

•349 

.19908 

•35 

.2 

•35i 

.20092 

.352 

.201  84 

•353 

V ersed 
Sine. 


Seg.  Area. 


,302  76 
.20368 
.2046 
•205  53 
.20645 
.207  38 


.20923 
.210  15 
.211  08 
.21201 
.212  94 
.21387 
.2148 
.215  73 
.21667 
.2176 
•21853 

.21947 
.220  4 
.221  34 
.222  28 
.22321 
.22415 
.22509 
.22603 
.226  97 
.22791 
.228  86 
.2298 
.230  74 
.231 69 
.23263 

.23358 
.234  53 
.23547 
.23642 
.23737 
.23832 
.23927 
.24022 
.241 17 
.242  12 
.24307 
.24403 
.244  98 
•24593 
.246  89 


•354 

•355 

.356 

•357 

.358 

•359 

.36 

.361 

.362 

•363 

.364 

•365 

.366 

.367 

.368 

.369 

•37 
.37i 
•372 
•373 
•374 
•375 
.376 
•377 
•378 
-379 
.38 
.381 
.382 

.383 

•384 

.385 

.386 

.387 

.388 

.389 

•39 

•39 1 

.392 

•393 

•394 

•395 

.396 

•397 

•398 

•399 

•4 

.401 
l|  .402 


Versed  ] 
Sine. 


.2488 

.24976 

.25071 

.25167 

.25263 

.25359 

-25455 

.255  51 

.25647 

-25743 

.258  39 

.25936 

.260  32 

.261  28 

.262  25 

,26321 

.264  18 

.265 14 

.266  II 

.267  08 

.268  04 

.269OI 

.26998 

.27095 

.271  92 

.27289 

.27386 

.27483 

.27580 

.27677 

-277  75 

.278  72 

.27969 

.28067 

.281 64 

.282  62 

.28359 
.28457 
.285  54 

.28652 

.2875 

.28848 

.28945 

.29043 

.291 41 

.29239 

.29337 

.29435 

.29533 


•403 

.404 

.405 

,406 

.407 

.408 

.409 

.41 

.411 

412 

413 

.414 

415 

.416 

.417 

.418 

419 

.42 

,421 

422 

.423 

.424 

425 

426 

427 


•43 


•44 


l 

Seg.  Area. 

f ersed  1 

Sine.  Seg.  Area. 

.29631 

.452  - 

-344  77 

.297  29 

-453 

■345  77 

.298  27 

•454 

.34676 

.29925 

•455 

•347  76 

.30024 

.456 

.348  75 

.3OI  22 

•457 

•349  75 

.302  2 

.458 

•350  75 

.30319 

•459 

.35i  74 

•30417 

.46 

•352  74 

.305  15 

.461 

•35374 

.30614 

.462 

•354  74 

•3°7  I2 

.463 

•355  73 

.308H 

.464 

.35673 

.30909 

.465 

•357  73 

.31008 

.466 

.35872 

.3HO7 

.467 

.359  72 

•3I2o5 

.468 

.360  72 

.31304 

.469 

.361  72 

•3I4°3 

•47 

.362  72 

.31502 

.471 

•36371 

.316 

.472 

.36471 

•3i699 

•473 

.36571 

*3I7  98 

•474 

.36671 

> -3i897 

•475 

.367  7i 

.31996 

.476 

.36871 

1 .32095 

•477 

.36971 

> .32194 

.478 

•37071 

.32293 

•479 

•37 1 7 

; .32391 

.48 

•372  7 

5 .3249 

.481 

•373  7 

$ .325  9 

.482 

•374  7 

[ .32689 

•483 

•375  7 

5 .32788 

.484 

•3767 

5 .32887 

.485 

•377  7 

7 .32987 

.486 

.378  7 

3 .33086 

•487 

•3797 

? .331 85 

.488 

.3807 

.332  84 

•489 

.3817 

1 .33384 

•49 

.3827 

2 .33483 

•491 

•383  7 

3 -33582 

.492 

.3847 

4 .33682 

•493 

•385  7 

5 .337  81 

•494 

.3867 

6 .3388 

•495 

•387  7 

7 -3398 

.496 

.3887 

8 .340  79 

, .497 

•389  7 

9 -34i  79 

1 .498 

.390  7 

; .342  78 

i -499 

.391  7 

;i  -34378 

1 1 *5 

.392  7 

To  Compute  Area  of  a Segment  of  a Circle  preceding 
Tat>le. 

Rule  —Divide  height  or  versed  sine  by  diameter  of  circle ; find  quotient  in 
column  of  versed  sines.  Take  area  for  versed  sine  opposite  to  it  m next  col- 
umn on  right  hand,  multiply  it  by  square  of  diameter,  and  it  will  gi\e  area. 


AREAS  OF  ZONES  OF  A CIRCLE. 


269 


Example  —Required  area  of  a segment  of  a circle,  its  height  being  10  feet  and 
diameter  of  circle  50.  6 

10  -f-  50  = . 2,  and  . 2,  per  table , = . 1 1 1 82 ; then  . 1 1 1 82  X 502  = 279. 55  feet. 

When,  in  Division  of  a Height  by  Base , Quotient  has  a Remainder  after 
I hird  Place  of  Decimals , and  great  Accuracy  is  required. 

Rule.— Take  area  for  first  three  figures,  subtract  it  from  next  following- 
area,  multiply  remainder  by  said  fraction,  add  product  to  first  area  and 
sum  will  give  area  for  whole  quotient. 

lieVh^nt' ? 1 iS  area  °f  a segraent  of  a circle>  diameter  of  which  is  10  feet,  and 

i.575ri°~i575;  tabular  area  for  .157  = .07892.  and  for  .158  = .070 6q  the  dif- 
ference between  which  is  .00073.  y 5 79  aij 

Then  .5  x .00073  = .000365. 

Hence!  .157  =.07892 

.0005  = .006  365 

of  circle  is  to  be  multiplied ; and  .079  285  = Square  °fdiameter 


Areas  of  Zones  of  a Circle. 

The  Diameter  of  a Circle  — 1,  and  divided  into  1000  equal  Parts. 


H’ght 

j Area. 

|[ 

Area. 

| H’ght 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

.OOI 

.002 

.003 

.OO4 

.005 

.006 

.007 

.008 

.OO9 

.OI 

.Oil 

.012 

.013 

.OI4 

.015 

.Ol6 

.OI7 

.018 

.OI9 

.02 

.021 

.022 

.023 

.024 

.025 

.026 

.027 

.028 

.O29 

•03 

.031 

.O32 

•033 

•034 

.OOI 
.002 
.003 
.004 
1 .005 
.006 
.007 
.008 
.009 

.OI 

.on 

.012 

.013 

.014 

.015 

.016 

.017 

.018 

.019 

.02 

.021 

.022 

.023 

.024 

.025 

.02599 

.02699 

.02799 

.028  98 

.02998  1 

.03098 

.03198 

.03298 

•03397  1 

•035 

.036 

•037 

.038 

•039 

.04 

.041 

.042 

•043 

.044 

•045 

.046 

.O47 

.048 

.049 

•05 

.051 

.O52 

•053 

•054 

•055 

.056 

•057 

.058 

•059 

.06 

.061 

.062 

.063 

.064 

.065 

.066 

.067 

.068 

•03497 

•03597 

.03697 

.03796 

.03896 

.03996 

•040  95 

.041  95 

.04295 

.04394 

.04494 

•04593 

.04693 

•04793 

.048  92 

.04992 

.05091 

.0519 

.0529 

•05389 

.05489 

.055  88 

.056  88 

•05787 
.05886 
.059  86 
.060  85 
.061  84 
.062  83 
.063  82 
.06482 
.0658 
.0668 
-0678 

.069 

•07 

.071 

.072 

•073 

.074 

•075 

.076 

•077 

.078 

•079 

.08 

.081 

.082 

.083 

.084 

.085 

.086 

.087 

.088 

.089 

.09 

.091 

.O92 

•093 

•°94 

•095 

.096 

•097 

.098 

•°99 

.1 

.IOI 

.102 

2 

.068  78 
.06977 
.070  76 
•07175 
.072  74 
•°73  73 

•074  72 

•075  5 

.07669 

.07768 

.07867 

.07966 

.080  64 

.081 63 

.082  62 

.0836 

.08459 

•085  57 

.086  56 

.08754 

.08853 

.08951 

.0905 

.091 48 

.09246 

•09344 

.09443 

.0954 

.09639 

•09737 

•09835 

•09933 
.10031 
.101  29 
:* 

.103 

.IO4 

.105 

.106 

.107 

.108 

.IO9 

.11 

.III 

.112 

•113 

.114 

•115 

.Il6 

.117 

.118 

•II9 

.12 

.121 

.122 

.123 

.I24 

.125 

.126 

.127 

.128 

.I29 

•L3 

.131 

.132 

•133 

•134 

•135 

•136 

.102  27 
•IO325 
. IO4  22 
.105  2 
.I0618 
.IO715 
.IO813 
.IO911 
.11008 
.III06 
.11203 
•113 
.II398 
•II495 
.II592 
.1169 
.II787 
.11884 
.II981 
.120  78 

• 121  75 
.122  72 
.12369 
.12469 
.125  62 
.12659 
•I27  55 
.12852 
.12949 

•13045 

.13141 

•13238 

•13334 

•1343 

•137 

.138 

•139 

.14 

.141 

.142 

•143 

.144 

•145 

.146 

.147 

.148 

.149 

•15 

•151 

•152 

•153 

•154 

•155 

.156 

•157 

.158 

•159 

.16 

.l6l 

.162 

.163 

.164 

.165 

.166 

.167 

.168 

.169 

•*7 

•I3527 

•t36  23 

•137  19 
•13815 

.13911 

.14007 

.14103 

.14198 

.14294 

•T439 

.14485 

.14581 

• i46  77 

.14772 
.14867 
.149  62 

.15058 

•15153 

.15248 
•I53  43 
•15438 
•155  33 
.15628 

•i5723 

.15817 

•15912 

.16006 

.16101 

.16195 

.1629 

.16384 

.16478 

.165  72 

.16667 

270 


AREAS  0E  ZONES  OF  A CIRCLE, 


H’ght. 

.171 

.172 

.173 

.174 

•175 

.176 

.177 

.178 

.179 

.18 

.181 

.182 

.183 

.184 

.185 

.186 

.187 

.188 

.189 

.19 

.191 

.192 

.193 

.194 

.195 

.196 

.197 

-198 

.199 

.2 

.201 

.202 

.203 

.204 

.205 

.206 


Area.  H’ght.  Area.  H’ght 

.21805 
.21894 
.21983 
.220  72 

.221  6l 
.222  5 
.22335 
.233  I .22427 
.234  .225 15 

.235  I .22604 
.236  I .22692 
.237  .2278 

.2^8  .22868 

.239 


.207 

.208 

.209 

.21 

.211 

.212 

.213 

.214 

.215 

.216 

.217 

.218 

.219 

.22 

.221 

.222 

.223 

.224 

.225 


,16761' 
,16855 
.16948 
.170  42 
.17136 

.1723 

.173  23 

.17417 

.1751 
.17603 
.17697 
.1779 
.17883 
.179  76 
.18069 
.181  62 
.182  54 
.i8347 
.1844 

.185  32 

.18625 

.18717 

.18809 

.18902 

.18994 

.19086 

.191  78 

.1927 

.193  61 

.194  53 

.19545 

.19636 

.197  28 

.198  19 

.1991 

.20001 

.20092 

.201  83 

.202  74 

.20365 

.204  56 

.205  46 

.20637 

.207  27 

.208  18 

.20908 

.20998 

.21088 

.211  78 

.21268 

.21358 
.21447 
-21537 
.21626 
.217  16 


.226 

.227 

.228 

.229 

.23 

.231 

.232 


.24 

.241 

.242 

•243 

.244 

•245 

.246 

.247 

.248 

.249 

.25 

.251 

.252 

.253 

.254 

.255 

.256 

•257 

.258 

.259 

.26 

.261 

.262 

.263 

.264 

.265 

.266 

.267 

.268 

.269 

.27 

<27 1 

.272 

•273 

.274 

.275 

.276 

•277 

.278 

•279 

.28 


.229  56 
.23044 

.231  3i 
.232  19 

.23306 
.23394 
.234  81 

.235  68 
.236  55 
.237  42 
.238  29 

.239 1 5 

.24002 
.240  89 
.241  75 
.242  61 
•243  47 
.244  33 
.245  19 
.246  04 
.2469 

.247  75 

.248  61 
.249  46 
.25021 
.251 16 
.25201 
.25285 
•253  7 
.254  55 
.25539 
.25623 
.25707 
.25791 

.25875 
.25959 
.26043 
.261  26 
.26209 
.26293 
^263  76 
.26459 


.281 
.282 
.283 
.284 
.285 
.286 
.287 
.288 
.289 
.29 
.291 
.292 
.293 
.294 
.295 
.296 

.297 

.298 
•299 
•3 
•301 
.302 
•303  I 
.304  I 
.305 

.306 

.307 

•3°8 

•309 

•31 
•311 
•312 
.313 
.314 
.315 
.316 
317 
.318 
■3i9 
•32 

.321 
.322 

.323 
.324 
.325 
.326 
.327 

.328 

•329 

•33 
.331 
.332 
•333 
•334 
•335 


,265  41 
.266  24 
,267  06 
.267  89 
.268  71 

.26953 
,27035 

.271 17 
.27199 
.272  8 
.273  62 
.27443 

.27524 

.27605 
.276  86 
.27766 

.27847 

.27927 
.280  07 
.28088 
.281  67 
.28247 
I .28327 
.28406 
I .28486 
I .28565 
.28644 
.28723 
.28801 
.2888 
.28958 
.29036 

.291 15 
.291  92 
.292  7 
.29348 
.294  25 

.295  02 
.2958 
.296  56 

.297  33 
.298  1 
.298  86 
.299  62 

•30039 

.301 14 

.3019 

.30266 

.30341 

•3°4 16 

.304  91 

.305  66 
.30641 

.307  15 


.336 
•337 
•338 
i *339 
•34 
.341 
.342 
•343 
•344 
•345 
•346 
•347 
.348 
•349 
•35 
I *351 
•352 
•353 
•354 
•355 
•356 
357 


•358 
•359 
•36 

.361 

.362 

.363 

.364 

.365 

.366 

•367 

.368 

•369 

•37 

•37i 

.372 

•373 

•374 

•375 

.376 

•377 

•378 

•379 

•38 

.381 

•382 

.383 

.384 

.385 

.386 

.387 

.388 

-389 


.3079  II  -39 


.308  64 
30938 

.310 12 
•3io85 
•311  59 
•3I232 

31305 

-3*3 1% 

•3145 

•31523 

■31595 
-3l667 
•3X7  39 
•3l8  11 
.31882 
•3*954 
,320  25 
.320  96 
.321  67 

32237 

.32307 

323  77 
•32447 

.325 17 
.32587 
.32656 
•32725 

.327  94 
.328  62 
.329  31 
.32999 

•33067 

•331 35 
•33203 

.332  7 
33337 
•33404 
334  7 1 
•335  37 
33604 
•336  7 
•337  35 
.33801 
.33866 
.33931 
.33996 
.34061 

.34125 
.3419 
.342  53 
.34317 
•3438 
•34444 
.34507 
.34569 


H’ght.  I 

Area. 

.391 

.34632 

•392  | 

.34694 

•393  1 

•347  56 

•394 

.348  18 

•395 

.348  79 

.396 

•3494 

•397 

.35001 

.398 

•35062 

•399 

•351  22 

•4 

.35182 

.401 

.35242 

.402 

•35302 

.403 
404 
.405 

.406 

.407 

.408 

.409 

.41 
.411 
,412 
•4i3 
.414 
■415 
.416 

.417 

.418 

.419 

.42 

.421 

.422 

•423 

.424 

.425 

.426 

•427 

.428 

•429 

•43 

.431 

.432 

•433 

•434 

•435 

.436 

•437 

.438 

•439 

•44 

.441 

.442 

•443 

.444 

•445 


.35361 

■3542 

•35479 
•35538 
•35596 
•35654 
•357II 
.35769 
.35826 
.35883 
•35939 
•35995 
.36051 
.361 07 
.361 62 

.362 17 

,362  72 
.36326 

•3638 

.36434 

.36488 

.36541 

.36594 

.36646 

.36698 

.3675 

.368  02 
.36853 
.36904 

.36954 
.37005 
.370  54 
•37 1 °4 
•37153 
.37202 
•372  5 
.37298 

.37346 
•37393 
•3744 
.37487 
•375  33 
•375  79 


AREAS  OF  ZONES  OF  A CIRCLE. 


271 


H’ght. 

Area. 

| H’ght. 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

H’ght. 

Area. 

.446 

.37624 

•457 

.38096 

.468 

•385  14 

•479 

.38867 

•49 

•391 37 

•447 

.37669 

•458 

•381 37 

.469 

•385  49 

.48 

•38895 

•491 

•39i  56 

.448 

! -37714 

•459 

•381  77 

•47 

•385  83 

.481 

•389  23 

.492 

•39i  75 

•449 

•377  58 

.46 

.382  16 

.471 

.38617 

.482 

•3895 

•493 

.391 92 

•45 

.37802 

.461 

•382  55 

.472 

.3865 

•483 

.389  76 

•494 

.39208 

•45i  I 

•37845 

.462 

.382  94 

•473 

.38683 

•484 

.390  01 

•495 

•39223 

.452  : 

.37888 

•463 

•38332 

•474 

•387  15 

•485 

.390  26 

.496 

•392  36 

♦453 

•37931 

.464 

•38369 

•475 

•38747 

.486 

•390  5 

•497 

•39248 

•454  ! 

•379  73 

•465 

.38406 

.476 

•387  78 

•487 

•390  73 

.498 

•392  58 

•455 

.380 14 

.466 

•384  43 

•477 

.38808 

.488 

•390  95 

•499 

.392  66 

•456: 

.38056 

.467 

•384  79  1 

•478 

.38838 

•489 

•39117 

•5 

•392  7 

This  Table  is  computed  only  for  Zones,  lonyest  Chord  of  which  is  Diam- 
eter. 


To  Compute  Area  of  a Zone  "by  preceding  Table. 

When  Zone  is  Less  than  a Semicircle . 

Rule— Divide  height  by  diameter,  find  quotient  in  column  of  heights. 
Take  area  for  height  opposite  to  it  in  next  column  on  right  hand,  multiply 
it  by  square  of  longest  chord,  and  product  will  give  area  of  zone. 

Example.— Required  area  of  a Zone,  diameter  of  which  is  50,  and  its  height  15. 

15  4- 50  = .3;  and  .3,  as  per  table,  — . 280  88. 

Hence  . 280  88  X 502  = 702.2  area. 

When  Zone  is  Greater  than  a Semicircle. 

Rule.— Take  height  on  each  side  of  diameter  of  circle,  and  ascertain,  by 
preceding  Rule,  their  respective  areas ; add  areas  of  these  two  portions  to- 
gether, and  sum  will  give  area. 

Example.  — Required  area  of  a zone,  diameter  of  circle  being  50,  and  heights  of 
zone  on  each  side  of  diameter  of  circle  20  and  15. 

20-h-  50  = . 4;  .4,  as  per  table,  ==.351  82;  and  .351  82  X so2  = 879.55. 
i£-^5o  = -3;  .3,  as  per  table,  =.28088;  and  .28088  X so2  = 702.2. 

Hence  879.554-702.2  = 1581.75  area. 


When,  in  Division  of  a Height  by  Chord,  Quotient  has  a Remainder  after 
Third  Place  of  Decimals,  and  great  Accuracy  is  required. 

Rule.— Take  area  for  first  three  figures,  subtract  it  from  the  next  follow- 
ing area,  multiply  remainder  by  said  fraction,  and  add  product  to  first  area ; 
sum  will  give  area  for  whole  quotient. 

Example.  — What  is  area  of  a zone  of  a circle,  greater  chord  being  100  feet,  and 
breadth  of  it  14  feet  3 ins.?  5 


14  feet  3 ins.  — 14-25,  and  14. 25 -=-100=3.1425;  tabular  length  for 
and  for  . 143  — . 141 03,  difference  between  which  is  .00096. 

Then  . 5 x .000  96  = .000  48.  Hence  .142  = . 140  07 
.0005  = .000  48 


.142  = .140  07, 


chord  is  to  be  multiplied  ; 


. 1 40  55 , mm  by  which  square  of  greater 
and  . 140  55  X 1002  ■=  1405.5  feet. 


[ lsh.it. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

12 

13 

14 

i5 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

4i 

42 

43 

44 

45 

46 

47 

48 

49 

50 

5i 

52 

53 

54 


SQUARES,  CUBES,  AND  ROOTS. 


es.  Cubes,  and  Square  and  Cube 
From  1 to  1600. 


Square. 


I 

4 

9 

16 

25 
36 
49 
64 
81 
1 00 
1 21 
144 
1 69 

1 96 
225 
256 

2 89 

324 

3 61 
400 
441 
484 

529 

5 76 
625 

6 76 

729 

784 
841 
900 
961 
1024 
1089 
11  56 
1225 
1296 
1369 
1444 
1521 
1600 
1681 
1764 
1849 
1936 
2025 
21  16 
2209 
2304 
2401 
2500 
2601 
2704 
2809 
29 16 


Cube. 

Square  Root. 

I 

8 

I 

I.4142136 

27 

I-732  0508 

64 

2 

125 

2.236068 

216 

2-449  489  7 

343 

2.645  751  3 

512 

2.828  427 1 

729 

3 

1 000 

3.1622777 

I331 

3.316624  8 

1 728 

3.464  101  6 

2197 

3 605  5513 

2 744 

3.741  6574 

3 375 

3.8729833 

4096 

4 A 

4 9*3 

4.123  1056 

5832 

4.242  640  7 

6859 

4*358  598  9 

8000 

4*472  136 

9261 

4.582  575  7 

10648 

4.6904158 

12  167 

4*795  831  5 

13  824 

4.898  9795 

15625 

5 

17  576 

5.0990195 

19683 

5.1961524 

21952 

5.291  5026 

24  389 

5.385  l648 

27000 

5.4772256 

29  791 

5.5677644 

32  768 

5.6568542 

35  937 

5.744  5626 

39  3°4 

5.8309519 

42  875 

5.9160798 

46656 

6 

50653 

6.082  762  5 

54872 

6.164414 

59319 

6.244  998 

64000 

6.324  555  3 

68  921 

6.4031242 

74  088 

6.480  740  7 

79  507 

6.557  438  5 

85184 

6.633  2496 

91  I25 

6.708  2039 

97  336 

6.782  33 

103  823 

6.8556546 

1 10  592 

6.928  203  2 

117649 

7 . 0 

125  000 

7.071  067  8 

132  651 

7.141  4284 

140  608 

7.21 1 1026 

148877 

7.280  1099 

157464 

7.3484692 

Numbe 

55 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

7i 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

9i 

92 

93 

94 

95 

96 

97 

-98. 

99 

100 

101 

102 

103 

104 

105 


SQUARES,  CUBES,  AND  ROOTS. 


273 


Square. 

f Cube. 

30  25 

166375 

3136 

175  616 

32  49 

185  193 

3364 

195  1 12 

34  81 

205  379 

36  OO 

216000 

3721 

226  981 

3844 

238  328 

3969 

250  047 

40  96 

262  144 

42  25 

274  625 

43  56 

287  496 

44  89 

300  763 

46  24 

314  432 

4761 

328  509 

4900 

343000 

5041 

357  911 

5184 

373  248 

53  29 

389017 

54  76 

4°5  224 

5625 

421  875 

57  76 

438  976 

59  29 

456533 

60  84 

474  552 

6241 

493  039 

64  00 

512000 

6561 

53i  44i 

6724 

551368 

68  89 

57i 787 

7056 

592  704 

72  25 

614  125 

7396 

636056 

7569 

658  503 

77  44 

681  472 

7921 

704  969 

81  00 

729000 

82  81 

753  571 

8464 

778688 

86  49 

804  357 

88  36 

830  584 

9025 

857  375 

92  16 

884  736 

9409 

912673 

96  04 

941  192 

9801 

970  299 

1 0000 

1 000000 

1 0201 

1 030  301 

1 0404 

1 061  208 

1 0609 

1 092  727 

1 08  16 

1 124864 

1 1025 

1 157625 

1 1236 

1 191 016 

1 1449 

1 225  043 

1 1664 

1 259712 

1 1881 

1 295  029 

1 21 00 

1 331 000 

Square  Root. 

Cube  Root. 

7.416  198  5 

3.802  952  5 

7-4833I4  8 

3.825  862  4 

7-549834  4 

3.848  501  I 

7-6i5  773  1 

3.8  708766 

7.681  145  7 

3.8929965 

7-745  9667 

3.9148676 

7.810249  7 

3.936  497  2 

7.874  0079 

3-957  89I5 

7-937  253  9 

3-979057I 

8.062  257  7 

4 

4.020  725  6 

8.1240384 

4.041  240  1 

8-185  352  8 

4.061  548 

8.246  21 1 3 

4.081  655  1 

8.306  623  9 

4.101  566  1 

8.3666003 

4.121  2853 

8.426  149  8 

4.140817  8 

8.485  281  4 

4.160  167  6 

8-544  003  7 

4-179  339 

8.602  3253 

4.1983364 

8.660  254 

4.217  1633 

8.7177979 

4-235  823  6 

8.774  9644 

4.254321 

8.831  7609 

4.272  6586 

8.888 194  4 

4.290  840  4 

8.9442719 

4.308  869  5 

9 

4.326  748  7 

9.0553851 

4.344  4815 

9-1104336 

4.362  070  7 

9.165  1514 

4-379  5I9I 

9.219  544  5 

4.396  829  6 

9.2736185 

4.414  004  9 

9.327  379  1 

4.4310476 

9.380831  5 

4.447  960  2 

9-433  98I  1 

4.464  745  1 

9.486833 

4.481 404  7 

9-539392 

4-497  94i  4 

9.591  663 

4.5H357  4 

9-643  650  8 

4-530  654  9 

9-695  359  7 

4-5468359 

9.746  794  3 

4.562  902  6 

9-797  959 

4-578  857 

9.848  8578 

4-5947009 

9.899  494  9 

4-6io  436  3 

9.9498744 

4.626065 

10 

4-641  588  8 

10.049  875  6 

4-6570095 

I0-099  504  9 

4.672  3287 

10.148  891  6 

4.687  548  2 

10.198039 

4.702  6694 

10.246  950  8 

4.717694 

10.295  630 1 

4.732  6235 

10.344  °8o  4 

4-747  4594 

IO-392  304  8 

4.762  203  2 

10.440  306  5 

4.776  8562 

10.488  088  5 

4-791  4199 

JMBEK. 

Ill 

1 12 

113 

114 

115 

Il6 

117 

Il8 

IT9 

120 

121 

122 

123 

124 

125 

126 

127 

128 

129 

130 

131 

132 

133 

134 

135 

136 

137 

138 

139 

140 

141 

142 

143 

144 

145 

146 

147 

148 

149 

150 

151 

152 

153 

154 

155 

156 

157 

158 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 

CUEE. 

I 23  21 

I 367  63I 

125  44 

I 4O4  928 

I 2769 

I 442  897 

I 2996 

i 481  544 

1 32  25 

1 520  875 

i 34  56 

1 560  896 

1 3689 

1 601  613 

139  24 

1 643  032 

1 41  61 

1 685  159 

1 4400 

1 728000 

1 4641 

1 771  561 

1 48  84 

1815848 

1 51  29 

1 860  867 

1 53  76 

1 906  624 

1 56  25 

1 953  125 

1 58  76 

2000376 

1 61  29 

2 048  383 

1 6384 

2097  152 

1 6641 

2 146689 

1 6900 

2 197000 

17161 

2 248  091 

1 74  24 

2 299  968 

1 7689 

2 352  637 

1 79  56 

2 406  104 

1 82  25 

2 460  375 

1 8496 

2 5t5  456 

1 87  69 

2 571  353 

19044 

2 628  072 

1 93  21 

2 685  619 

1 9600 

2 744000 

1 9881 

2803221 

201  64 

2 863  288 

20449 

2 924  207 

20736 

2 985  984 

2 1025 

3 048  625 

2 13  16 

3 1 12  136 

2 1609 

3 1 76523 

2 19  04 

3241  792 

2 2201 

3 307  949 

2 2500 

3375000 

2 2801 

3 442  951 

23104 

3511008 

23409 

3 581577 

23716 

3652  264 

24025 

3 723  875 

2 43  36 

3796416 

24649 

3 869  893 

24964 

3944312 

2 52  81 

4019679 

2 5600 

4096000 

25921 

4 173  281 

262  44 

4251  528 

26569 

4 330  747 

26896 

4410944 

2 72  25 

4492  125 

2 75  56 

4574296 

Square  Root. 

10.535  653  8 

IO.583OO52 

10.630  145  8 
10.6770783 
10.723  805  3 
10.770  3296 
10.8166538 
10.862  7805 
10.908  7121 

10.954  4512 

11 

11.045  361 
11.090  5365 
11. 135  5287 
11.1803399 
11.224972  2 
11.269427  7 
iI*3I3  708  5 
11.3578167 
1 1. 401  7543 
11.445  5231 
11.489 125  3 
II-532  5626 
n-575  8369 
11.61895 
1 1. 661  9038 
11.7046999 
n-747  340  1 
11.789  826  1 
11.832  1596 
11.874  342  1 
11.9163753 
11.958  260  7 
12 

12.041  5946 
12.083046 
12:124355  7 
12.165  5251 
,12.2065556 
12.247448  7 
12.288  205  7 
12.328  828 
12.3693169 
12.4096736 
12.449  8996 
12.489996 
12.5299641 
12.569  805  1 
12.6095202 
12.649  1106 
12.6885775 
12.727  922  1 
12.767  1453 
12.806  248  5 
12.845  232  6 
12.884098  7 


SQUARES,  CUBES,  AND  ROOTS. 


275 


Number.  | Square.  | Cube. 


167 

168 


t 


169 


170 

171 

172 

173 

174 

175 


176 


2 78  89 
2 82  24 
28561 
2 89  OO 
29241 
295  84 
29929 
302  76 
30625 
30976 


177 

178 

179 

180 

181 

182 

183 

184 

185 

186 

187 

1 88 

189 

190 

191 

192 

193 

194 

195 

196 

197 

198 

199 

200 

201 

202 

203 

204 

205 

206 

207 

208 

209 

210 

211 

212 

213 

214 

215 

216 

217 

218 

219 

220 

221 

222 


31329 

3 1684 
32041 
32400 
32761 
33124 
3 34  89 
3 38  56 
3 42  25 
3450 
3 4969 
3 53  44 
3 57  21 
361  00 
36481 
36864 
3 72  49 
37636 

380  25 
38416 
38809 

3 92  04 
39601 
40000 
40401 
40804 

4 12  09 
4 16  16 
420  25 

42436 

42849 
43264 
4 36  8i 
441  00 
4 45  21 
4 4944 
4 53  69 
4 57  96 
462  25 
46656 
4 70  89 
4 75  24 
4 796i 
4 8400 
48841 
4 92  84 


4657463 

4 74i  632 

4 826  809 
4913000 

5 000  21 1 
5 088  448 
5 177  717 
5 268  024 
5 359375 
5 45i  776 
5 545  233 

5639  752 
5 735  339 
5 832  000 

5 929  74i 

6 028  568 
6 128487 
6 229  504 
6331625 
6434  856 
6 539  203 
6 644  672 
6751  269 

6 859  000 
6967  871 
7077888 
7189057 

7301384 

7 4i4  875 
7 529  536 
7 645  373 
7762  392 

7 880  599 

8 000  000 
8 120601 
8 242  408 
8365  427 
8489664 
8 615  125 
8 741  816 

8 869  743 
8998912 

9 129329 
9 261  000 
9393  931 
9528128 

9663597 
9800344 
9938  375 
10077696 
10218313 
10  360  232 
10  503  459 
10  648  000 
10  793  861 
10  941  048 


Square  Root. 

12.922  848 
12.961  481  4 
13 

13.038  404  8 
i3*°76  6968 
13.114877 

13-1529464 

13.190906 
13.228  7566 
13.2664992 
I3*3°4  134  7 
13-34I  664  1 
I3-379  °88  2 
13.4164079 
*3-453  624 
I3-49°  737  6 
i3-527  7493 

*3-564  66 

13.601  4705 
13.638  181  7 
*3-674  794  3 
I3-711  3°9  2 
I3-747  727  1 
13.784  0488 
13.820275 
13-8564065 

i3-892  44 

13- 9283883 
13.96424 

14 

*4-°35  668  8 
14.0712473 
14.106  736 

14- I42  1356 
I4-I77  4469 
14.212  6704 
14.247  806  8 
14.282  856  9 
14.317821 1 
I4-352  700  1 
14.387  494  6 
14.422  205  1 
i4-456  832  3 
I4-49I  376  7 
I4-525  839 
14.560219  8 

*4-594  5*9  5 
14.628  738  8 
14.662  878  3 
14.6969385 
*4-730  919  9 
14.764823  1 
*4-798  648  6 
*4-832  397 
14.866  068  7 
148996644  i 


Cube  Root. 


5-5068784 
5.5178484 
5.5287748 
5-539658  3 
5.550  4991 
5.561  2978 
5-572  0546 
5.582  7702 
5-593  444  7 
5.604078  7 
5.6146724 
5.625  2263 
5.635  7408 
5.6462162 
5.6566528 
5.667  051  1 
5-6774114 
5.687  734 
5.6980192 
5.708  2675 

5-7*8479* 

5.728  6543 
5.738  793  6 
5.748  897  1 
5.758  9652 
5.768  9982 
5.778  9966 
5.788  9604 
5.798  89 
5.808  785  7 
5.8186479 
5.8284767 
5.8382725 
5-8480355 
5.857  766 
5.8674643 
5.877  1307 
5.886  765  3 
5.896  3685 
5.9059406 

5.9*54817 

5.924  9921 
5.934472  1 
5.943  922 

5- 953  34*8 
5.962  732 
5.972  092  6 
5.981  424 
59907264 
6 

6.009  245 
6.018  461  7 
6.027  650  2 
6.036  810  7 

6- 045  943  5 
6 055  048  9 


UMBEE. 

223 

224 

225 

226 

227 

228 

229 

23° 

231 

232 

233 

234 

235 

236 

237 

238 

239 

240 

241 

242 

243 

244 

245 

246 

247 

248 

249 

250 

251 

252 

253 

254 

255 

2^6 

257 

258 

259 

260 

261 

262 

263 

264 

265 

266 

267 

268 

269 

270 

271 

272 

273 

274 

275 

276 

277 

278 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 

Cube. 

4 97  29 

II089567 

501  76 

1 1 239  424 

50625 

II390625 

5 IO  76 

II543176 

51529 

11697083 

5 1984 

II  852352 

52441 

12  008  989 

52900 

12  1670OO 

5 33  6i 

12326391 

5 38  24 

12  487  l68 

5 42  89 

12649337 

54756 

12812904 

5 52  25 

I2977875 

5560 

13  144  256 

56169 

13312  053 

56644 

I3481  272 

5 71  21 

I3651919 

5 7600 

I3824  OOO 

5 8081 

13  997  521 

585  64 

14  172  488 

5 90  49 

14348907 

59536 

14  526  784 

60025 

14  706  125 

605  16 

14886936 

6 1009 

15069223 

6 1504 

15252992 

6 2001 

15  438  249 

6 25  00 

15  625000 

63001 

15813251 

63504 

16  003  008 

64009 

16194277 

645  16 

16  387  064 

65025 

16581375 

65536 

16  777  216 

66049 

16974  593 

66564 

I7I735I2 

6 7081 

17  373  979 

6 7600 

17  576000 

681  21 

17  779581 

68644 

17984728 

691  69 

18  191  447 

69696 

18  399  744 

702  25 

18  609  625 

7°7  56 

18821 096 

7 12  89 

19034163 

7 18  24 

19  248  832 

7 2361 

19  465  109 

7 2900 

19  683  OOO 

7 34  41 

19902  511 

7 39  84 

20  123648 

7 45  29 

20346417 

7 50  76 

20  570  824 

75625 

20  796875 

7 61  76 

21  024  576 

767  29 

21  253933 

7 72  84 

21  484  952 

Square  Root. 

14.9331845 
14.966  629  5 

*5  . 

15-033  2964 
15.0665192 
15.0996689 
i5-*32  746 
i5-i65  7509 
15.198  6842 
15-231  546  2 
15264  337  5 
15.297  0585 
I5-329  7°9  7 
15.362  291  5 
15-394  804  3 
15.4272486 
15.459  6248 

I5-49I  933  4 
15.5241747 
I5-556349  2 
I5-588  457  3 
15.6204994 
15.6524758 
15.684  3871 
15-7162336 
15.748015  7 
15-779  733  8 
15.811  3883 
15.842  9795 
I5-874  507  9 
15-9059737 

15-937  377  5 
15.9687194 
16 

16.031  2195 
16.062  378  4 
16.0934769 

16.1245155 
16.155  4944 
16.186414  1 
16.217  274  7 
16.248  0768 
16.278  8206 
16.309  5064 
16.3401346 
16.370  705  5 
16.401  219  5# 
16.431  676  7 
16.462  077  6 
16.492  422  5 
16.522  71 1 6 

16.5529454 

16.583124 
16.613247  7 
16.643317 

16.678  332 


N UMBE 

279 

280 

28l 

282 

283 

284 

285 

286 

287 

288 

289 

290 

291 

292 

293 

294 

295 

296 

297 

298 

299 

300 

301 

302 

303 

304 

305 

306 

307 

308 

309 

310 

31 1 

312 

313 

314 

315 

316 

317 

318 

319 

320 

321 

322 

323 

324 

325 

326 

327 

328 

329 

330 

331 

332 

333 

334 


SQUARES,  CUBES,  AND  ROOTS. 


277 


Square. 

Cube. 

Square  Root. 

I Cube  Root. 

77841 

21  7I7639 

16.703  293  I 

6.534  335  I 

7 8400 

21  952OOO 

16.733  200  5 

6.542  132  6 

7 8961 

22  188041 

16.7630546 

6.549911  6 

7 95  24 

22  425  768 

16.792  8556 

6.5576722 

80089 

22  665  187 

16.822  603  8 

6.5654144 

806  56 

22  906  304 

16.8522995 

6.573  138  5 

8 12  25 

23  I49  125 

16.881  943 

6.580  8443 

8 17  96 

23  393  656 

16.9115345 

6.588  532  3 

823  69 

23  639  903 

16.9410743 

6.596  202  3 

82944 

23  887  872 

16.970  562  7 

6.603  854  5 

83521 

24  137  569 

17 

6.61 1 489 

841  00 

24  389  OOO 

17.029  3864 

6.619  106 

84681 

24642  1 71 

17.058  722  1 

6.626  705  4 

85264 

24  897  088 

17.088  007  5 

6.634  287  4 

85849 

25  153  757 

17.117  242  8 

6.641  852  2 

86436 

25412  184 

17.1464282 

6.649  399  8 

870  25 

25  672  375 

I7-I75  564 

6.656  930  2 

8 76 16 

25  934  336 

17.2046505 

6.664  443  7 

8 82  09 

26  198  073 

17.233  6879 

6.6719403 

8 88  04 

26  463  592 

17.262  6765 

6.679  42 

8 9401 

26  730899 

17.291  6165 

6.686  883  1 

90000 

27000000 

17.320  508  1 

6.6943295 

90601 

27  270  901 

I7*34935i6 

6.701  7593 

9 1204 

27543608 

17.378  1472 

6.709 1729 

9 1809 

27  818  127 

17.406  895  2 

6.71657 

924  16 

28  094  464 

17*435  595  8 

6.723  9508 

930  25 

28372  625 

17.4642492 

6.731  3i5  5 

93636 

28  652616 

I7-492  8557 

6.738  664  1 

942  49 

28934443 

17.521  415  5 

6.745  9967 

948  64 

29218  112 

17.5499288 

6.753  3134 

9 54  8i 

29  503629 

I7-578395  8 

6.7606143 

961  00 

29  791  OOO 

17.6068169 

6.767  8995 

967  21 

30080  231 

17*635  192  1 

6.775  169 

9 7344 

30371328 

17.663521  7 

6.782  4229 

97969 

30664297 

17.691  806 

6.789  661  3 

98596 

30959 144 

17.720045  1 

6.796  8844 

992  25 

31 255  875 

17.748  2393 

6.804  °92  1 

998  56 

31 554496 

17.7763888 

6.811  284  7 

10  04  89 

31855013 

17.8044938 

6.818462 

10  11  24 

32157432 

17*832  554  5 

6.825  624  2 

10  1761 

32  461  759 

17.860571  1 

6.832  771  4 

102400 

32  768  OOO 

17.8885438 

6.839  9°3  7 

10  3°  41 

33076161 

17.916472  9 

6.847  02 1 3 

10  36  84 

33386248 

. I7-944  3584 

6.854  124 

1043  29 

33  698  267 

17.972  2008 

6.86l  212 

1049  76 

34012224 

18 

6.868  285  5 

10  56  25 

34328125 

18.027  7564 

6.875  344  3 

10  62  76 

34645976 

18.055  47°  1 

6.882  388  8 

10  69  29 

34  965  783 

18.083  Hi  3 

6.889418  8 

10  75  84 

35  287552 

18. no  770  3 

6.896  434  5 

108241 

35611  289 

18.1383571 

6.903435  9 

108900 

35  937  000 

18.165  902  1 

6.910423  2 

10  95  61 

36  264  691 

18.1934054 

6.9173964 

1 1 02  24 

36  594  368 

18.220  867  2 

6.924  3556 

1 1 08  89 

36926037 

18.248  287  6 

6.9313088 

n 1556 

37  259  704 

A A 

18.275  6669 

6.938  232  1 

337 

33$ 

339 

340 

34i 

342 

343 

344 

345 

34^ 

347 

34§ 

349 

350 

35i 

352 

353 

354 

355 

35^ 

357 

358 

359 

360 

361 

362 

3^3 

364 

365 

366 

367 

368 

3^9 

37c 

37i 

372 

37: 

374 

37! 

37; 

37* 

37< 

38c 

38: 

38: 

38. 

38. 

38 

38 

38 

38 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 


II  22  25 
1 1 28  96 
1135  69 
1 1 42  44 

II  49  21 
1 1 56  OO 
II  62  8l 

1 1 69  64 
II  7649 
H83  36 
119025 

11  97  16 
1204  09 

12  11  04 
12  1801 
12  2500 
123201 
123904 
124609 
1253  16 
126025 
12  67  36 

127449 

12  81  64 
128881 
129600  J 

13  03  21  j 
13  1044 

13  I7  69 

132496 

1332  25 
13  39  56 
134689 
13  54  24 
1361  61 
136900 
137641 

1383  84 

13  91  29 
1398  76 
1406  25 

14  J3  76 

1421  29 
14  28  84 
143641 
144400 
i4  5l61 
14  59  24 
14  66  89 
147456 
148225  j 
14  89  96 

14  97  69 

15  05  44 
15  13  21 

15  21  00  I 


Cube. 


37  595  375 

37  933056 

38  272  753 
38614472 
38958219 

39  304000 

39  651821 

40  001  688 

40  353607 

40  707  584 

41  063  625 
41  421  736 

41  781  923 

42  144  192 
42  508  549 

42  875  000 

43  243  551 
43614208 

43  986977 
44361  864 

44  738  875 

45  118016 
45  499  293 

45  882712 

46  268  279 

46  656  000 

47  °45  831 
47  437  928 

47  832  147 

48  228  544 

48  627  125 
49027  896 

49  430  863 
49836032 

50  243  409 
50653000 

51  064  81 1 
51478  848 

51895117 

52  3T3  624 

52  734  375 

53  157  376 

53  582  633 
54010152  j 

54  439939  ! 

KA  872  OOO 

55  306  341  | 

55  742  968  ; 

56  181  887 
56  623  104 
57066625 
57512456 
57960603 
58411072 
58  863  869 

593i900° 


Square  Root. 


18.3030052 
18.330  302  8 
18.357  559  8 
18.3847763 
18.4H  952  6 
18.439  088  9 
18.466  185  3 
18.493  242 
18.5202592 
18.547  237 

18.574175  6 
18.601  O75  2 
18.627936 
18.6547581 
18.681  541  7 
18.708  2869 

18.734994 

18.761 663 
18.788  2942 
18.814887  7 
18.841  4437 
18.867  962  3 
18.894  4436 
18.920  887  9 
18.947  2953 
18.973  666 

19 

19.026  297  6 

19-0525589 

19.078  784 
19.1049732 
19.131  1265 
19.157  244  1 
19.183  3261 
19.209  372  7 
19-235  384  1 

19.2613603 
19.287  301  5 | 

i9-3I3  207  9 
I9-3390796  j 
19.364  9l6  7 | 

19.3907194 
19.4164878 
19.442  222  1 
19.467  922  3 
19-493  588  7 
X9-5I9  221  3 
19.544  820  3 
19-57°  3858 
19-595  9*7  9 
19.621  4169 
19.646  882  7 
19.6723156 

19.697  7156 
19.723  0829 

19.7484177  I 


SQUARES,  CUBES,  AND  ROOTS. 


279 


Number 

Square. 

Cube. 

Square  Root. 

1 Cube  Root. 

391 

152881 

59776471 

I9-773  7I9  9 

7.3123828 

392 

15  3664 

60  236  288 

19.798  9899 

7.3186114 

393 

15  44  49 

60698  457 

19.824  227  6 

7.324  8295 

394 

15  52  36 

6l  162  984 

i9-849433  2 

7*33 1 0369 

395 

156025 

6l  629  875 

19.874  6069 

7*337  233  9 

396 

15  68  16 

62  099  136 

19.899  748  7 

7.343  420  5 

397 

15  7609 

62  570  773 

19.924  858  8 

7.349  596  6 

398 

158404 

63  044  792 

J9-949  937  3 

7*355  762  4 

399 

15  92  01 

63  521  199 

I9-974  984  4 

7.3619178 

400 

16  0000 

64  000  000 

20 

7.368  063 

401 

16  08  01 

64481  201 

20.024  984  4 

7.374  1979 

402 

16  1604 

64  964  808 

20.049937  7 

7.380322  7 

403 

! 162409 

65  450  827 

20.074  859  9 

7.386  437  3 

404 

1632  16 

65  939  264 

20.099  75 1 2 

7.392  541  8 

405 

1640  25 

66  430125 

20.124611  8 

7.3986363 

406 

1648  36 

66  923  416 

20.149441  7 

7.404  7206 

407 

1656  49 

67419  i43 

20.174  241 

7.410795 

408 

1664  64 

67917312 

20.1990099 

7.4168595 

409 

16  72  81 

68417929 

20.223  748  4 

7.422  914  2 

410 

16  8l  OO 

68  921  000 

20.248  456  7 

7.428  958  9 

4ir 

168921 

69426  531 

20.273  134  9 

7.434  993  8 

412 

16  97  44 

69934528 

20.297  783  1 

7.441  018  9 

4r3 

1705  69 

70  444  997 

20.322  401  4 

7*447  °34  2 

414 

171396 

70  957  944 

20.346  989  9 

7-453  0399 

4r5 

17  22  25 

7M73  375 

20-371  548  8 

7.459  035  9 

416 

1730  56 

71  991  296 

20.396  078  1 

7.465  022  3 

417 

1738  89 

72511713 

20.4205779 

7.470999  1 

418 

174724 

73  034  632 

20.445  048  3 

7.476  966  4 

419 

i7  55  6i 

73  560059 

20.469  489  5 

7.482  924  2 

420 

1 7 64  00 

74  088  000 

20.493  901  5 

7.488  872  4 

421 

17  72  41 

74618461 

20.518  2845 

7.494811  3 

422  j 

17  80  84 

75  151  448 

20.542  638  6 

7.500  740  6 

423  ; 

1 7 89  29 

75  686  967 

20.566  963  8 

7.506660  7 

424 

1797  76 

76  225  024 

20.591  260  3 

7.512  571  5 

425 

18  06  25 

76  765  625 

20.615  528  1 

7.518473 

426 

18  14  76 

77308776 

20.639  767  4 

7.524  365  2 

427 

18  23  29 

77854483 

20.663  978  3 

7.530  248  2 

428  | 

183184 

78  402  752 

20.688  1609 

7.536  122  1 

429 

18  4041 

78  953  589 

20.712  315  2 

7.541  986  7 

430 

18  4900 

79  507  000 

20.736441  4 

7.547  842  3 

43i 

185761 

80  062  991 

20.7605395 

7*553  688  8 

432 

18  66  24 

80621  568 

20.784609  7 

7.5595263 

433 

18  74  89 

81  182  737 

20.808  652 

7*565  354  8 

434 

18  83  56 

81  746504 

20.832  666  7 

7.571 1743 

435 

18  92  25 

82312875 

20.856  653  6 

7.576  9849 

436 

190096 

82  88 1 856 

20.880613 

7.582  786  5 

437 

190969 

83  453  453 

20.904545 

7*5885793 

438 

19  18  44 

84  027  672 

20.928  449  5 

7.594  363  3 

439 

19  27  21 

84  604519 

20.952  326  8 

7.600 1385 

440 

193600 

85  184000 

20.976 177 

7.605  904  9 

441 

194481 

85  766  121 

21 

7.611  662  6 

442 

195364 

86350888 

21.023  796 

7.617  411  6 

443 

1962  49 

86938  307 

21.0475652 

7.623  151  Q 

444 

19  71  36 

87528384 

21.071307  5 

7.628  883  7 

445 

19  80  25 

88  121  125 

21.095  023  1 

7.634  606  7 

446  1 

198916 

88  716  536 

21.1187121 

7.6403213 

28o 


SQUARES,  CUBES,  AND  ROOTS. 


Number. 

Square. 

447 

19  98  09 

448 

20  Q7  04 

449 

20  l6oi 

450 

20  25  OO 

45i 

20  34  OI 

452 

20  43  04 

453 

20  52  09 

454 

20  61  l6 

455 

20  70  25 

456 

20  79  36 

457 

20  88  49 

458 

20  97  64 

459 

21  0681 

460 

21 1600 

461 

21  25  21 

462 

21  3444 

463 

21  43  69 

464 

21  52  96 

465 

21  62  25 

466 

21  71  56 

467 

21  80  89 

468 

21  90  24 

469 

21  9961 

470 

22  0900 

47i 

22  18  41 

472 

22  27  84 

473 

22  37  29 

474 

22  46  76 

475 

22  56  25 

476 

22  65  76 

477 

22  75  29 

478 

22  84  84 

479 

22  94  41 

480 

230400 

481 

23  1361 

482 

23  23  24 

483 

23  32  89 

484 

2 3 42  56 

485 

23  52  25 

486 

23  61  96 

487 

23  71  69 

488 

23  81  44 

489 

2391  21 

490 

2401  00 

49 1 

24  1081 

492 

24  20  64 

493 

243049 

494 

24  40  36 

495 

24  50  25 

496 

24  60  16 

497 

24  70  09 

498 

24  80  04 

499 

24  90  01 

5°o 

25  00  00 

501 

25  1001 

502 

25  20  04 

Cube.  I Square  Root. 


89  314  623 
89915  392 
90518849 
91  125  000 

91  73385i 

92  345  408 
92  959  677 
93576664 

94  ^375 

94818816 

95  443993 
96071  912 

96  702  579 

97  336000 
97972  181 
98611 128 
99252847 
99897  344 

100  544  625 

101  194696 

101  847  563 

102  503  232 

103  16 1 709 
103  823  000 
104487  hi 

105  154048 

105  823817 

106  496  424 

107  171  875 

107  850  176 

108  531333 

109  215  352 
109  902  239 
no  592  OOO 
III  284  641 
III  980  168 
112678  587 

113379904 

114084  125 
1 14  791 256 

115501303 

116214272 
116930  169 
1 1 7 649  000 
118370771 
1 19  095  488 

119823157 

120553  784 

121  287375 

122023936 

122  763  473 

123  505  992 

124  251499 

125  000000 

125  751  501 

126  506  008 


21.1423745 

21.166010  5 
21.189  620  1 
21.2132034 
21.236  7606 
21.260  291  6 
21.283  7967 
21.307  275  8 
21.330  729 

21.3541565 

21.377  558  3 
21.4009346 
21.424  285  3 
21.447  6106 

21.4709106 

21.494  1853 
2i«5I7  434  8 
21.5406592 

21.5638587 
21.587  033  1 

21.610  182  8 
21.633307  7 
21.656407  8 
21.679  4834 
21.702  5344 
2i.725  561 
21.748  5632 
21.771  541 1 
21-794  494  7 
21.8174242 
21.840  3297 
21.863211  1 
21.8860686 

21.908  9023 
21.931  7122 
21.954498  4 
21.977  261 
22 

22.022  715  5 
22.045  407  7 
22.0680765 
22.090  722 

22.1133444 
22.135  9436 

22.1585198 
22.181  073 
22.2036033 
22.226  no  8 
22.248  595  5 
22.271 057  5 

22.2934968 
22  StSW6 
22.3383079 
22.3606798 
22.383  0293 
22.405  356  5 


Cube  Root. 

7.646  027  2 
7.651  724  7 
7.6574138 
7.6630943 
7.668  766  5 
7.6744303 
7.680085  7 
7.685  732  8 
j.691  371  7 
7.697  002  3 
7.702  6246 
7.708  238  8 
7.7138448 
7.7194426 
7.7250325 
7.7306141 
7.736  187  7 
7-741  7532 
7-747  310  9 
7.752  8606 
7.7584023 
7-763  936i 
7.769  462 
7.7749801 
7.780  4904 
7.7859928 
7-791  487  5 
7.7969745 
7.802  453  8 
7.807  9254 

7.8133892 

7.818  845  6 
7.824  294  2 
7-829  735  3 
7.835  1688 
7.840  594  9 
7.8460134 
7.8514244 
7.856  828  1 
7.862  224  2 
7.867  613 
7.8729944 
7.878  3684 
7-883  735  2 

7.8890946 
7.894  446  8 
7.899  791  7 
7.905  1294 
7-9io4599 
7-9I5  7832 
7.921  0994 
7.926  408  5 
7-931  7i°  4 
7-937  005  3 
7.942  293  1 

7-947  5739 


Ncmbe 

503 

504 

505 

506 

507 

508 

509 

510 

511 

512 

513 

514 

515 

516 

517 

518 

519 

520 

521 

522 

523 

524 

525 

526 

527 

528 

529 

530 

531 

532 

533 

534 

535 

536 

537 

538 

539 

540 

54i 

542 

543 

544 

545 

546 

547 

548 

549 

550 

55i 

552 

553 

554 

555 

556 

557 

558 


SQUARES,  CUBES,  AND  ROOTS. 


28l 


Square. 

Cube. 

25  3009 

127263527 

25  40  16 

128  024  064 

25  50  25 

128  787  625 

25  60  36 

129  554216 

25  70  49 

130  323  843 

25  80  64 

131  096512 

259081 

I31  872  229 

26  01  00 

132651  OOO 

26  11  21 

I33  432831 

2621  44 

134  217  728 

2631  69 

135  005  697 

2641  96 

*35  796  744 

26  52  25 

136590875 

26  62  56 

J37  388  096 

26  72  89 

138188413 

26  83  24 

138991  832 

269361 

x39  798  359 

270400 

140  608000 

27  1441 

141  420  761 

27  24  84 

142  236  648 

27  35  29 

*43  055667 

2745  76 

I43  877  824 

2756  25 

144  703  125 

27  66  76 

*45  53i  576 

27  7729 

146363  183 

27  87  84 

r47  197  952 

2798  41 

148  035  889 

28  09  00 

148  877  OOO 

28  1961 

149  721  291 

28  30  24 

150  568  768 

28  40  89 

I5I  4X9  437 

285156 

I52  273  304 

28  62  25 

*53  I3°375 

28  72  96 

^3990656 

28  83  69 

J54  854153 

289444 

155  720872 

29  05  21 

156  590819 

29  16  00 

157464000 

29  2681 

J58  340421 

29  37  64 

159  220  088 

294849 

160  103007 

29  5936 

160989  184 

29  70  25 

161  878  625 

2981  16 

l62  771  336 

29  92  09 

163667323 

30  03  04 

164566  592 

30  1401 

165  469  I49 

30  2500 

166  375  OOO 

303601 

167  284151 

304704 

168  196  608 

36  58  09 

169112377 

3069  16 

170031464 

30  80  25 

1 70  953  875 

309136 

171  879616 

3102  49 

172  808693 

3i  1364 

173  74i  1 12 

A A* 


Square  Root. 

22.427  661  5 
22.4499443 
22.472  205  I 
22.494  443  8 

22.5166605 

22- 538  855  3 
22.561  0283 
22.583  1796 
22.605  309  1 
22.627  417 
22.649  503  3 
22.671  568  1 
22.693  61 1 4 
22.715633  4 
22.737634 

22.7596134 
22.7815715 
22  8035085 
22.825  4244 

22  8473193 
22.8691933 

22  891  0463 
22.912  8785 
22.934  6899 
22.956  4806 
22.978  2506 

23 

. 23.021  7289 

23- 043  437  2 
23.065  125  2 
23.086  792  8 
23.10844 
23.130067 
23.1516738 
23.173  2605 
23.194  827 
23.2163735 

^3-237  900  1 
23.2594067 
23.280  8935 
23-302  3604 

23-323  8076 
23-345  235  1 
23.366  6429 
23.388031  1 
23-409  399  8 
23-430  749 
23-452  078  8 
23-473  389  2 
23.494  680  2 
23-5I5  952 
23-5372046 
23  558  438 
23-5796522 
23.600  847  4 
23.622  023  6 I 


j Cube  Root. 

7.952  847  7 
7.9581144 
7-963  374  3 
7.968  627  I 
7-973  873  I 

7- 979  112  2 
7.9843444 
7.989  569  7 
7.994  7883 
8 

8.005  204  9 
8.010  403  2 
80155946 
8.020  7794 
8.025  957  4 
8.031  128  7 
8.036  293  5 
8041  451  5 
8.046603 

8- 051  747  9 
8.056  8862 
8.062  018 
8.067  14  3 2 
8.072  262 
8.077  374  3 
8.082  48 
8.0875794 
8.092  672  3 
8 097  7589 
8.102  839 
8.107  912  8 
8.112  9803 
8.118041 4 
8.123  0962 
8.128  144  7 
8.133187 
8.138  223 
8.1432529 
8.148  276  5 
8-153  2939 
8.158  305  1 
8.1633102 
8.168  3092 
8.173  302 

8.1782888 
8.1832695 
8.188  244  1 
8.193212  7 

8-1981753 

8.203  131  9 
8.208  082  5 
8.213027  1 
8.2179657 
8.222  898  5 
8.227  8254 
8.232  7463 


2!  282 


SQUARES,  CUBES,  AND  ROOTS. 


jfu  Number.  I 


Square. 


559 

560 

561 

562 

563 

564 

565 

566 

567 

568 

569 

57° 

571 

572 

573 

574 

575 

576 

577 

578 

579 

580 

581 

582 

583 

584 

585 

586 

587 

588 

589 

59° 

591 

592 

593 

594 

595 

596 

597 

598 

599 

600 

601 

602 

603 

604 

605 

606 

607 

608 

609 

610 

61 1 

612 

613 

614 


31  24  81 
31  3600 
31  47  21 
31  58  44 
31  6969 
31  80  96 

31  92  25 

32  03  56 
32  14  89 

32  26  24 
32  37  61 
32490° 

32  60  41 
32  71  84 
32  83  29 

32  94  76 

33  06  25 
33  17  76 
33  29  29 
334084 
33  52  4i 
336400 
33  75  61 
338724 

33  98  89 

34  i°  56 
3422  25 
34  33  96 
3445  69 
34  57  44 
346921 
3481  00 

34  92  81 

35  04  64 
35  i649 
35  28  36 
35  40  25 
35  52  16 
3564°9 
35  7604 

35  8801 

36  00  00 
36  12  01 
36  24  04 
363609 
36  48  16 
36  60  25 
36  72  36 

368449 

36  96  64 

37  08  81 
47  21  00 
37  33  21 
37  45  44 
37  57  69 

376996 


Square  Root. 


Cube  Root. 


174  676879 
175616000 

176558  481 
177  504  328 
178453  547 
179406  144 
180  362 125 
181 321 496 

182  284  263 

183  250  432 
184220009 
185 193000 
186  169411 

187 149  248 
188 132517 

189 1 19  224 
190  109  375 
191 102  976  1 

192  100  033 

193  100  552 

194  104  539 

195  112000 

196  122  941 

197  *37  368 
198155287 

199  176  704 

200  201  625 

201  230  056 

202  262  003 

203  297  472 

204  336  469 

205  37900° 
206425  071 

207  474  688 

208  527  857 

209  584  584 
210644  875 

21 1 708  736 

212  776  173 

213847  I92 
214921  799 

216000000 

217  081  801 

218  167  208 
219256  227 

220  348  864 

221  445  I25 

222  545  016 

223  648  543 

224  755  712 

225  866  529 
226981  000 
228099  131 

229  220  928 

230  346  397 
231 475  544 


23.643  180  8 
23.664  319  1 
23.685  4386 
23.706  539  2 
23.727  621 
23.748  6842 
23.769  728  6 
23-79°  754  5 
23.811  761  8 
23.832  750  6 
23.853  7209 

23.874672  8 
23.8956063 
23.916521  5 
23.937  41 8 4 
23.958  2971 
23-979  J57  6 
24 

24.020  824  3 
24.041  630  6 
24.062  418  8 
24.083  189 1 
24.103  94 1 6 

24.1246762 

24.1453929 

24.166091  9 
24.1867732 
24.207  436  9 
24.228  082  9 
24.2487113 
24.269  322  2 
24.289915  6 
24.310491  6 
24  331  °5°  1 
24-351  591  3 
24-372  n5  2 
24-392  621  8 
24  413  in  2 
24-433  583  4 
24.454  038  5 
24.474  476  5 
24.494  897  4 
24-5 1 5 3° 1 3 
24-535  688  3 

24.5560583 
24.576  41 1 5 
24- 596  747  8 
24.617067  3 
24-637  37 
24.657  656 
24.677  925  4 

24.698  178  1 
24.7184142 
24.738  6338 
24.758  8368 
24.779  0234 


8.237  661  4 
8.2425706 
8.247  474 
8.252  3715 
8.257  263  3 
8.262  1492 
8.267  0294 
8.2719039 
8.2767726 

8.281  625  5 
8.2864928 
8.291  344  4 
8.2961903 
8.301  0304 
8.305  865  1 
8.310  694  1 
8.315  517  5 
8.320  335  3 

3.325  147  5 

8.329954  2 
8.334  755  3 
8.339  550  9 
8-344  34i 

8-349  I25  6 

8.353  904  7 
8.3586784 
8.3634466 
8.368  209  5 
8.372  9668 
8-377  7l8  8 
8.382  465  3 
8.387  2065 
8.391  942  3 

8.3966729 

8.401  398 1 
8.406  1 18 

8.4108326 

| 8.415  54i  9 

8.420246 

8.424  944  8 
8.429  6383 
8.434  326  7 

8.439009  8 

8.4436877 
8.448  360  5 
8.4530281 
8.4576906 

8.462  347  9 
8.467 

8.471  647  1 
8.476  289  2 
8.480926 1 

8.4855579 

8.490  1848 
8.494  806  5 
8.4994233 


i 


!>  UMJ 

6ij 

6i< 

6i' 

6i< 

6i< 

62( 

62] 

62: 

62^ 

624 

62= 

62^ 

62  y 

628 

62g 

63O 

631 

632 

633 

634 

635 

636 

637 

638 

639 

64O 

64I 

642 

643 

644 

645 

646 

647 

648 

949 

650 

651 

652 

653 

654 

655 

656 

657 

658 

659 

660 

661 

662 

663 

664 

665 

666 

667 

668 

669 

670 


SQUARES,  CUBES,  AND  ROOTS. 


37  82  25 

37  94  56 
380689 

38  19  24 
383161 
384400 
38  5641 

38  68  84 

38  81  29 
3893  76 
390625 

39  18  76 
393129 
3943  84 

395641 

39  69  00 
3981  61 

39  94  24 

40  06  89 
40  19  56 
40  32  25 
40  44  96 
405769 

40  70  44 
4083  21 
409600 

41  08  81 
41  21  64 
413449 
41  47  36 
416025 
41  73  16 

41  86  09 

; 419904 

42  1201 

| 42  25  00 

42  38  01 
425104 
42  6409 

42  77  16 
429025 
430336 

43  16  49 
432964 
43  4281 
43  56oo 
43  6921 
43  82  44 

43  95  69 
4408  96 

44  22  25 
4435  56 
4448  89 
4462  24 
44  75  61 
448900 


232  608  375 

233  744  896 
234885  1 13 
236  029  032 
237176659 

238  328  000 

239  483  061 

240  641  848 

241  804  367 
242970  624 

244  140625 

245  134  376 

246  491  883 

247  673  152 

248  858  189 

250  047  000 

251  239  591 

252  435  968 

253  636  137 

254  840  104 
256047875 
257  259456 

258474853 

259  694  072 
260917  119 
262  144  000 
263374  721 

264  609  288 

265  847  707 

267  089  984 

268  336  125 

269585  136 

270  840023 

272  097  792 

273  359  549 

274  625000 

275  894  451 
277  167  808 
278445077 
279  726  264 

281  on  375 

282  300416 

283  593393 

284  890312 

286  191  179 

287  496000 

288  804  781 
290117528 

291  434  247 

292  754  944 
294079625 
295  408  296 
296740963 
298  077  632 
299418309 
300  763  000 


24.799  19  3 5 
24.8193473 
24.8394847 
24.859  605  8 
24.879  7106 
24.899  799  2 
24.919871  6 

24- 939927  8 
24.959  9679 
24.979992 

25 

25.019992 
25.039  968  1 
25.0599282 
25.0798724 
25.099  8008 
25.1197134 

25.1396102 

25- I5949I3 

25-I79356  6 
25.199  2063 
25.2190404 
25-238  858  9 
25.258  661  9 
25.2784493 
25.298  221  3 
25-3i7  977  8 
25-337  7*8  9 

25-357  444  7 
25-377  *55  * 
25-396  8502 

25-4*65301 

25.436  194  7 
25-455  8441 
25475  478  4 
25-495  097  6 
25.514701  6 
25-534  2907 
25  553  8647 
25-573  423  7 
25.592  9678 
25.6124969 
25.632011  2 
25-6515107 
25.6709953 
25.6904652 
25-7099203 
25.729  3607 
25-748  7864 
25.768  1975 
25-787  593  9 
25.8069758 
25.8263431 

25-845  696 

25-8650343  i 
25-8843582  | 


283 

Cube  Root. 

8.504  035 
8.508641  7 
8.513  243  5 
8.5178403 
8.522  432  I 
8.5270189 
8.5316009 
8-536I78 
8.540  750  I 
8-545  3*7  3 
8.5498797 
8-554  437  2 
8.5589899 
8.563  537  7 
8.568  080  7 
8.572  6189 
8-577I523 
8.581  6809 
8.586  204  7 
8-590  723  8 
8-595  238 
8.599  747  6 
8.604  252  5 
8.608  752  6 
8.613  248 
8.617  7388 
8.622  224  8 
8.626  706  3 
8.631  183 

8.635  655  1 

8.640  122  6 
8.644  585  5 
8.6490437 
8.6534974 
8.6579465 
8.662  391 1 
8.666  831 
8.671  2665 
8.675  6974 
8.680 123  7 
8.684  545  6 
8.688  963 
8-6933759 
8.697  7843 
8.702  1882 
8.7065877 
8.710982  7 
8-7*5  3734 
8.7197596 
8.724  141  4 
8.7285187 
8.732  891  8 
8.7372604 
8.741  6246 
8.745  9846 

8.7503401 


284 

SQUARES,  CUBES,  AND  ROOTS. 

Number. 

Square. 

Cube. 

Square  Root. 

671 

672 

673 

674 

675 

676 

677 

678 

679 

680 

681 

682 

683 

684 

685 

686 
687 

• 688. 

689 

690 

691 

692 

693 

694 

695 

696 

697 

698 

699 

700 

701 

702 
7°3 
7°4 
7°5 

706 

707 

708 

7°9 

710 

711 

712 

713 

714 

715 

716 

717 

718 

719 

720 

721 

722 

723 

724 

725 

726 

45  02  41 
45  15  84 
45  29  29 
45  42  76 
45  56  25 
45  69  76 
45  83  29 

45  9684 

46  1041 
46  24  00 
463761 
465124 
46  64  89 

46  78  56 
4692  25 

47  °5  96 
47  x969 
47  3344 
47  47  2i 
47  61 00 
47  748i 

47  88  64 

48  02  49 
48  16  36 

483025 
48  44  16 
48  5809 
48  72  04 

48  8601 
490000 

49  1401 
49  28  04 

4942  09 

49  56  16 
49  7°  25 

49  8436 
4998  49 

50  12  64 
50  26  81 
5041  00 
50  55  21 
50  69  44 

50  83  69 
509796 

51  1225 
51  26  56 
51  4089 
5X55  24 
51  6961 
51  8400 

51  98  41 

52  1284 
52  27  29 
52  41  76 
52  5625 
52  7°  76 

302  III  711 

303  464  448 
304821  217 

306  182  024 

307  546  875 

308  915  776 

310288  733 

311665  752 
313046839 
314  432  OOO 
315821241 
317214568 
318611  987 
320013504 

321  419  125 

322  828  856 

324  242  703 

325  660  672 

327  082  769 

328  509  OOO 
329939371 
331  373  888 
332812557 
334255  384 
335  7°2  375 
337153530 

338608873 

340  068  392 

34 1 S32  099 
343  000  000 
344472101 

345  948  408 
347428927 
348913664 
350  402  625 

351895816 

353393  243 
354  894  91 2 
356  400  829 

3C7  QII  000 

359425  43x 
360944  128 

362  467  097 

363  994  344 

365  525  875 

367  061  696 
368601  813 
370 146  232 
37 1 694  959 

373  248  000 

374  805  361 

376  367  048 

377  933067 
379  503  424 
381  078  125 

1 382657176 

25.903  667  7 
25.922  962  8 
25.942  243  5 
25.961  51 
25.980  762  1 
26 

26.0192237 
26.038  433  1 
26.057  628  4 

26.0768096 

26.095  976  7 
26.115  1297 
26.134  268  7 
26.153  393  7 
26.172  5047 
26.191  601  7 

26.2106848 

26.229  754  1 
26.248  809  5 
26.267  851 1 
26.2868789 
26.305  892  9 
26.324  893  2 
26.3438797 
26.362  852  7 
26.381  81 1 9 
26.400  757  6 
26.419  689  6 
26.438  608  1 
26.457  5X3X 
26.476  404  6 
26.495  282  6 
26.5141472 
26.532  998  3 
26.5518361 
26.5706605 
26.589  47 16 
26.608  269  4 
26.6270539 
26.645  825  2 
26.664  583  3 
26.683  328  1 
26.702  0598 
26.720  7784 
26.739  4839 
26.7581763 
26.776  855  7 
26.795  522 
26.8i4x75  4 
26.8328157 
26.851  443  2 
26.8700577 
26.888  659  3 
26.907  248  1 
26.925  824 

26.944  387  2 

Cube  Root. 


8.754  691  3 
8.759  °38  3 
8.763  3809 

8.7677192 

8.7720532 

8.776  383 
8.780  7084 
8.785  0296 
8.789  3466 
8.793  659  3 
8.797  967  9 
8.802  272  1 
8.8065722 
8.810868  1 

8.8151598 
8.819447  4 
8.823  730  7 
8.8280099 
8.832  285 

8.8365559 

8.840  822  7 
8.845  085  4 
8.849  344 
8.853  598  5 
8.8578489 
8.862  095  2 
8.8663375 
8.8705757 
8.874  809  9 
8.87904 
8.883  266  1 
8.887  488  2 
8.891  7063 
8.895  9204 
8.9001304 
8.904  336  6 
8.908  538  7 
8.9127369 
8.916931 1 
8.921  1214 
8.925  3°7  8 
8.929  49°  2 
8.9336687 
8.937  8433 
8.942014 
8.946  1809 
8.950  343  8 
8-954  5029 
8.958  658  1 
8.962  809  5 
8.966  957 
8.971  1007 
8.975  240  6 

8.979  3766 

8.983  5089 

8.9876373 


i 

i 

1 

j 

i 

1 

I 


UMBI 

727 

728 

729 

730 

731 

732 

733 

734 

735 

736 

737 

738 

739 

740 

74i 

742 

743 

744 

745 

746 

747 

748 

749 

750 

75i 

752 

753 

754 

755 

756 

757 

758 

759 

760 

761 

762 

763 

764 

765 

766 

767 

768 

769 

770 

771 

772 

773 

774 

775 

776 

777 

778 

779 

780 

781 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 


Cube. 


Square  Root. 


52  85  29 

52  99  84 

53  i4  4i 
53  2900 
53  436i 
53  58  24 
53  72  89 

53  87  56 
5402  25 

54  16  96 
54  31  69 
54  46  44 
5461  21 

54  76  00 
549081 
550564 

55  2049 
55  35  36 
55  50  25 
55  65  16 
55  80  09 

55  95  04 

56  1001 
56  2500 
56  4001 
565504 
56  70  09 

56  85  16 
570025 
5715  36 

57  30  49 
57  45  64 
576081 

57  7600 
5791  21 
580644 

58  21  69 

583696 

58  52  25 
586756 
58  82  89 

58  98  24 

59  1361 
59  29  00 
5944  41 
59  5984 

59  75  29 
5990  76 
6006  25 

60  21  76 
603729 
6052  84 
6068  41 
60  84  00 
609961 
61 1524 


384  240583 

385  828  352 
387420489 
389017000 
390617891 
392  223  168 
393832  837 
395  446904 
397065375 
398  688  256 
400315553 
40 1 947  272 

403  583  419 

405  224  000 

406  869  021 
408518488 

410  172  407 

41 1 830  784 
413493625 
415  160936 
416832  723 
418508  992 
420489  749 
421  875000 

423  564751 

425  259  008 
426957777 
428  661  064 
430368875 

432  081  216 

433  798093 
435  5i9  512 

437  245  479 

438  976  000 
440  71 1 081 
442  450  728 

444  194  947 

445  943  744 

447697  125 

449  455  096 

45 1 217663 

452  984  832 
454  756  609 
456  533ooo 
458314011 

460  099  648 

461  889917 
463  684  824 
465  484  375 
467  288  576 
469097433 
470910952 

472  729  139 

474  552  000 
476379  54i 
478  21 1 768 


26.962  937  z 
26.981  475  1 
27 

27.018  512  2 
27.037011  7 
27.055  498  5 
27-073  972  7 

27.092  434  4 
27.1108834 
27.1293199 
27-I47  743  9 
27.1661554 
27-184  5544 
27.202  941 
27.2213152 
27.239  676  9 
27.258  026  3 
27.2763634 
27.294688  1 
27.3130006 
27-33I  300  7 
27*3495887 
27.367  8644 
27.3861279 
27.4043792 

27.422  6184 
27.440  845  5 
27-4590604 
27.4772633 
27.495  454  2 

27-513633 

27-531  7998 
27-549  954  6 
27.568  097  5 
27.5862284 
27.604  347  5 
27.622  4546 
27-6405499 
27.6586334 
27.676  705 
27.694  764  8 
27.7128129 
27.7308492 
27.7488739 
27.7668868 
27.784  888 
27.802  877  5 
27.820  8555 
27.838  821  8 
27.856  7766 
27.8747197 
27.8926514 
27-9105715 
27.928480  1 
27-9463772 

27.964  262  9 


285 

I Cube  Root. 

8.991  762 
8.995  882  9 
9 

9.0041134 
9.008  222  9 
9.012  3288 
9.0164309 
9.0205293 
9.024  623  9 
9.028  7149 
9.032  802  1 
9.036  885  7 
9.040  965  5 
9.045  041  7 
9.049  1142 
9-053  183  1 
9.0572482 
9.061  309  8 
9.065  367  7 
9.069  422 
9.0734726 
9-077  5I9  7 
9.081  563 1 
9.085  603 
9.089  639  2 
9.0936719 
9.097  701 
9.101  7265 
9-io5  7485 
9.109  7669 
9.113  7818 

9.1177931 

9.121  801 
9-125  8053 
9.129  806 1 
9-133  8034 
9-I37  797I 

9-141  7874 

9-I45  774  2 
9.149  757  6 

9-153  737  5 
9*I57  7U39 
9.161  6869 
9.165  6565 
9.169622  5 
9-I73  585  2 
9-177  544  5 
9.181  5003 

9-i85  4527 
9.189401  8 
9-193  3474 
9-197  2897 
9.201  2286 
9.205  164  1 
9.2090962 
9.213025 


286 


SQUARES,  CUBES,  AND  ROOTS. 


783 

784 

785 

786 

787 

788 

789 

790 

791 

792 

793 

794 

795 

796 

797 

798 

799 

800 

801 

802 

803 

804 

805 

806 

807 

808 

809 

810 

811 

812 

813 

814 

815 

816 

817 

818 

819 

820 

821 

822 

823 

824 

825 

826 

827 

828 

829 

830 

831 

832 

833 

834 

835 

836 

837 

838 


..L 


Square  Root. 


6l  30  89 
6l  46  56 
6l  62  25 

61  77  96 
61 93  69 

62  09  44 
62  25  21 
62  41  00 
62  5681 
62  72  64 

62  88  49 
630436 

63  20  25 
63  36 16 
63  52  09 
63  68  04 

63  84  01 

64  00  00 
64  1601 
64  32  04 
64  48  09 
64  64  16 
64  80  25 

64  96  36 

65  1249 
65  28  64 
65  4481 
65  61 00 
65  77  21 

65  9344 
660969 

66  25  96 
66  42  25 
66  58  56 
66  74  89 

66  91  24 

67  07  61 
67  24  00 
67  40  41 
67  56  84 
67  73  29 

67  89  76 

68  06  25 
68  22  76 
6839  29 

685584 

68  72  41 

68  89  00 
6905  61 

69  22  24 
69  38  89 
695556 
69  72  25 

69  88  96 

70  05  69 
70  22  44 


480048  687 
481  890  304 

483  736625 
485  587  656 

487  443  403 

489303872 

491 169  069 
493039  ooo 
494  9i367i 
496  793  °88 
498  677257 
500  566  184 
502459875 

504  358336 
506  261  573 
508  169  592 
510082  399 

512  000000 

513  922  401 
515  849  608 
517  781627 
519718464 
521  660  125 
523  606  616 
525  557  943 
527  5*4  112 
529475  I29 
531  441 000 
533  4«  73J 
535  387  328 
537  367  797 
539353  x44 
541  343  375 
543  338490 
545  338  513 
547  343432 
549353  259 
K C I 368  OOO 
553387661 
555  412  248 
557  441  767 
559476  224 
561515625 
563  559976 
565  609  283 
567  663  552 
569  722  789 
571  787  ooo 
573856  I91 
575  930368 
578009537 
580093  704 
582  182  875 
584  277  056 
586  376  253 
588  480  472 


27.982  137  2 
28 

28.017  851  5 
28.035  691  5 
28.0535203 
28.071  337  7 
28.089  1438 

28.1069386 

28.124  722  2 

28.1424946 

28.160255  7 

28.1780056 

28.195  7444 
28.213472 
28.231  1884 
28.248  893  8 
28.266  588  1 
28.284271  2 
28.301  943  4 

28.3196045 

28.337  2546 
28.354  8938 

28.372  521  9 
28.390  139  1 
28.407  745  4 
28.425  340  8 
28.442  925  3 
28.460  498  9 
28.478061  7 

28.495  613  7 
28.5131549 

28.530  685  2 
28.548  2048 
28.565  713  7 
28.5832119 
28.600  699  3 
28.618  176 
28.635  642  1 
28.653  0976 
28.670  542  4 
28.687  976  6 

28.7054002 

28.7228132 

28.740215  7 
28.757  607  7 
28.774  989  1 
28.792  360  1 
28.809  7206 
28.8270706 
28.844  410  2 

28.861  739  4 
28.879  0582 
28.896  366  6 

28.9136646 

28.9309523 
28.948  229  7 


9.2169505 
9.2208726 
9.224  791  4 
9.228  706  8 
9.2326189 
9.236527  7 
9.2404333 
9-244  335  5 
9.248  234  4 

9-252  13 

9.256  022  4 

9.259  91 1 4 

9.263  797  3 
9.267  679  8 
9-271  5592 
9-275  435  2 
9.279  308  1 

9.2831777 

9.287  044 
9.290907  2 

9.294  767  1 
9.298  623  9 
9.302  477  5 

9.3063278 
9.310175 
9-3I4OI9 
9-3I7  8599 
9.3216975 
9-325  532 
9-329  363  4 
9-333I9I  6 
9-337  016  7 
9-34°  838  6 
9344  657  5 

9.3484731 

9.352  285  7 
9-3560952 
9-359  901  6 
9-363  704  9 
9-367  505  1 
9-37 1 3°2  2 
9-375  096  3 
9.378  8873 
9.382  675  2 

9.38646 

9.390  241  9 

9.3940206 
9.397  7964 
9.401  569 1 
9-405  338  7 
9.409  105  4 
9.412  869 
9.4166297 
9.420  387  3 
9.424  142 

9.4278930 


Numbe 

839 

840 

841 

842 

843 

844 

845 

846 

847 

848 

849 

850 

851 

852 

853 

854 

855 

856 

857 

858 

859 

860 

861 

862 

863 

864 

865 

866 

867 

868 

869 

870 

871 

872 

873 

874 

875 

876 

877 

878 

879 

880 

881 

882 

883 

884 

885 

886 

887 

888 

889 

890 

891 

892 

893 

894 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 


Cube. 


Square  Root. 


70  3921 
70  5600 
70  72  8l 

70  89  64 

71  0649 
71  2336 
714025 
715716 
71  7409 

71  91 04 

72  08  01 
72  25  00 
72  4201 
72  5904 
72  7609 

72  93  16 

73  10  25 
732736 
73  44  49 
7361  64 

73  78  81 
739600 

74  13  21 
74  30  44 
744769 
746496 
74  82  25 

74  99  56 

75  1689 
75  34  24 

75  5i6i 
75  69  00 

75  8641 
760384 

76  21  29 
763876 
76  56  2 5 

76  73  76 
7691  29 
7708  84 

77  2641 
774400 
7761  61 
77  79  24 

77  96  89 

78  14  56 

78  32  25 

784996 

78  67  69 

78  85  44 
790321 

79  21  00 
7938  8i 
79  5664 
79  74  49 
7992  36 


590589719 
592  704  000 
594  823  321 
596  947  688 
599077107 
601  21 1 584 
603351 125 
605  495  736 
607  645  423 
609800  192 
61 1 960049 
614  125  000 
616295  051 
618470  208 
620  650477 
622  835  864 
625  026375 
627  222  016 
629  422  793 
631  628  712 
633  839  779 
636  056  000 
638277381 
640  503  928 
642  735  647 
644  972  544 
647214625 
649  461  896 
651714363 
653  972  032 
656  234  909 
658  503  000 
660  776311 
663  054  848 
665  338617 
667627  624 
669921  875 
672  221  376 
674  526  133 
676  836  152 
679  151  439 
681  472  000 
683  797  841 
686 128  968 
688  465  387 
690  807  104 
693  i54  125 
695  506456 
697  864  103 
700  227072 
702  595  369 
704  969  000 
707  347  97i 
709  732  288 
712  121  957 
714516984 


28.965  496  7 
28.982  753  5 

29 

29.0172363 
29.034  462  3 
29.051  678  1 
29^68  883  7 
29.086  079  1 
29.103  2644 
29.120  4396 
29.1376046 
29-I54  759  5 
29.1719043 
29.189039 
29.206 163  7 
29.223  278  4 
29.246383 
29-257  477  7 
29.274  562  3 
29.291  637 
29.308  701  8 

29-325  7566 
29.342  801  5 
29-3598365 
29.376  861  6 
29-393  876  9 
29.410882  3 
29.427  877  9 
29.444  863  7 
29.461  839  7 
29.478  805  9 
29.495  762  4 
29.512  709  1 
29.529  646  1 
29-546573  4 

29-563491 

29-5803989 
29-597  297  2 
29.614  185  8 
29.631 064  8 
29.647  934  2 
29.664  793  9 
29.681  644  2 
29.698  484  8 
29-7I5  3I5  9 
29-732  1375 

29.7489496 

29*765  752  1 

29.782  545  2 
29.7993289 
29.816 103 
29.832  867  8 
29.849  623  1 
29.866  369 
29.883  105  6 
29.899  832  8 


287 

Cube  Root. 
9.431  642  3 
9-435  38 
9.439  130  7 
9.4428704 
9.446  607  2 
9-450  34i 
9.4540719 

9 457  799  9 
9.461  5249 
9.465  247 
9.468  966  1 
9.472  682  4 
9476395  7 
9.480  106  1 
9.483  813  6 
9.4875182 
9.491  22 
9.4949188 
9.498  614  7 
9.502  307  8 
9-505  998 
9.5096854 

9-5133699 

9-5170515 

9.520  7303 
9.5244063 
9.5280794 
9-53i  749  7 
9-535  4I7  2 
9-539  081  8 
9-542  743  7 
9.546  402  7 
9-5500589 

9-553  7I2  3 

9-557  363 
9.561  0108 

9-5646559 

9.568  298  2 
9-57r937  7 
9-575  574  5 
9-579  2085 
9.582  839  7 
9.586  468  2 
9-590  093  7 
9-593  7i69 

9-597  337  3 
9.600  954  8 
9.604  569  6 
9.608  181  7 
9.611  791  1 
9-6i5  397  7 
9.619001  7 
9.622  603 
9.626  201  6 
9.629  797  5 
9-633  390  7 


SQUARES,  CUBES,  AND  ROOTS. 


2 

Ni 


288 


Square. 


895 

896 

897 

898 

899 

900 

901 

902 
9°3 
9°4 
9°5 
906 

9°7 

908 

9°9 

910 

911 

912 

9T3 

9*4 

915 

916 

917 

918 

919 

920 

921 

922 

923 

924 

925 

926 

927 

928 

929 

93° 

931 

932 

933 

934 

935 

936 

937 

938 

939 

940 

941 

942 

943 

944 

945 

946 

947 

948 

949 

950 


80  10  25 
80  28  16 
804609 
80  64  04 

80  82  01 
81 0000 
81 1801 

81  3604 
81  5409 
81  72  16 

81  9025 
820836 

82  26  49 
82  44  64 
82  62  81 
82  81  00 

82  9921 

831744 

833569 

835396 

83  72  25 

839056 

840889 

84  27  24 

84  45  61 
84  6400 

84  82  41 

85  00  84 
85  19  29 

85  37  76 
85  56  25 
85  74  76 

85  93  29 

86  1 1 84 
86  30  41 
86  4900 
86  67  61 

86  86  24 

87  04  89 
87  23  56 
87  42  25 
87  6096 

87  7969 
879844 

88  1721 
88  36  00 
88  5481 
88  73  64 

88  92  49 

89  11  36 
893025 
89  49  16 
89  6809 

89  87  04 

90  06  01 
902500 


Square  Root. 


716917375 
7i9323  *36 
721  734  273 
724150  792 
726  572  699 
729000000  I 
731  432  701 
733  870  808 
73^3X4  327 
738  763  264 
741  217625 
743  677416 
746  142  643 
748613312 
751089  429 
753571000 
756058031 
758550528 
761  048  497 
763  55i  944 
766  060  875 
768  575  296 
771  095213 
773  620  632 
776  151  559 
778  688  000 
781  229961 

783  777  448 

786330467 
788  889  024 

791  453  I25 
794  022  776 
796  597  983 
799178  752 
801  765  089 
804357000 
806  954  491 
809557568 
812  166237 
814780  504 
817400375 
820  025  856 
822  656  953 
825  293  672 
827936019 
830  584  000 
833237621 
835  896  888 
838  561  807 
841  232  384 
843  908  625 
846590536 
849278  123 

851  97i  392 
854670349 
857  375  000 


29.916  5506 
29-933  259 1 

29.9499583 

29.966  648  1 
29.983  3287 
30 

30.016  662 

30.0333148 

30.0499584 
30.066  592  8 
30.0832179 
30.099  8339 

30.1164407 

30-1330383 

30.149  6269 
30.1662063 
30.1827765 
30.199  337  7 
30.215  8899 
30.232  432  9 
30.248  966  9 
30.265  491  9 
30.282  007  9 
30.298  514  8 
30.3150128 
30-331  501  8 

30.3479818 
30.364  452  9 
30.3809151 
30-397  368  3 

30.4138127 
30.430  248  1 
30.446  674  7 
30.463  092  4 

30.479  5oi  3 
30.495  9° 1 4 
30.512  292  6 
30.528  675 
30.545  048  7 

30.5614136 

30.577  769  7 
30.5941171 

30.6104557 
30.626  785  7 
30.643  1069 
30.6594194 

30.675  723  3 

30.692  018  5 
30.708  305  1 
30.724  583 
30.7408523 
30.757 JI3 
30.773  365  l 
30.789  608  6 
30.805  843  6 
30.822  07 


Cube  Root. 

9.636  981  2 
9.640  569 
9.6441542 
9.647  736  7 
9.651  3166 
9.654  893  8 
9.658  468  4 
9.662  040  3 
9.6656096 
9.669  1 76  2 
9.672  74O  3 
9.676  3OI  7 
9.679  860  4 
9.6834166 
9.686  97O  I 
9.690  521  I 
9.694  069  4 
9.6976151 
9.7OI  1583 
9.704  6989 
9.708  2369 
9.7H  7723 
9-7I5  305I 
9-7l8  835  4 
9.722  363  1 
9.725  888  3 
9.7294109 
9'732  930  9 
9.7364484 
9-739963  4 
9-743  475  8 
9-746985  7 . 
9-750  493 
9-753  997  9 

9-757  500  2 ; 

9.7610001  ' 

9.7644974  * 

9.7679922  j 
9-771  484  5 | 
9-774  9743  j 

9.7784616  - 
9.7819466  j 
9.7854288 
9.7889087  1 

9.7923861 
9.795861 1 J 
9-7993330 
9.8028036 
9.806271 1 
9.8097362 

9.813  1989 
9.816659 1 
9.820  1169 

9-8235723 

9.827  025  2 

I 9-830  475  7 


UMBER 

951 

952 

953 

954 

955 

956 

957 

958 

959 

960 

961 

962 

963 

964 

965 

966 

967 

968 

969 

970 

971 

972 

973 

974 

975 

976 

977 

978 

979 

980 

981 

982 

983 

984 

985 

986 

987 

988 

989 

990 

991 

992 

993 

994 

995 

996 

997 

998 


SQUARES,  CUBES,  AND  ROOTS. 


289 


Square.  I 

Cube. 

' Square  Root.  | 

Cube  Root. 

9044OI 

860085  351 

30.838  287  9 

9.8339238 

906304 

862  8oi  408 

30.854  4972 

9-837  369  5 

9082  09 

865  523177 

30.870  698  I 

9.840812  7 

91  OI  16 

868  250  664 

30.886  890  4 

9.8442536 

91  20  25 

870983875 

30.903  074  3 

9.847  692 

9!3936 

873  722  816 

30.919  247  7 

9.851 128 

91  5849 

876467  493 

30.935  4l6  6 

9.854  561  7 

91  7764 

879217912 

30.951  575  1 

9.8579929 

91  9681 

881  974079 

30.967  725  1 

9.861  421  8 

92  1600 

884  736  000 

30.983  866  8 

9.864  848  3 

92  3521 

887  503  681 

3i 

9.868  272  4 

92  54  44 

890  277  128 

31.016  1248 

9.871  694  1 

92  7369 

893056  347 

31.032  2413 

9.875  1135 

Q2  Q2  Q6 

895  841  344 

31.048  3494 

9-878  530  5 

93  12  25 

898632  125 

31.0644491 

9.881  945  1 

93  3i  56 

901  428  696 

31.0805405 

9-885  357  4 

93  5089 

904  231  063 

31.096  6236 

9.888  767  3 

93  70  24 

907  039  232 

31.112  698  4 

9.8921749 

93  89  61 

909853  209 

31.128  7648 

9.895  580  1 

940900 

912  673000 

31.144  823 

9.898  983 

942841 

915498611 

31.160  872  9 

9.902  383  5 

94  47  84 

918330048 

31.1769145 

9.905  781  7 

94  67  29 

921  167317 

31.1929479 

9.909  177  6 

94  86  76 

924010424 

31.208973  1 

9.912  571  2 

950625 

926  859375 

31.224  99 

9.9159624 

95  25  76 

929  714  176 

31.2409987 

9-9I935I3 

95  45  29 

932  574  833 

31.2569992 

9.922  7379 

95  64  84 

935  441352 

31.2729915 

9.926 122  2 

95  8441 

938  31 3 739 

31.2889757 

9.929  5042 

960400 

941  192  000 

31.304951  7 

9.932  8839 

96  23  61 

944076 141 

31.3209195 

9.936  261  3 

9643  24 

946  966  168 

31.3368792 

9.9396363 

96  62  89 

949  862  087 

31.3528308 

9.943  009  2 

96  82  56 

952  763  904 

31.3687743 

9-946379  7 

97  02  25 

955671625 

31.384  7097 

9.949  7479 

97  21 96 

958  585  256 

31.4006369 

9.9531138 

974169 

961  504  803 

31.4165561 

9956477  5 

9761  44 

964  430  272 

31432  4673 

9-9598389 

97  81  21 

967  361  669 

31.4483704 

9.963  198  1 

9801  00 

970  299  000 

31.4642654 

9.9665549 

98  2081 

97 3 242271 

31.4801525 

9.9699095 

98  40  64 

976  191  488 

31.4960315 

9.973  261  9 

98  6049 

979  146657 

31.5119025 

9.976612 

98  80  36 

982  107  784 

31.5277655 

9-9799599 

990025 

985  074  875 

31.5436206 

9.983  305  5 

99  20  16 

988  047  936 

31.5594677 

9.986  648  8 

994009 

991  026  973 

3i-575  3o68 

9.98999 

996004 

994  01 1 992 

31.591  138 

9.993  3289 

998001 

997  002999 

31.606961  3 

9.996665  6 

1 000000 

1 000000000 

31.622  7766 

10 

1 000201 

1 003003001 

31.638  584 

10.003  322  2 

1 00  40  04 

1 006012008 

31.6543836 

10.006662  2 

1 006009 

1 009027027 

31.6701752 

10.009  989  9 

1 00  80  16 

x 012  048  064 

31.685  959 

10.013  315  5 

1 01  00  25 

1 015075  125 

31.701  7349 

10.016  638  9 

1 01  20  36 

1 018  108216 

B B 

3i-7i7  5o  3 

10.019960  1 

290 


SQUARES,  CUBES,  AND  ROOTS, 


Number. 

Square. 

Cube. 

Square  Root.  | 

Cube  Root. 

IOO7 

I OI  40  49 

I 021  147  343 

31.733  2633 

IO.023  279  I 

1008 

I OI  60  64 

I O24  192  512 

31.7490157 

10.026  595  8 

IOO9 

I 01  8081 

I 027  243  729 

31.764  7603 

IO.029  910  4 

IOIO 

I 02  OI  OO 

I O30  30I  OOO 

31.7804972 

10.033  222  8 

IOII 

I 02  21  21 

IO33  364  331 

31.796  2262 

IO.O36  533 

IOI2 

I 02  41  44 

IO36433  728 

31.8119474 

IO.O39  841 

1013 

I 02  6l  69 

I O39  509  197 

31.8276609 

IO.O43  ^69 

IOI4 

I 02  8l  96 

I 042  590  744 

31.8433666 

10.046  450  6 

1015 

I 03  02  25 

I 045  678  375 

31.8590646 

10.049  752  1 

IOl6 

I 03  22  56 

I 048  772  096 

31.874  7549 

10.053  0514 

1017 

I 03  42  89 

I 051  871  913 

31.8904374 

10.056  348  5 

IOl8 

I 03  63  24 

IO54977832 

31.9061123 

10.0596435 

1019 

I 03  83  6l 

I 058  089  859 

31.921  7794 

10.062  936  4 

1020 

I 04  04  OO 

I 061  208  OOO 

3I-937  438  8 

10.066  227  1 

1021 

I 04244I 

I 064  332  261 

31.9530906 

10.0695156 

1022 

I 04  44  84 

I 067  462  648 

31.968  7347 

10.072  802 

1023 

I 04  65  29 

I 070  599  167 

31.9843712 

10.076  0863 

1024 

I 04  85  76 

I 073  74I  824 

32 

10.079  368  4 

1025 

105  06  25 

I 076  890  625 

32.OI5  621  2 

10.082  648  4 

1026 

I 05  26  76 

1080045576 

32.O31  2348 

10.085  926  2 

1027 

I 05  47  29 

I 083  206  683 

32.046  840  7 

10.089  201  9 

1028 

I 05  67  84 

I 086  373  952 

32.062  439  I 

10.092  475  5 

1029 

I 05  884I 

I 089  547  389 

32.078  0298 

10.095  7469 

1030 

I 06  09  OO 

I 092  727  OOO 

32.0936131 

10.0990163 

1031 

I 06  2961 

1 095  912  791 

32.IO9  188  7 

10.102  2835 

1032 

I 06  50  24 

1 099  104  768 

32.124  7568 

10.105  5487 

1033 

I 06  70  89 

I 102  302  937 

32.1403173 

10.108  81 1 7 

1034 

I 06  91  56 

I IQ5  507  304 

32.1558704 

10.1120726 

1035 

I 07  12  25 

I 108  717875 

32.1714159 

10.1155314 

IO36 

I 07  32  96 

I III934656 

32.1869539 

10.118  5882 

IO37 

107  53  69 

I II5157653 

32.202  4844 

10.121  842  8 

IO38 

107  7444 

I H8386872 

32.2180074 

10.126  095  3 

1039 

1 07  95  21 

I 12 1 622  319 

32.2335229 

10.128  3457 

IO4O 

1 08  16  00 

I I24864OOO 

32.24903I 

10.131  5941 

IO41 

1 08  36  81 

I I28  III  921 

32.264  5316 

10.134  8403 

IO42 

1085764 

I 13 1 366088 

32.280  0248 

10.138  0845 

1043 

1 08  78  49 

I I34  626  507 

32.2955105 

10.141  3266 

IO44 

1 08  99  36 

I I37  893  184 

32.310988  8 

10.144  5667 

1045 

1 09  20  25 

I I4I  l66  125 

32.326  4598 

10.147  804  7 

IO46 

1 0941 16 

I 144445  336 

32.341  92  3 3 

10. 15 1 0406 

IO47 

1 09  62  09 

I I47  73O  823 

32.357  3794 

10.1542744 

IO48 

1 09  83  04 

I 15 1 022  592 

32.372  828  1 

10.157  5062 

IO49 

1 10  04  01 

I 154320649 

32.388  2695 

10.160  7359 

1050 

1 102500 

I 157  625  OOO 

32.403  7035 

10.163  9636 

1051 

1 104601 

1 160935  651 

32.419  130 1 

10.167  1893 

1052 

1 106704 

1 164  252  698 

32.434  5495 

10.170  412  9 

1053 

1 10  88  09 

1 167  575  877 

32.4499615 

10.1736344 

1054 

1 1 1 09  16 

1 170905464 

32.465  3662 

10.1768539 

1055 

1 11  3025 

1 174  241  375 

32.480  763  5 

10.180071 4 

IO56 

1 11  5136 

1 177583616 

32.4961536 

10.1832868 

*057 

1 11  7249 

1 180  932  193 

32.5115364 

1 10.1865002 

1058 

1 119364 

1 184287112 

32.5269119 

10.1897116 

1059 

1 12  1481 

1 187  648  379 

32.542  2802 

10.1929209 

,1060 

1 12  3600 

1 191  016000 

32.5576412 

10.196  1283 

1061 

1 12  5721 

1 1943899S1 

32.572  9949 

10.1993336 

1062 

1 12  7844 

1 197  770328 

32.5883415 

10.202  5369 

SQUARES,  CUBES,  AND  ROOTS. 


29I 


Number. 

Square, 

Cube. 

Square  Root. 

Cube  Root. 

1063 

I 129969 

I 201  157047 

32.603  680  7 

10.205  738  2 

1064 

I 13  2096 

I 204  550  144 

32.6190129 

10.208  937  5 

1065 

I 1342  25 

I 207  949  625 

32.6343377 

10.212  I34  7 

1066 

i 13  63  56 

1211355496 

32.6496554 

10.215  33 

1067 

1 138489 

I 214  767  763 

32.6649659 

IO.2185233 

1068 

1 14  06  24 

I 218  186  432 

32.680  2693 

10.221  7146 

1069 

1 142761 

I 221  6l  I 509 

32.695  565  4 

10.224  9039 

IO7O 

1 14  49  00 

I 225  043  OOO 

32.7108544 

IO.22809I  2 

IO71 

1 14  70  41 

I 228480911 

32.726  1363 

IO.23I  2766 

IO72 

1 1491  84 

I 231  925  248 

32.741  41 1 I 

IO.2344599 

1073 

1 15  *3  29 

1235  376017 

32.7566787 

IO.237  641  3 

IO74 

1 15  34  76 

I 238  833  224 

32. 771  939  2 

IO.24O  820  7 

IO75 

1 155625 

I 242  296  875 

32.787  1926 

IO.243  998  I 

IO76 

1 15  77  76 

I 245  766976 

32.802  4389 

IO.2471735 

IO77 

1 159929 

I 249  243  533 

32.817  6782 

10.250  347 

IO78 

1 16  20  84 

I 252  726  552 

32.8329103 

IO.2535186 

IO79 

1 1642  41 

I 256216039 

32.8481354 

10.256  688  1 

1080 

1 166400 

I 259  712  OOO 

32.8633535 

10.259  8557 

Io8l 

1 168561 

I 263  214  441 

32.8785644 

10.263021 3 

1082 

1 1707  24 

I 266  723  368 

32.8937684 

10.266  185 

1083 

1 17  28  89 

I 27O  238  787 

32.9089653 

10.269  346  7 

IO84 

1 175056 

I 273  760  704 

32.9241553 

10.272  5065 

1085 

1 1772  25 

I 277  289  125 

32.9393382 

10.275  6644 

1086 

1 179396 

I 280  824  056 

32.9545141 

10.278  8203 

1087 

1 18  15  69 

I 284  365  503 

32.969683 

10.281  9743 

1088 

1 18  37  44 

I 287913472 

32.984  845 

10.285  1264 

1089 

1 18  5921 

I 291  467  969 

33 

10.288  2765 

IO9O 

1 1881  00 

1 295  029  OOO 

33.015  148 

10.291  424  7 

IO9I 

1 1902  81 

1 298  596  571 

33.030  289  1 

10.2945709 

IO92 

1 19  24  64 

1 302  170688 

33.045  4233 

10.2977153 

IO93 

1 194649 

1305  751357 

33-o6o  550  5 

10.300  8577 

IO94 

1 19  68  36 

1309338  584 

33-075  670  8 

10.303  998  2 

1095 

1 19  90  25 

1312  932  375 

33.090  784  2 

10.307  136  8 

IO96 

1 20  12  16 

1 316  532  736 

33.105  8907 

10.3102735 

IO97 

1203409 

1 320  139  673 

33.1209903 

10.313  408  3 

IO98 

1 20  56  04 

1 323  753  192 

33.136083 

10.316541 1 

IO99 

1 20  78  01 

1 327  373  299 

33.151  1689 

10.319672  1 

I IOO 

1 21 0000 

1 331  000000 

33.1662479 

10.322  801  2 

IIOI 

1 21  2201 

1334  633301 

33.181  32 

10.325  9284 

1102 

1 21  4404 

1 338  273  208 

33.1963853 

10.3290537 

1103 

1 21  6609 

i34i  9i9  727 

33.2114438 

10.332  177 

IIO4 

1 21  88  16 

1 345  572  864 

33.2266955 

10.335  2985 

1105 

1 22  1025 

1 349  232  625 

33.241  5403 

10.3384181 

II06 

1 22  32  36 

1 352  899  016 

33-2565783 

10.341  535  8 

1107 

1 225449 

1356572043 

33.2716095 

10.344651  7 

IIO8 

1 22  76  64 

1 360  251  7J2 

33.286  633  9 

10.347  765  7 

IIO9 

1 22  98  81 

1 363  938  029 

3 3-3° 1 651  6 

10.350  8778 

iiio 

1 2321 00 

1 367  631  000 

33.3166625 

10.353988 

IIII 

1 23  4321 

1 371  330631 

33-331  6666 

10.3570964 

1112 

1 23  65  44 

1375036928 

33.346664 

10.360  202  9 

1113 

1 23  87  69 

1378  749897 

33.361  654  6 

10.363  307  6 

1114 

1 240996 

1382469544 

33-3766385 

10.3664103 

1115 

1 24  32  25 

1 386  195  875 

33-391  615  7 

10.3695113 

1116 

124  5456 

1 389  928  896 

33.406  586  2 

10.372  610  3 

1117 

1 24  76  89 

1393668613 

33.4215499 

10.375  707  6 

1118 

1 24  99  24 

1 397  4*5  °32 

33-436  507 

10.378  803 

292 


SQUARES,  CUBES,  AND  ROOTS. 


Number. 

Square. 

III9 

I 25  21  6l 

1120 

I 25  44  OO 

1121 

I 256641 

1122 

i 25  88  84 

1123 

1 26  1 1 29 

1124 

1 2633  76 

1125 

1 26  56  25 

1126 

1 26  78  76 

1127 

1 2701  29 

1128 

1 27  23  84 

1129 

1 27  4641 

1130 

1 27  6900 

II3I 

1 27  91  61 

1132 

i 28  14  24 

1 133 

1 28  36  89 

1134 

1 28  59  56 

1135 

1 28  82  25 

II36 

1 29  04  96 

1137 

1 29  27  69 

II38 

1 29  5044 

1139 

1 29  7321 

II4O 

1 29  96  00 

II4I 

1 30  18  81 

II42 

1304164 

1143 

130  6449 

1144 

1308736 

1145 

1 31  1025 

II46 

1 31  33  16 

1147 

1315609 

II48 

1 31  7904 

1149 

1 32  02  01 

1150 

1 32  25  00 

II5I 

1 32  48  01 

1152 

1 32  71  04 

1153 

132  9409 

1154 

133  17  16 

1155 

1334025 

II56 

13363  36 

1157 

1338649 

1158 

1340964 

1159 

1343281 

Il6o 

1 34  5600 

Il6l 

134  7921 

.El62 

1 35  02  44 

I163 

1 35  25  69 

I164 

1 35  48  96 

I165 

1 35  72  25 

Il66 

1 35  95  56 

I167 

1 36  18  89 

Il68 

13642  24 

I169 

1366561 

1170 

1 36  89  00 

II7I 

1371241 

1172 

1 37  35  84 

1173 

137  59  29 

1174 

1 37  82  76 

Cube. 


I 401  168  159 
I 404  928  OOO 
I 408  694  561 
I 412  467  848 
I 416  247  867 
I 420  034  624 
I 423  828  125 
I 427  628  376 
i 43 1 435  383 
I 435  249  152 
i 439  069  689 
1 442  897  000 
1 446  731  091 
1450571968 

i454  4i9637 
1 458  274  104 
1462135  375 
1 466  003  456 
1469878353 
1 473  760  072 
1 477  648  619 
1 481  544000 
1 485  446  221 

1489355  288 

1 493  271  207 
1497  193984 
1 501 123625 
1 505  060 136 
1 509  603  523 
1512953  792 
1 516910949 
1 520  875  OOO 
1524845  951 
I 528  823  808 
I 532  808  577 
1 536  800  264 
1540  798875 
1 544804416 
1 548816893 
1552  836312 
1 556  862  679 
1 560  896  OOO 
1564936  281 
1 568  983528 
1573037  747 
1577098944 
1 581  167  125 
1 585  242  296 

1589324463 

*593  4*3  632 
1 597  509809 
1 601 613  OOO 
I 605  723  21 1 
I 609  84O  448 
1613964717 
I 618096024  1 


Square  Root. 


33-451457  3 
33.4664OI  I 
33.481  338  I 
33.496  2684 
33.511  1921 
33.5261092 
33.5410196 
33-555  923  4 
33.570  8206 

33-585  7II2 

33.600  595  2 

33-6i5  4726 

33-630  3434 
33.645  207  7 
33.6600653 
33.674  9i6  5 
33.689  761 

33-704  599 1 
33.7194306 
33-734255  6 
33-749074I 
33.7638860 
33.7786915 
33-793  490  5 
33.808  283 
33.823  069  1 
33-8378486 
37.852  621  8 
33.8673884 
33.882  148  7 
33.8969025 
33.9116499 
33.9263909 
33.941 1255 
33-955  853  7 

33- 970  575  5 
33.985  291 
34 

34.014  702  7 
34.029399 
34.044  089 
34.058  772  7 

34- 073  450  1 
34.088  12 1 1 
34.102  7858 
34.1174442 
34.132  0963 
34.146  7422 
34.161381  7 
34.176015 
34.190642 
34.205  262  7 
34.2198773 
34.2344855 
34.2490875 
34.2636834 


Cube  Root. 

IO.381  8965 
IO.384  988  2 
IO.388  078  I 
IO.39I  l66  I 
IO.394  252  3 

10.397  336  6 

IO.4OO  419  2 
IO.4034999 
IO.406  578  7 
IO.409  655  7 
IO.412  73I 
IO.415  8044 
IO.418  876 
10.421  945  8 
10.425  013  8 
10.428  08 

10.431  1443 

10.434  2069 
10.437  267  7 
10.4403267 

10.443  3839 

10.4464393 
10.4494929 
10.452  5448 
10.455  594  8 
10.458  643  1 
10.4616896 
10.4647343 
10.4677773 
to.4708185 
10.4738579 
10.476  895  5 ♦ 

10.47993*4 
10.482  965  6 
10.485  998 
10.489  028  6 
10.4920575 
10.495  084  7 
10.498  no  1 
10.501 133  7 
10.504  1556 

10.5071757  I 

10.510  194  2 
IO.513  2109  ; 

IO.5162259  j 
IO.519  239  I | 
IO.5222506 
IO.525  260  4 ' 

IO.528  268  5 

10.531 2749  ; 

10.5342795 
10.537  282  5 
10.540  283  7 
10.543  283  2 
10.546  281 
10.5492771 


SQUAKES,  CUBES,  AND  BOOTS. 


293 


Number. 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

H75 

I 38  06  25 

I 622  234  375 

34.278  273 

IO.552271  5 

1176 

I 38  29  76 

I 626  379  776 

34.292  856  4 

10.555  264  2 

1177 

i 38  53  29 

I 630  532  233 

34-307  4336 

IO.558  255  2 

1178 

138  76  84 

I 634  691  752 

34.3220046 

10.561  2445 

1179 

I 390041 

1638  858  339 

34-336  5694 

IO.564  232  2 

Il8o 

139  2400 

I 643  032  OOO 

34.351  1281 

IO.567  2l8  I 

Il8l 

13947  6l 

I 647  212  741 

34.365  6805 

IO.57O  202  4 

1182 

139  7124 

I 651  4OO  568 

34.380  226  8 

IO.5731849 

1183 

139  94  89 

I 655  595  487 

34.394  767 

10.576  165  8 

1184 

I 40  18  56 

I 659  797  504 

34.409301  1 

IO.5791449 

1185 

I 40  42  25 

I 664  006  625 

34.4238289 

IO.582  122  5 

Il86 

I 40  65  96 

i 668  222  856 

34.438  3507 

IO.585  098  3 

1187 

I 40  89  69 

1 672  446  203 

34.452  866  3 

IO.5880725 

Il88 

141  1344 

1 676676672 

34.467  375  9 

IO.59I  O45 

H89 

I 41  37  21 

1 680  914  269 

34.481  8793 

IO.594OI58 

II90 

I 41  6l  OO 

1 685  159000 

344963766 

IO.596  985 

II91 

I 41  8481 

1 689410871 

34.5108678 

IO.5999525 

II92 

I 42  08  64 

1 693  669  888 

34-525  353 

IO.6029184 

1193 

I 42  32  49 

1697936057 

34-539  8321 

10.605  882  6 

II94 

I 42  56  36 

1 702  209  384 

34554  3051 

10.608  845  1 

1195 

I 42  80  25 

1 706  489  875 

34.568  772 

10.61 1 806 

1196 

I430416 

1 710777536 

34.583  232  9 

IO.614  765  3 

1197 

i 43  28  09 

.1715072373 

34.597  687  9 

10.617  722  8 

II98 

1435204 

1 719374  392 

34.6121366 

10.6206788 

II99 

143  7601 

1723683599 

34.6265794 

10.623  633  1 

1200 

1 44  00  00 

1 728  000  000 

34.641  016  2 

10.626  585  7 

1201 

1 44  24  01 

1 732  323  601 

34.6554469 

10.629  536  7 

1202 

1 44  48  04 

1 736654408 

34.669871  6 

10.632  486 

1203 

1 44  72  09 

1740992  427 

34.684  290  4 

10.635  433  8 

1204 

1 44  96  16 

1 745  337664 

34.698  703  1 

10.638  379  9 

1205 

1 45  20  25 

1 749690  125 

34.713  1099 

10.641  3244 

1206 

1 45  44  36 

1 754049816 

34-7275107 

10.644  267  2 

1207 

145  6849 

1 758416743 

34.741  905  5 

10.647  208  5 

1208 

1 45  92  64 

1 762  790912 

34.756  2944 

10.650  148 

1209 

1 46  1681 

1 767  172  329 

34.7706773 

10.653  086 

1210 

1 46  41  00 

1 771  561  000 

34.785  0543 

10.656  022  3 

I2II 

1466521 

1775  956931 

34.7994253 

10.658957 

1212 

| 1 46  89  44 

1 780360  128 

34.8137904 

10.661  890  2 

1213 

1 1471369 

1 784  770  597 

34.828  149  5 

10.664821  7 

1214 

| 1473796 

1 789  188  344 

34.842  502  8 

10.667  751 6 

1215 

1 47  62  25 

1793  613  375 

34.856  8501 

10.6706799 

I2l6 

1 1 47  86  56 

1 798  045  696 

34.871 191  5 

10.673  606  6 

1217 

1 48  10  89 

1 802  485  313 

34.885  527  1 

10.676531  7 

1218 

! 1483524 

1 806  932  232 

34.8998567 

10.6794552 

1219 

j 1485961 

1 81 1 386459 

34.9141805 

10.682  377  1 

1220 

1 1 48  84  00 

1 815  848000 

34.928  4984 

10.685  297  3 

1221 

1 49  08  41 

1 820316861 

34.9428104 

10.688  216 

1222 

14932  84 

1 824  793  048 

34.9571166 

10.691  133 1 

1223 

1495729 

1 829  276  567 

34.9714169 

10.694  048  6 

1224 

1 4981 76 

1 833  767  424 

34.9857114 

10.696  962  5 

1225 

1 500625 

1 838  265  625 

35 

10.699  874  8 

1226 

1 5030  76 

1 842  771  176 

35.0142828 

10.702  785.5 

1227 

15055  29 

1 847  284  083 

35-0285598 

10.7056947 

1228 

150  7984 

1 851  804352 

35.042  8309 

10.708  6023 

1229 

1 51  0441 

1 856  331  989 

35-0570963 

10.71 1 5083 

J230 

I 51  2900 

1 860  867  000 

35-07I355  8 

10.714  412  7 

B B* 


1237 

1238 

1239 

1240 

1241 

1242 

1243 

1244 

1245 

1246 

1247 

1248 

1249 

1250 

1251 

1252 

1253 

1254 

1255 

1256 

1257 

1258 

1259 

1260 

1261 

1262 

1263 

1264 

1265 

1266 

1267 

1268 

1269 

1270 

1271 

1272 

1273 

1274 

1275 

1276 

1277 

1278 

1279 

1280 

1281 

1282 

1283 

1284 

1285 

1286 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 

15153  61 
I 51  78  24 
I 52  02  89 
I 52  27  56 
I 52  52  25 
I 527696 
I 5301  69 

153  2644 

i 53  5i  21 
1 53  7600 
1 540081 
1 54  25  64 

154  50  49 
1 54  75  36 
1 55  00  25 
1 55  25  16 

155  5009 
i 55  75  04 
1 56  00  01 
1 56  25  00 
1 56  5001 
1 56  75  04 
1 570009 
1 57  25  16 
157  50  25 
1 57  75  36 
1 580049 
1 58  25  64 
1585081 
1 58  76  00 
1 5901  21 
159  2644 
1595169 

159  7696 
1 60  02  25 

1 60  27  56 
1 60  52  89 
1 60  78  24 
1 61  03  61 
1 61  2900 
1 61  5441 
1 61  79  84 
1 62  05  29 
1 62  30  76 
1 62  56  25 
1 62  81  76 
1 63  07  29 
1 63  32  84 
1635841 
1 63  84  00 
1 640961 
16435  24 
1 64  60  89 
1 64  86  56 
1 65  1225 
1653796 


Cube. 

I 865  409  391 
I 869  959  168 
1874516337 
I 879  080  904 
1883  652  875 
i 888  232  256 
1 892819053 
1897413272 
1 902  014  919 
1 906  624  000 
1 91 1 240521 
1 915  864488 
1920495  907 
1 925  134  784 
1 929  781  125 
1934  434  936 
1939096223 
1 943  764  992 
1 948  441  249 
1953  125000 
1957816251 
1 962  515  008 
1 967  221  277 
1971935064 
1976  656375 
1 981  385  216 

1 986  121  593 
1990865512 
1995616979 

2 000  376  000 
2005  142  581 
2009916  728 
2 014  698  447 
2019487  744 
2 024  284  625 
2 029  089  096 
2033901  163 
2 038  720  832 
2 043  548  109 
2 048  383  000 
2053  225  51 1 
2 058  075  648 

2062  933417 

2 067  798  824 
2 072  671  875 
2077552576 
2 082  440  933 
2087  336952 
2 092  240  639 
2097  152000 
2 102  071  041 
2 106  997  768 
2 hi  932  187 
2 116874304 
2 121  824  125 
2 126  781  656 


Square  Root. 


35.0856096 
35.0998575 
35.II40997 
35.128  3361 
35.142  5568 
35.156  791  7 
35.1710108 
35.185  2242 
35.1994318 
35.2136337 
35.2278299 
35.242  0204 
35.256  2051 
35.270  3842 
35.2845575 
35.298  725  2 
35-312  887  2 
35.3270435 
35-341  194  1 
35-355  339 1 
35-3694784 
35.383612 

35-397  74 
35.4118624 
35.425  9792 
35.4400903 
35.454  195  8 
35.468  295  7 
35.482  39 
35.4964787 
35.5105618 
35.5246393 
35-538  7II3 
35-552  777  7 
35.566  8385 
35  5808937 

35-594  943  4 
35.608  9876 
315.6230262 
35-637  0593 
35.6510869 
35.665  109 
35.679  125  5 
35-693I366 
35.707  142  1 
35.721  1422 
35-735  136  7 
35.7491258 
35-763  109  5 
35.7770876 
35.791  0603 
35.805  0276 
35.8189894 
35-832  945  7 
35.846  8966 
35.860  842  1 


Number. 

1287 

1288 

1289 

1290 

1291 

1292 

1293 

I294 

1295 

1296 

I297 

1298 

1299 

1300 

I3°i 

1302 

1303 

1304 

1305 

1306 

1307 

1308 

1309 

1 310 

1311 

1312 

1313 

1314 

1315 

1316 

1317 

1318 

1319 

1320 

1321 

1322 

1323 

1324 

1325 

1326 

1327 

1328 

1329 

1330 

1331 

1332 

*333 

z334 

*335 

1336 

I337 

1338 

*339 

1340 

I34i 

1342 


SQUARES,  CUBES,  AND  ROOTS. 


295 


Square. 

Cube. 

Square  Root. 

Cube  Root. 

I 65  63  69 

2 13 1 746903 

35.874  782  2 

IO.877427  I 

165  8944 

2 136  719  872 

35.888  7169 

10.880  243  6 

i 66  15  21 

2 141  7OO  569 

35.902  646  I 

10.883  058  7 

1 6641  00 

2 I46  689  OOO 

35.9165699 

IO.885  8723 

166  6681 

2 151685  I7I 

35.9304884 

10.888  684  5 

1 66  92  64 

2 156689  088 

35.9444OI  5 

10.891  495  2 

1 67  1849 

2 l6l  7OO  757 

35-958  309  2 

10.894  304  4 

167  4436 

2 l66  720  184 

35.972  211  5 

10.897  112  3 

1 67  70  25 

2 I7I  747  375 

35.9861084 

10.8999186 

1 67  96  16 

2 176  782  336 

36 

10.902  723  5 

1 68  22  09 

2 181  825  073 

36.OI38862 

10.905  5269 

1 68  48  04 

2 186875  592 

36.027  767  I 

10.908  329 

1 68  74  01 

2 191  933  899 

36.041  6426 

10.911  1296 

1 690000 

2 197  OOO  OOO 

36.055  512  8 

10.913928  7 

1 69  26  01 

2 202  O73  9OI 

36.0693776 

10  916  7265 

1 69  52  04 

2 207  I55  608 

36.083  237  I 

10.919  522  8 

1 69  78  09 

2 212  245  I27 

36.097  091  3 

10.922  317  7 

1 7004  16 

2 217  342  464 

36. 1 IO  94O  2 

10.925  hi  1 

1 7030  25 

2 222  447  625 

36.124  7837 

10.927  903  1 

1 705636 

2 227  560616 

36.138  622 

10.930  693  7 

1 70  82  49 

2 232  68l  443 

36.152455 

10.933  482  9 

1 71  08  64 

2 237  8lO  1 12 

36.1662826 

10.936  270  6 

1 713481 

2 242  946  629 

36.180  105 

ro.9390569 

1 71 61  00 

2 248  09I  OOO 

36.I93922I 

10.941  841  8 

1 71  87  21 

2 253  243  231 

36.207  734 

10.9446253 

1 72  13  44 

2 258  403  328 

36.221  5406 

10.9475074 

1 723969 

2263571  297 

36.235  3419 

10.950  188 

1 72  65  96 

2 268  747  I44 

36.2491379 

10.952  967  3 

1 7292  25 

2 273  930  875 

36.262  928  7 

10.955  745  1 

1731856 

2 279  122  496 

36.2767143 

10.958  521  5 

1 734489 

2 284  322  013 

36.290  4946 

10.961  296  5 

1 73  71  24 

2289529  432 

36.304  269  7 

10.964  070 1 

1 7397  6i 

2 294  744  759 

36.3180396 

10.966  842  3 

i 74  24  00 

2 299  968  000 

36.3318042 

10.969  613  1 

1 745041 

2 305  199  161 

36.345  563  7 

10.972  382  5 

1 74  7684 

2310438  248 

36.359317  9 

10.9751505 

1 75  03  29 

2 315  685  267 

36.373  067 

10.977  917  1 

1 75  29  76 

2 320  940  224 

36.3868108 

10.980  682  3 

1 75  56  25 

2 326  203  125 

36.4OO5494 

10.983  446  2 

1 75  82  76 

233M73  976 

36.4142829 

10.986  208  6 

1 76  09  29 

2 336  752  783 

36.4280II  2 

10.988  969  6 

17635  84 

2342039552 

36.441  7343 

10.991  7293 

1 766241 

2 347  334  289 

36.455  452  3 

10.994  487  6 

1 76  89  00 

2 352  637  000 

36.469  165 

10.997  2445 

1771561 

2357  947  69i 

36.482  872  7 

11 

1 7742  24 

2 363  266  368 

36.4965752 

11.002  754 1 

1 77  68  89 

2368  593  037 

36.5IO2725 

11.005  5069 

1 77  95  56 

2373  927  704 

36.523  9647 

11.008  2583 

1 78  22  25 

2379  270  375 

36.537  6518 

1 1. 01 1 008  2 

1 78  48  96 

2 384621  056 

36.551333  8 

11.0137569 

1 78  75  69 

2389979  753 

36  565  010  6 

1 1. 016  504  1 

1790244 

2 395  346  472 

36  5786823 

11. 019  25 

1 79  2921 

2 400  721  219 

36.592  3489 

11.021  9945 

1 79  5600 

2 406  104000 

36.6060104 

11.024  7377 

1 7982  81 

2411  494821 

36.6196668 

11.0274795 

1 800964 

2 416  893  688 

36.6333181 

11.030  2199 

296 

Number. 

1343 

1344 

1345 

1346 

1347 

1348 

1349 

1350 

1351 

1352 

1353 

1354 

1355 

1356 

1357 

1358 

1359 

1360 

1361 

1362 

1363 

1364 

1365 

1366 

1367 

1368 

1369 

1370 

i37i 

1372 

1373 

1374 

1375 

1376 

1377 

1378 

1379 

1380 

1381 

1382 

1383 

1384 

1385 

1386 

1387 

1388 

1389 

1390 

i39i 

1392 

1393 

1394 

1395 

1396 

1397 

1398 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 


I 80  36  49 
I 80  63  36 
I 80  90  25 
I 8l  17  16 
I 8l  44  09 
I 8l  71  04 
I 8l  9801 
I 82  25  OO 
I 82  52  OI 
I 82  79  04 
I 830609 
I 83  33  16 
1836025 
1838736 
I 84  14  49 
I 84  41  64 
i 84  68  81 
1 84  96  00 
1 85  23  21 

185  5044 
1 85  77  69 
1 86  04  96 

186  32  25 

1 86  59  56 
1 86  86  89 
1 87  14  24 
1 87  41  61 
1 87  69  00 

1 87  9641 
1 88  23  84 
1 88  51  29 
1 88  78  76 
1 89  06  25 
1 89  33  76 
1 8961  29 
1 89  88  84 
1 90  1641 
1 904400 
1 90  71  61 
1909924 
1 91  26  89 
1 91  54  56 

1 91  82  25 
1 92  09  96 
1 92  37  69 

192  6544 

192  9321 

1 93  2 1 00 
193  4881 
193  7664 
1940449 
1943236 
1 94  60  25 
1 94  88  16 
1 95  1609 
195  4404 


Cube. 

2 422  300  607 
2427715584 
2 433  x38  625 
2 438  569  736 
2 444  008  923 
2449456  192 
2 454  911549 
2 460  375  000 
2465  846551 
2 471  326  208 
2476813977 
2 482  309  864 
2487813875 
2493326016 
2 498  846  293 
2 504  374  712 

2 509  91 1 279 
2 C15  456000 
2 521  008  881 
2 526  569  928 

2 532  139  147 
2537716544 
2 543  302  125 
2 548  895  896 

2 554  497  863 

2 560  108032 
2565  726409 
2 571  353000 
2 576  987  81 1 
2 582  630  848 
2 588  282  117 
2 593  94 1 624 
2599609375 
2 605  285  376 
2 610969633 
2 616662  152 
2 622  362  939 
2 628  072  000 
2633  789  341 
2639514968 
2 645  248  887 
2 650  991  104 
2 656  741  625 
2 662  500  456 
2 668  267  603 
2 674  043  072 
2 679  826  869 
2 685  619000 
2 691  419  471 
2 697  228  288 
2 703  045  457 
2 708  870  984 
2 714704875 
2 720  547  136 
2726397773 
2 732  256  792 


Square  Root. 


36.6469644 
36.660  605  6 
36.674  241  6 
36.687  872  6 
36.701  4986 
36.715  1195 
36.728  7353 
36.742  346 1 

36- 755  9519 

36-769  552  6 

36.783  1483 
36.796  739 
36.8103246 
36.823  9053 
36.8374809 
36.851  051  5 
36.864  617  2 
36.8781778 
36.891  733  5 
36.905  284  2 
36.918  8299 
36.932  3706 

36.9459064 
36.959  437  2 
36.972  963  1 
36.986  484 
37 

37- oi3  5ii 
37.027  017  2 
37.0405184 
37.0540146 
37.067  506 
37.089  9924 
,37  094  474 

37-io7  95o6 
37.121  4224 
37-134  8893 
37.1483512 
37.161  8084 
37.175  2606 
37.188  7079 
37.202  1505 
37.2155881 
37.229  0209 
37.242  4489 
37-255  872 
37.269  2903 
37.282  703  7 
37.2961124 

37-309  5J6  2 

37.3229152 
37-3363094 
37-349698  8 
37.3630834 
37.376  4632 
37-3898382 


Number 

1399 

1400 

1401 

1402 

1403 

1404 

1405 

1406 

1407 

1408 

1409 

1410 

14H 

1412 

1413 

1414 

I4I5 

1416 

1417 

1418 

1419 

1420 

1421 

1422 

1423 

1424 

1425 

1426 

1427 

1428 

1429 

1430 

1431 

1432 

1433 

1434 

1435 

i436 

1437 

1438 

1439 

1440 

1441 

1442 

1443 

1444 

1445 

1446 

1447 

1448 

1449 

1450 

i45i 

1452 

1453 

1454 


SQUARES,  CUBES,  AND  ROOTS. 


297 


Square. 


1 95  72  OI 

I 96  OOOO 

I 96  28  OI 

1 96  56  04 
I 96  8409 
i 97  12  16 

1 97  40  25 
1 97  68  36 
197  9649 
1 98  24  64 
1 98  52  81 
1 98  81  00 

1 9909  21 
19937  44 
19965  69 
1999396 

2 00  22  25 
2 00  50  56 
2 00  78  89 
2 01  07  24 
201  3561 
2 01  64  00 

2 OI  92  41 

2 02  20  84 
2 02  49  29 
2 02  77  76 
2 03  06  25 
2 03  34  76 
20363  29 
2039184 
2 04  20  41 

2 04  49  OO 

2 04  7761 
2 05  06  24 
2053489 
2 05  63  56 
2 05  92  25 
2 06  20  96 
2 06  49  69 
2 06  78  44 
207O72I 
2 07  36  OO 
2076481 
2079364 
2 08  22  49 
2 08  51  36 
2 08  80  25 
2 O909  l6 
2 09  38  09 
2 09  67  04 
209960I 
2 10  25  00 
2 IO  54OI 
2 IO  83  04 
2 II  12  09 
2 II  41  l6 


Square  Root. 


2 738  124  199 

2 744  OOO  OOO 

2 749  884  201 
2 755  776  808 
2 761  677  827 
2 767  587  264 
2 773  505  125 
2 779  431  4i6 
2 785  366  143 
2791  30  9312 
2 797  260  929 
2 803  221  000 
2809  189  531 
2 815  166  528 
2 821  151  997 
2 827  145  944 
2 833  148  375 
2 839  159  296 
2845  178713 
2 851  206  632 
2857243  059 
2 863  288  000 
2 869341  461 
2 875  403  448 
2881473967 
2 887  553  024 
2 893  640  625 
2 899  736  776 
.2  905  841  483 
2911  954  752 
2918076589 
2 924  207  000 
2 930  345  99i 

2936  493  568 
2 942  649  737 
2948814  504 
2954987875 
2961  169856 
2967360453 
2 973  559672 
2 979  767519 
2 985  984  000 
2992  209  121 

2 998  442  888 
3004685  307 

3 010  936  384 
3017  196  125 
3 023  464  536 
3 029  741  623 
3036027392 
3042  321  849 
3 048  625  000 
3054936851 
3061  257408 
3067  586677 
3073924664 


37.4032084 
37-4165738 
37.4299345 
37.443  2904 
37.4566416 
37.469988 
37-483  3296 
37.4966665 
37-509  9987 
37.523  326  1 
37-5366487 
37.5499667 
37-563  2799 
37.5765885 
37.589  8922 
37-603  191  3 
37.6164857 
37.629  775  4 
37.6430604 
37*656  340  7 
37.6696164 
37.682  887  4 

37  696  1536 
37.7094153 
37.722  672  2 
37-735  924  5 
37.749  1722 
37.7624152 
37-775  653  5 
37.788  8873 
37.802  1163 
37.8153408 
37.828  5606 
37.841  7759 
37.854  9864 
37.868  192  4 
37-88I393  8 
37.894  5906 
37.907  782  8 
37.9209704- 
37-934  153  5 
37-947  33i  9 
37.960  505  8 

37- 973  675  1 
37.9868398 

38 

38.0131556 
38.026  306  7 
38.0394532 
38.052  595  2 
38.065  732  6 
38.078  8655 
38.0919939 
38.105  1178 
38.1182371 

38- I31  35 1 9 


Cube  Root. 

II.184  225  2 
II.  186  889  4 
II.1895523 
II. 192  2139 
II.1948743 

II-I97  533  4 
11.200  191  3 
11.202  847  9 
11.205  5032 
11.208  1573 
11.210810  1 
11.213461  7 
11.216  112 
11.218  761  1 
1 1. 221  408  9 
11.224005  4 
11.226  700  7 
11.229  344  8 
11. 231  987  6 
11.234  629  2 
11.2372696 
11.239908  7 
11.242  5465 
11.245  183.1 
11.247  818  5 
11.250  4527 
11.2530856 

11.2557173 

11.258  3478 
11.260977 
11.263605 
11.266  231  8 
11.268  8573 
11. 271  481  6 
11.274  io4  7 
11.276  7266 
11.279  347  2 
11.281  966  6 
11.2845849 
11.287  201  9 
11.289  817  7 
11.292  4323 
11.295  0457 
11.297  657  9 
1 1 .300  268  8 
11.302  878  6 
11.305  4871 
11.308  094  5 
1 1. 310  7006 
11.3133056 

II-3I5  9°94 
11.3185119 
11.321  1132 
II-323  7I3  4 
11.3263124 
11.3289102 


298 

Number. 

1455 

1456 

1457 

1458 

1459 

1460 

1461 

1462 

1463 

1464 

1465 

1466 

1467 

1468 

1469 

1470 

147I 

1472 

1473 

1474 

1475 

1476 

1477 

1478 

1479 

1480 

1481 

1482 

1483 

1484 

1485 

i486 

1487 

1488 

1489 

149O 

1491 

1492 

1493 

1494 

1495 

1496 

1497 

1498 

1499 

1500 

1501 

1502 

1503 

1504 

1505 

1506 

1507 

1508 

1509 

1510 


SQUARES,  CUBES,  AND  ROOTS. 


Square. 

Cube. 

Square  Root. 

2 1 1 70  25 

3080271375 

38.144462  2 

2 II9936 

3086  626816 

38.1575681 

2 12  28  49 

3 092  990  993 

38.1706693 

2 125764 

3099363912 

38.183  7662 

2 12  86  8l 

3 105  745  579 

38.1968585 

2 13  1600 

3 112  136000 

38.2099463 

21345  21 

3 MS  53s  181 

38.223  029  7 

2 13  74  44 

3124  943128 

38.236  108  5 

2 14  03  69 

3I31  359  847 

38.249  182  9 

2 143296 

3 137  785  344 

38.262  252  9 

2 14  62  25 

3144  219625 

38.2753184 

2 I49I  56 

3 150  662  696 

38.2883794 

2 15  2089 

3157114563 

38.301  436 

2 15  50  24 

3 163  575  232 

38.3144881 

2 15  7961 

3170044  709 

38-327  5358 

2 l6  09  OO 

3176523000 

38.340579 

2 16384I 

3 183010  hi 

38.3536178 

2 1667  84 

3189  506048 

38.366  6522 

2 1697  29 

3 196010817 

38.379  682  1 

2 1 7 26  76 

3 202  524  424 

38.392  707  6 

2175625 

3 209  046  875 

38.405  728  7 

2 1785  76 

3215578176 

38.418  745  4 

2 l8  15  29 

3222  118333 

38.431  757  7 

2 1844  84 

3 228  667  352 

38.444  765  6 

2 l8  744I 

3 235  225  239 

38.457  769  I 

2 19  04  OO 

3 241  792  000 

38.470  768  I 

2 193361 

3248  367641 

38.483  762  7 

2 196324 

3 254  952  168 

38.496  753 

2 I992  89 

3 261  545  587 

38-509  739 

2 20  22  56 

3 268  147  904 

38.522  7206 

2 20  52  25 

3274  759125 

38.5356977 

2 20  8l  96 

3 281  379  256 

38.548  670  5 

2 21  1169 

3 288  008  303 

38.5616389 

2 21  41  44 

3 294  646  272 

38.574603 

2 21  71  21 

3 301  293  169 

38.587  562  7 

2 22  01  OO 

3 307  949  000 

38.600  518  1 

2 22  3081 

3314613  771 

38.6134691 

2 22  60  64 

3321287488 

38.626415  8 

2 22  90  49 

3327970157 

38.639  358  2 

2 23  20*36 

3 334661  784 

38.652  296  2 

2 23  50  25 

3341362375 

38.665  229  9 

2 23  80  l6 

3348071936 

38.6781593 

2 24  IOO9 

3 354  790  473 

38.691  084  3 

2 24  40  O4 

3361517992 

38.704005 

2 24  70  OI 

3368  254499 

38.7169214 

2 25  OO  OO 

3 375oooooo 

38.7298335 

2 25  30  OI 

3 38i  754  501 

38.742  741  2 

2 25  60  04 

3388518008 

38.755  644  7 

2 25  90  09 

3 395  290527 

38.768  5439 

2 2620  l6 

3 402  072  064 

38.7814389 

2 26  50  25 

3 408  862  625 

38.7943294 

2 26  80  36 

3415662216 

38.8072158 

2 27  IO  49 

3422  470843 

38.820097  8 

2274064 

3429  288512 

38.8329757 

2 27  70  8l 

3436115229 

38.845  849  1 

2 28  oI  OO 

3442  951000 

38.8587184 

Number. 

I5II 

1512 

1513 

1514 

1515 

1516 

1517 

1518 

1519 

1520 

1521 

1522 

1523 

1524 

1525 

1526 

1527 

1528 

1529 

1530 

1531 

1532 

T533 

1534 

1535 

1536 

1537 

1538. 

1539 

1540 

1541 

1542 

1543 

1544 

1545 

1546 

1547 

1548 

T549 

1550 

i55i 

1552 

1553 

1554 

1555 

1556 

1557 

1558 

1559 

1560 

1561 

1562 

1563 

1564 

1565 

1566 


SQUARES,  CUBES,  AND  ROOTS. 


299 


Square. 

22831  21 
2 28  6l  44 
22891  69 
2 299I  96 
2 29  52  25 
2 29  82  56 

230  12  89 
2 30  43  24 
2 30  73  6l 

231  O4OO 

2 31  34  41 
2316484 
23195  29 
2 32  25  76 
2 32  56  25 

232  86  76 
2331729 

2334784 
2 33  7841 
2 340900 
2343961 
2347024 
2 35  00  89 
2 35  31  56 
2 35  62  25 
2359296 
2 36  23  69 
236  54  44 

236  8521 

237  1600 
2 37  46  81 
2 37  77  64 
2380849 

238  3936 
2 38  70  25 
2 39  01  16 
2393209 

2396304 

2399401 
2 40  25  00 
2405601 
2 40  87  04 
241  1809 

241  49  16 
2 41  80  25 

242  11  36 
2 42  42  49 

242  7364 
2430481 

243  3600 
2436721 
2439844 
2442969 
2 44  60  96 
24492  25 
2 45  23  56 


Cube. 


3 449  795  831 
3456649  728 
3463512697 
3 470  384  744 
3 477  265  875 
3484156096 

3 49i  055  4i3 

3497963832 
3 504  881  359 
3511  808000 

3518743  761 
3 525  688  648 
3 532  642  667 
3 539  6°5  824 
3546578  125 
3 553  559  576 
3560558  183 
3 567  549  952 
3 574  558889 
3581  577000 
3 588  604  291 
3 595  640  768 
3602  686437 
3609741  304 
3616805  375 
3 623  878  656 
3630961  153 
3 638  052  872 
3645153819 
3 652  264  000 

3659383  421 

3666512  088 
3673  650  007 
3680797  184 
3687953  625 
3695119336 
3 702  294  323 
3 709478  592 
3716672  149 
3 723  875  000 
3731087  151 
3 738  308  608 
3 745  539  377 
3 752  779464 
3 760  028  875 
3 767  287  616 
3 774  555  693 
3781833  1 12 
3789119879 
3 796  416  000 
3803721481 
3811036  328 
3818360  547 
3 825  641  144 

3833037  125 

3 840  389  496 


Square  Root. 

38.8715834 
38.884  444  2 
38.8973OO6 
38.9IO  1529 
38.923OOO9 
38.935  844  7 
38.948  684  I 
38.961  5194 
38.974  350  5 
38.9871774 
39 

39.012  8184 
39.025  632  6 
39.038  442  6 
39.051  2483 
39.0640499 
39.076  8473 
39.089  640  6 
39.102  4296 
39.115  2144 

39-I27  995  1 

39.140771  6 

39-x53  5439 
39.166312 
39.179076 
39-I9I  835  9 
39.204  591  5 
39.2 1 7 343  1 
39.2300905 
39.242  833  7 
39-255  572  8 
39.268  307  8 
39.2810387 
39.2937654 
39.306  488 
39.319  206  5 
39.331  920  8 
39.344631  1 

39-357  337  3 
39.3700394 
39.382  737  3 
39-395*431  2 
39.408  121 
39.420  806  7 
39-433  4883 
39.446  165  8 
39.4588393 
39.471  508  7 
39.484 174 
39.4968353 
39.5094925 
39.522  145  7 
39-534  794  8 
39-547  4399 
39.5600809 
39.5727179 


Cube  Root. 

JI-475  0562 

11.4775871 

11.480  1169 
11.4826455 
11.485 1731 
11.4876995 
11.4902249 
11.492  7491 
11.4952722 
H.497  7942 
11.500  315 1 
11.502  8348 
11.505  353  5 
11.507871  1 
11.5103876 
11.512903 
11.5154173 
n.5I7  9305 
11.5204425 
11.522  9535 
11.525  4634 
i 1.527  972  2 
11.5304799 
11.532  9865 
11.535  492 
11.5379965 
11.5404998 
11.543002  1 
11.545  503  3 
11.548  0034 
11.550  5025 
11.5530004 
n-555  497  3 
11 -557  993  1 
11.560487  8 
11.562  981  5 
11.565  474 
11.567  9655 
11.570  455  9 
11.572  9453 
n-575  433  6 
11.5779208 
11.5804069 
11.582  891 9 
11.585  375  9 
11.5878588 
11.590  3407 
11.592  821  5 
11.595  3013 
n-597  7799 
11.600  257  6 
11.602  7342 
11.605  209  7 
11.607  684 1 
11.6101575 
11.6126299 


300 


SQUARES,  CUBES,  AND  ROOTS. 


Number. 

Square. 

Cube. 

Square  Root. 

Cube  Root. 

1567 

2 45  54  89 

3 847  75i  263 

39-585  350  8 

II.615  IOI  2 

1568 

2 45  86  24 

3855  123  432 

39-597  979  7 

II.617571  5 

1569 

2 46  17  61 

3 862  503  009 

39.6106046 

1 1 .620  040  7 

1570 

2 46  49  00 

3 869  893  000 

39.623  225  5 

11.622  508  8 

I571 

2 46  8041 

3877292  41 1 

39.635  842  4 

11.624  9759 

I572 

2 47  1 1 84 

3 884  701  248 

39.648  455  2 

11.627  442 

1573 

2 47  43  29 

3892  119517 

39.661  064 

11.629  907 

1574 

2 47  74  76 

3 899  547  224 

39.673  668  8 

11.632  371 

1575 

2 48  06  25 

3 906  984  375 

39.686  269  6 

11.634  8339 

1576 

2 48  37  76 

3914430  976 

39.698  866  5 

11.637  2957 

1577 

2 48  69  29 

3921  887033 

39.7114593 

11.639  7566 

1578 

2 49  00  84 

3 929  352  552 

39-724  048  1 

11.642  216  4 

1579 

2 49  32  41 

3 936  827  539 

39-736  632  9 

11.644675  1 

1580 

2 49  64  00 

3 944312000 

39.7492138 

11.647  1329 

1581 

249  9561 

3 951  805  941 

39.761  7907 

n.649  5895 

1582 

2 50  27  24 

3 959  309  368 

39-774  363  6 

11.652045  2 

1583 

2 50  58  89 

3 966  822  287 

39.7869325 

11.6544998 

1584 

2 50  90  56 

3 974  344  7°4 

39.799  497  5 

11.656  9534 

1585 

2 51  22  25 

3 981  876  625 

39.8120585 

11.659  4059 

1586 

2 51  53  96 

3 989  418  056 

39.8246155 

1 1. 661  857  4 

1587 

251  8569 

3 996  969  003 

39.8371686 

11.6643079 

1588 

2 52  1744 

4 004  529  472 

39.8497177 

11.666  757  4 

1589 

2524921 

4 012  099  469 

39.862  262  8 

1 1 .669  205  8 

159° 

2 52  81  00 

4 019  679000 

39.874  804 

11.671  6532 

1591 

2 53  12  81 

4 027  268  071 

39.887  341  3 

11.674  099  6 

i592 

2 53  44  64 

4 034  866  688 

39.899  874  7 

11.676  5449 

1593 

2 53  76  49 

4 042  474  857 

39.912  404  1 

11.678  9892 

1594 

2 54  08  36 

4 050  092  584 

39.924  929  5 

1 1. 681  432  5 

1595 

2 54  40  25 

4057  719875 

39-937  451  1 

11.683  874  8 

1596 

2 54  72  16 

4065  356  736 

39.9499687 

11.686  316  i 

1597 

2 55  04  09 

4073003  173 

39.962  482  4 

11.688  7563 

1598 

2 55  36  04 

4 080  659  192 

39.974  992  2 

11.691  1955 

1599 

255  6801 

4 088  324  799 

39.987  498 

11.6936337 

1600 

2 560000 

4 096  000  000 

40 

11.6960709 

Uses  of  preceding  table  may  be  greatly  extended  by  aid  of  following 
Rides : 

To  Compute  Square  or  Cube  of  a liiglier  iNTnm'ber  than, 
is  contained,  in  Table. 

When  Number  is  divisible  by  a Number  without  leaving  a Remainder . 

Rule.— If  number  exceed  by  2,  3,  or  any  other  number  of  times,  any  number 
contained  in  table,  multiply  square  or  cube  of  that  number  in  table  by  square  of  2, 
3,  etc.,  and  product  will  give  result. 

Example. — Required  square  of  1700. 

1700  is  10  times  170,  and  square  of  170  is  2 8900. 

Then,  2 89  00  X 102  = 2 89  00  00. 

2. — What  is  cube  of  2400? 

2400  is  twice  1200,  and  cube  of  1200  is  1 728000000. 

Then  1 728  000  000  X 23  = 13  824  000  000. 

When  Number  is  an  Odd  Number. 

Rule.— Take  the  two  numbers  nearest  to  each  other,  which,  added  together, 
make  that  sum;  then  from  sum  of  squares  or  cubes  of  these  twro  numbers,  multi- 
plied by  2,  subtract  1,  and  remainder  will  give  result. 


SQUARES,  CUBES,  AND  ROOTS, 


301 


Example. — What  is  square  of  1745? 

Two  nearest  numbers  are  j g73  } — 1745. 

Then,  per  table,  fffig  ggg 

I 52  25  13  x 2 = 3 045  026  — I = 3 04  50  25. 

To  Compute  Square  or  Cube  Root  of  a,  liiglier  NTumber 
tban  is  contained,  in  Table. 

When  Number  is  divisible  by  4 or  8 without  leaving  a Remainder. 

Rule. — Divide  number  by  4 or  8 respectively,  as  square  or  cube  root  is  required; 
take  root  of  quotient  in  table,  multiply  it  by  2,  and  product  will  give  root  required. 

Example. — What  are  square  and  cube  roots  of  3200? 

3200  -f-  4 = 800,  and  3200  -7-  8 — 400. 

Then,  square  root  for  800,  per  table,  is  28.28  42  71  2,  which,  being  X 2 = 56.56  85  424 
root 

Cube  root  for  40x3,  per  table,  is  7.368  063,  -which,  being  X 2 = 14.736 126  root. 

When  the  Root  (which  is  taken  as  Number ) does  not  exceed  1600. 

The  Numbers  in  table  are  roots  of  squares  or  cubes,  which  are  to  be  taken 
as  numbers. 

Illustration. —Square  root  of  6400  is  80,  and  cube  root  of  512000  is  80. 

When  a Number  has  Three  or  more  Ciphers  at  its  right  hand. 

Rule. — Point  off  number  into  periods  of  two  or  three  figures  each,  according  as 
square  or  cube  root  is  required,  until  remaining  figures  come  within  limits  of  table; 
then  take  root  for  these  figures,  and  remove  decimal  point  one  figure  for  every  pe- 
riod pointed  otf. 

Example. — What  are  square  or  cube  roots  of  1 500000? 

1 500  000  = 150,  remaining  figure,  square  root  of  which= 12. 247  45 ; hence  1224. 745, 
square  root 

1 500000  = 1500,  remaining  figures,  cube  root  of  which  = 11.447  14  ; hence 
1 14. 47 14,  cube  root 

To  Ascertain.  Cxi  "be  Root  of  any  Number  over  1GOO. 

Rule.— Find  by  table  nearest  cube  to  number  given,  and  term  it  assumed  cube; 
multiply  it  and  given  number  respectively  by  2 ; to  product  of  assumed  cube  add 
given  number,  and  to  product  of  given  number  add  assumed  cube. 

Then,  as  sum  of  assumed  cube  is  to  sum  of  given  number,  so  is  root  of  assumed 
cube  to  root  of  given  number. 

Example. — What  is  cube  root  of  224  809? 

By  table,  nearest  cube  is  216000,  and  its  root  is  60. 

216  000  X 2 + 224  809  = 656  809, 

And  224809  X 24-216000  = 665618. 

Then  656809  : 665618  60  : 60.804-j-,  root 

To  Ascertain  Square  or  Cube  Root  of*  a NTumber  con- 
sisting of  Integers  and.  Decimals. 

Rule. — Multiply  difference  between  root  of  integer  part  and  root  of  next  higher 
integer  by  decimal,  and  add  product  to  root  of  integer  given;  the  sum  will  give  root 
of  number  required. 

This  is  correct  for  Square  root  to  three  places  of  decimals,  and  for  Cube  root  to  seven. 

C c 


202  SQUARES,  CUBES,  AND  ROOTS. 

Example.— What  is  square  root  of  53.75,  and  cube  root  of  843.75? 


V54  =7.3484 

^844  =9.4503 

V 53  =7.2801 

V843  =9.4466 

.0683 

.0037 

•75 

•75 

.051  225 

.002775 

a/53  =7-2801 

^843  =9.4466 

V53.75  = 7.33i325 

^843.75  = 9.449  375 

When  the  Square  or  Cube  Root  is  required  for  Numbers  not  exceeding  Roots 
given  in  Table. 

Numbers  in  table  are  squares  and  cubes  of  roots. 

Rule  —Find  by  table,  in  column  of  numbers  that  number  representing  figures 
of  integer  and ’decimals  for  which  root  is  required,  and  point  it  off  decimally  by 
Diaces  of  2 or  3 figures  as  square  or  cube  root  is  required;  and  opposite  to  it,  in 
column  of  roots,  take  root  and  point  off  1 or  2 additional  places  of  decimals  to  those 
in  root,  as  square  or  cube  root  is  required,  and  result  is  root  required. 

Example  i.— What  are  square  roots  of  .15, 1.50,  and  15.00? 

In  table,  15  has  for  its  root  3.87  29  8 ; hence  .38  72  98  = square  root  for  . 15. 

1 ko  has  for  its  root  12. 24  74  5 ; hence  1. 22  47  45  = square  root  for  1.50. 

1500  has  for  its  root  38.72  98;  hence  3.87  29  8 = square  root  for  15. 

2._What  are  cube  roots  of . 15, 1. 50,  and  15.00 ? 

Add  a cipher  to  each,  to  give  the  numbers  three  places  of  figures,  as  .150, 1.500, 
and  15.000. 

In  table  150  has  for  its  root  5.3133;  hence  .531  33  = cube  root  of.  15. 

r coo  has  for  its  root  11.447;  hence  1.1447  = culje  root  °f  x*5°*  . _ , 

15  has  for  its  root  2.4662;  and  15.000,  by  addition  of  3 places  of  figures,  has 
24.662 ; hence  2.4662  = cube  root  of  15.00. 

To  Ascertain  Square  or  Cu/be  Roots  of  Decimals  alone. 

Rule.— Point  off  number  from  decimal  point  into  periods  of  two  or  three  figures 
each,  as  square  or  cube  root  is  required.  Ascertain  from  table  or ,b^  c^c^^10n 
root  of  number  corresponding  to  decimal  given,  the  same  being  read  off  by ^remo^ 
’ng  the  decimal  point  one  place  to  left  for  every  period  of  2 figures  if  square  root  is 
required,  and  one  place  for  every  period  of  3 figures  if  cube  root  is  required. 

Example. — What  are  square  and  cube  roots  of  .810,  .081,  and  .0081  ? 

.810,  when  pointed  off  = . 81,  and-v/.8i  — -9* 

>0gi5  “ “ =.081,  “ a/*0^1  =*2846. 

.0081,  “ “ “ =.0081,  “ V- 0081  = .09. 

.810,  when  pointed  off  = .810,  and  a/.  810  = 932 17. 

.081,  “ “ “ = .081,  “ ^.081  =.43267. 

.0081,  “ “ “ = .0081,  “ ty.QoZl  =.20083. 

To  Compute  Root  of*  a,  Number. 

Rule.— Take  square  root  of  its  square  root. 

Example. — What  is  the  -fy  of  1600? 

^1600  = 40,  and  V4°  = 6.32 45  55  3. 

To  Compute  6tli  Root  of*  a Number. 

Rule.— Take  cube  root  of  its  square  root. 

Example. — What  is  the  of  441  ? 

V44X  = at,  and  ^21  = 2. 7 589  243- 


FOURTH  AND  FIFTH  POWERS  OF  NUMBERS.  303 
4 tli  and.  5 th.  Powers  of  Numbers. 


From  1 to  150. 


Number. 

4th  Power. 

5th  Power. 

Number. 

4th  Power. 

5 th  Power. 

■I 

1 

1 

64 

16  777  216 

1 073  741  824 

2 

16 

32 

65 

17  850  625 

1 160  290  625 

3 

81 

243 

66 

18974736 

1 252  332  576 

4 

256 

1 024 

$7 

20  151  121 

1 350125  107 

5 

625 

3125 

68 

21  381  376 

1453  933  568 

6 

I 296 

7 776 

69 

22  667  121 

1 564  031  349 

7 

2 401 

16  807 

70 

24  010000 

1 680  700  000 

8 

4096 

32768 

71 

25  411  681 

1 804  229351 

9 

6561 

59  °49 

72 

26  873  856 

1 934917632 

10 

10000 

100  000 

73 

28  398  241 

2073071  593 

11 

14  641 

161 051 

74 

29  986  576 

2219006624 

12 

20736 

248  832 

75 

31  640  625 

2373046  875 

13 

28  561 

37i  293 

76 

33362  176 

2 535  525  376 

14 

38416 

537824 

77 

35153041 

2706  784157 

i5 

50625 

759  375 

78 

37015056 

2887174368 

16 

65  536 

1 048  576 

79 

38950081 

3077056  399 

17 

83  521 

1419857 

80 

40  960  000 

3 276  800000 

18 

104976 

1 889  568 

81 

43046721 

3 486  784  401 

J9 

130  321 

2 476  099 

82 

45  212  176 

3 707  398  432 

20 

160000 

3 200  000 

83 

47  458  321 

3 939  °4°  643 

21 

194  481 

4084  IOI 

84 

49787136 

4 182  119  424 

22 

234256 

5153632 

85 

52  200625 

4437053125 

23 

279841 

6436  343 

86 

54708016 

4704270  176 

24 

331  776 

7 962  624 

87 

57  289761 

4 984  209  207 

25 

390625 

9 765  625 

88 

59969536 

5277319168 

26 

456976 

11  881  376 

89 

62  742  241 

5 584  059  449 

27 

53i 441 

14348907 

90 

65  610000 

5 904  900  000 

28 

614  656 

17  210368 

91 

68  574  961 

6 240  321 451 

29 

707  281 

20  51 1 149 

92 

71  639  296 

6590815  232 

30 

810000 

24  300  000 

93 

74  805  201 

6956883693 

31 

923  52i 

28  629 151 

94 

78  074  896 

7 339  040  224 

32 

1 048  576 

33  554  432 

95 

81  450625 

7 737  809375 

33 

1 185  921 

39  393 

96 

84  034656 

8153726  976 

34 

1 336  336 

45  435  424 

97 

88  529  281 

8587  340257 

35 

1 500625 

52521875 

98 

92  236  816 

9 039  207  968 

36 

1 679616 

60  466  176 

99 

96  059  601 

9 509  900  499 

37 

1 874  161 

69  343  957 

100 

100  000  000 

10  000  000  000 

38 

2 085 136 

79235168 

IOI 

104  060  401 

10  510 100  501 

39 

2313441 

90224  199 

102 

108  243  216 

11 040  808  032 

40 

2 560  000 

102  400000 

103 

112  550881 

11  592  740743 

4i 

2 825  761 

1 15  856  201 

104 

116  985  856 

12  166  529024 

42 

3 1 1 1 696 

130  691  232 

105 

121  550625 

12  762  815  625 

43 

3 418  801 

147  008  443 

106 

126  247  696 

13382  255776 

44 

3748096 

164916224 

107 

131  079601 

14025517307 

45 

4 100625 

184  528125 

108 

136  048  896 

14693  280768 

46 

4 477  456 

205  962  976 

109 

141  158  161 

15386  239549 

47 

4 879681 

229345007 

no 

146  410  000 

16  105  100000 

48 

5 308  416 

254  803  968 

III 

151  807  041 

16850581  551 

49 

5 764  801 

282  475  249 

112 

I57  35i936 

17  623  416  832 

5o 

6 250000 

312  500000 

H3 

163  047  361 

18424351793 

5i 

6 765  201 

345025251 

114 

168  896016 

19  254  *45  824 

52 

7311616 

380  204  032 

H5 

174900625 

20113581875 

53 

7890  481 

418195493 

Il6 

181  063  936 

21  003  416  576 

54 

8 503  056 

459165  024 

117 

187  388  721 

21924480357 

55 

9 i5o625 

503  284  375 

Il8 

*93  877  776 

22  877577568 

56 

9 834  496 

550  731776 

119 

200  533  921 

23  863  536  599 

57 

10  556  001 

601  692  057 

120 

207  360  000 

24  883  200  000 

58 

11  316496 

656356768 

121 

214358881 

25  937  424  601 

59 

12  117  361 

714924299 

122 

22i  533  456 

27027  081  632 

60 

12960000 

777  600  000 

123 

228  886  641 

28153056  843 

61 

13  845  841 

844  596  3GI 

124 

236  421  376 

29  316  250  624 

62 

14776336 

916  132  832 

125 

244  140625 

30517578125 

63 

15752  961 

992  436  543 

126 

252047  376 

31  757  969  376 

POWERS  OF  NUMBERS. — RECIPROCALS. 


4th  Power. 

5th  Power, 

] Number.  | 

4th  Power. 

5th  Power. 

260  144  641 
268  435  456 
276  922  881 
285610000 
294  499  921 

3°3  595  776 
312900721 
322417936 
332150625 
342  102016 

352  275  361 

362673936 

33  °38  369  4°7 

34  359  738  368 
35723051649 
37  129  300000 
38579  489  651 
40074642432 
41615795893 

43  204  003  424 

44  840  334  375 
46  525  874  W6 
48261724457 
50049003168 

I 139 
140 
1 Hi 

142 

143 

144 

145 

146 

147 

148 

149 
1 150 

373  3CI  641 
384  160000 
395254161 
406  586  896 
418  161 601 
429  981  696 
442  050  6?5 
454  37i856 
466  948  881 
479785216 
492  884  401 
506250000 

51  888  844  699 
53  782  400000 
55  730  836  701 
57  735  339232 
59  797  108943 
61917364224 
64  097  340  625 
66  338  290976 
68  641  485  507 
71  008  211  968 
73  439  775  749 
75  937  500000 

greater  than  is 


304 

Number. 

127 

128 

129 

130 

132 

133 

134 

135 

136 

137 

138 

To  Compute  4:th  Power  of  a Number 
contained,  in  Table. 

Rule.— Ascertain  square  of  number  by  preceding  table  or  by  calculation,  and 
square  it;  product  is  power  required. 

Example.— What  is  4th  power  of  1500? 

15002  = 2 250  000,  and  2 250 ooo2  = 5 062  500 000  000. 

To  Compute  5 th.  Power  of  a Number  greater  than  is 
contained  in  Table. 

Rule.— Ascertain  cube  of  number  by  preceding  table  or  by  calculation,  and  mul- 
tiply it  by  its  square;  product  is  power  required. 

To  Compute  4th  and  5th  Powers  by  another  NLethod. 

rule  —Reduce  number  by  2 until  it  is  one  contained  within  table.  Take  power 
which  is  required  of  that  number,  and  multiply  it  by  16,  162,  or  i63  respectively 
for  each  Son,  by  2 for  4th  poier,  and  by  32,  32=,  or  323  respectively  for  each 
division  by  2 for  5th  power. 

Example.— What  are  the  4th  and  5th  powers  of  600? 

600  -r-  2 = 300,  and  300  -T-  2 ==  150. 

The  4th  power  of  150,  per  table,  = 506  250000,  which  x 162,  multiplier  for  a second 
division  256  = 129  600000  000,  4 th  power. 

Again,  the  5th  power  of  150  = 75  937  500000,  which  X 32J>  multiplier  for  a second 
division  1024  — 77  760000000000  — power. 

To  Compute  €>th  Power  of*  a Number. 

Rule.— Square  its  cube. 

Example.— What  is  the  6th  power  of  2? 

2 

23  = 64. 

To  Compute  4tli  or  5th  Root  of  a Number  per  Table. 

Rule— Find  in  column  of  4th  and  5th  powers  number  given,  and  number  from 
which  that  power  is  derived  will  give  root  required. 

Example.— What  is  the  5th  root  of  3 200000? 

3 200000  in  table  is  5th  power  of  20;  hence  20  is  root  required. 


RECIPROCALS. 

Reciprocal  of  a number  is  quotient  arising  from  dividing  1 by  number;  thus,  re- 
ciprocal of  2 is  1 -7-  2 = • 5 

Product  of  a number  and  its  reciprocal  is  always  equal  to  1 ; thus,  2 X .5  = *• 
Reciprocal  of  a vulgar  fraction  is  denominator  divided  by  numerator ; thus,  - = • 5- 


LOGARITHMS. 


305 


LOGARITHMS. 


Logarithms  of  Numbers. 

Logarithms  are  a series  of  numbers  adapted  to  facilitate  the  operation  of 
numerical  computation, 

Addition  being  substituted  for  Multiplication,  Subtraction  for  Division, 
Multiplication  for  Involution,  and  Division  for  Evolution. 

The  Logarithm  of  a number  is  the  exponent  of  a power  to  which  10 
must  be  raised  to  give  that  number. 

It  is  not  necessary,  however,  that  the  base  should  be  10,  it  may  be  any  other  num- 
ber; but  Tables  of  Logarithms,  in  common  use,  are  computed  with  10  as  the  base. 

Thus,  Number  100  Log.  = 2,  as  io2  base  and  exponent  = 100. 

“ 10000  u = 4,  u io4  u “ u = 10000. 

The  Unit  or  Integral  part  of  a Logarithm  is  termed  the  Index , and  the  Decimal 
part  the  Mantissa ; the  sum  of  the  index  and  mantissa  is  the  Logarithm. 

The  Index  of  the  Logarithm  of  any  number,  Integral  or  Mixed , when  the  base  is  10, 
is  equal  to  the  number  of  digits  to  the  left  of  the  decimal  point  less  1.  From  o to 
9,  it  is  o;  from  10  to  99,  it  is  1,  and  from  100  to  999,  it  is  2,  etc. 

Thus,  logarithm  of  3304  = 3.51904,  3 being  the  index  and  .51904  the  mantissa. 

The  Index  of  the  Logarithm  of  a Decimal  Fraction  is  a negative  number,  and  is 
equal  to  the  number  of  places  which  the  first  significant  figure  of  the  decimal  is  re- 
moved from  the  place  of  units. 

Thus,  index  of  logarithm  .005  is  3 or  — 3,  the  first  significant  figure,  5,  being  re- 
moved three  places  from  that  of  units.  The  bar  or  minus  sign  is  placed  over  an 
index  to  indicate  that  this  alone  is  negative,  while  the  decimal  part  is  positive. 

The  Difference  is  the  tabular  difference  between  the  two  nearest  logarithms. 

The  Proportional  Part  is  the  difference  between  the  given  and  the  nearest  less 
tabular  logarithm. 

The  Arithmetical  Complement  of  a number  is  the  remainder  after  subtracting  it 
from  a number  consisting  of  1,  with  as  many  ciphers  annexed  as  the  number  has 
integers.  When  the  index  of  a logarithm  is  less  than  10,  its  complement  is  ascer- 
tained by  subtracting  it  from  10. 


Number. 
4743 
474-3- •• 
47-43- 
4-743' 


Illustrations. 


Logarithm. 

3.676053 

2.676053 

1.676053 
.676053 


I Number. 
•4743  •• 
•°47  43  • 
.004  743 


Logarithm. 

^.676053 

2.676053 

3-676053 


Computation  of  Negative  Indices. 

To  add  two  Negative  Indices.  Add  them  and  put  the  sum  negative.  As  5 -j-  3 = 8*. 

To  add  a Positive  and  Negative  Index.  Subtract  the  less  from  the  greater  and 
to  remainder  give  the  positive  or  negative  sign,  according  as  the  positive  or  nega- 
tive index  is  the  greater.  As  6 + 2"—  4,  and  6 + 2 = 4. 

Illustration. — Add  6.387  57  and  2.924  59.  6.387  57 

2-  924  59 
5.31216 

Here  the  excess  of  1 from  13  in  the  first  decimal  place,  being  positive,  is  carried 
to  the  positive  6,  which  makes  7,  and  7 — 2 = 5. 

To  Subtract  a Negative  Index.  Change  its  sign  to  plus  or  positive,  and  then  add 
it  as  in  addition.^  As  3 from  2,  = 3 + 2 = 5.  And  5 from  2,  = 5 + 2 = 3 ; also 
3 from  5,  =3  + 5 = 2. 


Illustration.— Subtract  5.765  52  from  2.346  74.  2.346  74 

5-76552 

2.581 22 

Here,  excess  of  1 in  the  first  decimal  place  used  with  the  .3  in  subtracting  the  .8 
from  the  1.3  is  to  be  subtracted  from  the  upper  number  2,  which  makes  it  3;  then 


logarithms. 


306 


To  Subtract  a Positive  Index ._  Change  its  sign  to  negative,  and  then  add  as  in 
addition.  As  2 — 2 = 2 -J-  2 = 4. 

To  Multiply  a Negative  Index.  Multiply  the  fractional  parts  by  the  ordinary  rule, 
then  multiply  the  negative  index,  which  will  give  a negative  product,  and  when  an 
excess  over  io  is  to  be  carried,  subtract  the  less  index  lrom  the  greater,  and  the  re- 
mainder gives  the  positive  or  negative  index,  according  as  the  positive  or  negative 
index  is  the  greater.  As  2 X 5 = 10,  and  1 to  be  carried  = 9. 


Illustration. — Multiply  2.3681  by  2,  and  3.7856  by  6. 


2.3681 


3.7856 

6 


4.7362  H-7I36 

Here  2X2  = 4,  also  3X6  = 18,  with  a positive  excess  of  4 = 14. 

To  Divide  a Negative  Index.  If  index  is  divisible  by  divisor, 
der,  put  quotient  with  a negative  sign.  If  negative  exponent .is  not  ^visible  by 
divisor  add  such  a negative  number  to  it  as  will  make  it  divisible,  and  prefix  an 
equal  positive  integer  to  fractional  part  of  logarithm;  then  d1  v ; de  mcreased  n ega- 
tive  exponent  and  the  other  part  of  logarithm  separately  by  ordinary  nd^and  for- 
mor  quotient,  taken  negatively,  will  be  index  to  fractional  part  of  quotient.  As 
6_^3==2.  ^ = 3 requires  2 to  be  addedor  2 to  be  subtracted,  to  make  it  divisible 

without  a remainder,  then  10  -f  2 = 12,  12-4-3  = 4)  and  2 (tlie  sum  subtracted)  -r- 
3 — .66,  the  quotient  therefore  is  4.66. 

Illustration  i. — Divide  6.324282  by  3. 

6. 324  282  -4-3  = 2. 108  094. 

2.— Divide  14.326745  by  9. 

14. 326  745  -4-  9 = 18  + 4-326  745  -4-  9 = 2. 480  749+. 

Here  4 is  added  toTZ,  that  the  sum  18  may  be  divided  by  9,  and  as  4 is  added,  4 
must  be4 prefixed  to  the  fractional  part  of  the^ logarithm,  and  thus  the  value  of  the 
logarithm  is  unchanged,  for  there  is  added  4,  and  4 = 0,  or  4 is  subtracted  and  4 
added. 

To  Ascertain  Logaritlirn.  of’  a Number  Toy  Tatole. 
When  the  Number  is  less  than  tot. 

Look  into  first  page  of  table,  and  opposite  to  number  is  its  logarithm  with  its 
index  prefixed. 

Illustration.— Opposite  7 is  .845098,  its  logarithm;  hence  70=1.84509  , .7  — 
I.845098,  and  .07  = 2.845098. 

When  the  Number  is  between  100  and  1000. 
r»mP,  Find  the  aiven  number  in  left-hand  column  of  table  headed  No.,  and  un- 
der  o fn^eiTcolumTTs  dectaal  part  of  its  logarithm,  to  which  is  to  be  prefixed  a 
whofe  Sumber  for  an  index,  of  i or  *,  according  as  the  number  consists  of  a or  3 
figures. 

Example.— What  is  logarithm  of  450,  and  what  of  .4^? 

Log.  450  = 2.653  213,  and  of  .45  = 1-653  213. 

When  the  Number  is  between  1000  and  10000. 

index. 

Example.— What  is  logarithm  of  4505,  and  what  of  .04505  ? 

Log.  4505  = 3.653  695,  and  of  .045  05  = 2.653  695. 


LOGARITHMS. 


307 


When  the  Number  consists  of  Five  Figures. 

Rule.— Find  the  logarithm  of  the  number  composed  of  the  first  four  figures  as 
preceding,  then  take  the  tabular  difference  from  the  right-hand  column  under  D 
and  multiply  it  by  the  fifth  figure;  reject  the  right-hand  figure  of  the  product  and 
add  the  other  figures,  which  are,  and  are  termed,  a proportional  part  to  the  logarithm 
found  as  above,  observing  that  the  right-hand  figure  of  the  proportional  part  is  to 
be  added  to  that  of  the  logarithm,  and  the  rest  in  order. 

Example. — Required  logarithm  of  83  407  ? 

Note.— When  the  number  consists  of  less  than  4 figures  conceive  a cipher  an- 
nexed to  make  it  four. 

Log.  of  8340  (83  407)  ==  4.921 1 66 

Tabular  difference  52,  which  x 7 (5th  figure)  = 364  = 364 

4. 92 1 202  4 logarithm. 

The  difference  of  the  numbers  is  nearly  proportionate  to  the  difference  of  their 
logarithms. 

Thus,  difference  between  the  numbers  8340  and  8341,  the  next  in  order,  is  1,  and 
the  difference  between  their  logarithms  or  tabular  difference  is  52. 

The  log.  of  this  1 in  the  4th  place  is  therefore  52.  The  correction  then,  for  the  7 
nf  the  5th  place,  which  is  .7  of  1 in  the  4th  place,  is  ascertained  by  the  proportion 
1 : 52  ::  .7  : 36.4. 

The  correction  is  obtained  by  multiplying  the  tabular  difference  by  7,  rejecting 
the  right  hand  figure  of  the  product,  if  the  log.  is  to  be  confined  to  six  decimal 
places. 

When  the  Number  consists  of  any  Number  over  Four  Figures. 

Rule. — Proceed  as  for  four  figures  for  the  first  four,  multiplying  the  tabular  dif- 
ference by  the  excess  of  figures  over  4 and  rejecting  one  right-hand  figure  of  the 
product  for  a number  of  five  figures,  and  two  for  one  of  six,  and  so  on. 

Example  i.— Required  logarithm  of  834079? 

Log.  of  8340  (834079)=  5.921 166 
Tabular  difference  52,  which  x 79  = 4108 


5.92120708  logarithm. 

2. — Required  logarithm  of  8340794? 

Log.  of  8340  (8  340  794)  = 6.921 166 

Tab.  diff.  52,  which  x 794  (5th,  6th,  and  7th  figures)  = 41  288 


Or,  Log.  of  8340  = 

“ “ 7 (5th  figure)  X 52  tab.  dif. 

“ “ 9 (6th  “ ) X 52  “ “ 

“ “ 4 (7th  “ ) X 52  “ 

Log.  with  index  for  7 figures  .... 


6.921  207  288  logarithm. 
.921 166 


208 


6.921  207  288 


To  Ascertain.  Logarithms  of  a Affixed  Number. 

Rule. — Take  out  logarithm  of  the  number  as  if  it  were  an  integer  or  whole  num- 
ber, to  wrhich  prefix  the  index  of  the  integral  part  of  the  number. 

Example.— What  is  logarithm  of  834.0794? 

Mantissa  of  log.  of  8 340  794  = 9 212  073 ; hence  log.  of  834.0794  = 2.921  207  3. 


To  Ascertain  Logarithm  of  a Decimal  If raction. 

Rule. — Take  logarithm  from  table  as  if  the  figures  were  all  integers,  and  prefix 
index  as  by  previous  rules. 

Example.— Logarithm  of  .1234=7.091  305. 

To  Ascertain  Logarithm  of  a ‘V'nlgar  Fraction. 

Rule. — Reduce  the  fraction  to  a decimal,  and  proceed  as  by  preceding  rule.  Or, 
subtract  logarithm  of  denominator  from  that  of  numerator,  and  the  difference  will 
give  logarithm  required. 

Example.— Logarithm  of 

^ = .1875.  Log.  .1875  = 1.273 001  logarithm. 

Or,  Log.  3 = .477 121 
u 16  = 1.20412 

1.273001  logarithm. 


logarithms. 


308 

To  _A.scertai.il.  tHe  Number  Corresponding  to  a Given 
Logaritlim. 

When  the  given  or  exact  Logarithm  is  in  the  Table. 

Operation.— Opposite  to  first  two  figures  of  logarithm,  neglecting  the  index  in 
column  o look  for  the  remaining  figures  of  the  log.  in  that  column  or  in  any  of  the 
n^ne  at  the' right  thereof;  the  first  three  figures  of  the  number  will  be  found  at  the 
fpft  in  column  under  No.,  and  the  fourth  at  top  directly  over  the  log 

The  number  is  to  be  made  to  correspond  to  index  of  logarithm,  by  pointing  off 
decimals  or  prefixing  ciphers. 

Illustration.— What  is  number  corresponding  to  log.  3.963977  ? 

Opposite  to  963977,  in  page  329,  is  920,  and  at  top  of  column  is  4;  hence,  num- 
ber = 9204. 

When  the  given  or  exact  Logarithm  is  not  in  the  Table. 

Operation  -Take  the  number  for  the  next  less  logarithm  from  table,  which  will 

giVTeofla^  logarithm  in  table  from  the  given 

Inlrithm  add  ciphers  and  divide  by  the  difference  in  column  D opposite  the 
KlthS  Annex  quotient  to  the  four  figures  already  ascertained,  and  place  deci- 


mal  point. 

Illustration  i.— What  is  number  corresponding  to  log.  5.921  207  . 

Given  log.  = 

5.921  207 

8340 

Next  less  in  table 

5.921 166 

D = 52)  4100  (78-f- 

78 

364 

834  078 

Hence,  number  = 834  078. 

460 

416 

44 

2. —What  is  number  corresponding  to 

log.  3.922853? 

Given  log.  = 

3.922853 

8372 

Next  less  in  table 

3.922  829 

D = 52)  2400  (46  -j- 

46 

208 

837  246 

Hence,  number  = 8372.46. 

320 

312 

8 

Multiplication. 

Rule.— Add  together  the  logarithms  of  the  numbers  and  the  sum  will  give  the 
logarithm  of  the  product. 

Example  1.— Multiply  345.7  by  2.581. 

Log-  345*7  =2.538699 

u 2.581=  .411788 

2.950487  log.  of  product.  Number  = 892. 251. 

2.— Multiply  .03902,  59.71,  and  .003 147. 

Log.  .03902  =2.591287 
u 59-  71  =1.776047 

“ .003147  = 3-497  897 

3.865  231  log.  of  product.  Number  = .007  332 15. 
Division. 

Rule. -From  logarithm  of  dividend  subtract  that  of  divisor,  and  remainder  wiU 
give  logarithm  of  the  quotient. 

Example.— Divide  371.4  by  52-  37- 

Log.  371.4  =2.569842 
“ 52-37  = i-7i9°83 

• 850759  log.  of  quotient.  Number  = 7.091 80. 


LOGARITHMS. 


309 


Rule  of  Three,  or  Proportion. 

Rule.— Add  together  the  logarithms  of  the  second  and  third  terms,  from  their 
sum  subtract  logarithm  of  the  first,  and  the  remainder  will  give  logarithm  of  the 
fourth  term. 

Or,  instead  of  subtracting  logarithm  of  first  term,  add  its  Arithmetical  Comple- 
ment, and  subtract  10  from  its  index. 

Example  1.— What  is  fourth  proportional  to  723.4,  .025  19,  and  3574? 

As  723.4  log.  — 2.859379 

Is  to  .02519  li  ==2.401228 
So  is  3574  “ = 3-553  155 

1-954  383 

First  term  “ 2.859379 

1.095  004  log.  of  4 th  term.  Number  = . 124  453. 

By  Arithmetical  Complement. 

Illustration.— As  723.4  log.  = 2.859 379>  Ar-  com.  = 7- 140621 

Is  to  .02519“  = 7.40x228 

So  is  3574  “ = 3-553  *55 

,T  1. 095  004  log.  of  \th  term. 

Number  = .124  453.  ^ 

2— If  an  engine  of  67  IP  cari  raise  57  600  cube  feet  of  water  in  a given  time,  what 
IP  is  required  to  raise  8 575  000  cube  feet  in  like  time  ? 

L°g-  8 575  000  = 6.933  234 
67  = 1.826  075 


u 


8-759  3°9 
57600  = 4.760422 


3.998  877  log.  of  4th  term.  Number  = 9974.4  cube  feet. 

3.  — If  14  men  in  47  days  excavate  5631  cube  yards,  what  time  will  it  require  to 
excavate  47  280  at  same  rate  of  excavation  ? 394. 626  days. 


Involution. 

Rule.— Multiply  logarithm  of  given  number  by  exponent  of  the  power  to  which 
it  is  to  be  raised,  and  the  product  will  give  the  logarithm  of  the  required  power. 
Example.— What  is  cube  of  30.71  ? 

Log.  30.71  = 1.487  28 
3 

4.461 84  log.  of  power.  Number  = 28  962.73. 


Evolution. 

Rule.— Divide  logarithm  of  given  number  by  exponent  of  the  root  which  is  to  be 
extracted,  and  quotient  will  give  logarithm  of  required  root. 

Example  1.— What  is  cube  root  of  1234? 

Log.  1234  = 3.091  315 

Divide  by  3 = 1.030438  log.  of  root.  Number  = 10.72601. 

2.— What  is  4th  root  of  .007654? 

Log.  .007654  = 3.883888 

Divide  by  4 (here  34-1  + 1)  = 1.470  972  log.  of  root.  Number  = .295  78. 

To  Ascertain  Reciprocal  of  a IN’uirn.ber. 

Rule.— Subtract  decimal  of  logarithm  of  the  number  from  .000000;  add  1 to  in- 
dex  of  logarithm  and  change  its  sign.  The  result  is  logarithm  of  the  reciprocal. 
Example.— Required  reciprocal  of  230? 

.000000 
Log.  230  = 2.361  728 

3.638  272  = log.  of  .004  348  reciprocal. 


3io 


LOGARITHMS. 


Simple  Interest. 

Rule  —Add  together  logarithm  of  principal,  rate  per  cent.,  and  time  in  years,  from 
the  sum  subtract  2,  and  the  remainder  will  give  logarithm  of  the  interest. 

Example.— What  is  interest  on  $ 500,  @ 6 per  cent.,  for  3 years? 

Log.  500  = 2.69897 

6 = .77815* 

3=  *477  *21 

3.954  242 

2 

1.954  242  log.  of  interest.  Number  = 90  dollars. 

Compound  Interest. 

rule. Compute  amount  of  $ . or  £ i,  etc,  at  the  given  rate  of  interest  for  one 

vpar  for  the  first  term,  which  is  termed  the  ratio.  . 

^ Multiply  logarithm  of  ratio  by  the  time,  add  to  product  logarithm  of  the  principal, 
and  sum  is  logarithm  of  the  amount. 

given  Hates  Her  Cent. 


Rate. 


j Log.  of  Ratio. 

Rate. 

Log.  of  Ratio. 

Rate. 

Log.  of  Ratio. 

j Rate. 

Log.  of  Ratio. 

.004321  4 
.005  395 

. 006  466 
.0075344 
.008  600  2 
.009  6633 
.010723  9 
.011 781 8 
.012  837  2 

3-25 

3-5 

3- 75 

4 

4- 25 
4-5 

4- 75 

5 

5- 25 

.013  890  1 
.0149403 
.015  988  1 
.0170333 
.018076  1 
.019  116  3 
. 020  1 54 
.021  189  3 

.022  222  I 

5-5 

5- 75 
6 

6.25 

6- 5 

6- 75 
7 

7'25 

7- 5 

.0232525 
.024  2804 
*025  305  9 
.026  328  9 
.027  3496 
.028  763  9 
.029  3838 
.030  3973 
.031  408  5 

I 7-75 
8 

8-25 

8.5 

8- 75 
9 

9.25 

9- 5 

1 9-75 

.0324373 
.033  4238 
.034  4279 
.035  4297 
.0364293 
.037  426  5 
.038  421  4 
.0394141 
.040404  5 

1.25 

i-5 

1- 75 

2 

2.25 

2.5 

2- 75 

3 

Example.— What  will  $364,  at  6 per  cent, 
to  in  23  years? 

Los  of  ratio  from  above  table  .025  3059 
23 


per  annum,  compounded  yearly,  amount 


364 


• 5420357 

2.561  IOI 

3.1031367  log.  of  amount.  Number  = 1268.05  doll. 


Miscellaneous  Illustrations. 

1.  What  is  area  and  circumference  of  a circle  of  21.72  feet  in  diameter? 
1.336  860 


Log.  of  21. 72s  =2-673720 
“ “ .7854  = 1.895091 

u « 2.568811  =37°- 54  feet  area. 

Log.  of  21.72  =2.33686 

41  “ 3.1416  =_1497£5 

u a j.839  71  = 68.236  feet  circum. 

2.  Sides  of  a triangle  are  564,  373,  and  747  feet;  what  is  its  area? 

Log.  of  sides  564 + 373 + 747  _ 2 g2$  3I2 

“ “ .5  side  — a = 842  — 564  = 2.444045 

“ “ .5  side  — 6 = 842  — 373  — 2.671 173 

“ “ .5  side  — c =842  — 747  =±9TTJ2± 

2)10.018  254 

Area  = Number  of  5.009 127  = 1021.24  feet. 


3.-_ What  is  logarithm  of  8 

r 8 x 36  36 

Log.  


,3-6  o 


_ t.  x log.  8 = 3.6  X .90309  = 3.251 124-  Number  = 1782.89. 


LOGARITHMS  OF  NUMBERS. 


311 


Logarithms  of  !NAimT>ers. 


From  1 to  10  000. 


No. 

j Logarithm. 

| No. 

| Logarithm. 

||  No. 

Logarithm. 

| No. 

Logarithm. 

1 

*° 

26 

*•414  973 

51 

I.70757 

76 

1.880814 

2 

.301  03 

27 

I-43I  364 

52 

I.716  003 

77 

1.886  491 

3 

1 -477  121 

28 

1.447  158 

53 

I.724  276 

78 

1.892095 

4 

1 .60206 

29 

1.462  398 

54 

I-732  394 

79 

1.897  627 

5 

.698  97 

30 

1.477  121 

55 

1.740363 

80 

1.90309 

6 

•778  151 

31 

1. 49 1 362 

56 

1.748  188 

81 

1.908  485 

7 

.845  098 

32 

1-505  15 

57 

1-755  875 

82 

1-913  814 

8 

.90309 

33 

1.518514 

58 

1.763  428 

83 

1.919  078 

9 

•954  243 

34 

I-53I  479 

59 

1.770  852 

84 

1.924279 

10 

1 

35 

1.544  068 

60 

1.778  151 

85 

1.929  419 

11 

1. 041  393 

36 

i-556  303 

61 

1-785  33 

86 

1.934  498 

12 

1.079  J8i 

37 

1.568  202 

62 

1.792  392 

87 

1.939  519 

13 

I*II3  943 

38 

1.579  784 

63 

1-799  34i 

88 

1.944  483 

14 

1.146  128 

39 

1. 591  065 

64 

1.806  18 

89 

1.949  39 

15 

1.176  091 

40 

1.602  06 

65 

1.812  913 

90 

i-954  243 

16 

1.204  12 

41 

1.612  784 

66 

1.819544 

91 

1.959  041 

17 

1.230  449 

42 

1.623  249 

67 

1.826  075 

92 

1.963  788 

18 

1-255273 

43 

1.633468 

68 

1.832  509 

93 

1.968  483 

19 

1.278  754 

44 

1-643  453 

69 

1.838  849 

94 

1.973  128 

20 

1. 301  03 

45 

1.653213 

70 

1.845098 

95 

1.977  724 

21 

1.322  219 

46 

1.662  758 

71 

1.851  258 

96 

1.982  271 

22 

i-342  423 

47 

1.672  098 

72 

I-857  332 

97 

1.986  772 

23 

1.361  728 

48 

1. 681  241 

73 

1-863323 

98 

1. 991  226 

24 

1.380  21 1 

49 

1.690  196 

74 

1.869232 

99 

1.995  635 

25 

1 -397  94 

50 

1.69897 

75 

1.875061 

100 

2 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

100 

00- 

0000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

346i 

3891 

432 

101 

102 

00- 

00- 

432i 

86 

475i 

9026 

5181 

945i 

5609 

9876 

6038 

6466 

6894 

7321 

7748 

8174 

428 

425 

102 

01- 

— 

— 

— 

03 

0724 

1147 

157 

T993 

2415 

424 

103 

01- 

2837 

3259 

368 

4i 

4521 

494 

536 

5779 

6197 

6616 

420 

104 

01- 

7033 

745i 

7868 

8284 

87 

9116 

9532 

9947 

— 

417 

104 

02- 

— 

— 

— 

— 

— 

0361 

0775 

416 

105 

_ 

02- 

1189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

412 

IOO 

TO7 

02- 

02- 

53o6 

9384 

5715 

9789 

6125 

6533 

6942 

735 

7757 

8164 

8571 

8978 

408 

405 

IO7 

T 

lo3- 

— 

— 

oi95 

06 

1004 

1408 

1812 

2216 

2619 

3021 

404 

IOo 

IO9 

°3- 

°3“ 

3424 

7426 

3826 

7825 

4227 

8223 

4628 

862 

5029 

9017 

543 

9414 

583 

9811 

623 

6629 

7028 

400 

398 

IO9  j 

04- 

— 

— 

— 

— 

— 

0207 

0602 

0998 

397 

110 

04- 

I393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

454 

4932 

393 

in 

112 

04- 

04- 

5323 

9218 

5714 

9606 

6105 

9993 

6495 

6885 

7275 

7664 

8053 

8442 

883 

389 

388 

112 

113 

05- 

05- 

3078 

3463 

3846 

038 

423 

0766 

4613 

ii53 

4996 

1538 

5378 

1924 

576 

2309 

6142 

2694 

6524 

386 

383 

114 
1 14 

05- 

06- 

6905 

7286 

7666 

8046 

8426 

8805 

9^5 

9563 

9942 

383 

2_ 

— 

— 

— 

— 

032 

379 

No.  | 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

312 


LOGARITHMS  OF  NUMBERS. 


No. 

0 

1 

2 ■ 

3 

4 I 

5 

6 

7 

8 

9 

D 

115 

06-  0698 

1075 

1452 

1829 

2206 

2582 

2958  3333 

3709  4083 

376 

116 

06-  4458 

4832 

5206  558 

5953 

6326 

6699 

7071 

7443 

7815 

373 

117 

06-  8186 

8557 

8Q28  9298  9668  j 

— 

— 

— 

— 

— 

380 

117 

07-  -- 

— 

— 

— 

— 

0038 

0407  0776 

1145 

1514 

370 

118 

07-  1882 

225 

2617 

2985 

3352 

3718  4085 

4451 

4816 

5182 

366 

119 

07-  5547 

5912 

6276  664 

7004 

7368 

773i 

8094  8457 

8819 

363 

120 

07-  9181 

9543 

9904 

362 

120 

08-  — 

— 

0266 

0626 

0987 

1347 

1707 

2067  2426 

360 

121 

08-  2785 

3144 

3503  3861 

4219 

4576  4934 

5291 

5647  6004 

357 

122 

08-  636 

6716  7071 

7426  7781 

8136  849 

8845 

9!98  9552 

355 

123 

08-  9905 

355 

123 

09-  — 

0258  0611 

0963 

1315 

1667 

2018 

237 

2721 

3071 

353 

124 

09-  3422 

3772 

4122 

4471 

482 

5169 

5518  5866  6215 

6562 

349 

125 

09-  691 

7257 

7604 

7951 

8298 

8644  899 

9335  9681 

— 

348 

125 

10-  — 

0026 

346 

126 

10-  0371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3ii9 

3462 

343 

127 

10-  3804  4146  4487 

4828 

5169 

55i 

5851 

619I 

6531 

6871 

34i 

128 

10-  721 

7549 

7888  8227 

8565 

8903  9241 

9579  99io 

— 

338 

128 

11-  — 

0253 

337 

129 

11-  059 

0926  1263 

1599 

1934 

227 

2605 

294 

3275 

3609 

335 

130 

11-  3943 

4277 

4611 

4944  5278 

5611 

5943 

6276  6608 

694 

333 

131 

11-  7271 

7603 

7934 

8265 

8595 

8926  9256  9586  9915 

— 

33i 

I3I 

12-  — 

0245 

330 

132 

12-  0574 

0903 

1231 

156 

1888 

2216 

2544  2871 

3198 

3525 

328 

133 

12-  3852 

4 Y?8 

4504  483 

5156 

5481 

5806  6131 

6456 

6781 

325 

134 

12-  7105 

7429 

7753  8076  8399 

8722  9045  9368  969 

— 

323  , 

134 

13- 

0012 

323 

135 

13-  0334 

0655 

0977 

1298 

1619 

1939 

226 

258 

29 

3219 

321 

136 

13-  3539 

3858  4177 

4496  4814 

5133 

5451 

5769 

6086 

6403 

318 

137 

13-  6721 

7037 

7354 

7671 

7987 

8303 

8618  8934 

9249 

9564 

316 

138 

13-  9879 

3i5 

138 

14-  — 

0194 

0508 

0822 

1136 

145 

1763 

2076 

2389 

2702 

3i4 

139 

14-  3oi5 

3327 

3639 

395i 

4263 

4574 

4885 

5196 

5507 

5818 

3ii 

140 

14-  6128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

89II 

309  • 

*4i 

14-  9219 

9527 

9835 

— 

— 

— 

— 7 

— 

— 

— 

308 

141 

15-  — 

— - 

— 

0142 

0449 

0756 

1063 

137 

1676 

1982 

307 

142 

15-  2288 

2594 

29 

3205 

35i 

3815 

412 

4424 

4728 

5032 

305 

143 

15-  5336 

564 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8o6l 

303 

144 

15-  8362 

8664 

8965 

9266 

9567 

9868 

— 

— 

— 

— 

302 

144 

16-  — 

— 

— 

— 

— 

— 

0168 

0469 

0769 

1068 

301 

145 

16-  1368 

1667 

1967 

2266 

2564 

2863 

3161 

346 

3758 

4055 

299  ■ 

146 

16-  4353 

465 

4947 

5244 

554i 

5838 

6134 

643 

6726 

7022 

297  \ 

147 

16-  7317 

7613 

7908 

8203 

8497 

8792 

9086 

938 

9674 

9968 

295  j 

148 

17-  0262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

293  l 

149 

17-  3186 

3478 

3769 

406 

435i 

4641 

4932 

5222 

5512 

5802 

291 

150 

17-  6091 

6381 

667 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

151 

17-  8977 

9264 

9552 

9839 

— 

— 

— 

— 

— 

— 

287  } 

151 

18-  — 

— 

— 

— 

0126 

0413 

0699 

0986 

1272 

I538 

287 

152 

18-  1844 

2129 

2415 

27 

2985 

327 

3555 

3839 

4123 

4407 

285 

153 

18-  4691 

4975 

5259 

5542 

582s 

6108 

6391 

6674 

6956 

7239 

283 

154 

18-  7521 

7803 

8084 

8366 

8647 

8928 

9209 

949 

9771 

— 

281 

154 

19- 

0051 

281 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

eT 

LOGARITHMS  OF  NUMBERS. 


313 


No. 

0 

r 

2' 

3 

4 

5 

6 

7 

8 9 

155 

19-  0332 

0612 

0892 

1171 

1451 

173 

201 

2289  2567  2846 

156 

19-  3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069  5346  5623 

157 

19-  59 

6176 

6453  6729 

7005 

7281 

7556 

7832 

8107  8382 

158 

19-  8657 

8932 

9206  9481 

9755 

— 

— — 

158 

20-  — 

— 

— 

— 

— 

0029 

0303 

0577  085  1124 

159 

20-  1397 

167 

1943 

2216 

2488 

2761 

3033 

3305 

3577  3848 

160 

20-  412 

439i 

4663 

4934 

5204 

5475 

5746  6016 

6286  6556 

161 

20-  6826 

7096 

7365 

7634 

7904 

8173  8441 

871 

8979  9247 

162 

20-  9515 

9783 

162 

21-  — 

— 

0051 

0319 

0586 

0853 

1121 

1388 

1654  1921 

163 

21-  2188 

2454 

272 

2986 

3252 

35i8 

378 3 

4049 

4314  4579 

164 

21-  4844 

5109 

5373 

5638 

5902 

6166 

643 

6694 

6957  7221 

165 

21-  7484 

7747 

801 

8273 

8536 

8798 

906 

9323 

9585  9846 

166 

22-  0108 

037 

0631 

0892 

ii53 

1414 

1675 

1936 

2196  2456 

167 

22-  2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792  5051 

168 

22-  5309 

5568 

5826 

6084 

6342 

66 

6858 

7ii5 

7372  763 

169 

22-  7887 

8144 

84 

8657 

8913 

917 

9426 

9682 

99 38  — 

3:69 

23-  — 

— 

— 

— 

— 

— 

— 

— 

— 0193 

170 

23-  0449 

0704 

096 

1215 

147 

1724 

!979 

2234 

2488  2742 

171 

23-  2996 

325 

3504 

3757 

4011 

4264 

4517 

477 

5023  5276 

172 

23-  5528 

578i 

6033 

6285 

6537 

6789 

7041 

7292 

7544  7795 

173 

23-  8046 

8297 

8548 

8799 

9049 

9299 

955 

98 

173 

24-  — 

005  03 

174 

24-  0549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541  279 

175 

24-  3038  3286  3534 

3782 

403 

4277 

4525 

4772 

5019  5266 

376 

24-  55i3 

5759 

6006  6252  6499 

6745 

6991 

7237 

7482  7728 

177 

24-  7973 

8219  8464  8709  8954 

9198  9443  9687 

9932  — 

177 

25-  — 

178  25- 

179  25- 

180  ! 25- 

181  2s- 


182 

183  I 

184 

185 

186 


042  0664  0908 

2853  3°96  3338 
5273  5514  5755 
7679  7918  8158 
0071  031  0548 

2451  2688  2925 
4818  5054  529 
7172  7406  7641 
9513  9746  998 


II5I  1395 
358  3822 
5996  6237 
8398  8637 
0787  1025 
3162  3399 
55 25  5761 
7875  81 1 


186 

187  27-  1842  2074  2306 

188  27-  4158  4389  462 

189  ! 27-  6462  6692  6921 

190  27-  8754  8982  9211 


190 

191 

192 

193 

194 

195 

196 

197 

198  j 

199  ! 
199 

No. 


27- 

27- 

27- 

27~ 

27- 

28- 
28- 
28- 
28- 
28- 
29- 
29- 
29- 
29- 

29- 

30- 


0213  0446 
2538  277 
485  5081 

7151  738 

9439  9667 


1033  1261 
3301  3527 
5557  5782 
7802  8026 
0035  0257 
2256  2478 
4466  4687 
6665  6884 

8853  9071 


1488  1715  1942 
3753  3979  4205 
6007  6232  6456 
8249  8473  8696 
048  0702  0925 
2699  292  3141 
4907  5127  5347 
7104  7323  7542 
9289  9507  9725 


1638 

4064 

6477 

8877 

1263 

3636 

5996 

8344 


1881  2125 
4306  4548 
6718  6958 
9116  9355 
1501  1739 
3873  4109 
6232  6467 
8578  8812 


2368  261 
479  5031 

7198  7439 

9594  9833 
1976  2214 
4346  4582 
6702  6937 
9046  9279 


0679  0912 
3001  3233 
5311  5542 
7609  7838 

9895  — 

— 0123 
2169  2396 
4431  4656 
6681  6905 
892  9143 
1147  1369 
3363  3584 
5567  5787 
7761  7979 


ii44  1377  1609 
3464  3696  3927 
5772  6002  6232 
8067  8296  8525 

035 1 0578  0806 
2622  2849  3075 
4882  5107  5332 
7i3  7354  7578 

9366  9589  9812 
1591  1813  2034 
3804  4025  4246 
6007  6226  6446 
8198  8416  8635 


D 


279 

278 

276 

275 

274 

272 

271 

269 

268 

267 

266 

264 

262 

261 

259 

258 

257 

256 

255 

253 

252 

251 

250 

249 

248 

246 

246 

245 

243 

242 

241 

239 

238 

237 

235 

234 

234 

233 

232 

230 

229 

228 

228 

227 

226 

225 

223 

222 

221 

220 

219 

218 

218 

T)~ 


3I4  LOGARITHMS  OF  NUMBERS. 


No. 

0 

1 

2 ■ 

3 

4 

5 

6 

7 

8 

9 

D 

200 

30-  io3 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764  298 

217 

201 

30-  3196 

3412 

3628  3844  4059 

4275 

4491 

4706 

4921 

5136 

216 

202 

30-  5351 

5566 

5781 

5996  6211 

6425  6639  6854 

7068 

7282 

215 

203 

30-  7496 

771 

7924  8137 

83s  1 

8564  8778 

8991 

9204 

9417  ! 

213 

204 

30-  963 

9843 

1 

213 

204 

31-  — 

— 

0056  0268  0481 

0693  0906 

1118 

133 

1542 

212 

205 

3i“  1754 

1966 

2177 

2389 

26 

2812 

3023 

3234 

3445 

3656 

211 

206 

3i“  3867 

4078 

4289  4499 

471 

492 

513 

534 

555i 

576  | 

210 

207 

31-  597 

618 

639 

6599  6809 

7018 

7227 

7436 

7646 

7854 : 

209 

208 

31-  8063 

8272 

8481 

8689  8898 

9106  9314 

9522 

973 

9938  ! 

208 

209 

32-  0146 

0354 

0562  0769  0977 

1184 

1391 

1598 

1805 

2012  1 

207 

210 

32-  2219 

2426 

2633  2839  3046 

3252 

3458 

3665 

3871 

4077 

206 

211 

32-  4282 

4488 

4694  4899  5105 

53i 

5516 

572i 

5926 

6131 

205 

212 

32-  6336  6541 

6745  695 

7155 

7359 

7563 

7767 

7972 

8176 

204 

213 

32-  838 

8583 

8787  8991 

9194 

9398  9601 

9805 

— 

— 

204 

213 

.33“ 

0008 

0211 

203 

214 

33“  0414  0617  0819 

1022 

1225 

1427 

163 

1832 

2034 

2236 

202 

215 

.33“  2438 

264 

2842 

3044  3246 

3447  3649  385 

4051 

4253 

202 

216 

33“  4454  4655 

4856  5057 

5257 

5458 

5658 

5859 

6059 

626 

201 

217 

33“  M 

666 

686 

706 

726 

7459 

7659  7858 

8058 

8257 

200 

2x8 

33-  8456'  8656  8855 

9054 

9253 

945i 

965 

9849 

— 

— 

200 

2x8 

34- 

— 

0047 

O246 

199 

2x9 

34-  0444 

0642 

0841 

1039 

1237 

1435 

1632 

183 

2028 

2225 

198 

220 

.34“  2423 

262 

2817 

3014 

3212 

3409 

3606  3802 

3999 

4196 

I97 

221 

.34“  4392 

4589 

478s 

4981 

5178 

5374 

557 

5766 

5962 

6157 

196 

.222 

34“  6353 

6549 

6744  6939 

7135 

7.33 

7525 

772 

79T5 

8ll 

*95 

223 

.34-8305 

85 

8694 

888; 

9083 

9278 

9472  9666 

986 

— 

194  ' 

223 

.224 

35- 

.35-0248 

0442 

0636  0829 

1023 

1216 

141 

1603 

1796 

0054 

1989 

I94 
!93  ; 

225 

35-  2x83 

2375 

2568 

2761 

2954 

,3I47 

3339 

3532 

3724 

39l6 

I93  • 

226 

35-  4io8 

4301 

4493 

4685 

4876 

5068 

526 

5452 

5643 

5834 

192  • 

227 

35-6026 

6217 

6408 

6599 

679 

6981 

7172 

7363 

7554 

7744 

I9I 

.228 

35“  7935 

8125 

8316 

8506 

8696 

8886 

9076  9266 

9456 

9646 

19a 

.229 

.35-  9835 

189 

229 

36-  — 

0025 

0215 

0404 

0593 

0.783 

0972 

1161 

135 

1539 

189  ; 

280 

36-  .1728 

I9I7 

2105 

2294 

2482 

2671 

2859 

304s 

3236 

3424 

188  ; 

231 

36-  3612 

38 

3988 

4176  4363 

4551 

4739 

4926 

5ii3 

5301 

188 

232 

36-  5488 

5675 

5862 

6049  6236 

6423 

661 

6796 

6983 

7169 

187 

233 

36-  7356 

7542 

7729 

79*5 

8101 

8287 

8473 

8659 

8845 

903 

186 

234 

36-  9216 

9401 

9587 

9772  9958 

— 

— 

— 

— 

— 

186 

235 

37-  “ 

— 

— 

— 

— 

0143 

0328 

0513 

0698 

0883 

185 

235 

37-  xo68 

1253 

1437 

1622 

1806 

1991 

2175 

236 

2544 

2728 

184  i 

236 

37-  2912 

3096 

328 

3464  3647 

3831 

4015 

4198 

4382 

4565 

184 

237 

37"  4748 

4932 

5ii5 

5298 

548i 

5664 

5846 

6029 

6212 

6394 

183  ! 

-238 

37-  ^577 

6759 

6942 

7124 

7306 

7488 

767 

7852 

8034 

8216 

182 

239 

37"  8398 

858 

8761 

8943 

9I24 

9306 

9487 

9668 

9849 

— 

l82  , 

239 

3S 

003 

,lSl 

240 

38-  0211 

0392 

0573 

0754 

0934 

11x5 

1296 

1476 

1656 

1837 

; 181  ; 

241 

38-  2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

242 

38-  3815 

3995 

4174 

4353 

4533 

4712 

4891 

507 

5249 

5428 

179 

243 

38-  5606 

5785 

5964  6142 

6321 

6499  6677  6856 

7034 

7212 

178 

244 

38-  739 

7568 

7746 

7923 

8101 

8279  8456  8634  88 1 1 

8989 

178 

1 0 

1 

2 

3 

4 

I 5 

6 

7 

8 

9 

| D 

logarithms  of  numbers. 


315 


No.  | 0 

245  38-  9l66 
245  39-  “ 
39-  °935 
39-  2697 
39"  4452 
39"  6i99 
39“  794 

39-  9674 

40-  — 
40-  1401 
40-  3121 
40-  4834 
40-  654 
40-  824 

40-  9933 

41-  — 
41-  162 
41-  33 


9343  952  9698  9S75 


246 

247 

248 

249 

250 

251 

251 

252 

253 

254 

255 

256 

257 

257 

258 

259 

260 

261 

262 

263 

263 

264 

265 

266 

267 

268 

269 

269 

270 

271 


1 1 12  1288  1464 
2873  3048  3224 
4627  4802  4977 
6374  6548  6722 
8114  8287  8461 

9S47  — 

— 002  0192 

1573  J745  X9X7 
3292  3464  3635 
5005  5176  5346 
671  6881  7°5r 

841  8579  8749 


41-  4973 
41-  6641 
41-  8301 

41-  9956 

42-  — 
42-  1604 
42-  3246 
42-  4882 
42-  6511 
42-  8135 

42-  9752 

43-  — 
43-  1364 

; 43-  2969 

272  1 43-  45^9 

273  43"  6i63 

274  j 43-  7751 

27  5 43"  9333 

275  ! 44"  — 

276  44-  0909 

277  44-  248 

278  44-  4045 

279  44“  5604 

280  I 44“  7158 

281  44-  8706 

281  45-  — 

282  45-  0249 

283  45“  j786 

284  45“  33i8 
235  45-  4845 

286  45-  6366 

287  45-  7882 

288  45-  9392 

288  46-  — 

289  46-  0898 

No.  I 0 


0102  027 1 044 
1788  1956  2124 

3467  3635  3803 
514  5307  5474 

6807  6973  7139 
8467  8633  8798 


0121  0286  0451 
1768  1933  2097 

34i  3574  3737 

5045  5208  5371 
6674  6836  6999 
8297  8459  8621 

99r4  — “ 

— 0075  0236 

1525  1685  1846 
3T3  329  345 

4729  4888  5048 
6322  6481  664 
7909  8067  8226 
9491  9648  9806 

1066  1224  1381 
2637  2793  295 
4201  4357  45T3 
576  59x5  6071 

73 1 3 7468  7623 
8861  9015  917 

0403  0557  0711 
194  2093  2247 

3471  3624  3777 

4997  5*5  5302 

6518  667  6821 

8033  8184  8336 
9543  9694  9845 


1641 

34 

5152 

6896 

8634 

0365 

2089 

3807 

55x7 

7221 

8918 

0609 

2293 

397 

5641 

7306 

8964 

0616 

2261 

39QI 

5534 

7161 

8783 


0051 

1817 

3575 

5326 

7071 

8808 

0538 

2261 

3978 

5688 

7391 

9087 

0777 

2461 

4*37 

5808 

7472 

9I29 

0781 

2426 

4065 

5697 

7324 

8944 


0398  ; 0559 
2007  I 2167 

361  ! 377 
5207  : 5367 
6799  6957 


8384 

9964 

1538 

3106 

4669 

6226 

7778 


8542 


0228  0405  0582 

1993  2169  2345 

375 1 3926  4101 
5501  5676  585 
7245  7419  7592 
8981  9154  9328 

0711  0883  1056 
2433  2605  2777 
4149  432  4492 

5858  6029  6199 
7561  773x  1901 
9257  9426  9595 

0946  3114  1283 
2629  2796  2964 
4305  4472  4639 
5974  6141  -6308 
7638  7804  797 
9295  946  9625 

0945  in  1275 
259  2754  2918 

4228  4392  45 55 
586  6023  6186 

7486  7648  7811 
9106  9268  9429 

072  0881  1042 

2328  2488  2649 

393  409  4249 

5526  5685  5844 
7116  7275  7433 
8701  8859  9017 


D 

177 
177 
176 
176 
175 
174 

9501  I x73 
— *73 

1228  173 
2949  I 172 


0759 

2521 

4277 

6025 

7766 


4663 

637 

807 

9764 


0122 

1695 

3263 

4825 

6382 

7933 


9324  j 9478 
086s  ; 1018 


24 

393 

5454 

6973 

8487 

9995 


1048  1198  1348  1499 


2553 

4082 

k6o6 

7I25 

8638 

0146 

1649 


0279  0437  0594 
1852  2009  2166 
34*9  3576  3732 
4981  5137  5293 
6537  6692  6848 
8088  8242  8397 
9633  9787  9941 

1172  1326  1479 
2706  2859  3012 
4235  4387  454 
5758  591  6062 

7276  7428  7579 
8789  894  9091 

0296  0447  0597 
1799  1948  2098 


171 

17 1 

170 
169 
169 
169 
168 
167 
167 
166 

165 
165 
165 

164 
164 
163 
162 
162 
162 
161 
161 
160 
159 
159 
158 
158 
158 
157 
157 
156 
155 
i55 
154 
154 
154 
153 
153 
152 

x52 
i5x 

151 

0748  151 

2248  | 150 

0 “nr 


1451 

3I32 

4806 

6474 

8135 

9791 

1439 

3082 

4718 

6349 

7973 

9591 

1203 

2809 

4409 

6004 

7592 

9*75 

0752 

2323 

3889 

5449 

7003 

8552 


0095 

1633 

3l65 

4692 

6214 

7731 

9242 


3 1 6 LOGARITHMS  OF  NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 9 

D 

290 

46-  2398 

2548 

2697 

2847 

2997 

3146  3296  3445 

3594  3744 

150 

291 

46-  3893 

4042 

4191 

434 

449 

4639  4788  4936  5085  5234 

149 

292 

40-  5383  5532 

568 

5829  5977 

6126  6274  6423  6571  6719 

149 

293 

46-  6868 

7016 

7164 

7312 

746 

7608 

7756 

7904 

8052  82 

148 

294 

46-  8347  8495 

8643  879 

8938 

9085 

9233  938 

9527  9675 

148 

295 

46-  9822  9969 

147 

295 

47-  — 

— 

0116 

0263 

041 

0557 

0704 

0851 

0998  1145 

147 

296 

47-  1292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464  261 

146 

297 

47-  2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925  4071 

146 

298 

47-  4216  4362 

4508 

4653 

4799 

4944 

509 

5235 

5381  5526 

146 

299 

47-  5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832  6976 

145 

300 

47-  7121 

7266 

7411 

7555 

77 

7844 

7989 

8i33 

8278  8422 

145 

30 1 

47-  8566  8711 

8855 

8999 

9H3 

9287 

9431 

9575 

9719,9863 

144 

302 

45-  0007 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156  1299 

144 

3°3 

48-  1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588  2731 

143 

304 

48-  2874 

3°l6 

3159 

3302 

3445 

3587 

373 

3872 

4015  4157 

143 

305 

48-  43 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437  5579 

142 

306 

48-  5721 

5863 

6005 

6147 

6289 

643 

6572 

6714 

6855  6997 

143 

307 

48-  7138 

728 

742i 

7563 

7704 

7845 

7986 

8127 

8269  841 

141 

308 

48-  8551 

8692 

8833 

8974 

9IJ4 

9255 

9396 

9537 

9677  9818 

141 

309 

48-  9958 

140 

309 

49-  — 

0099 

0239  038 

052 

0661 

0801 

0941 

1081  1222 

140 

310 

49-  1362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481  2621 

140 

3ii 

49-  276 

29 

304 

3179 

33i9 

3458 

3597 

3737  3876  4015 

139 

312 

49-  4155 

4294 

4433 

4572 

4711 

485 

4989  5128  5267  5406 

139 

3i3 

49-  5544 

5683 

5822  596 

6099 

6238  6376  6515  6653  6791 

139 

3i4 

49-  693 

7068 

7206 

7344 

7483 

7621 

7759 

7897  8035  8173 

138 

315 

49-  8311 

8448 

8586  8724 

8862 

8999 

9r37 

9275 

9412  955 

138 

3l6 

49-  9687 

9824 

9962 

. — 

— 

— 

137 

3l6 

50-  — 

— 

— 

0099 

0236 

0374 

0511 

0648  0785  0922 

137  ' 

3i7 

50-  1059 

1196 

*333 

147 

1607 

1744 

188 

2017 

2154  2291 

137 

318 

50-  2427 

2564 

27 

2837 

2973 

3io9  3246  3382  3518  3655 

136 

3i9 

50-  379i 

3927 

4063 

4199 

4335 

4471 

4607  4743  4878  5014 

136  i 

320 

50-  515 

5286 

542i 

5557 

5693 

5828  5964  6099  6234  637 

136 

321 

50-  6505  664 

6776  6911 

7046 

7181 

7316 

745i 

7586  7721 

135 

322 

50-  7856 

7991 

8126 

826 

8395 

853 

8664  8799  8934  9068 

135 

323 

50-  9203 

9337 

9471 

9606 

974 

9874 

— 

— 

— — 

134 

323 

51-  — 

— 

— 

— 

— 

— 

0009 

0143 

0277  0411 

J34 

324 

51-  0545  0079 

0813 

0947 

1081 

1215 

1349 

1482 

1616  175 

134 

325 

51-  1883 

2017 

2151 

2284 

2418 

2551 

2684  2818 

2951  3084 

133  ,■ 

326 

51-  3218  3351 

3484  3617 

375 

3883 

4016  4149  4282  4415 

133 

327 

51-  4548  4681 

4813  4946 

5079  1 

5211 

5344  5476  5609  5741 

133  \ 

328 

51-  5874  6006 

6139  627j 

6403  : 

6535  6668 

68 

6932  7064 

132  S 

329 

51-  7196  7328 

746 

7592 

7724 

7855 

7987  8119  8251  8382 

132  \ 

330 

51-  8514  8646 

8777  8909 

904 

9171 

9303 

9434  9566  9697 

131 

33i 

51-  9828  9959 

— 

— 

— 

— 

— 

— 

— — 

131 

33i 

52-  — 

— 

009 

0221 

0353 

0484  0615 

0745  0876  1007 

131 

332 

52-  1138 

1269 

14 

153 

1661 

1792 

1922 

2053 

2183  2314 

131 

333 

52-  2444 

2575 

2705 

2835 

2966 

3096  3226  3356  3486  3616 

130 

334 

52-  3740  3870 

4006  4136 

4266 

4396  4526  4656  4785  4915 

130 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 9 

LOGARITHMS  OF  NUMBERS. 


317 


No. 


335 

336 


337 

338 

338 

339 


340 

341 

342 

343 

344 


345 

346 

346 

347 

348 

349 


350 


351 

352 

353 

354 

354 

355 


356 

357 

358 

359 


360 

361 

362 

363 

363 

364 


365 

366 

367 

368 

369 


370 

371 

371 

372 

373 

374 


375 

376 

377 

378 

379 


No. 


0 

1 

2 

3 

4 

52-  5045 

5i74 

5304 

5434 

5563 

52-  6"nq  6469  6598  6727  6856 

52-  763 

7759 

7888  8016  8145 

52-  8917 

9045 

9X74 

9302 

943 

53 

53“  02 

0328  0456  0584  0712 

53-  *479 

1607 

1734 

1862 

199 

53-  2754 

2882 

3°°9 

3136  3264 

53-  4026  4153  428 

4407 

4534 

53-  5294 

542i 

5547 

5074  58 

53"  655§  6685 

6811 

6937 

7063 

53"  7819 

7945 

8071 

8197 

8322 

53“  9076  9202 

9327 

9452 

9578 

54- 

54-  0329 

0455 

058 

0705 

083 

54-  1579 

1704 

1829 

1953 

2078 

54-  2825 

295 

3074 

3199 

3323 

54-  4068  4192  4316 

444 

4564 

54-  5307 

543i 

5555 

5678 

5802 

54-  6543  6666  6789  6913 

7036 

54-  7775 

7898 

8021 

8144  8267 

54-  9003 

9126  9249 

9371 

9494 

55" 

55-  0228 

0351 

0473 

0595 

0717 

55-  145 

1572 

1694 

1816 

1938 

55-  2668 

279 

2911 

30 33 

3X55 

55-  3883 

4004 

4126 

4247 

4368 

55-  5094 

5215 

5336  5457 

5578 

55-  6303 

6423 

6544  6664 

6785 

55-  7507 

7627 

7748 

7868 

7988 

55-  8709 

8829 

8948 

9068  9188 

55-  9907 

56-  — 

0026 

0146 

0265 

0385 

56-  IIOI 

1221 

i34 

1459 

1578 

56-  2293 

2412 

2531 

265 

2769 

56-  3481 

36 

37i8 

3837 

3955 

56-  4666 

4784 

4903 

5021 

5i39 

56-  5848 

5966 

6084 

6202 

632 

56-  7026 

7*44 

7262 

7379 

7497 

56-  8202 

8319 

8436 

8554 

8671 

56-  9374 

9491 

9608 

9725 

9842 

57“  — 
57-  0543 

066 

0776 

0893 

IOI 

57-  1709 

1825 

1942 

2058 

2174 

57-  2872 

2988 

3104 

322 

3336 

57-  4031 

4147 

4263 

4379 

4494 

57-  5i88 

5303 

5419 

5534 

565 

57"  6341 

6457 

6572 

6687 

6802 

57-  7492 

7607 

7722 

7836 

795i 

57-  8639 

8754 

8868 

8983 

9097 

0123 


56789 

5693  5822  5951  6081  621 

6985  71 14  7243  7372  7501 

8274  8402  8531  866  8788 

9559  9687  9815  9943  — 

— — — — 0072 

084  0968  1096  1223  1351 

2117  2245  2372  25  2627 

3391  3518  3645  3772  3899 

4661  4787  4914  5041  5167 

5927  6053  618  6306  6432 

7189  7315  7441  7567  7693 

8448  8574  8699  8825  8951 

9703  9829  9954  — — 

— — — 0079  0204 

0955  108  1205  133  1454 

2203  2327  2452  2576  2701 

3447  3571  3696  382  3944 

4688  4812  4936  506  5183 

5925  6049  6172  6296  6419 

7159  7282  7405  7529  7652 

8389  8512  8635  8738  8881 

9616  9739  9861  9984  — 

— — — — 0106 

084  0962  1084  1206  1328 

206  2181  2303  2425  2547 

3276  3398  3519  364  3762 

4489  461  4731  4852  4973 

5699  582  594  6061  6182 

6905  7026  7146  7267  7387 

8108  8228  8349  8469  8589 

9308  9428  9548  9667  9787 


0504  0624  0743  0863  0982 

1698  1817  1936  2035  21 74 

2887  3006  3125  3244  3362 

4074  4192  4311  4429  4548 

5257  5376  5494  5612  573 

6437  6555  667 3 6791  6909 

7614  7732  7849  7967  8084 

8788  8905  9023  914  9257 

9959  — — — — 

— 0076  0193  0309  0426 

H26  1243  1359  1476  1592 

2291  2407  2523  2639  2755 

3452  3568  3684  38  39*5 

461  4726  4841  4957  5072 

5765  588  5996  61 1 1 6226 

6917  7032  7147  7262  7377 

8066  8181  8295  841  8525 

9212  9326  9441  9555  9669 

6 7 8 9 


D 


129 

129 

129 

128 

128 

128 

128 

127 

127 

126 

126 

126 

126 

125 

125 

125 

124 

124 

124 

123 

123 

123 

123 

122 

122 

121 

121 

121 

120 

120 

120 

120 

119 

119 

ng 
119 
118 
1 18 

115 

117 

117 

117 

117 

11 6 
116 

1 id 

1 IS 
*15 
1 14 

D 


4 I 5 

D D* 


Nol 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

380 

080 

57- 

58- 

9784 

9898 

0012 

0126 

0241 

0355 

0469 

0583 

0697 

0811 

114 

114 

381 

58- 

0925 

1039 

ii53 

1267 

1381 

1495 

1608 

1722 

1836 

195 

114 

382 

58- 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

383 

58- 

3X99 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

ZI3 

384 

58- 

433 1 

4444 

4557 

467 

4783 

4896 

5009 

5122 

5235 

5348 

**3 

385 

58- 

5461 

5574 

5686 

5799 

59X2 

6024 

6137 

625 

6362 

6475 

i*3 

386 

58- 

6587 

67 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

387 

58- 

77H 

7823 

7935 

8047 

816 

8272 

8384 

8496 

8608 

872 

1 12 

388 

58- 

8832 

8944 

9056 

9167 

9279 

939 1 

9503 

96i5 

9726 

9838 

1 12 

389 

389 

58- 

59- 

995 

0061 

0173 

0284 

0396 

0507 

0619 

073 

0842 

0953 

1 12 
1 12 

390 

59“ 

1065 

1176 

1287 

1399 

151 

1621 

x732 

1843 

1955 

2066 

III 

39 1 

59- 

2177 

2288 

2399 

251 

2621 

2732 

2843 

2954 

3064 

3*75 

III 

392 

59- 

3286 

3397 

3508 

36*8 

3729 

384 

395 

4061 

4I7I 

4282 

III 

393 

59“ 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

no 

394 

59- 

5496 

5606 

57x7 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

no 

395 

59“ 

6597 

6707 

6817 

6027 

7037 

7146 

7256 

7366 

7476 

7586 

no 

396 

59- 

7695 

7805 

79*4 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

no 

397 

59- 

8791 

89 

9009 

9**9 

9228 

9337 

9446 

9556 

9665 

9774 

109 

398 

-208 

59- 

60- 

9883 

9992 

0101 

021 

0319 

0428 

0537 

0646 

0755 

0864 

109 

109 

399 

60- 

0973 

1082 

1191 

1299 

1408 

i5x7 

1625 

1734 

1843 

i95i 

109 

400 

60- 

206 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3°36 

108 

401 

60- 

3144 

3253 

336* 

3469 

3577 

3686 

3794 

39°2 

401 

4118 

108 

402 

60- 

4226 

4334 

4442 

455 

4658 

4766 

4874 

4982 

5089 

5X97 

108 

4°3 

60- 

5305 

54*3 

5521 

5628 

5736 

5844 

595x 

6059 

6166 

6274 

108 

404 

60- 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7 *33 

7241 

7348 

107 

405 

60- 

7455 

7562 

7669 

7777 

7884 

799i 

8098 

8205 

8312 

8419 

107 

406 

60- 

8526 

8633 

874 

8847 

8954 

9061 

9i67 

9274 

9381 

9488 

107 

40  7 

60- 

9594 

9701 

9808 

9914 

— 

— 

— 

— 

107 

AO  7 

61- 



— 

0021 

0128 

0234 

034x 

0447 

0554 

107 

408 

61- 

066 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

106 

409 

61- 

1723 

1829 

1936 

2042 

2148 

2254 

236 

2466 

2572 

2678 

106 

410 

61- 

2784 

289 

2996 

3 102 

3207 

33x3 

34x9 

3525 

363 

3736 

106 

411 

61- 

3842 

3947 

4053 

4159 

4264 

437 

4475 

458i 

4686 

4792 

106 

4.12 

61- 

4897 

5003 

5X08 

5213 

5319 

5424 

5529 

5634 

574 

5845 

105 

T" 

413 

61- 

595 

6055 

6l6 

6265 

637 

6476 

6581 

6686 

679 

6895 

105 

4i4 

61- 

7 

7105 

721 

73*5 

742 

7525 

7629 

7734 

7839 

7943 

105 

415 

61- 

8048 

8i53 

8257 

8362 

8466 

857x 

8676 

878 

8884 

S989 

105 

416 

416 

417 

61- 

62- 
62- 

9°93 

0136 

9*98 

024 

9302 

0344 

9406 

0448 

95ii 

0552 

96i5 

0656 

9719 

076 

9824 

0864 

9928 

0968 

0032 

1072 

*05 

104 

104 

418 

62- 

1176 

128 

X384 

1488 

1592 

1695 

I799 

I9°3 

2007 

211 

104 

419 

62- 

• 2214 

2318 

2421 

2525 

2628 

2732 

283s 

2939 

3042 

3x46 

104 

4*20 

62- 

• 3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4X79 

103 

421 

62- 

- 4282 

4385 

4488 

459 1 

4695 

4798 

49°! 

5004 

5107 

521 

103 

422 

62- 

■ 5312 

54*5 

55*8 

5621 

5724 

5827 

5929 

6032 

6i35 

6238 

103 

423 

62- 

634 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

103 

424 

62- 

■ 7366 

7468 

757i 

7673 

7775 

7878 

798 

8082 

8185 

8287 

102 

No. 

0 

1 . 

2 

3 

4 

5 

6 

7 

8 

9 

D 

LOGARITHMS  OF  NUMBERS.  3X9 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

425 

62- 

8389 

8491 

8593 

8695 

8797 

89 

9002 

9io4 

9206 

9308 

102 

426 

62- 

94i 

9512 

9613 

9715 

9817 

9919 

— 

— 

— 

102 

426 

63- 

— 

— 

— 

— 

— 

0021 

0123 

0224 

0326 

102 

427 

63‘ 

0428 

053 

0631 

0733 

0835 

0936 

1038 

ii39 

1241 

1342 

102 

428 

63- 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

IOI 

429 

63- 

2457 

2559 

266 

2761 

2862 

2963 

3064 

3i65 

3266 

3367 

IOI 

430 

63- 

3468 

3569 

367 

3771 

3872 

3973 

4074 

4i75 

4276 

4376 

IOI 

43 1 

63- 

4477 

4578 

4679 

4779 

488 

4981 

5081 

5182 

5283 

5383 

IOI 

432 

63- 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

IOO 

433 

63- 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

729 

739 

IOO 

434 

63- 

749 

759 

769 

779 

789 

799 

809 

819 

829 

8389 

IOO 

435 

63- 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

IOO 

436 

63- 

9486 

9586 

9686 

9785 

9885 

9984 

— 

0183 

— 

— 

IOO 

436 

64- 

— 

— 

— 

— 

— 

— 

0084 

0283 

0382 

99 

437 

64- 

0481 

0581 

068 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

99 

438 

64- 

1474 

1573 

1672 

1771 

1871 

197 

2069 

2168 

2267 

2366 

99 

439 

64- 

2465 

2563 

2662 

2761 

286 

2959 

3058 

3156 

3255 

3354 

99 

440 

64- 

3453 

355i 

365 

3749 

3847 

3946 

4044 

4i43 

4242 

434 

99 

44 1 

64- 

4439 

4537 

4636 

4734 

4832 

493i 

5029 

5127 

5226 

5324 

98 

44  2 

64- 

5422 

552i 

5619 

5717 

5815 

5913 

6011 

611 

6208 

6306 

98 

443 

64- 

6404 

6502 

66 

6698 

6796 

6894 

6992 

7089 

7i87 

7285 

98 

444 

64- 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

98 

445 

64- 

836 

8458 

8555 

8653 

875 

8848 

8945 

9°43 

9*4 

9237 

97 

446 

64- 

9335 

9432 

953 

9627 

9724 

9821 

9919 

— 

— 

— 

97 

446 

65- 

— 

— 

— 

— 

— 

— 

— 

0016 

0113 

021 

97 

447 

65- 

0308 

0405 

0502 

0599 

0696 

0793 

089 

0987 

1084 

1181 

97 

448 

65- 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

215 

97 

449 

65- 

2246 

2343 

244 

2536 

2633 

273 

2826 

2923 

3OI9 

3116 

97 

450 

65- 

32i3 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

408 

96 

45i 

65- 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

485 

4946 

5042 

96 

452 

65- 

5138 

5235 

533i 

5427 

5523 

5619 

5715 

581 

5906 

6002 

96 

453 

65- 

6098 

6194 

629 

6386 

6482 

6577 

6673 

6769 

6864 

696 

96 

454 

65- 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

782 

7916 

96 

455 

65- 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

887 

95 

456 

65- 

8965 

906 

9r55 

925 

9346 

9441 

9536 

9631 

9726 

9821 

95 

457 

65- 

99l6 

95 

457 

66- 

0011 

0106 

0201 

0296 

°39I 

0486 

0581 

0676 

0771 

95 

458 

66- 

0865 

096 

1055 

ii5 

1245 

1339 

1434 

1529 

1623 

1718 

95 

459 

66- 

1813 

1907 

2002 

2096 

2191 

2286 

238 

2475 

2569 

2663 

95 

460 

66- 

2758 

2852 

2947 

3041 

3135 

323 

3324 

34i8 

3512 

3607 

94 

461 

66- 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

436 

4454 

4548 

94 

462 

66- 

4642 

4736 

483 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

463 

66- 

558i 

5675 

5769 

5862 

5956 

605 

6143 

6237 

6331 

6424 

94 

464 

66- 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7i73 

7266 

736 

94 

465 

66- 

7453 

7546 

764 

7733 

7826 

792 

8013 

8106 

8199 

8293 

93 

466 

66- 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9L3i 

9224 

93 

467 

66- 

9317 

941 

9503 

9596 

9689 

9782 

9875 

9967 

— 

— 

93 

467 

67- 

006 

0153 

93 

468 

67- 

0246 

0339 

0431 

0524 

0617 

071 

0802 

0895 

0988 

108 

93 

469 

67- 

ii73 

1265 

1358 

i45i 

1543 

1636 

1728 

1821 

1913 

2005 

93 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

320 


LOGARITHMS  OF  NUMBERS. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 j 

D 

470 

67-  2098 

219 

2283 

2375 

2467 

256 

2652 

2744 

2836 

2929 

92 

471 

67-  3021 

3ii3 

3205 

3297 

339 

3482 

3574 

3666 

3758 

385 

92 

472 

67-  3942 

4034 

4126  4218 

43i 

4402 

4494 

4586 

4677 

4769 

92 

473 

67-  4861 

4953 

5045 

5137 

5228 

532 

5412 

5503 

5595 

5687 

92 

474 

67-  5778 

5«7 

5962  6053  61 45 

6236  6328 

6419 

6511 

6602  i 

92 

475 

67-  6694  6785  6876  6968 

7059 

7i5i 

7242 

7333 

7424 

75i6 

91 

476 

67-  7607 

7698 

7789 

7881 

7972 

8063 

8i54 

8245 

8336 

8427 

9T 

477 

67-  8518  8609  87 

8791 

8882 

8973  9064 

9I55 

9246 

9337  | 

9i 

478 

67-  9428  9519  961 

97 

9791 

9882  9973 

— 

— 

— I 

9i 

478 

68-  — 

0063 

oi54 

0245 

91 

479 

68-  0336 

0426 

0517 

0607 

0698 

0789 

On 

CO 

O 

097 

106 

1151  | 

9i 

480 

68-  1241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

90 

481 

68-  2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

482 

68-  3047 

3*37 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

68-  3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

484 

68-  4845 

4935 

5025 

5ii4 

5264 

5294 

5383 

5473 

5563 

5652 

90 

485 

68-  5742 

5831 

592i 

601 

61 

6189 

6279 

6368 

6458 

6547 

89 

486 

68-  6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

735i 

744 

89 

487 

68-  7529 

7618 

7707 

7796 

7886 

7975 

8064 

8i53 

8242 

8331 

89 

488 

68-  842 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9J3i 

922 

89 

489 

68-  9309 

9398 

9486 

9575 

9664 

9753 

9841 

993 

— 

— 

89 

489 

69- 

0019 

0107 

89 

490 

69-  0196 

0285 

0373 

0462 

055 

0639 

0728 

0816 

0905 

0993 

89 

49 1 

69-  1081 

1 17 

1258 

1347 

1435 

1524 

1612 

17 

1789 

1877 

88 

492 

69-  1965 

2053 

2142 

223 

2318 

2406 

2494 

2583 

2671 

2759 

88 

493 

69-  2847 

2935 

3023 

3111 

3i99 

3287 

3375 

3463 

355i 

3639 

88 

494 

69-  3727 

3815 

39°3 

399i 

4078 

4166 

4254 

4342 

443 

4517 

88 

495 

69-  4605 

4693 

4781 

CO 

0 

00 

4956 

5044 

5131 

5219 

5307 

5394 

88 

496 

69-  5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

497 

69-  6356 

6444 

6531 

6618 

6706 

6793 

688 

6968 

7055 

7142 

87 

498 

69-  7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

499 

69-  8101 

8188 

8275 

8362 

8449 

853s 

8622 

8709 

8796 

8883 

87 

600 

69-  897 

9057 

9T44 

923x 

9377 

9404 

9491 

9578 

9664 

975i 

87 

501 

69-  9838 

9924 

87 

501 

70-  — 

— 

0011 

0098 

0184 

0271 

0358 

0444 

0531 

0617 

87 

502 

70-  0704 

079 

0877 

0963 

105 

1136 

1222 

1309 

1395 

1482 

86 

503 

70-  1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

504 

70-  2431 

2517 

2603 

2689 

2775 

2861 

2947 

30  33 

3ii9 

3205 

S6 

505 

70-  3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

506 

70-  4151 

4236 

4322 

4408 

4494 

4579 

4665 

475i 

4837 

4922 

86 

507 

70-  5008 

5094 

5i79 

5265 

535 

5436 

5522 

5607 

5693 

5778 

86 

508 

70-  5864 

5949 

6035 

612 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 

70-  6718 

6803 

6888 

6974 

7059 

7144 

7229 

73i5 

74 

7485 

85 

510 

70-  757 

7655 

774 

7S26 

7911 

7996 

8081 

8166 

8251 

8336 

85 

5ii 

70-  8421 

8506 

8591 

8676 

8761 

S846 

8931 

9OI5 

9i 

9185 

85 

512 

70-  927 

9355 

944 

9524 

9609 

9694 

9779 

9863 

9948 

— 

85 

512 

71- 

0033 

85 

5i3 

71-  0117 

0202 

M 

0 

00 

0 

0456 

054 

0625  071 

0794 

0879 

85 

514 

71-  0963 

1048 

1132 

1217 

1301 

1385 

147 

1554 

1639 

1723 

84 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

eT 

LOGARITHMS  OF  [NUMBERS.  32 1 


No. 

0 

2 

3 

4 

5 

6 

7 

8 

9 

D 

515 

71-  1807  1892 

1976  206 

2144 

2229 

2313 

2397 

2481 

2566 

84 

5i6 

71-  265  2734 

2818 

2902 

2986 

307 

3154  3238 

3323 

3407 

84 

5i7 

71-  3491  3575  3659 

3742 

3826 

391 

3994  4078  4162 

4246 

84 

5i8 

71-  433  4414 

4497 

458i 

4665 

4749  4833 

49l6 

5 

5084 

84 

519 

71-  5167  5251 

5335 

5418 

5502 

5586  5669  5753  5836 

592 

84 

520 

71-  6003  6087  617 

6254  6337 

6421 

6504  6588  6671 

6754 

83 

521 

71-  6838  6921 

7004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

83 

522 

71-  7671  7754 

7837 

792 

8003 

8086 

8169  8253  8336 

8419 

83 

523 

71-  8502  8585 

8668  8751 

8834 

8917 

9 

9083  9165 

9248 

83 

524 

71-  9331  9414 

9497  958 

9663 

9745 

9828  9911 

9994 

— 

83 

524 

72-  — — 

0077 

83 

525 

72-  0159  0242 

0325 

0407 

049 

0573 

0655 

0738 

0821 

0903 

83 

526 

72-  0986  1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

82 

527 

72-  1811  1893 

1975 

2058 

214 

2222 

2305 

2387 

2469 

2552 

82 

528 

72-  2634  2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

529 

72-  3456  3538 

362 

3702 

3784 

3866 

3948 

403 

4112 

4194 

82 

530 

72-  4276  4358 

444 

4522 

4604 

4685 

4767 

4849 

493i 

5013 

82 

53i 

72-  5095  5176 

5258 

534 

5422 

5503 

5585 

5667 

5748 

583 

82 

532 

72-  5912  5993 

6075 

6156 

6238 

632 

6401 

6483 

6564 

6646 

82 

533 

72-  6727  6809 

689 

6972 

7053 

7134 

7216 

7297 

7379 

746 

81 

534 

72-  754i  7623 

7704 

7785 

7866 

7948 

8029 

811 

8191 

8273 

81 

535 

72-  8354  8435 

8516 

8597 

8678 

8759 

8841 

8922 

9°°3 

9084 

81 

536 

72-  9165  9246 

9327 

9408 

9489 

957 

9651 

9732 

9813 

9893 

81 

537 

72-  9974  — 

81 

537 

73-  — 0055 

0136 

0217 

0298 

0378 

0459 

054 

0621 

0702 

81 

538 

73-  0782  0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

539 

73-  1589  1669 

175 

183 

1911 

1991 

2072 

2152 

2233 

2313 

81 

540 

73-  2394  2474 

2555 

2635 

2715 

2796 

2876 

2956 

3037 

3ii7 

80 

54i 

73-  3J97  3278 

3358 

3438 

35i8 

3598 

3679 

3759 

3839 

3919 

80 

542 

73-  3999  4079 

416 

424 

432 

44 

448 

456 

464 

472 

80 

543 

73-  48  488 

496 

504 

512 

52 

5279 

5359 

5439 

5519 

80 

544 

73-  5599  5679 

5759 

5838 

59j8 

5998 

6078 

6157 

6237 

6317 

80 

545 

73-  6397  6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7ii3 

80 

546 

73-  7i93  7272 

7352 

743i 

75ii 

759 

767 

7749 

7829 

79 °8 

79 

547 

73-  7987  8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

548 

73-  8781  886 

8939 

9018 

9097 

9I77 

9256 

9335 

9414 

9493 

79 

549 

73-  9572  965* 

973i 

981 

9889 

9968 

— 

79 

549 

74-  — — 

— 

— 

— 

— 

0047 

0126 

0205 

0284 

79 

550 

74-  0363  0442 

0521 

06 

0678 

0757 

0836 

0915 

0994 

1073 

79 

55i 

74-  1152  123 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

186 

79 

552 

74-  1939  2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

79 

553 

74-  2725  2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

343 1 

78 

554 

74"  35i  3588 

3667 

3745 

3823 

3902 

398 

4058 

4136 

4215 

78 

555 

74-  4293  437i 

4449 

4528 

4606 

4684 

4762 

484 

4919 

4997 

78 

556 

74-  5075  5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

557 

74-  5855  5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

! 558  ! 

74-  6634  6712 

679 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

78 

559  ; 

74-  7412  7489 

7567 

7645 

7722 

78 

7878 

7955 

8033 

811 

78 

No.  j 

0 1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

322 


LOGARITHMS  OF  NUMBERS. 


No.  | 

0 

1 

2 

3 

4 

5 6 

7 

8 

9 j 

D 

5C0 

74-  8188 

8266 

8343 

8421 

8498 

8576  8653 

8731 

8808 

8885 

77 

561 

74-  8963 

904 

9118 

9X95 

9272 

935  9427 

9504 

9582  9659 

77 

562 

74-  97 36 

9814 

9891 

9968 

— 

— — 

— 

— 

— 

77 

562 

75-  — 

— 

— 

— 

0045 

0123  02 

0277 

0354 

0431 

77 

563 

75*  0508 

0586 

0663 

074 

0817 

0894  0971 

1048 

1125 

1202 

77 

564 

75-  1279 

1356 

1433 

x5i 

1587 

1664  1741 

1818 

1895 

1972 

77 

505 

75-  2048 

2125 

2202 

2279 

2356 

2433  2509 

2586  2663 

274 

77 

566 

75-  2816 

2893 

297 

3047 

3X23 

32  3277 

3353 

343 

3506 

77 

567 

75-  3583 

366 

3736 

3813 

3889 

3966  4042 

4119 

4195 

4272 

77 

568 

75-  4348 

4425 

4501 

4578 

4654 

473  4807 

4883  496 

5036 

76 

569 

75-  5112 

5189 

5265 

534i 

54x7 

5494  557 

5646  5722 

5799 

76 

570 

75-  5875 

595i 

6027 

6103 

618 

6256  6332 

6408  6484  656 

76 

57i 

75-  6636 

6712 

6788 

6864 

694 

7016  7092 

7168 

7244 

732 

76 

572 

75"  7396 

7472 

7548 

7624 

77 

7775  7851 

7927 

8003 

8079 

76 

573 

75-  8155 

823 

8306 

8382 

8458 

8533  8609 

8685 

8761 

8836 

76 

574 

75-  8912 

8988 

9063 

9*39 

9214 

929  9366 

9441 

9517 

9592 

76 

575 

75-  9668 

9743 

9819 

9894 

997 

_ __ 

— 

— 

— 

76 

575 

76-  — 

— 

— 

— 

— 

0045  0121 

0196 

0272 

0347 

75 

576 

76-  0422 

0498 

0573 

0649 

0724 

0799  0875 

095 

1025 

IIOI 

75 

577 

76-  1176 

1251 

1326 

1402 

1477 

1552  1627 

1702 

1778 

1853 

75 

578 

76-  1928 

2003 

2078 

2153 

2228 

2303  2378 

2453 

2529 

2604 

75 

579 

76-  2679 

2754 

2829 

2904 

2978 

3053  3128 

3203 

3278  3353 

75 

580 

76-  3428 

3503 

3578 

3653 

3727 

3802  3877 

3952 

4027 

4101 

75 

581 

76-  4176 

4251 

4326 

44 

4475 

455  4624 

4699  4774  4848 

75 

582 

76-  4923 

4998 

5072 

5X47 

5221 

5296  537 

5445 

552 

5594 

75 

583 

76-  5669 

5743 

5818 

5892 

5966 

6041  6115 

619 

6264  6338 

74 

584 

76-  6413 

6487 

6562 

6636 

671 

6785  6859 

6933  7007 

7O0  2 

74 

585 

76-  7156 

723 

7304 

7379 

7453 

7527  7601 

7675 

7749 

7823 

74 

586 

76-  7898 

7972 

8046 

812 

8194 

8268  8342 

8416  849 

8564 

74 

587 

76-  8638 

8712 

8786 

886 

8934 

9008  9082 

9156  923 

9303 

74 

588 

76-  9377 

945i 

9525 

9599 

9673 

9746  982 

9894  9968 

— 

74 

588 

77 

0042 

74 

589 

77"  01I5 

0189  0263 

0336 

041 

0484  0557 

0631 

0705 

0778 

74 

590 

77-  0852 

0926  0999 

1073 

1146 

122  1293 

1367 

144 

1514 

74 

591 

77-  x587 

1661 

1734 

1808 

1881 

1955  2028 

2102 

2175 

2248 

, 73 

592 

77"  2322 

2395 

2468 

2542 

2615 

2688  2762 

2835 

2908 

2981 

; 73 

593 

77-  3055 

3128  3201 

3274 

3348 

342i  3494 

3567 

364 

3713 

: 73 

594 

77"  3786 

386 

3933 

4006 

4079 

4152  4225 

4298 

4371 

4444 

; 73 

595 

77-  45T7 

459 

4663 

4736 

4809 

4882  4955 

5028 

51 

5173 

73 

596 

77-  5246 

53x9 

5392 

5465 

5538 

561  5683 

5756 

5829  5902 

73 

597 

77-  5974 

6047 

612 

6193 

6265 

6338  6411 

6483 

6556  6629 

73 

598 

77-  6701 

6774  6846 

69t9 

6992 

7064  7137 

7209 

7282 

7354 

73 

599 

77-  7427 

7499 

7572 

7644 

7717 

7789  7862 

7934 

8000  8079 

72 

000 

77-  8151 

8224  8296 

8368 

8441 

8513  8585 

8658 

873 

8802 

72 

601 

77-  8874 

8947 

9019 

9091 

9i63 

9236  9308 

938 

9452 

9524 

72 

602 

77-  9596 

9669  9741 

9813 

9885 

9957  — 

— 

— 

— • 

! 72 

602 

78-  — 

— 

— 

— 

— 

— 0029 

0101 

oi73 

0245 

| 72 

603 

78-  0317 

0389  0461 

0533 

0605 

0677  0749 

1 0821 

0893  0965 

72 

604 

78-  1037 

1109 

1 1181 

1253 

1324 

1396  1468 

154 

1012 

1084 

72 

No. 

0 

1 

2 

3 

4 

5 6 

7 

8 

9 

D 

LOGARITHMS  OP  NUMBERS. 


323 


No. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

605 

78-  1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

606 

78-  2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

607 

78-  3189 

326 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

7i 

608 

78-  3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

7i 

609 

78-  4617 

4689 

476 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

7i 

610 

78-  S33 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

597 

7i 

611 

78-  6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

668 

7i 

612 

78-  6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

739 

7i 

613 

78-  746 

753i 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

7i 

614 

78-  8l68 

8239 

831 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

7i 

615 

78-  8875 

8946 

9016 

9087 

9T57 

9228 

9299 

9369 

944 

95i 

7i 

616 

78-  9581 

965i 

9722 

9792 

9863 

9933 

— 

— 

— 

— 

70 

616 

79" 

0004 

0074 

0144 

0215 

70 

617 

79-  0285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

70 

618 

79-  0988 

1059 

1129 

1199 

1269 

134 

141 

148 

155 

162 

70 

619 

79-  1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

79-  2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

79-  3092 

3162 

3231 

3301 

337i 

344i 

35ii 

358i 

3651 

3721 

70 

622 

79~  379 

386 

393 

4 

407 

4139 

4209 

4279 

4349 

4418 

70 

623 

79"  4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5ii5 

70 

624 

79-  5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

574i 

5811 

70 

625 

79-  588 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

69 

626 

79-  6574 

6644 

6713 

6782 

6852 

6921 

699 

706 

7129 

jigS 

69 

627 

79-  7268 

7337 

7406. 

7475 

7545 

7614 

7683 

7752 

7821 

789 

69 

628 

79-  70 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69 

629 

79"  8651 

872 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

79-  934i 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

69 

631 

80-  0029 

0098 

0167 

0236 

0305 

0373 

0442 

0511 

058 

0648 

69 

632 

80-  0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1 335 

69 

633 

80-  1404 

1472 

i54i 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

69 

634 

80-  2089 

2158 

2226 

2295 

2363 

2432 

25 

2568 

2637 

2705 

69 

635 

80-  2774 

2842 

291 

2979 

3047 

3116 

3184 

3252 

33  21 

3389 

68 

636 

80-  3457 

3525 

3594 

3662 

373 

3798 

3867 

3935 

4003 

4071 

68 

637 

80-  4139 

4208 

4276 

4344 

4412 

448 

4548 

4616 

4685 

4753 

68 

638 

80-  4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

639 

80-  5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

68 

640 

80-  618 

6248 

6316 

6384 

6451 

6519 

6587 

6655 

6723 

679 

68 

641 

80-  6858 

6926 

6994 

7061 

7129 

7197 

7264 

7332 

74 

7467 

68 

642 

80-  7535 

7603 

767 

7738 

7806 

7873 

7941 

8008 

8076 

8i43 

68 

643 

80-  8211 

8279 

8346 

8414 

8481 

8549 

8616 

8684 

8751 

8818 

67 

644 

80-  8886 

8953 

9021 

9088 

9156 

9223 

929 

9358 

9425 

9492 

67 

645 

80-  956 

9627 

9694 

9762 

9829 

9896 

9964 

— 

— 

— 

67 

645 

81-  — 

0031 

0098 

0165 

67 

646 

81-  0233 

g>3 

0367 

0434 

0501 

0569  0636 

0703 

077 

0837 

67 

647 

81-  0904 

0971 

1039 

1106 

11 73' 

124 

1307 

1374 

1441 

1508 

67 

648 

81-  1575 

1642 

1709 

1776 

1843 

191 

1977 

2044 

2111 

2178 

67 

649 

81-  2245 

2312 

2379 

2445 

2512 

2579  2646 

2713 

278 

2847 

67 

650 

81-  2913 

298 

3047 

3114  3181 

3247 

3314  338i 

3448 

3514 

67 

651 

81-  3581 

3648  3714  3781 

3848 

39i4 

398i 

4048 

4114 

4181 

67 

652 

81-  4248  4314  4381 

4447 

4514 

4581 

4647 

4714 

478 

4847 

67 

653 

81-  4913  498 

5046 

5ii3 

5i79 

5246 

5312 

5378 

5445 

55ii 

66 

654 

81-  5578 

5644 

57ii 

5777 

5843 

59i 

5976  6042 

6109 

6i75 

66 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

"IT 

324 


LOGARITHMS  OF  NUMBERS. 


No.  | 0 12.  3 4 

655  ! 8i-  6241  6308  6374  644  6506 

656  ! 81-  6904  697  7036  7102  7169 

657  I 81-  7565  7631  7698  7764  783 

658  81-  8226  8292  8358  8424  849 

659  81-  8885  8951  9017  9083  9149 

660  81-  9544  961  9676  9741  9807 

660  82-  — — — — — 

661  82-  0201  0267  0333  0399  0464 

662  82-  0858  0924  0989  1055  1 12 

663  82-  1514  1579  1645  171  1775 

664  82-  2168  2233  2299  2364  243 

665  82-  2822  2887  2952  3018  3083 

666  82-  3474  3539  3605  367  3735 

667  82-  4126  4191  4256  4321  4386 

668  82-  4776  4841  4906  4971  5036 

669  82-  5426  5491  5556  5621  5686 

670  82-  6075  614  6204  6269  6334 

671  82-  6723  6787  6852  6917  6981 

672  82-  7369  7434  7499  7563  7628 

673  82-  8015  808  8144  8209  8273 

674  82-  866  8724  8789  8853  8918 

67  5 82-  9304  9368  9432  9497  9561 

676  82-  9947  — — — — 

676  83-  — oori.  0075  0139  0204 

677  83-  0589  0653  0717  0781  0845 

678  83-  123  1294  1358  1422  i486 

679  83-  187  1934  1998  2062  2126 

680  83-  2509  2573  2637  27  2764 

681  83-  3147  3211  3275  3338  3402 

682  83-  3784  3848  3912  3975  4039 

683  83-  4421  4484  4548  4611  4675 

684  83-  5056  512  5183  5247  531 

685  83-  5691  5754  5817  5881  5944 

686  83-  6324  6387  6451  6514  6577 

687  83-  6957  702  7083  7146  721 

688  83-  7588  7652  7715  7778  7841 

689  83-  8219  8282  8345  8408  8471 

690  83-  8849  8912  8975  9038  9101 

691  83-  9478  9541  9604  9667  9729 

691  84-  — — — — — 

692  84  0106  0169  0232  0294  0357 

693  84-  0733  0796  0859  0921  0984 

694  84-  1359  1422  1485  1547  161 

695  84-  1985  2047  21 1 2172  2235 

696  84-  2609  2672  2734  2796  2859 

697  84-  3233  3295  3357  342  3482 

69S  84-  3855  3918  398  4042  4104 

699  84-  4477  4539  4601  4664  4726 

700  84-  5098  516  5222  5284  5346 

701  84-  5718  578  5842  5904  5966 

702  84-  6337  6399  6461  6523  6585 

703  84-  6955  7017  7079  7141  7202 

704  84-  7573  7634  7696  7758  7819 

No. 


5 6 7 8 9 

6573  6639  6705  6771  6838 

7235  7301  7367  7433  7499 

7896  7962  8028  8094  816 

8556  8622  8688  8754  882 

9215  9281 9346  9412  9478 

9873  9939  — — — 

0004  007  0136 

053  °595  0661  0727  0792 

1186  1251  1317  1382  1448 

1841  1906  1972  2037  2103 

2495  256  2626  2691  2756 

3148  3213  3279  3344  3409 

38  3865  393  3996  4061 

4451  4516  4581  4646  4711 

5101  5166  5231  5296  5361 

575i  5815  588  5945  601 

6399  6464  6528  6593  6658 

7046  71 1 1 7175  724  7305 

7692  7757  7821  7886  7951 

8338  8402  8467  8531  8595 

8982  9046  91 1 1 9175  9239 

9625  969  9754  9818  9882 


0268  0332  0396  046  0525 

0909  0973  1037  1102  1166 
155  1614  1678  1742  1806 

2189  2253  2317  2381  2445 
2828  2892  2956  302  3083 

3466  353  3593  3^57  372i 

4103  4166  423  4294  4357 

4739  4802  4866  4929  4993 
5373  5437  55  55^4  5627 

6007  6071  6134  6197  6261 
6641  6704  6767  683  6894 

7273  7336  7399  7462  7525 
7904  7967  803  8093  8156 

8534  8597  866  8723  8786 


66 

66 

66 

66 

66 

66 

66 

66 

66 

65 

65 

65 

65 

65 

65 

65 

65 

65 

65 

64 
64 
64 
64 
64 
64 
64 
64 
64 
64 
64 
64 
I 63 
63 
: 63 

63 
; 63 


9164  9227  9289  9352  9415  ; 63 
9792  9855  9918  9981  — 63 

— — — — 0043  63 

042  0482  0545  0608  0671  ; 63 
1046  II09  1172  1234  1297  63 

1672  1735  1797  186  1922  | 63 

2297  236  2422  2484  2547  62 

2921  2983  *3046  3 108  31 7 62 

3544  3606  3669  373i  3793  62 

4166  4229  4291  4353  44*5  62 

4788  485  4912  4974  5036  62 

5408  547  5532  5594  5656  62 

6028  609  6151  6213  6275  62 

6646  6708  677  6832  6894  62 

7264  7326  7388  7449  75ii  62 

7881  7943  8004  8066  8127  ] 62^ 

| c 6 7 8 9 I D 


LOGARITHMS  OF  NUMBERS. 


325 


No. 

0 

\ 

2 

3 

4 

5 

6 

7 

8 

9 

D 

705 

84- 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

862 

8682 

8743 

62 

706 

84- 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61 

707 

84- 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

708 

85- 

0033 

0095 

0156 

0217 

0279 

034 

0401 

0462 

0524 

0585 

61 

709 

85- 

0646 

0707 

0769 

083 

0891 

0952 

1014 

IO75 

1136 

1197 

61 

710 

85- 

1258 

132 

1381 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

61 

711 

85- 

187 

i93i 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

712 

85- 

248 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

7i3 

85- 

309 

3i5 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

714 

85- 

3698 

3759 

382 

3881 

394i 

4002 

4063 

4124 

4185 

4245 

61 

715 

85- 

4306 

4367 

4428 

4488 

4549 

461 

467 

4731 

4792 

4852 

61 

716 

85- 

49i3 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

7*7 

85- 

55i9 

558 

564 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

718 

85- 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

719 

85- 

6729 

6789 

685 

691 

697 

7031 

7091 

7152 

7212 

to 

to 

60 

720 

85- 

7332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

60 

721 

85- 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

722 

85- 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

723 

85- 

9138 

9*98 

9258 

9318 

9379 

9439 

9499 

9559 

96i9 

9679 

60 

724 

85- 

9739 

9799 

9859 

9918 

9978 

— 

— 

— 

— 

— 

60 

724 

86- 

— 

— 

— 

— 

— 

0038 

0098 

0158 

0218 

0278 

60 

725 

86- 

0338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

60 

726 

86- 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

727 

86- 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

728 

86- 

2131 

2191 

2251 

231 

237 

243 

2489 

2549 

2608 

2668 

60 

729 

86- 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3i44 

3204 

3263 

60 

730 

86- 

3323 

3382 

3442 

3501 

356i 

362 

368 

3739 

3799 

3858 

59 

73i 

86- 

39i7 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

732 

86- 

45ii 

457 

463 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

59 

733 

86- 

5104 

5163 

5222 

5282 

534i 

54 

5459 

5519 

5578 

5637 

59 

734 

86- 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

61 1 

6169 

6228 

59 

735 

86- 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

676 

6819 

59 

736 

86- 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

735 

7409 

59 

737 

86- 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

788 

7939 

7998 

59 

738 

86- 

8056 

8115 

8174 

8233 

8292 

835 

8409 

8468 

8527 

8586 

59 

739 

86- 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

86- 

9232 

929 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

976 

59 

741 

86- 

9818 

9877 

9935 

9994 

— 

— 

— 

— 

— 

— 

59 

74i 

87- 

— 

— 

— 

— 

0053 

OIII 

017 

0228 

0287 

0345 

59 

742 

87- 

0404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

093 

58 

743 

87- 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

744 

87- 

1573 

1631 

169 

1748 

1806 

1865 

1923 

1981 

204 

2098 

58 

745 

87- 

2156 

2215 

2273 

233i 

2389 

2448 

2506 

2564 

2622 

2681 

58 

746 

87- 

2739 

2797 

2855 

2913 

2972 

303 

3088 

3146 

3204 

3262 

58 

747 

87- 

332i 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

748 

87- 

39°2 

396 

4018 

4076 

4i34 

4192 

425 

4308 

4366 

4424 

58 

749 

87- 

4482 

454 

4598 

4656 

47i4 

4772 

483 

4888 

4945 

5003 

58 

750 

87- 

5061 

5ii9 

5177 

5235 

5293 

535i 

5409 

5466 

5524 

5582 

58 

75i 

87- 

564 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

616 

58 

752  j 

87- 

6218 

6276 

6333- 

6391 

6449 

6507 

6564 

6622 

668 

6737 

58 

753  ; 

87- 

6795 

6853 

691 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

754  j 

87- 

737i 

7429 

00 

*■ 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

No.  | 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D ' 

E E 


No. 

755 

756 

757 

758 

758 

759 

760 

761 

762 

763 

764 

765 

766 

767 

768 

765 

77C 

771 

77" 

77: 

77, 

771 

77< 

771 

77 

77 

77 

78 

78 

78 

78 

'78 


logarithms  of  numbers. 


7947  8004  8062  8119  8177 

8522  8579  8637  8694  8752 

9096  9153  92 11  9268  9325 

9669  9726  9784  9841  9898 


i- 

1- 

v 

)■ 


0242  0299  0356  0413  0471 
0814  0871  0928  0985  1042 

1385  1442  1499  1556  1613 

1955  2012  2069  2120  2183 

2525  2581  2638  2695  2752 

3093  3*5  32°7  3264  3321 

3661  3718  3775  3832  3888 

4229  4285  4342  4399  4455 

4795  4852  49°9  49^5  5022 

536x  5418  5474  5531  55§7 

5926  5983  6039  6096  6152 

6491  6547  6604  666  6716 

• 7054  71 1 1 7i67  7223  728 

■7617  7674  773  7786  7842 

- 8179  8236  8292  8348  8404 

- 8741  8797  8853  8909  8965 

- 9302  9358  9414  947  9526 

9862  9918  9974  — — 

— — 003  0086 

0421  0477  0533  0589  0645 

O98  1035  1091  1147  ^03 

1537  J593  i649  i7°5  i7° 

2095  215  2206  2262  2317 

2651  2707  2762  2818  2873 
3207  3262  3318  3373  3429 
3762  3817  3873  3928  3984 
4316  4371  4427  4482  4538 
487  4925  498  5°36  5091 

5423  5478  5533  5588  5644 

• 5975  603  6085  614  6195 

- 6526  6581  6636  6692  6747 

• 7°77J  7I32  7i87  7242  7297 

- 7627  7682  7737  7792  7847 

- 8176  8231  8286  8341  8396 

- 8725  878  8835  889  8944 

- 9273  9328  9383  9437  9492 

- 9821  9875  993  9985  — 

— 0039 

- 0367  0422  0476  0531  0586 

- 0913  0968  1022  1077  II3I 

- 1458  I5i3  1567  1622  1676 
2003  2057  2112  2166  2221 

>-  2547  2601  2655  271  2764 

>-  309  3144  3X99  3253  33°7 

)-  3633  3687  374i  3795  3849 
>-  4174  4229  4283  4337  439 1 
>-  4716  477  4824  4878  4932 

>-  5256  531  5364  54l8  5472 


8234  8292  8349  8407  8464 
8809  8866  8924  8981  9039 
9383  944  9497  9555  9612 
9956  — — — 

— 0013  007  0127  0185 
0528  0585  0642  0699  0756 
1099  1156  1213  1271  1328 
167  1727  1784  i84i  ^98 
224  2297  2354  2411  2468 
2809  2866  2923  298  3037 
3377  3434  3491  3548  36°5 
3945  4002  4059  4115  4X72 
4512  4569  4625  4682  4739 

5078  513s  5192  5248  5305 
5644  57  5757  5813  587 

6209  6265  6321  6378  6434 
6773  6829  6885  6942  6998 
7336  7392  7449  75°5  75^1 
7898  7955  8011  8067  8123 
846  8516  8573  8629  8685 
9021  9077  9134  9*9  9246 
9582  9638  9694  975  9806 


0141  0197  0253  0309  0365 
07  0756  0812  0868  0924 

1259  1314  137  T4s6  1482 
1816  1872  1928  1983  2039 
2373  2429  2484  254  2595 

2929  2985  304  3096  315^ 

3484  354  3595  3^5 1 37°6 

4039  4094  415  42°5  4261 

4593  4648  47°4  4759  4814 
5146  5201  5257  5312  5367 
5699  5754  5809  5864  592 
6251  6306  6361  6416  6471 
6802  6857  6912  6967  7022 
7352  7407  7462  75 1 7 7572 
7902  7957  8012  8067  8122 
8451  8506  8561  8615  867 
8999  9°54  9io9  9i64  9218 
9547  9602  9656  97 11  9766 


57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

57 

5: 

5' 

5* 

5< 

5< 

5' 

51 

5 

5 

5 

5 

5 

q 

c 

: 

c 

: 


l 


OO94  0149  0203 
064  0695  0749 

Il86  124  1295 

I73I  I785  184 
2275  2329  2384 
28l8  2873  2927 

3361  3416  347 
3904  3958  4° 12 
4445  4499  4553 
4986  504  5094 

5526  558  5634 


0258  0312 
0804  0859 
1349  1404 
1894  1948 
2438  2492 
2981  3036 

3524  3578 
4066  412 
4607  4661 
5148  5202 
5688  5742 


5 


6 


7 


LOGARITHMS  OF  NUMBERS. 


327 


No 

f 0 1 2 3 4|5  6 7 8 9 

D 

80s 

806 

807 

808 

809 

810 

81 1 

812 

812 

813 

814 

815 

816 

817 

818 

819 

820 

821 

822 

823 

824 

825 

826 

827 

828 

829 

830 

831 

831 

832 

833 

834 

835 

836 

837 

838 

839 

840 

841 

842 

843 

844 

845 

846 

847 

848 

849 

850 

851 

851 

852  < 

853  ' 

854  ' 

90-5796  585  5904  5958  601: 
90-  6335  6389  6443  6497  6551 
90-  6874  6927  6981  7035  7085 
90-  7411  7465  7519  7573  762 6 
90-  7949  8002  8056  81 1 8163 

90-  8485  8539  8592  8646  8699 
90-  9021  9074  9128  9181  9235 

90-  9556  9609  9663  9716  977 

91-  0091  0144  0197  0251  0304 

91-  0624  0678  0731  0784  0838 
91-  1158  1211  1264  1317  1371 
91-  169  1743  1797  185  1903 
91-  2222  2275  2328  2381  2435 
91*  2753  2806  2859  2913  2966 
91-  3284  3337  339  3443  3496 

91-  3814  3867  392  3973  4026 

9i-  4343  4396  4449  45°2  4555 
9i-  4872  4925  4977  503  5083 

9r~  54  5453  55o 5 5558  5611 

9i-  5927  598  6033  6085  6138 

9J-  6454  6507  6559  6612  6664 
91-  698  7033  7085  7138  719 

gi-  7506  7558  7611  7663  7716 
91-  803  8083  8135  8188  824 

9i-  8555  8607  8659  8712  8764 
91-  9078  913  9183  9235  9287 

91-  9601  9653  9706  9758  981 

92-  0123  0176  0228  028  0332 

92-  0645  0697  0749  °8oi  0853 
92-  1166  1218  127  1322  1374 

92-  1686  1738  179  1842  1894 

92-  2206  2258  231  2362  2414 

92-  2725  2777  2829  2881  2933 
92-  3244  3296  3348  3399  345i 
92-  3762  3814  3865  3917  3969 

92-  4279  433i  4383  4434  4486 
92-  4796  4848  4899  4951  5003 
92-  5312  5364  5415  5467  5518 
92-  5828  5879  5931  5982  6034 
92-  6342  6394  6445  6497  6548 
92-  6857  6908  6959  7011  7062 
92-  737  7422  7473  7524  7576 

92-  7883  7935  7986  8037  8088 
92-  8396  8447  8498  8549  8601 
92-  8908  8959  901  9061  9112 

92-  9419  947  9521  9572  9623 

92-  993  998i  — — — 

93-  — — 0032  0083  0134 

93-  044  0491  0542  0592  0643  , 

93“  0949  1 1051  1102  1153 

93"  1458  1509  156  161  1661 

j 6066  6119  6173  6227  6281 
6604  6658  6712  6766  682 
> 7143  7196  725  7304  7358 
' 7o8  7734  7787  7841  7895 

; 8217  827  8324  8378  8431 

1 8753  8807  886  8914  8967 

9289  9342  9396  9449  9503 

9823  9877  993  9984  — 

— — — — 0037 

0358  0411  0464  0518  0571 

0891  0944  0998  1051  1104 

1424  1477  153  1584  1637 

1956  2009  2063  2116  2169 

2488  2541  2594  2647  27 
3019  3072  3125  3178  3231 
3549  3602  3655  3708  3761 
4079  4132  4184  4237  429 
4608  466  4713  4766  4819 

5136  5189  5241  5294  5347 

5664  5716  5769  5822  5875 

6191  6243  6296  6349  6401 

6717  677  6822  6875  6927 

7243  7295  7348  74  7453 

7768  782  7873  7925  7978 

8293  8345  8397  845  8502 

8816  8869  8921  8973  9026 

934  9392  9444  9496  9549 

9862  9914  9967  — — 

— — 0019  0071 

0384  0436  0489  0541  0593 

0906  0958  101  1062  1 1 14 

1426  1478  153  1582  1634 

i946  1998  205  2102  2154 

2466  2518  257  2622  2674 

2985  3037  3089  314  3192 

3503  3555  3607  3658  371 
4021  4072  4124  4176  4228 
4538  4589  4641  4693  4744 
5°54  5io6  5157  5209  5261 
557  5621  5673  5725  5776 

6085  6137  6188  624  6291 

66  6651  6702  6754  6805 

7114  7165  7216  7268  7319 
7627  7678  773  7781  7832 

814  8191  8242  8293  8345 

8652  8703  8754  8805  8857 
9163  9215  9266  9317  9368 
9674  9725  9776  9827  9879 

0185  0236  0287  0338  0389 
0694  0745  0796  0847  0898 
1203  1254  1305  1356  1407 
1712  1763  1814  1865  1915 

: 54 
54 
1 54 
54 
54 
54 
54 
54 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
52 
52 
52 

"52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

5r 

5i 

5i 

5i 

5i 

5 1 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

No. 

0 ‘ 234 

56789 

D 

No. 

855 

856 

857 

858 

859 

860 

861 

862 

863 

864 

865 

866 

867 

868 

865 

87C 

871 


logarithms  of  numbers. 


V 

\ 


1966  2017  2068 
2474  2524  2575 
2981  3031  3082 

3487  3538  3589 

3993  4044  4094 

4498  4549  4599 
5003  5054  5io4 
5507  5558  56°8 
6011  6061  61 1 1 

- 6514  6564  6614 

- 7016  7066  7117 

- 7518  7568  7618 

- 8019  8069  8119 

- 852  857  862 

- 902  907  912 

- 9519  9569  9^9 

- 0018  0068  0118 

- 0516  0566  0616 

- 1014  1064  1 1 14 

- 1511  i561  1611 

- 2008  2058  2107 
2504  2554  2603 

r-  3 3049  3°99 

^ 3495  3544  3593 
(.-  3989  4038  4080 

[-  4483  4532  458i 
[-  4976  5025  5074 
t-  5469  55i8  5567 
5961  601  6059 

6452  6501  6551 
4.-  6943  6992  7041 

4“  7434  7483  7532 
4-  7924  7973  8022 
4-  8413  8462  8511 
4-  8902  8951  8999 

4-  939  9439  9488 

,4-  9878  9926  9975 

►5-  — — ~ 

>5"  0365  0414  0402 
>5-  0851  09  0949 

)5-  1338  1386  1435 
>5-  1823  1872  192 
)5-  2308  2356  2405 
)5_  2792  2841  2889 
)5~  3276  3325  3373 
}5-  376  3808  3856 

}5-  4243  4291  4339 
15-  4725  4773  4821 
?5~  5207  5255  5303 
?5-  5688  5736  5784 
95-  6168  6216  6265 


8 


2118  2169 
2626  2677 

3T33  3i83 
3639  369 
4145  4195 
465  47 

5154  5205 
5658  5709 
6162  6212 
6665  6715 

7167  7217 
7668  7718 
8169  8219 
867  872 

917  922 

9669  9719 

0168  0218 
0666  0716 
1163  1213 
166  171 


2157  2207 
2653  2702 
3148  3198 
3643  3692 

4137  4186 

4631  468 
5124  5173 
5616  5665 
6108  6157 
66  6649 

709  714 

7581  763 
807  8119 

856  8609 

9048  9097 

9536  9585 


222  2271 

2727  2778 

3234  3285 
374  3791 

4246  4296 

475 1 4801 
5255  5306 
5759  5809 
6262  6313 
6765  6815 

7267  7317 
7769  7819 
8269  8319 
877  882 

927  932 

9769  9819 
0267  0317 
0765  0815 
1263  1313 
176  1809 

2256  2306 
2752  2801 

3247  3297 

3742  3791 
4236  4285 

4729  4779 
5222  5272 
5715  5764 
6207  6256 
6698  6747 
7189  7238 
7679  7728 
8168  8217 
8657  8706 
9146  9195 

9634  9683 


2322  2372 
2829  2879 
3335  3386 
3841  3892 
4347  4397 

4852  49°2 
5356  5406 
586  59 1 

6363  6413 
6865  6916 

7367  7418 
7869  7919 
837  842 

887  892 

9369  9419 
9869  9918 
0367  0417 
0865  0915 
1362  1412 
1859  1909 

2355  2405 
2851  2901 
3346  3396 
3841  389 
4335  4384 
4828  4877 

532i  537 

5813  5862 

6305  6354 

6796  6845 

7287  7336 
7777  7826 
8266  8315 
8755  8804 
9244  9292 
9731  978 


2423  5i 
293  5i 


3437 

3943 

4448 


4953  5C 
5457  5< 

596 


0024  0073 
0511  056 
0997  1046 
i483  J532 
1969  2017 
2453  2502 
2938  2986 
342i  347 
39°5  3953 

4387  4435 
4869  4918 
535 1 5399 
5832  588 
6313  636 1 


0121  017 
0608  0657 
1095  1143 
158  1629 

2066  2114 

255  2599 
3034  3°83 
3518  3566 
4001  4049 
4484  4532 
4966  5014 
5447  5495 
5928  5976 
6409  6457 


6463 
6966 

7468 
7969 
847 
897 
9469 

9968 
0467 
0964 
1462 
1958 
2455 
295 
3445 
3939 
4433 
4927 
5419  1 
?912  I 
6403  I 

6894 

7385 
7875 
8364 

8853 

934i 
9829 

0219  0267  0316 
0706  0754  0803 
1192  124  1289 

1677  1726  1775 
2163  2211  226 
2647  2696  2744 
3131  3i8  3228 
3615  3^3  37i 1 
4098  4146  4194 
458  4628  4677 

5062  511  5I58 

5543  5592  564 
6024  6072  612 
6505  6553  6601 


5< 

5< 

5( 

5( 

5' 

5 

5 

5 

5 

5 

c 

e 

c 

{ 

: 

2 

. 


LOGARITHMS  OF  NUMBERS.  329 


No.  | 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

905 

95- 

6649 

6697 

6745 

6793 

684 

6888 

6936 

6984 

7032 

708 

48 

qo6 

95- 

7128 

7176 

7224 

7272 

732 

7368 

741.6 

7464 

7512 

7559 

48 

907 

95- 

7607 

7655 

7703 

775T 

7799 

7847 

7894 

7942 

799 

8038 

48 

qo8 

95~ 

8086 

8134 

8l8l 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

909 

95- 

8564 

8612 

8659 

8707 

8755 

8803 

885 

8898 

8946 

8994 

48 

910 

95- 

9°4I 

9089 

9*37 

9i85 

9232 

928 

9328 

9375 

9423 

947i 

48 

911 

95- 

95i8 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

99 

9947 

48 

912 

912 

95- 

96- 

9995 

0042 

009 

0138 

0185 

0233 

028 

0328 

0376 

0423 

48 

48 

913 

96- 

0471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48 

914 

96- 

0946 

0994 

1041 

1089 

11 36 

1184 

1231 

1279 

1326 

1374 

47 

915 

96- 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

916 

96- 

1895 

1943 

199 

2038 

2085 

2132 

218 

2227 

2275 

2322 

47 

9X7 

96- 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

96- 

2843 

289 

2937 

2985 

3032 

3079 

3126 

3J74 

3221 

3268 

47 

9T9 

96- 

33l6 

3363 

34i 

3457 

3504 

3552 

3599 

3646 

3693 

374i 

47 

920 

96- 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

47 

921 

96- 

426 

4307 

4354 

4401 

4448 

4495 

4542 

459 

4637 

4684 

47 

922 

96- 

473i 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

47 

923 

96- 

5202 

5249 

5296 

5343 

539 

5437 

5484 

553i 

5578 

5625 

47 

924 

96- 

5672 

57i9 

5766 

5813 

586 

5907 

5954 

6001 

6048 

6095 

47 

925 

96- 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

647 

6517 

6564 

47 

926 

96- 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

47 

927 

96- 

708 

7127 

7l13 

722 

7267 

7314 

7361 

7408 

7454 

75oi 

47 

928 

96- 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

47 

929 

96- 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

839 

8436 

47 

930 

96- 

8483 

853 

8576 

8623 

867 

8716 

8763 

881 

8856 

8903 

47 

93 1 

96- 

895 

8996 

9°43 

909 

9^6 

9183 

9229 

9276 

9323 

9369 

47 

932 

96- 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47 

933 

96- 

9882 

9928 

9975 

0068 

0161 

47 

933 

91- 

— 

— 

— 

0021 

0114 

0207 

0254 

03 

47 

934 

97- 

0347 

0393 

044 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

935 

97- 

0812 

0858 

0904 

0951 

0997 

1044 

109 

ii37 

1183 

1229 

46 

936 

97- 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46 

937 

97- 

i74 

1786 

1832 

1879 

1925 

I97i 

2018 

2064 

211 

2157 

46 

938 

97- 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

939 

97- 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

46 

940 

97- 

3128 

3174 

322 

3266 

3313 

3359 

3405 

345i 

3497 

3543 

46 

941 

97- 

359 

3636 

3682 

3728 

3774 

382 

3866 

39i3 

3959 

4005 

46 

942 

97- 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

442 

4466 

46 

943 

97- 

4512 

4558 

4604 

465 

4696 

4742 

4788 

4834 

488 

4926 

46 

944 

97“ 

4972 

5018 

5064 

511 

5156 

5202 

5248 

5294 

534 

5386 

46 

945 

97- 

5432 

5478 

5524 

557 

5616 

5662 

5707 

5753 

5799 

5845 

46 

946 

97- 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

947 

97- 

635 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

948 

91- 

6808 

6854 

69 

6946 

6992 

7037 

7083 

7129 

7175 

722 

46 

949 

91- 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

97" 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8i35 

46 

95i 

97- 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

85 

8546 

8591 

46 

952 

97- 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

953 

97- 

9093 

9138 

9i84 

923 

9275 

9321 

9366 

9412 

9457 

9503 

46 

954 

97- 

9548 

9594 

9639 

9685 

973 

9776 

9821 

9867 

9912 

9958 

46 

.No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

Ee* 


330 


LOGARITHMS  OF  NUMBERS. 


No. 

955 


956 

957 

958 

959 


960 

961 

962 

963 

964 


965 

966 

967 

968 

969 


970 


971 

972 

973 

974 


975 

976 

977 

977 

978 

979 


980 

981 

982 

983 

984 


985 

986 

987 

988 

989 


990 

991 

992 

993 

994 

995 

996 

997 

998 

999 


01234 

98-  0003  0049  0094  014  0185 

98-  0458  0503  0549  0594  064 
98-  0912  0957  1003  1048  1093 
98-  1366  1411  1456  1501  I547 
98-  1819  1864  1909  1954  2 

98-  2271  2316  2362  2407  2452 
98-  2723  2769  2814  2859  2904 
98-  3J75  322  3265  33i  3356 

98-  3626  3671  3716  3762  3807 
98-  4077  4122  4167  4212  4257 

98-  4527  4572  4617  4662  4707 
98-  4977  5022  5067  5112  5*57 
98-  5426  5471  5516  5561  5606 
98-  5875  592  5965  601  6055 
98-  6324  6369  6413  6458  6503 

98-  6772  6817  6861  6906  6951 
98-  7219  7264  7309  7353  7398 
98-  7666  7711  7756  78  7845 

98-  8113  8157  8202  8247  8291 
98-  8559  8604  8648  8693  8737 

98-  9005  9049  9094  9138  9183 
98-  945  9494  9539  95 83  9628 

98-  9895  9939  9983  “ — 

99-  — — — 0028  0072 

99-  °33 9 0383  0428  0472  0516 
99-  0783  0827  0871  0916  096 

99-  1226  127  1315  1359  I4°3 

99-  1669  1713  1758  1802  1846 
99-  21 11  2156  22  2244  2288 

99-  2554  2598  2642  2686  273 
99-  2995  3039  3083  3127  3172 

99-  3436  348  3524  3568  3613 

99"  3877  392i  3965  4009  4053 
99-  43 1 7 4361  4405  4449  4493 
99-  4757  4801  4845  4889  4933 
99-  5196  524  5284  5328  5372 

99-  5635  5679  5723  5767  5811 
99-  6074  6117  6161  6205  6249 
99-  6512  6555  6599  6643  6687 
99-  6949  6993  7037  708  7124 

99-  7386  743  7474  75*7  75^i 

99-  7823  7867  791  7954  7998 

99-  8259  8303  8347  839  8434 

99-  8695  8739  8782  8826  8869 
99-  9131  9174  9218  9261  9305 
99-  9565  9^  9652  9696  9739 


5 6 7 8 9 D 


0231  0276  0322  0367  0412 

0685  073  0776  0821  0867 

1139  1184  1229  1275  132 

1592  1637  1683  1728  1773 

2045  209  2135  2181  2226 


45 

45 

45 

45 

45 


2497 

2949 

3401 

3852 

4302 


2543 

2994 

3446 

3897 

4347 


2588 
304 
349 1 
3942 
4392 


2633 

3085 

3536 

3987 

4437 


2678 

3*3 

3581 

4032 

4482 


45 

45 

45 

45 

45 


4752  4797  4842  4887  4932 

5202  5247  5292  5337  5382 

5651  5696  5741  5786  583 

61  6144  6189  6234  6279 

6548  6593  6637  6682  6727 


45 

45 

45 

45 

45 


6996  704  7085  713  7j75 

7443  7488  7532  7577  7622 

789  7934  7979  8024  8068 

8336  8381  8425  847  8514 

8782  8826  8871  8916  896 

9227  9272  9316  9361  9405 

9672  9717  9761  9806  985 


0117  0161  0206  025  0294 

0561  0605  065  0694  0738 

1004  1049  1093  1137  1182 

1448  1492  1536  158  1625 

189  1935  *979  2023  2067 

2333  2377  2421  2465  2509 

2774  2819  2863  2907  2951 

3216  326  3304  3348  3392 


45 

45 

45 

45 

45 

45 


3657  3701  3745  3789  3833 
4097  4141  4i85  4229  4273 
4537  458i  4625  4669  4713 
4977  5021  5065  5108  5152 
5416  546  5504  5547  5591 


5854  5898  5942 
6293  6337  638 
6731  6774  6818 
7168  7212  7255 
7605  7648  7692 

8041  8085  8129 
8477  8521  8564 
8913  8956  9 
9348  9392  9435 
9783  9826  987 


5986  603 
6424  6468 
6862  6906  j 
7299  7343  ; 
7736  7779  i 
8172  8216 
8608  8652 
9043  9087  I 
9479  9522  44 

99*3  9957  | 43 

8 


6 


7 


9 | D 


***  *****  *****  *****  ***** 


HYPERBOLIC  LOGARITHMS  OF  NUMBERS, 


331 


Hyperbolic  XjOgaritliixLS  of'  nSTiam'bers. 

From  1 .01  to  30. 

In  following  table,  the  numbers  range  from  1.01  to  30,  advancing  by  .01, 
up  to  the  whole  number  10 ; and  thence  by  larger  intervals  up  to  30.  ” The 
hyperbolic  logarithms  of  numbers,  or  Neperian  logarithms,  as  they  are  some- 
times termed,  are  computed  by  multiplying  the  common  logarithms  of  num- 
bers by  the  constant  multiplier,  2.302585. 

The  hyperbolic  logarithms  of  numbers  intermediate  between  those  which 
are  given  in  the  table  may  be  readily  obtained  by  interpolating  proportional 
differences. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

I. OI 

.OO99 

1. 41 

•3436 

I.8l 

•5933 

2.21 

•793 

2.6l 

•9594 

1.02 

.OI98 

I.42 

•3507 

1.82 

.5988 

2.22 

•7975 

2.62 

.9632 

I.03 

.0296 

i-43 

•3577 

I.83 

.6043 

2.23 

.802 

2.63 

.967 

I.04 

.0392 

1.44 

.3646 

I.84 

.6098 

2.24 

.8065 

2.64 

.9708 

1.05 

.0488 

i-45 

.3716 

1.85 

.6152 

2.25 

.8109 

2.65 

.9746 

1.06 

.0583 

1.46 

•3784 

1.86 

.6206 

2.26 

.8154 

2.66 

•9783 

I.07 

.0677 

1.47 

•3853 

1.87 

.6259 

2.27 

.8198 

2.67 

.9821 

1.08 

.077 

1.48 

•392 

1.88 

•6313 

2.28 

.8242 

2.68 

.9858 

I.09 

.0862 

i-49 

.3988 

1.89 

.6366 

2.29 

.8286 

2.69 

•9895 

1. 1 

•0953 

i-5 

•4055 

1.9 

.6419 

2-3 

.8329 

2.7 

•9933 

I. II 

.IO44 

I*5I 

.4121 

1.91 

.6471 

2.3I 

•8372 

2.71 

.9969 

1. 12 

•1133 

1.52 

.4187 

1.92 

•6523 

2.32 

.8416 

2.72 

1.0006 

I*I3 

.1222 

i-53 

•4253 

x-9  3 

•6575 

2-33 

.8458 

2.73 

1.0043 

1.14 

•13  * 

i-54 

.4318 

1.94 

.6627 

2-34 

.8502 

2.74 

1.008 

I*I5 

.1398 

i-55 

•4383 

i-95 

.6678 

2-35 

•8544 

2-75 

1.0116 

1. 16 

.1484 

1.56 

•4447 

1.96 

.6729 

2.36 

.8587 

2.76 

1.0152 

1. 17 

•157 

i-57 

•45ii 

1.97 

.678 

2-37 

.8629 

2.77 

1. 0188 

1. 18 

•1655 

1.58 

•4574 

1.98 

.6831 

2.38 

.8671 

2.78 

1.0225 

1. 19 

.174 

i-59 

•4637 

I*99 

.6881 

2.39 

•8713 

2.79 

1.026 

1.2 

.1823 

1.6 

•47 

2 

.6931 

2.4 

•8755 

2.8 

1.0296 

1. 21 

.1906 

1. 61 

.4762 

2.01 

.6981 

2.4I 

.8796 

2.81 

1.0332 

1.22 

.1988 

1.62 

.4824 

2.02 

.7031 

2.42 

.8838 

2.82 

1.0367 

1.23 

.207 

1 63 

.4886 

2.03 

.708 

2-43 

.8879 

2.83 

1.0403 

1.24 

.2151 

1.64 

•4947 

2.04 

.7129 

2-44 

.892 

2.84 

1.0438 

1.25 

.2231 

1.65 

.5008 

2.05 

.7178 

2-45 

.8961 

2.85 

1-0473 

1.26 

.2311 

1.66 

.5068 

2.06 

.7227 

2.46 

.9002 

2.86 

1.0508 

1.27 

.239 

1.67 

.5128 

2.07 

•7275 

2.47 

.9042 

2.87 

1-0543 

1.28 

.2469 

1.68 

.5188 

2.08 

•7324 

2.48 

.9083 

2.88 

1.0578 

1.29 

.2546 

1.69 

•5247 

2.09 

•7372 

2.49 

.9123 

2.89 

1.0613 

i-3 

.2624 

i-7 

•5306 

2.1 

.7419 

2-5 

•9i63 

2.9 

1.0647 

I*3I 

.27 

I*7I 

•5365 

2. 11 

.7467 

2.51 

.9203 

2.91 

1 .0682 

1.32 

.2776 

1.72 

•5423 

2.12 

•7514 

2.52 

•9243 

2.92 

1.0716 

i-33 

.2852 

i-73 

.5481 

2.13 

•756i 

2-53 

.9282 

2-93 

1-075 

i-34 

.2927 

1.74 

•5539 

2.14 

.7608 

2-54 

.9322 

2.94 

1.0784 

i-35 

.3001 

i-75 

•550 

2.15 

•7655 

2-55 

.9361 

2-95 

1.0818 

1.36 

•3075 

1.76 

•5653 

2.16 

.7701 

2.56 

•94 

2.96 

1.0852 

I*37 

.3148 

i-77 

•57i 

2.17 

•7747 

2.57 

•9439 

2.97 

1.0886 

1.38 

.3221 

1.78 

.5766 

2.18 

•7793 

2.58 

.9478 

2.98 

1. 0919 

i-39 

•3293 

1.79 

.5822 

2 19 

•7839 

2-59 

•9517 

2-99 

I-°953 

1.4 

•3365 

1.8 

.5878 

2.2 

.7885  , 

2.6 

•9555 

3 

1.0986 

332 


HYPERBOLIC  LOGARITHMS  OP  NUMBERS. 


No.  1 

Log.  | j ’ 

No. 

Log.  | 

No. 

Log. 

No.  | 

Log. 

No.  | : 

3.OI 

1. 1019 

3-5i 

1.2556  i 

4..OI 

I.3888  . 

4.51  : 

I.5063 

5.01  | 1. 

3.02 

1 .1053 

3-52 

I.2585  ' 

4.02 

I*39I3 

4-52  : 

1.5085 

5.02  1. 

3-°3 

1. 1086 

3-53 

I.2613  1 - 

4.03 

1.3938 

4.53 

1*5107 

5-03  1 

3-04 

I.III9 

3-54 

I.2641 

4.04 

1.3962 

4-54 

1.5129 

5-04  i 1 

3-°5 

I.II5I 

3*55 

I.2669 

4.05 

1.3987 

4-55 

I.5I5I 

5*05  | 1 

3.06 

I.I184 

3-56 

I.2698  1 

4.06 

1 .4012 

4-56 

I-5I73 

5.06  1 

3.07 

I. 1217 

3-57 

I.2726  1 

4.07 

1 .4036 

4-57 

I.5I95 

5-07  1 

3.08 

I. 1249 

3.58 

1.2754 

4.08 

1.4061 

4-58 

1*5217 

5.08  1 

3.09 

1.1282 

3-59 

I.2782 

4.09 

1.4085 

4-59 

1.5239 

5-09  1 

3-1 

I.I3H 

3*6 

I.2809 

4.I 

1. 41 1 

4.6 

I.5261 

5-i  1 

3. 11 

1.1346 

3.61 

I.2837 

4.II 

I.4I34 

4.61 

I.5282 

5.11  1 

3.12 

1.1378 

3.62 

I.2865 

4.12 

I.4I59 

4.62 

1.5304 

5.12  1 

3.13 

1.141 

3-63 

I.2892 

4.13 

1.4183 

4-63 

1.5326 

5-13  i 

3.14 

1. 1442 

3-64 

I.292 

4.14 

1.4207 

4.64 

1-5347 

5-14  1 

3-i  S 

1. 1474 

3-65 

I.2947 

4.15 

1. 4231 

4-65 

1.5369 

5.15  3 

3*16 

1.1506 

3.66 

1.2975 

4.16 

1.4255 

4.66 

1-539 

5. 16  i 

3-17 

i*i537 

3.67 

I.3OO2 

4.I7 

1.4279 

4.67 

i*54i2 

5-17  ] 

3.18 

1.1569 

3.68 

I.3029 

4.18 

1-4303 

4.68 

1-5433 

5. 18  ] 

3.19 

1. 16 

3-69 

1.3056 

4.19 

1.4327 

4.69 

1.5454 

5.i9  : 

3-2 

1.1632 

3-7 

1.3083 

4.2 

1. 435 1 

4-7 

I.5476 

5-2 

3.21 

1.1663 

3-71 

I-311 

4.21 

1-4375 

4.71 

1-5497 

5.21 

3.22 

1.1694 

3-72 

I*3I37 

4.22 

1.4398 

4.72 

i-55i8 

5-22 

323 

1. 1725 

3-73 

1.31:64 

4.23 

1.4422 

4-73 

1-5539 

5.23 

3.24 

1.1756 

3-74 

i*3i9i 

4.24 

1.4446 

4-74 

1.556 

5.24 

3*25 

1.1787 

3-75 

1.3218 

4.25 

1.4469 

4-75 

i.558i 

5.25 

3.26 

1.1817 

3-76 

1.3244 

4.26 

1-4493 

4.76 

1.5602 

5.26 

3.27 

1.1848 

3-77 

1.3271 

4.27 

1.4516 

4.77 

1.5623 

5.27 

3.28 

1.1878 

3-78 

1.3297 

4.28 

1-454 

4.78 

1.5644 

5.28 

3-29 

1. 1909 

3-79 

1-3324 

4.29 

14563 

4-79 

1.5665 

5.29 

33 

1 .*939 

3-8 

i-335 

4*3 

1.4586 

4.8 

1.5686 

5-3 

3.31 

1.1969 

3.81 

I-3376 

4.31 

1.4609 

4.81 

1.5707 

5-31 

3.32 

1. 1999 

3 82 

1-3403 

4.32 

1-4633 

4.82 

1.5728 

5.32 

3-33 

1.203 

3-83 

1.3429 

4*33 

1.4656 

4.83 

1.5748 

5-33 

3-34 

1.206 

384 

1-3455 

4*34 

1.4679 

4.84 

1-5769 

5-34 

3-35 

1.209 

3-85 

1.3481 

4-35 

1.4702 

4-85 

1-579 

5-35 

3.36 

1. 2119 

3.86 

1.3507 

4.36 

1.4725 

4.86 

1.581 

5.36 

3*37 

1. 2149 

3*87 

1-3533 

4-37 

1.4748 

4.87 

1.5831 

5-37 

3-38 

1. 2179 

3.88 

1.3558 

4.38 

1-477 

4 88 

1.5851 

5.38 

3.39 

1.2208 

3-89 

13584 

4-39 

1-4793 

4.89 

1.5872 

5-39 

3-4 

1.2238 

3-9 

1.361 

4-4 

1.4816 

4.9 

1.5892 

5-4 

3.41 

1.2267 

3-91 

1.3635 

4.41 

1.4839 

4.91 

I-59I3 

541 

3-42 

1.2296 

3-92 

1.3661 

4.42 

1.4861 

4.92 

1-5933 

5-42 

3-43 

1.2326 

393 

1.3686 

4-43 

1.4884 

4.93 

1-5953 

5-43 

3*44 

1*2355 

3-94 

1.3712 

4-44 

1.4907 

4.94 

1-5974 

5-44 

3-45 

1.2384 

3-95 

1-3737 

4*45 

1.4929 

4-95 

1-5994 

5-45 

3.46 

' 1.2413 

396 

1.3762 

4.46 

» 1.4951 

4.96 

1 1.6014 

1 

3.47 

1.2442 

3-97 

1.3788 

4-47 

1.4974 

4-97 

1.6034 

1 5-47 

1.48 

1 1.247 

3-98 

1.3813 

4.48 

; 1.4996 

4.98 

: 1.6054 

|i  5 48 

3*49  1.2499 

3-99 

1 1.3838 

4*49 

) i*5OI9 

4-99 

1 1.6074 

5-49 

3-5 

1 1.2528 

4 

1 1.3863 

4-5 

1.5041 

5 

1.0094  5.5 

Log. 


I.6487 

1.6506 

I.6525 

I.6514 

I.6563 

I.6582 

1. 6601 

1.662 

I.6639 

1.6658 

I.6677 

I.6696 

1.6715 

1.6734 

I.6752 

I.677I 

I.679 
I.6808 
I.6827 
I.6845 
1 .6864 

1.6882 

I.69OI 

I.6919 

1.6938 

I.6956 

I.6974 
1.6993 
1. 7011 
I.7029 
1.7047 


HYPERBOLIC  LOGARITHMS  OF  NUMBERS.  333 


No. 

Log. 

No. 

Log.  | 

No. 

Log. 

No. 

Log. 

No. 

Log. 

5*51 

I.7066 

6.01 

1-7934 

6.51 

1.8733 

7.OI 

1-9473 

7-51 

2.0162 

5-52 

1.7084 

6.02 

I-795I  ! 

6.52 

I.8749 

7.02 

1.9488 

7.52 

2.0176 

5-53 

1. 7102 

6.03 

1.7967 

6-53 

I.8764 

7-03 

1.9502 

7-53 

2.0189 

5-54 

1. 712 

6.04 

1.7984 

6-54 

1.8779 

7-04 

1.9516 

7-54 

2.0202 

5-55 

I.7138 

6.05 

1. 8001 

6.55 

1.8795 

7-05 

1-953 

7-55 

2.0215 

5-56 

1.7156 

6.06 

1.8017 

6.56 

I.881 

7.06 

1-9544 

7.56 

2.0229 

5-57 

i*7I74 

6.07 

1.8034 

6.57 

1.8825 

7-07 

1-9559 

7-57 

2.0242 

5-58 

1. 7192 

6.08 

1.805 

6.58 

1.884 

7.08 

1-9573 

7-58 

2.0255 

5-59 

1. 721 

6.09 

I.8066 

6-59 

I.8856 

7.09 

1.9587 

7-59 

2.0268 

5-6 

1.7228 

6.1 

1.8083 

6.6 

I.8871 

7-i 

1.9601 

7-6 

2.0281 

5.61 

1.7246 

6. 1 1 

1.8099 

6.61 

1.8886 

7.11 

1.9615 

7.61 

2.0295 

5.62 

1.7263 

6.12 

1. 8116 

6.62 

1.8901 

7.12 

1.9629 

7.62 

2.0308 

5-63 

1.7281 

6.13 

1.8132 

6.63 

1.8916 

7.13 

1.9643 

7-63 

2.0321 

5*64 

1.7299 

6.14 

1.8148 

6.64 

1.8931 

7.14 

i.9657 

7.64 

2.0334 

5-65 

I-73I7 

6.15 

1.8165 

6.65 

1.8946 

7.15 

1.9671 

7.65 

2.O347 

5.66 

1-7334 

6.16 

1. 8181 

6.66 

1.8961 

7.16 

1.9685 

7.66 

2.036 

5-67 

1-7352 

6.17 

1.8197 

6.67 

1.8976 

7.17 

1.9699 

7.67 

2.0373 

5.68 

i-737 

6.18 

1.8213 

6.68 

1.8991 

7.18 

I.97I3 

7.68 

2.0386 

5-69 

I-7387 

6.19 

1.8229 

6.69 

1.9006 

7.19 

1.9727 

7.69 

2.O399 

5-7 

1-7405 

6.2 

1.8245 

6.7 

1. 902 1 

7.2 

1.9741 

7-7 

2.0412 

5-7i 

1. .7422 

6.21 

1.8262 

6.71 

1.9036 

7.21 

1*9755 

7-71 

2.O425 

5-72 

1.744 

6.22 

1.8278 

6.72 

1. 905 1 

7.22 

1.9769 

7.72 

2.0438 

5-73 

1-7457 

6.23 

1.8294 

6-73 

1.9066 

723 

1.9782 

7-73 

2.O45I 

5-74 

1*7475 

6.24 

1.831 

6.74 

1.9081 

7.24 

1.9796 

7-74 

2.0464 

5*75 

1.7492 

6.25 

1.8326 

6-75 

1.9095 

7-25 

1.981 

7-75 

2.O477 

5-7^ 

1-7509 

6.26 

1.8342 

6.76 

1.911 

7.26 

1.9824 

7.76 

2.O49 

5-77 

1.7527 

6.27 

1.8358 

6.77 

!-9I25 

7.27 

1.9838 

7-77 

2.0503 

5*78 

1-7544 

6.28 

1.8374 

6.78 

1*91.4 

7.28 

1.9851 

7.78 

2.0516 

5-79 

1.7561 

6.29 

1.839 

6.79 

I-9I55 

7.29 

1.9865 

7-79 

2.0528 

5-8 

1-7579 

6-3 

1.8405 

6.8 

1.9169 

7-3 

1.9879 

7.8 

2.O54I 

5.81 

1.7596 

6.31 

1.8421 

6.81 

1.9184 

7.3i 

1.9892 

7.81 

2.0554 

5.82 

1.7613 

6.32 

1.8437 

6.82 

1. 9199 

7-32 

1.9906 

7.82 

2.0567 

5.83 

1-763 

6-33 

1.8453 

6.83 

1-9213 

7.33 

1.992 

7.83 

2.058 

5-84 

1.7647 

6-34 

1.8469 

6.84 

1.9228 

7*34 

1 ‘9933 

7.84 

2.0592 

5.85 

1.7664 

6.35 

1.8485 

6.85 

1.9242 

7.35 

1 -9947 

7.85 

2.0605 

5.86 

1.7681 

6.36 

1.85 

6.86 

1.9257 

7-36 

1.9961 

7.86 

2.o6l8 

5-87 

1.7699 

6.37 

1.8516 

6.87 

1.9272 

7-37 

1.9974 

7.87 

2.0631 

•5-88 

1.7716 

1 6.38 

I-8532 

6.88 

1.9286 

7.38 

1.9988 

7.88 

2.0643 

5-89 

1-7733 

6-39 

1.8547 

6.89 

1. 9301 

7-39 

2.0001 

7-89 

2.0656 

5-9 

1-775 

6.4 

1.8563 

6.9 

i.93i5 

7-4 

2.0015 

7-9 

2.0669 

5-9i 

1.7766 

6.41 

1-8579 

6.91 

1-933 

7.41 

2.0028 

7.91 

2.o68l 

5*92 

1.7783 

6.42 

1.8594 

6.92 

1-9344 

7.42 

2.0042 

7.92 

2.0694 

5-93 

1.78 

6-43 

1. 861 

6.93 

1-9359 

7-43 

2.0055 

7-93 

2.0707 

5-94 

1.7817 

6.44 

1.8625 

6.94 

1-9373 

7-44 

2.0069 

7-94 

2.0719 

5-95 

I-7834 

6.45 

I.8641 

6-95 

1.9387 

7-45 

2.0082 

7-95 

2.0732 

5-96 

1.7851 

6.46 

1.8656 

6.96 

1.9402 

7.46 

2.0096 

7.96 

2.O744 

5-97 

1.7867 

6.47 

1.8672 

6.97 

1.9416 

7-47 

2.0109 

7-97 

2.0757 

5-98 

1.7884 

6.48 

1.8687 

6.98 

1-943 

7-48 

2.0122 

7.98 

2.0769 

5-99 

1.7901 

6.49 

1.8703 

6.99 

1-9445 

7-49 

2.0136 

7-99 

2.0782 

6 

1.7918 

! 6.5 

1.8718 

7 

1-9459 

7-5 

2.0149 

8 

2.0794 

334 


HYPERBOLIC  LOGARITHMS  OF  NUMBERS. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

8.01 

2.0807 

8.41 

2.1294 

8.81  : 

2.1759 

9.21  : 

2.2203 

9.61 

8.02 

2.0819 

8.42 

2.1406 

8.82  : 

2.177 

9.22  : 

2.2214 

9.62 

8.03 

2.0832 

8-43  ' 

2.1318 

8.83  : 

2.1782 

9.23  i 

2.2225 

9-63 

8.04 

2.0844 

8.44 

2.133 

8.84  : 

2.1793 

9.24  : 

2.2235 

9.64 

8.05 

2.0857 

8-45 

2.1342 

8.85  ! 

2.1804 

9.25  : 

2.2246 

9-65 

8.06 

2.0869 

8.46 

2.1353 

8.86  : 

2.1815 

9.26  : 

2.2257 

9.66 

8.07 

2.0882 

8.47 

2.1365 

8.87 

2.1827 

9.27 

2.2268 

9.67 

8.08 

2.0894 

8.48 

2.1377 

8.88 

2.1838  | 

9.28 

2.2279 

9.68 

8.09 

2.0906 

8.49 

2.1389 

8.89 

2.1849 

9.29 

2.2289 

9.69 

8.1 

2.O919 

8-5 

2. 14OI 

8.9 

2.l86l 

9-3 

2.23 

9.7 

8.11 

2.O93I 

8.51 

2. 1412 

8.91 

2.1872 

9-31 

2.23II 

9.71 

8.12 

2.O943 

8.52 

2.I424 

8.92 

2.1883 

9-32 

2.2322 

9.72 

8.13 

2.0956 

8-53 

2.1436 

8-93 

2.1894 

9-33 

2.2332 

9-73 

8.14 

2.0968 

8-54 

2.1448 

8.94 

2.1905 

9-34 

2.2343 

9-74 

8.15 

2.098 

8-55 

2.1459 

8-95 

2. 1917 

9-35 

2.2354 

9-75 

8.16 

2.0992 

8.56 

2.I47I 

8.96 

2.1928 

9-36 

2.2364 

9.76 

8.17 

2.1005 

8-57 

2.1483 

8-97 

2.I939 

9-37 

2.2375 

9-77 

8.18 

2. 1017 

8.58 

2.1494 

8.98 

2.195 

9-38 

2.2386 

9.78 

8.19 

2.1029 

8-59 

2.1506 

8.99 

2.1961 

9-39 

2.2396 

9-79 

8.2 

2. IO4I 

8.6 

2.1518 

9 

2.I972 

9.4 

2.2407 

9.8 

8.21 

2.1054 

8.61 

2.1529 

9.01 

2.1983 

9.41 

2.2418 

9.81 

8.22 

2.1066 

8.62 

2.1541 

9.02 

2.I994 

9.42 

2.2428 

9.82 

8.23 

2.IO78 

8.63 

2.1552 

9-°3 

2.2006 

9-43 

2.2439 

9-83 

8.24 

2.109 

8.64 

2.1564 

9.04 

2.2017 

9.44 

2.245 

9.84 

8.25 

2. 1102 

8.65 

2.1576 

9-°5 

2.2028 

9-45 

2.246 

9-85 

8.26 

2.III4 

8.66 

2.1587 

9.06 

2.2039 

9.46 

2.2471 

9.86 

8.27 

2.1126 

8.67 

2.1599 

1 9-°7 

2.205 

1 9-47 

2.2481 

9.87 

8.28 

2.II38 

8.68 

2.l6l 

9.08 

2.2061 

1 9.48 

2.2492 

9.88 

8.29 

2.II5 

8.69 

2.l622 

9.09 

2.2072 

9.49 

2.2502 

9.89 

8-3 

2.H63 

8.7 

2.1633 

9.1 

2.2083 

9-5 

2.2513 

9.9 

8.31 

2.II75 

8.71 

2.1645 

9.11 

2.2094 

9-5i 

2.2523 

9.91 

8.32 

2.H87 

8.72 

2.1656 

9.12 

2.2105 

9-52 

2.2534 

9.92 

8.33 

2. II99 

8.73 

2.l668 

9-i3 

2.2Il6 

9-53 

2.2544 

9-93 

8-34 

2.I2II 

8.74 

2.1679 

9-I4 

2.2127 

9-54 

2.2555 

9.94 

8-35 

2.1223 

8-75 

2.1691 

9-i5 

2.2138 

9-55 

2.2565 

9-95 

8.36 

2.1235 

8.76 

2.1702 

9.16 

2.2148 

9-56 

2.2576 

9.96 

8-37 

2.1247 

8.77 

2.I7I3 

9.17 

2.2159 

9-57 

2.2586 

9-97 

8.38 

2.1258 

8.78 

2.1725 

9.18 

2.217 

9-58 

2.2597 

9-9* 

8.39 

2.127 

8.79 

2.1736 

9.19 

2.2l8l 

9-59 

2.2607 

9-99 

8.4 

2.1282 

8.8 

2.I748 

9.2 

2.2192 

9.6 

2.2618 

10 

10.25 

2.3279 

12.25 

1 2.5052 

14.25 

; 2.6567 

17-5 

2.8621 

23 

10.5 

2.3513 

12.5 

2.5262 

14-5 

2.674 

18 

2.8904 

24 

10-75 

; 2.3749 

12.75 

: 2.5455 

; 14-75 

i 2.6913 

18.5 

2.9173 

25 

11 

2.3979 

» 13 

2.5649 

1 15 

2.7081 

19 

2.9444 

. 26 

11.25 

; 2.4201 

13*25 

> 2.584 

i5-5 

2.7408 

19-5 

2.9703 

;l  27 

11.5 

2-443 

13-5 

2.6027 

1 16 

2.7726 

1 2 

2-9957 

» 28 

ii*7! 

> 2.4636 

> 13-75 

; 2.6211 

: 16.5 

2.8034 

. 21 

3-044= 

; 29 

12 

2.4845 

> 1 14 

2.6391 

[ 17 

2.8332 

: 22 

3-°9I] 

: U 

Log- 

2.2628 

2.2638 

2.2649 

2.2659 

2.267 

2.268 
2.269 
2.2701 
2. 27II 
2.2721 

2.2732 

2.2742 

2.2752 

2.2762 

2.2773 

2.2783 

2.2793 

2.2803 

2.2814 

2.2824 

2.2834 

2.2844 

2.2854 

2.2865 

2.2875 

2.2885 

2.2895 

2.2905 

2.2915 

2.2925 

2.2935 

2.2946 

2.2956 

2.2966 

2.2976 

2.2986 

2.2996 

2.3006 

2.3016 

2.3026 

3-1355 

3.1781 

32189 

3-258i 

3-2958 

3-3322 

33673 

3-4°12 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  335 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


Parallelograms. 


Definition— Quadrilaterals,  having  their  opposite  sides  parallel. 

To  Compute  Area  of*  a,  Square,  Rectangle,  Plioxxi'bns,  or 
!Rliom.looid..— Figs.  1,2,  3,  and.  4r. 

Rule. — Multiply  length  by  breadth  or  height. 

Or,  lxb  = area , l representing  length , and  b breadth. 


Note  i. — Opposite  angles  of  a Rhombus  and  a Rhomboid  are  equal. 

2.  — In  any  parallelogram  the  four  angles  equal  360°. 

3.  —Side  of  a square  multiplied  by  1.52  is  equal  to  side  of  an  equilateral  triangle 
of  equal  area. 


Definition. — Space  included  between,  the  lines  forming  two  similar  parallel- 
ograms, of  which  smaller  is  inscribed  within  larger,  so  that  one  angle  in  each  is 
common  to  both,  as  shown  by  dotted  lines,  Fig.  1. 

To  Compute  Area  of*  a G-nomon.-Fig.  1. 

Rule. — Ascertain  areas  of  the  two  parallelograms,  and  subtract  less  from 


Example. — Sides  of  a gnomon  are  10  by  10  and  6 by  6 ins. ; what  is  its  area  ? 
10X10  = 100,  and  6 X 6 = 36.  Then  100  — 36  = 64  square  ins. 


Definition.— Plain  superficies  having  three  sides  and  angles. 

To  Compute  Area  of  a Triangle.—  Pigs.  5,  G,  and  V . 
Rule. — Multiply  base  by  height,  and  divide  product  by  2. 


Note  i. — Hypothenuse  of  a right  angle  is  side  opposite  to  right  angle. 

2. — Perpendicular  height  of  a triangle  = twice  its  area  divided  by  its  base. 

3.  — Perpendicular  height  of  an  equilateral  triangle  = a side  X .866. 

4. — Side  of  an  equilateral  triangle  x .658  255  = side  of  a square  of  equal  area, 

Or  — 1.3468  = diameter  of  a circle  of  equal  area. 

Fig.  5.  Fig.  6.  Fig.  7. 


Fig.  1. 


Fig.  3- 


a 


l 


>G 


Fig.  2. 


h 


Example. — Sides  ab,bc,  Fig.  1,  are  5 feet  6 ins. ; what  is  area? 

5. 5 X 5-  5 = 30-  25  square  feet. 


GriioinorL, 


greater. 


Or,  a — a'  = area,  a and  a'  representing  areas. 


Triangles 


c 


G 


c 


Example.  — Base  a b,  Fig. 
5,  is  4 feet,  and  height  c b,  6 ; 
what  is  area  ? 


4 X 6 = 24,  and  24-=  2 — 12 
square  feet. 


d bad 


a 


b a 


To  Compute  Area  of  a Triangle  t)y  Length  of  its  Sides.- 
ITigs.  6 and  *7, 

Kuut.-From  half  sum  of  the  three  sides  subtract  each  side  separately; 
then  multiply  half  sum  and  the  three  remainders  continually  together,  and 
take  square  root  of  product. 

0r  V(«-«)X  («-■-&)  s = area,  a,  6,  c representing  sides,  and  S half  sum 

of  the  three  sides.  *•»«**«« 

Example.— Sides  of  a triangle,  Figs.  6 and  7,  are  30, 40,  and  50  feet;  what  is  area . 

— 00  = ao 


60—30  = 30) 

60  — 40  = 20  J remainders. 


30  + 40+50  ^ _ 60  or  half  sum  of  sides.  — -r-  . 

2 2 60  — 50  = 107 

Whence,  30  X 20  X 10  X 60  = 360000,  and  f36o  000  = 600  square  feet. 

When  all  Sides  are  Equal.  Bulk.— Square  length  of  a side,  and  multi- 
ply  product  by  .433. 

Or,  S2  X *433  — areai  S representing  length  of  a side. 


To  Compute  Length  of  One  Side  of  a Right-Angled 
Triangle. 


When  Length  of  the  other  Two  Sides  are  given. 

To  Ascertain  Hypothenuse.-Fig.  £5. 

Rule.— Add  together  squares  of  the  two  legs,  and  take  square  root  of 


sum. 


Or  Va  62  + b <?  = hypothenuse.  Or,  Vb2  + h2. 

Example.  — Base,  a ft,  Fig.  5,  is  30  ins.,  and  height,  6 c,  40;  what  is  length  of  hy- 
pothenuse  ? ^ + ^ = ^ and  V2$00=so  ins. 


To  Ascertain  otlier  Leg. 

When  TTmothenuse  and  One  of  the  Legs  are  given.— Fig.  5.  Rule.— Sub- 
trartsquK  given  leg  from  Square  of  hypothenuse,  and  take  square  root 
of  remainder. 


r TT2 t>  / I ( ab2  = 5 c. 

Or,  \J hyp-2—  {/t2  = ^ 0riyac  ~~\bc2  = ab. 


7 /^/  ^ (l  — */•  y \ 

Example. -Base  of  a triangle,  aft,  Fig.  5,  is  3°  feet,  and  hypothenuse,  a c,  30; 
what  is  height  of  it? 


so2  __  302  — j6oo,  and  Vl6o°  = 4°  feet- 
To  Compute  Length.  of  a Side. 

When  Hypothenuse  of  a Right-angled  Triangle  of  Equal  Sides  alone  is 
g{ven. — Fig.  8.  Rule.— Divide  hypothenuse  by  1.414213. 


Or,  -- — — = length  of  a side. 


Fig.  8. 


1.414213 


Example.  — Hypothenuse  acofa  right-angled  triangle,  Fig.  8,  is 
300  feet;  what  is  length  of  its  sides  ? ; 


300-4- 1. 414  213  = 212. 1321  feet. 


Ur  O 

To  Compute  Perpendicular  or  Height  of  a Triangle. 

When  Base  and  Area  alone  are  given.— Tig.  9.  Rule.— Divide  twice 
area  by  its  base.  Or,  20,-7-  b = h. 


irea  by  its  oase.  or,  2 r — . 

Example. -Area  of  a triangle,  Fig.  9,  is  10  feet,  and  length  of  its  base,  a 6,  5, 
what  is  its  perpendicular,  cd? 


10  X 2 = 20,  and  20  -T-  5 = 4 feet- 


MENSURATION  OP  AREAS,  LINES,  AND  SURFACES.  337 


To  Compute  Perpendicular  or  Height  of  a Triangle. 

When  Base  and  Two  Sides  are  given.  Kule.-As  base  is  to  sum  t)f  the 
sides  so  is  difference  of  sides  to  difference  of  divisions  of  base.  Halt  this 
difference  bebu?  added  to  or  subtracted  from  half  base  will  give  the  two  di- 
v S thereof;  Hence,  as  the  sides  and  their  opposite  division  of  base  con- 
stitute a right-angled  triangle,  the  perpendicular  thereof  is  readily  ascertained 
by  preceding  rules. 


Or, 


b c + c a X b c a,  c a 


b a 


— bd'X/da. 


Or, 


a c- -}- a b2  b c __a^.  -whence  V a c2  — ad2z=dc. 


2db 


Fig.  9. 


Example.— Three  sides  of  a triangle,  a be,  Fig.  9,  are  9.928, 
8,  and  5 feet;  what  is  length  of  perpeudicular  on  longest  side? 


’ As  9.928  : 8 + 5 : : 8 5 : 3.928  = difference  of  divisions  of 

the  base. 


Q.Q28  . 

Then  3.928  -f-  2 = 1.964,  which,  added  to  — — — 4-964  + 


!.  964  = 6 928  = length  of  longest  division  of  base. 


^TTpnre  there  is  a right-angled  triangle  with  its  base  6.928,  and  its  hypothenuse  8; 
consequenOyits  remainingside  or  perpendicular  is  V(82  - 6. 928*)  = 4 


When  anv  Two  of  the  Dimensions  of  a Triangle  and  One  of  the  corresponding 
Dimensions  of  a similar  Figure  are  given  and  it  is  required  to  ascertain 
the  other  corresponding  Dimensions  of  the  last  r igure. 

Fig.  10.  Fig.  ix.  , 

. Let  a be,  a ' b'  c',  be  two  similar  triangles,  Figs.  10 

and  11. 

Then  ab  :bc  ::  a'b'  : V c’ , or  a'b ' : b' c'  V.  ab  : be. 


Note  — Same  proportion  holds  with  respect  to  the 
similar  lineal  parts  of  any  other  similar  figures,  whether 


a V a plane  or  solid. 

Example. Shadow  of  a vertical  stake  4 feet  in  length  was  5 feet;  at  same  time, 

Shadow  of  a tree,  both  on  level  ground,  was  83  feet;  what  was  height  of  tree . 

5 a'  b’  : 4 b'  c* ::  83  ab  : 66.4  feet. 


To  Compute  Acreage. 

Divide  area  into  convenient  triangles,  and  multiply  base  of  each  triangle 
in  links  bv  half  perpendicular  in  links ; cut  off  5 figures  at  the  right,  remain- 
ing figures  will  give  acres ; multiply  the  5 figures  so  cut  off  by  4,  and  aga 
cut  off  5,  and  remainder  will  give  roods ; multiply  the  5 by  40,  and  abam 
cut  off  5 for  perches. 

Tr  ap  e z i um . 

Definition.— A Quadrilateral  having  unequal  sides  of  which  no  two  are  parallel. 


To  Compute  Area  of  a Trapezium.-Fig.  13. 

Rule. — Multiply  diagonal  by  sum  of  the  two  perpendiculars  falling  upon 
it  from  the  opposite  angles,  and  divide  product  by  2. 


db x a+c 
Or, • = area. 


Example. — Diagonal  d b,  Fig.  12,  is  125  feet,  and  perpen- 
— diculars  a and  c 50  and  37;  what  is  area? 


I2$  x 50  4-  37  = 10  875,  and  10  875  -4-  2 = 5437-  5 square  feet. 

F F 


338  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


When  the  Two  opposite  Angles  are  Supplements  to  each  other,  that  is,  when 
a Trapezium  can  be  inscribed  in  a Circle,  the  Sum  of  its  opposite  Angles 
being  equal  to  Two  Right  Angles , or  i8o°.  Rule. — From  half  sum  of  the 
four  sides,  subtract  each  side  severally ; then  multiply  the  four  remainders 
continually  together,  and  take  square  root  of  product. 

Example.— In  a trapezium  the  sides  are  15,  13,  14,  and  12  feet;  its  opposite  an- 
gles being  supplements  to  each  other,  required  its  area. 

15  -f  13  + 14  + 12  = 54>  and  — — 27. 

27  27  27  27  2 

15  13  14  12 

12X14X13X  15  = 32760,  and  -^32760  = 180.997  square  feet. 


Trapezoid. 

Definition. — A Quadrilateral  with  only  one  pair  of  opposite  sides  parallel. 

To  Compute  Area  of  a Trapezoid.— Fig.  13. 

Rule. — Multiply  sum  of  the  parallel  sides  by  perpendicular  distance  be- 
tween them,  and  divide  product. 

^ ab~\-dcxah  s-\-s'  xh 

Or, ' . Or,  — 1 = area , s and  s!  representing  sides. 

2 2 

Fig.  13.  a e p Example. — Parallel  sides  ab,cd , Fig.  13,  are  100  and  132 

feet,  and  distance  between  them  62. 5 feet ; what  is  area  ? 
100 -f- 132  x 62.5  = 14  500,  and  14  500  -4-  2 = 7250  square 
1 f^t. 

Polygons. 

Definition.— Plane  figures  having  three  or  more  sides,  and  are  either  regular  or 
irregular,  according  as  their  sides  or  angles  are  equal  or  unequal,  and  they  are  named 
from  the  number  of  their  sides  and  angles. 


li 


c h 


Regular  Polygons. 

To  Compute  Area  of*  a,  Itegixlar  iPolygon.—  Fig.  14. 
Rule.— Multiply  length  of  a side  by  perpendicular  distance  to  centre; 
multiply  product  by  number  of  sides,  and  divide  it  by  2. 

Or,  aJ>  * ce  = area , n representing  number  of  sides. 


Example.— What  is  area  of  a pentagon,  side  ab , Fig.  14,  being 
5 feet,  and  distance  ce  4. 25  feet ? 

5 X 4.25  X 5 (»)  = 106.25  — product  of  length  of  a side , dis- 
tance to  centre , and  number  of  sides. 

Then,  106.25-4-2  = 53-i25  square  feet. 


To  Compute  Radius  of  a Circle  that  contains  a Given 
Polygon. 

When  Length  of  a Perpendicular  from  Centre  alone  is  given.  Rule.— 
Multiply  distance  from  centre  to  a side  of  the  polygon,  by  unit  in  column  A 
of  following  Table. 

Example  — What  is  radius  of  a circle  that  contains  a hexagon,  distance  to  centre 
being  4.33  inches?  < 

4.33  X 1.156  = 5 mw- 


To  Compute  Length,  of  a Side  of  a Polygon  that  is  con-  \ 
tained  in  a Given  Circle. 

When  Radius  of  Circle  is  given.  Rule.— Multiply  radius  of  circle,  by 
unit  in  column  B of  following  Table. 

Example.— What  is  length  of  side  of  a pentagon  contained  in  a circle  8.5  feet  in 
diameter? 

8.5-4-2  = 4.25  radius,  and  4.25  X 1.1756  = 5 feet. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  339 


To  Compute  Radius  of*  a,  Circumscribing  Circle. 
When  Length  of  a Side  is  given.  Rule.— Multiply  length  of  a side  of  the 
polygon,  by  unit  in  column  C of  following  Table. 

Example. -What  is  radius  of  a circle  that  will  contain  a hexagon,  a side  being  5 
inches? 

5X1  — 5 ™s- 

To  Compute  Radius  of  a Circle  tliat  can.  be  Inscribed 
in  a Gfiven  Polygon. 

When  Length  of  a Side  is  given.  Rule.— Multiply  length  of  a side  of 
polygon,  by  unit  in  column  D of  following  Table. 

Example.— What  is  radius  of  the  circle  that  is  bounded  by  a hexagon,  its  sides 
being  5 inches? 

5 X- 866  = 4.33  ms. 


To  Compute  Area  of  a Ttegnlar  Polygon. 


When  Length  of  a Side  only  is  given.  Rule. — Multiply  square  of  side, 
by  multiplier  opposite  to  term  of  polygon  in  following  Table : 


No.  of 
Sides. 

Polygon. 

Area. 

A. 

Radius  of 
Circumscribed 
Circle. 

B. 

Length  of  a 
Side. 

C. 

Radius  of 
Circumscrib- 
ing Circle. 

D. 

Radius  of 
Inscribed 
Circle. 

3 

Trigon 

•43301 

2 

1 732 

-5773 

.2887 

4 

Tetragon 

1 

I-4I4 

1. 4142 

.7071 

• 5 

5 

Pentagon 

1.720  48 

1.238 

1.1756 

.8506 

.6882 

6 

Hexagon 

2. 598  08 

I.I56 

1 

1 

.866 

7 

Heptagon 

3-^33  91 

1. 11 

.8677 

I.I524 

1.0383 

8 

Octagon 

4.828  43 

1.083 

•7653 

1.3066 

1. 2071 

9 

Nonagon 

6. 181  82 

1.064 

.684 

1.4619 

1.3737 

10 

Decagon 

7.69421 

1.051 

.618 

1. 618 

1.5388 

11 

Un  decagon 

9.36564 

1.042 

•5634 

1-7747 

1,7028 

12 

Dodecagon 

11.196  15 

1037 

•5176 

I-93I9 

1.866 

Example.  — What  is  area  of  a square  (tetragon)  when  length  of  its  sides  is 
7.071067  8 inches? 

7.071 067  82  = 50,  and  50X1  = 50  square  ins. 


To  Compute  Length  of  a Side  and  If  adii  of  a Regular 
[Polygon. 

When  Area  alone  is  given.  Rule. — Multiply  square  root  of  area  of  poly- 
gon  by  multiplier  in  column  E of  the  following  table  for  length  of  side ; by 
multiplier  in  column  G for  radius  of  circumscribing  circle,  and  by  multiplier 
in  column  H for  radius  of  inscribed  circle  or  perpendicular. 


No.  of 
Sides. 

Polygon. 

E. 

Length  of 
Side. 

G. 

Radius  of 
Circumscrib- 
ing Circle. 

H. 

Radius  of 
Inscribed 
Circle. 

Angle. 

Angle  of 
Polygon. 

Tangent. 

3 

Trigon 

I-5I97 

.8774 

•4387 

120° 

6o° 

•5774 

4 

Tetragon 

1 

.7071 

-5 

90 

90 

1 

5 

Pentagon 

.7624 

.6485 

•5247 

72 

108 

1-3764 

6 

Hexagon 

.6204 

-6204 

•5373 

60 

120 

1.7321 

7 

Heptagon 

.5246 

.6045 

•5446 

51  25.71' 

128  34.29' 

2.0765 

8 

Octagon 

•4551 

.5946 

•5493 

45 

135 

2.4142 

9 

Nonagon 

.4022 

.588 

•5525 

40 

140 

2-7475 

10 

Decagon 

3605 

•5833 

.5548 

36 

144  . „ 

3.0777 

11 

Undecagon 

.3268 

•5799 

•5564 

32  43-64' 

147  16.36' 

3-4057 

12 

Dodecagon 

.2989 

•5774 

•5577 

30 

150 

3-7321 

Example  1.— Area  of  a square  (tetragon)  is  16  inches;  what  is  length  of  its  side? 
f 16  = 4,  and  4X1=4  ins. 

2.— Area  of  an  octagon  is  70.698  yards;  what  is  diameter  of  its  circumscribing 
circle? 

V70.698  x .5946  = 5,  and  5X2  = 10  yards. 


340  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 

Additional  Uses  of  foregoing  Table.—  6th  and  7 th  columns  of  table  facilitate  con- 
struction of  these1  figures  with  aid  of  a sector.  Thus,  if  it  is  required  to  describe  an 
octagon,  opposite  to  it  in  column  6th,  is  45;  then,  with  chord  of  60  on  sector  as 
radius,  describe  a circle,  taking  length  45  on  same  line  of  sector;  mark  this  dis- 
tance off  on  the  circumference,  which,  being  repeated  around  the  circle,  will  give 
points  of  the  sides. 

7th  column  gives  angle  which  any  two  adjoining  sides  of  the  respective  figures 
make  with  each  other;  and  8th  gives  tangent  of  the  angle  in  column  6th. 

To  Compute  Radius  of  Inscribed  or  Circumscribed 
Circles. 

When  Radius  of  Circumscribing  Circle  is  given.  Rule.— Multiply  radius 
given  by  unit  in  column  E,  in  following  Table,  opposite  to  term  of  polygon 
for  which  radius  is  required. 

When  Radius  of  Inscribed  Circle  is  given.  Rule.— Multiply  radius  given 
by  unit  in  column  F,  in  following  Table,  opposite  to  term  of  polygon  for 
which  radius  is  required. 

To  Compute  Area. 

When  Radii  of  Inscribed  or  Circumscribing  Circles  are  given.  Rule. — 
Square  radius  given,  and  multiply  it  by  unit  in  columns  G or  If,  in  following 
Table,  and  opposite  to  term  of  polygon  for  which  area  is  required. 

When  Length  of  a Side  is  given.  Rule.  — Square  length  of  side  and 
multiply  it  by  unit  in  column  I,  in  following  Table,  opposite  to  term  of 
polygon  for  which  area  is  required. 

To  Compute  Length.  of  a Side. 

When  Radius  of  Inscribed  Circle  is  given.  Rule. — Multiply  radius  given 
by  unit  in  column  K,  in  following  Table,  and  opposite  to  term  of  polygon  for 
which  length  is  required. 


No.  of 
Sides. 

Polygon. 

E. 

Radius  of 
Inscribed 
by  Circum- 
sci-ibing 
Circle. 

F. 

Radius  of 
Circumscrib- 
ing by 
Inscribing 
Circle. 

G. 

Area. 

By  Radius 
of  Inscribed 
Circle. 

H. 

Area. 
By  Radius 
of  Circum- 
scribing 
Circle. 

I. 

Area. 
By  Length 
of  Side. 

K. 

Length  of 
Side. 

By  Radius 
of  Inscribed 
Circle. 

3 

Trigon 

•5 

2 

5- 196  1 5 

1.299  04 

•433  01 

3.4641 

4 

Tetragon 

. 707  1 1 

1. 414  21 

4 

2 

1 

2 

5 

Pentagon 

. 809  02 

1.23607 

3.63272 

2.37764 

1.72048 

1.45308 

6 

Hexagon 

. 866  02 

I-I54  7 

3.464  1 

2.598  08 

2. 598  08 

I-I54  7 

7 

Heptagon 

.90097 

1. 109  92 

3.371  02 

2.73641 

3-633  91 

•96315 

8 

Octagon 

.923  88 

1.082  39 

3-3I37I 

2.828  42 

4.81843 

.82843 

9 

Nonagon 

•939  69 

1.064  1 8 

3-275  73 

2.89254 

6.182  82 

.72794 

10 

Decagon 

.95106 

1. 051  46 

3.2492 

2-938  93 

7.69421 

. 649  84 

11 

Undecagon 

•959  49 

1.042  22 

3.229  89 

2-973  53 

9-365  64 

• 587  25 

12 

Dodecagon 

•965  93 

1.03528 

3-215  39 

3 

11.196 15 

•535  9 

Regnlar  Bodies. 


To  Compute  Surface  or  Linear  Edge  of  Regular  Body. 

Rule. — Multiply  square  of  linear  edge,  or  radius  of  circumscribed  or  in- 
scribed sphere,  by  units  in  following  table,  under  head  of  dimension  used : 


No.  of 
Sides. 

Body. 

Surface  by 
Linear  Edge. 

Radius  of 
Circumscribed 
Sphere. 

Radius  of 
Inscribed 
Sphere. 

Linear  Edge 
by  Surface. 

4 

Tetrahedron 

1.73205 

1.632  99 

4. 898  98 

•759  84 

6 

Hexahedron 

6 

i-I54  7 

2 

. 408  25 

8 

Octahedron 

3.4641 

1. 414  21 

2.449  49 

•537  29 

12 

Dodecahedron 

20.64578 

•71364 

.89806 

. 220  08 

20 

Icosahedron 

8.66025 

1. 051  46 

1-323  i7 

•339  81 

Example. — What  is  surface  of  a hexahedron  or  cube,  having  sides  of  5 inches? 
52  x 6 = 25  X 6 = 150  square  ins. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  241 

To  Compute  Linear  Edge. 

When  Surface  alone  is  given.  Rule.— Multiply  square  root  of  surface, 
by  multiplier  opposite  to  term  of  body  under  head  of  Linear  Edge  by  Sur- 
face in  preceding  Table. 

Example. -What  is  linear  edge  of  a hexahedron,  surface  being  6 inches? 

V6  X .40825  = 1 inch. 

When  Radius  of  an  Inscribed  or  Circumscribed  Sphere  is  given.  Rule.— 
Multiply  radius  given,  by  multiplier  opposite  to  term  of  body  in  preceding 
Table,  under  head  of  the  Radius  given. 

Example.— Radius  of  circumscribing  sphere  of  a hexahedron  is  10  inches;  what 
is  its  linear  edge  ? 

10  X 1. 1547  = h-  547  ins- 
To  Compute  Surface. 

When  Linear  Edge  is  given.  Rule— Multiply  square  of  edge,  by  multi- 
plier opposite  to  term  of  body  in  preceding  Table,  under  head  of  Surface. 

Example.— Linear  edge  of  a hexahedron  is  1 inch;  what  is  its  surface? 
z2  X 6 = 6 square  ins. 

Irregular  Polygons. 

Definition.— Figures  with  unequal  sides. 

To  Compute  Area  of  an  Irregular  Lolygon.-Eigs.  IS 
and.  16. 


Fig.  15- 


Rule.— Draw  diagonals  and  per-  Fig.  16. 

pendiculars,  as  df  dg , a.  and  c,  Fig. 

/ 15,  and/ d,  gd,gb,gu,  and  i,  or  r,  and  j 
Fig.  16,  to  divide  the  figures  into 
triangles  and  quadrilaterals:  ascer- 
0 tain  areas  of  these  separately,  and 
take  their  sum.  g 

Note  —To  ascertain  area  of  mixed  or  compound  figures,  or  6> 

such  as  are  composed  of  rectilineal  and  curvilineal  figures  to- 

gether  computeareas  of  the  several  figures  of  which  the  whole  is  composed,  then 
add  them  together,  and  the  sum  will  give  area  of  compound  figure.  In  this  manner 
any  irregular  surface  or  field  of  land  may  be  measured  by  dividing  it  into  trapeziums 
and  triangles,  and  computing  area  of  each  separately. 

When  any  Part  of  a Figure  is  bounded  by  a Curve  the  Area  may  be  ascer- 
tained as  follows : 

Erect  any  number  of  perpendiculars  upon  base,  at  equal  distances,  and 

ascertain  their  lengths.  . . ,,  ... 

Add  lengths  of  the  perpendiculars  thus  ascertained  together,  mid  their 
sum,  divided  by  their  number,  will  give  mean  breadth ; then  multiply  mean 
breadth  by  length  of  base. 

To  Compute  Area  of  a.  Long,  Irregular  Eiguire.— Eig.  V7. 


Fig.  17. 


Rule.— Take  mean  breadths  at  several  places,  at  equal 
distances  apart,  as  1,  2,  3,  b d,  etc. ; add  them  together, 
divide  their  sum  by  number  of  breadths  for  total  mean 
breadth,  and  multiply  quotient  by  length  of  figure. 

+ etc. 

Or,  — - — A — l = area. 


F F* 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


342 


To  Concipixte 


Fig.  18. 


a 1 2 3 4 5 


a,x 1 Area  "bonxided  by  a Curve.-Fig.  18. 
(Simpson's  Rule.) 

Operation.— Divide  line  a b into  any  number  of  equal  parts, 
by  perpendiculars  from  base,  as  1,  2,  3)  etc.,  which  will  give 
an  odd  number  of  points  of  division.  Measure  lengths  of 
_ 5 these  perpendiculars  or  ordinates,  and  proceed  as  follows : 

To  sum  of  lengths  of  first  and  last  ordinates,  add  four  times  sum  of  lengths  of  all 
even  numbered  ordinates  and  twice  sum  of  odd;  multiply  their  sum  by  one  third 
of  distance  between  ordinates,  and  product  will  give  area  required. 

Illustration.— Water-line  of  a vessel  has  a length  of  90  feet,  and  ordinates  o,  1, 
1.2, 1.5,  2,  1.9,  1.5,  1. 1,  and  o,  each  10  feet  apart;  what  is  its  area? 

Ordinates. 

Even.  Odd.  Sums. 


1-5 
1.9 
1. 1 

ITs  x 4 = 22. 


i-5 


4-7  X 2 = 9.4 


first 
last  o 
even  22 
odd  9.4 

31.4  X 10  = 


: 314,  which  -f-  3 = 104.66  square  feet. 


Circle. 

Diameter  is  a right  line  drawn  through  its  centre,  bounded  by  its  periphery. 
Radius  is  a right  line  drawn  from  its  centre  to  its  circumference. 

Circumference  is  assumed  to  be  divided  into  360  equal  parts,  termed  degrees; 
each  degree  is  divided  into  60  parts,  termed  minutes ; each  minute  into  60  parts, 
termed  seconds  ; and  each  second  into  60  parts,  termed  thirds , and  so  on. 


To  Compute  Circumference  of  a Circle. 

Rule— Multiply  diameter  by  3.1416. 

Or,  as  7 is  to  22,  so  is  diameter  to  circumference. 

Or’  as  1 13  is  to  355,  so  is  diameter  to  circumference. 

Example.— Diameter  of  a circle  is  1.25  inches;  what  is  its  circumference? 

1.25  X 3.1416  = 3.927  ins. 

To  Coxxxpxxte  Diameter  of*  a Circle. 

Rule. — Divide  circumference  by  3.1416. 

Or,  as  22  is  to  7,  so  is  circumference  to  diameter. 

Note.  —Divide  area  by  .7854,  and  square  root  of  quotient  will  give  diameter  of  circle. 


To  Compute  Area  of*  a Circle. 

Rule.— Multiply  square  of  diameter  by  .7854. 

Or,  multiply  square  of  circumference  by  .079  58. 

Or,  multiply  half  circumference  by  half  diameter. 

Or,  multiply  square  of  radius  by  3.1416. 

Or,  p r2  = area , r representing  radius. 

Example.— The  diameter  of  a circle  is  8 inches;  what  is  the  area  of  it? 

82  = 64,  and  64  X -7854  = 50.2656  ins. 

Proportions  of*  a Circle,  its  ICcqxial,  IixscriUecl,  and  Cii  — 
cumscribed  Scpxares. 

CIRCLE. 

1.  Diameter  X .8862)  _ gide  0f  an  pqUal  Square. 

2.  Circumference  X .2821) 

3.  Diameter  X •7°7I) 

4.  Circumference  x .2251  > = Side  of  Inscribed  Square, 
e.  Area  X .9003  -4-  diam.  ) 

6.  Diameter  X 1.3468  = Side  of  an  Equilateral  Triangle. 


7.  A Side 

8. 


SQUARE. 

X 1. 1442  = Diameter  of  its  Circumscribing  Circle. 

X 4.443  = Circumference  of  its  Circumscribing  Circle. 

0.  ••  x 1.128  = Diameter  ) 

jo.  “ X 3-545  = Circumference  > of  an  Equal  Circle. 

11.  Square  inches  X 1-273  = Circle  inches  ) 

Note. Square  described  within  a circle  is  one  half  area  of  one  described  without  it. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  343 


To  Compute  Side  of  Greatest  Sqtiare  that  can  "be  In- 
scribed in  a Circle. 

Rule. — Multiply  diameter  by  .7071,  or  take  twice  square  of  radius. 
TJsefn.1  Factors. 

In  wliicli  p or  n represents  Circumference  of*  a Circle. 


Diameter  — 1. 


p=  3.141592653589+ 
2 p — 6.283 185  307  179+ 
4 p — 12. 566  370  614  359+ 
pz=.  1.570796326794-f 
p—  .785398163397+ 


|p  = 4.i88  79+ 
14  p=.  .523598-f 
%p=z  .392699+ 
T2  P=  .261  799+ 
■3  60 -P—  •' 008 726+ 

Diameter  = 10. 


Vp- 

1-772453 

s/i= 

.797884 

Log.  p = 

.49714987 

Y>y/P  = 

.886  226+ 

36  P — 

11 3-097  335+ 

1.  Chord  of  arc  of  semicircle 

2.  Chord  of  half  arc  of  semicircle 

3.  Versed  sine  of  arc  of  semicircle. 

4.  Versed  sine  of  half  arc  of  semicircle 

5.  Chord  of  half  arc,  of  half  of  arc  of  semicircle 

6.  Half  chord,  of  chord  of  half  arc 

7.  Length  of  arc  of  semicircle 

8.  Length  of  half  arc  of  semicircle 

9.  Square  of  chord,  of  half  arc  of  semicircle  (2) 

10.  Square  root  of  versed  sine  of  half  arc  (4) 

11.  Square  of  versed  sine  of  half  arc  (4) 

12.  Square  of  chord  of  half  arc,  of  half  arc  of  semicircle  (5) 

13.  Square  of  half  chord,  of  chord  of  half  arc  (6) 


=s  7.071067 
= 5 

= 1.464466 
— 3.82683 
= 3-535  533 
= 15-797  963 
= 7-853  9Sl 
= 50 

==  1.210151 
=3  2.144664 

==  14-644  67 

= 12.5 


Note.— In  all  computations  p is  taken  at  3.1416,  % p at  .7854,  % p at  .5236;  and 
whenever  the  decimal  figure  next  to  the  one  last  taken  exceeds  5,  one  is  added. 
Thus,  3.141  59  for  four  places  of  decimals  is  taken  as  3.1416. 


To  Compute  Length,  of  an  Arc  of  a Circle.— Fig.  19. 

When  Number  of  Degrees  and  Radius  are  given.  Rule  i.  — Multiply 
number  of  degrees  in  the  arc  by  3.1416  times  the  radius,  and  divide  by  180. 


2.— Multiply  radius  of  circle  by  .01745329,  and  product  by  degrees  in 
the  arc. 

If  length  is  required  for  minutes,  multiply  radius  by  .000  290  889 ; if  for 
seconds,  by  .000  004  848. 


Fig.  19. 


o 


0 


Example  i.—  Number  of  degrees  in  an  arc,  0 ah,  Fig.  19,  are 
90,  and  radius,  0 &,  5 inches;  what  is  length  of  arc? 

90  X (3.1416  X 5)  = 1413.72,  which  -4- 180  = 7.854  ins. 

2. — Radius  of  an  arc  is  10,  and  measure  of  its  angle  440  30' 
30";  what  is  length  of  arc? 

10  X .017  453  29  = . 174  5329,  which  X 44  = 7.6794476,  length 
for  44°. 


10  X .000  290  889  = .002  908  89,  which  x 30  = .087  266  7 , length  for  30'. 
10  X .000004  848  = .000048  48,  which  x 30  = .001  454  4,  length  for  30". 


Then  7.679  447  6 1 

.087  266  7 / ==  7.768  168  7 ins. 

.0014544) 

Or,  reduce  minutes  and  seconds  to  decimal  of  a degree,  and  multiply  by  it. 

See  Rule,  page  93.  30'  30"  = .5083,  and  .1745329  from  above  X 44.5083=: 

7.768 163  ins. 


344 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


When  Chord  of  Half  A rc  and  Chord  of  A ro  are  given.  Rule. — F rom  eight 
times  chord  of  half  arc  subtract  chord  of  arc,  and  one  third  of  remainder  will 
give  length  nearly. 


Or, 


8 c'  - 


, c'  representing  chord  of  half  arc,  and  c chord  of  arc. 


Example.— Chord  of  half  arc,  a c,  Fig.  19,  is  30  inches,  and  chord  of  arc,  ah,  48; 
what  is  length  of  arc  ? 

30  x 8 = 240  = 8 times  chord  of  half  arc  ; 240  — 48  = 192,  and  192  -7-  3 = 64  ins. 

When  Chord  of  Arc  and  Versed  Sine  of  Arc  are  given.  Rule.  — Mul- 
tiply square  root  of  sum  of  square  of  chord,  and  four  times  square  of  the 
versed  sine  (equal  to  twice  chord  of  half  arc),  by  ten  times  square  of  versed 
sine  • divide  this  product  by  sum  of  fifteen  times  square  of  chord  and  thirty- 
three  times  square  of  versed  sine ; then  add  this  quotient  to  twice  chord  of 
half  arc*  and  sum  will  give  length  of  arc  very  nearly. 


Or, 


^c2A~  4 v-  s^n-  X 10  v.  sm.  2 v s^n  representing  versed  sine. 


Hence 


: 7. 1599,  and  7.1599  + 100,  or  twice  chord  of  half  arc- 


15  c2+33  sin -2 

Example.— Chord  of  an  arc  is  80,  and  its  versed  sine,  c r,  30;  what  is  length  of  arc? 
g02  _ 6400  = square  of  chord  ; 302  = 900  = square  of  versed  sine. 

+(6400  + 900  X 4)  = 100  = square  root  of  square  of  chord  and  four  times  square 
of  versed  sine  ==  twice  chord  of  half  arc. 

Then  100  X 302  X 10  = 900  000  == product  of  10  times  square  of  versed  sine  and  root 
above  obtained. 

And  802  X 1 5 = 96  000  = 1 5 tiries  square  of  chord. 

302  x 33  = 29  700  = 33  times  square  of  versed  sine. 

125  700 

100  X 900000 
125  700 

107. 1599  length. 

When  Diameter  and  Versed  Sine  are  given . Rule.— Multiply  twice  chord 
of  half  the  arc  by  10  times  versed  sine ; divide  product  by  27  times  versed 
sine  subtracted  from  60  times  diameter,  add  quotient  to  twice  chord  o±  halt 
arc,  and  the  sum  will  give  length  of  arc  very  nearly. 

0r  ^ xiqt>.j^  + 2C>  = c, 

’ 60 d — 27  v.  sm. 

Example.— Diameter  of  a circle  is  100  feet,  and  versed  sine,  cr,  of  arc  25  ; what 
is  length  of  arc? 

V25  X 100  = 50  = chord  of  half  arc.  See  Rule,  page  345. 

50X2X25X10  = 25  000  = twice  chord  of  half  arc  by  10  times  versed  sine. 
VooXbo  — ~2sX2. 7 = 5325  = 27  times  versed  sine  from  60  times  diameter. 


Then 


25  000 

5325 


4.6948,  and  4.6948  + 50  X 2 = 104.6948  feet. 


To  Compute  CLord  of  an  Arc. 

When  Chord  of  Half  the  Arc  and  Versed  Sine  are  given.  Rule.— From 
square  of  chord  of  half  arc  subtract  square  of  versed  sine,  and  take  twice 
square  root  of  remainder. 

Or,  + (c*2  — v.  sin.2)  X 2 = c. 

Example.— Chord  of  half  arc,  a c,  is  60,  and  versed  sine,  c r,  36;  what  is  length 
of  chord  of  arc  ? 

602  — 36s  = 2304,  and  f 2304  X 2 = 96. 

* Square  root  of  sum  of  square  of  ehord  and  four  times  square  of  the  versed  sine  is  equal  to  twice 
«hord  of  half  arc. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  345 


When  Diameter  and  Versed  Sine  are  given . Multiply  versed  sine  by  2, 
and  subtract  product  from  diameter;  subtract  square  of  remainder  from 
square  of  diameter,  and  take  square  root  of  that  remainder. 

Or,  a/ d'2  — (d  — v.  sin.  X 2)2  = c. 

Example. — Diameter  of  a circle  is  100,  and  versed  sine  of  half  arc  is  36;  what  is 
length  of  chord  of  arc  ? 

(36  X 2 — 100)2  — 1002  =z  9216,  and  y/9216  = 96. 

To  Compute  Chord  of  FXalf  an  Arc. 

When  Chord  of  the  Arc  and  Versed  Sine  are  given.  Rule  i. — Divide 
square  root  of  sum  of  square  of  chord  of  the  arc  and  four  times  square  of 
versed  sine  by  two. 

2 —Take  square  root  of  sum  of  squares  of  half  chord  of  arc  and  versed 
sine. 


Or, 


■\/ c2  -j-  4 v.  sin . 2 


Or, 


a/(t 


-J-  v.  sin.2  = cr. 


When  Diameter  and  Versed  Sine  are  given.  Rule.— Multiply  diameter 
by  versed  sine,  and  take  square  root  of  their  product. 

Or,  a fd  X v.  sin.  — c'. 

To  Compute  Diameter. 

Rule  i. — Divide  square  of  chord  of  half  arc  by  versed  sine. 

Or,  c'2-rv.  sin.  = diameter. 

2 —Add  square  of  half  chord  of  arc  to  the  square  of  versed  sine,  and  divide 


this  sum  by  versed  sine. 


Or, 


(C  -r  2)2  -f  v.  sin.2  _ 


To  Compute  ~VersecL  Sine. 
Rule.— Divide  square  of  chord  of  half  arc  by  diameter. 


When  Chord  of  the  Arc  and  Diameter  are  given.  Rule.— From  square 
of  diameter  subtract  square  of  chord,  and  extract  square  root  of  remainder ; 
subtract  this  root  from  diameter,  and  divide  remainder  by  2. 

d — Vd2  — c2 
Or, = v.  sin. 

2 

When  it  is  greater  than  a Semidiameter.  Rule. — Proceed  as  before,  but 
add  square  root  of  remainder  (of  squares  of  diameter  and  chord)  to  diam- 
eter, and  halve  the  sum. 

^ d 4- Vd2  — c2 

Or,  — — = v.  sm. 

2 

Example. — Diameter  of  a circle  is  100,  and  chord  of  arc  97.9796;  what  is  its  versed 
sine? 


IOO  + V IOO2 97. 9796s  TOO  20 


60. 


To  Compute  Ordinate  of*  a Circnlar  Curve.—Fig.  SO. 

20‘  a/V2  — x2  — (r  — v)  = ordinate. 

Illustration. — Radius  of  circle  5 ins.,  versed  sine 
\ 2,  and  distance  x 2 ; what  is  length  of  ordinate  0 ? 

\ V'52  — 22  — (5  — 2)  = 4. 58  — 3 — i.  58  ins. 


346  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 
Sector  oF  a Circle. 

Definition. —A  part  of  a circle  bounded  by  an  arc  and  two  radii. 

To  Compute  Area  of  a Sector  of  a Circle. 

When  Degrees  in  the  Arc  are  given . — Fig.  21.  Rule. — As  360  is  to  num- 
ber of  degrees  in  a sector,  so  is  area  of  circle  of  which  sector  is  a part  to  area 
of  sector. 

Fig.  21.  . Or,  ==  area,  d representing  degrees  in  arc , and  a area 

v 7 , 306 

'u  of  circle. 

Example.  — Radius  of  a circle,  o a,  Fig.  21,  is  5 ins.,  and 
number  of  degrees  of  sector,  a b 0,  is  220  30' ; what  is  area  ? 

Area  of  a circle  of  5 ins.  radius  = 78. 54  ins. 

Then,  as  360°  : 220  30'  : 78-54  : 4-9o875  «*• 

When  Length  of  the  Arc , etc.,  are  given.  Rule.— Multiply  length  of  arc 
by  half  length  of  radius,  and  product  is  area. 

Or,  6 x r -f-  2 = area,  b representing  arc , and  r radius. 

Segment  of  a Circle. 

Definition.— A part  of  a circle  bounded  by  an  arc  and  a chord. 

To  Compute  Area  of  a Segment  of  a Circle. 

When  Chord  and  Versed  Sine  of  Arc,  and  Radius  or  Diameter  of  Circle  are 
given. 

When  Segment  is  less  than  a Semicircle , as  ah  c,  Fig.  21.  Rule.  Ascer- 
tain area  of  sector  having  same  arc  as  segment ; then  ascertain  area  of  tri- 
angle formed  by  chord  of  segment  and  radii  of  sector,  and  take  difference  of 
these  areas. 

Note.— Subtract  versed  sine  from  radius;  multiply  remainder  by  one  half  of 
chord  of  arc,  and  product  will  give  area  of  triangle. 

Or,  a — a'  — area , a and  a'  representing  areas  of  sector  and  triangle. 

When  Segment  is  greater  than  a Semicircle.  Rule.  Ascertain,  by  pre- 
ceding rule,  area  of  iesser  portion  of  circle  5 subtract  it  from  area  of  whole 
circle,  and  remainder  will  give  area. 

Or,  a — a'  — area , a and  a'  representing  areas  of  circle  and  lesser  portion. 

See  Table  of  Areas  of  Segments,  page  267. 

Fig.  22.  , Example.  — Chord,  a c , Fig.  22,  is  14.142;  diameter,  h e,  is  20 

ins. ; and  versed  sine,  h r,  is  2.929;  "what  is  area  of  segment  r 
14. 142  -r-  2 = 7.071  = half  chord  of  arc. 

V7.Q7i2+'2W  = 7.654  = square  root  of  sum  of  squares  of 
half  chord  of  arc  and  versed  sine , which  is  chord  ah  of  half  arc 
ah  c. 

By  Rule,  page  346,  . , 

7.654  X 2 X 2.929  X 10  = 448.371  = twice  chord  of  half  arc  hy  xo 
times  versed  sine. 

20  ><"6^2.929  x 27  = 1x20.917  = 60  times  diameter  subtracted  from  27  times  I 

versed  sine.  _ 

Then  448. 371  - 1120.917  = .4,  and  .4  added  to  7.654  x 2 (twice  chord  of  half  arc) 

= 15.708  inches , length  of  arc. 

By  Rule  above,  15.708  X ” = 78.54  = ®*  arc  multiplied  by  half  length  of  radius, 

— area  of  sector.  . . . 

10  — 2.929  — 7. 071  = versed  sine  subtracted  from  a radius , which  is  height  oftn - 

angle  aoc,  and  7.071  X 50  = area  of  triangle. 

Consequently,  78. 54  — 50  = 28. 54. 


mensuration  of  areas,  lines,  and  surfaces.  347 


When  the  Chords  of  Arc,  and  of  half  of  Arc,  and  Versed  Sine  are  given. 
Rule.— To  chord  of  whole  arc  add  chord  of  half  arc  and  one  third  of  it 
more ; multiply  this  sum  by  versed  sine,  and  this  product,  multiplied  by 
.404  26,  will  give  area  nearly. 

Or,  c _j_  c'  — u.  sin.  X • 404  26  = area  nearly. 

Example.— Chord  of  a segment,  a c,  Fig.  22,  is  28  feet;  chord  of  half  arc,  a b,  is 
15;  and  versed  sine,  b r,  6;  what  is  area  of  segment? 

20  _p  l5  — = chord  of  arc  added  to  chord  of  half  arc  and  one  third  of  it  more. 

4g  x 6 = 288=:  product  of  above  sum  and  versed  sine.  Hence  288  X . 4°4  26  = 1 l6-  427 
square  feet. 

When  the  Chord  of  A rc  and  Versed  Sine  only  are  given.  Rule.— Ascer- 
tain chord  of  half  arc,  and  proceed  as  before. 

r_p0  Compute  Cliorcl  and.  Ileiglit  of  a Segment  of  a Circle. 

When  A rea  is  given.  Rule.— Divide  area  by  square  of  diameter  of  circle, 
take  tab.  height  for  area  from  table  of  Areas  of  Segments  of  a Circle,  p.  267, 
multiply  it  by  diameter,  and  product  will  give  required  height. 

From  diameter  subtract  height,  multiply  remainder  by  height,  take  square 
root  of  product  and  multiply  it  by  2 for  required  chord. 

Or  — = (tab.  area  for  height)  X d =4,  and  Vd  — h X h X 2 = c. 

’ d2 

Circular  Measure.  (See  Rule,  page  1 13.) 

Sphere. 

Definition. — A figure,  surface  of  which  is  at  a uniform  distance  from  centre. 

To  Compute  Convex  Surface  of  a Spliere.— !Kig.  S3. 

Rule.— Multiply  diameter  by  circumference,  and  prod- 
uct will  give  surface. 

Or,  4 p r 2 = surface.  * Or,  pd2  = surface.  • 
Example.— What  is  convex  surface  of  a sphere,  Fig.  23,  hav- 
ing a diameter,  a b,  of  10  ins? 

10  X 31-416  = 314.16  square  ins. 

Segment  of  a Spliere. 

Definition.— A section  of  a sphere. 

To  Compute  Surface  of  a Segment  of  a Spliere.— If  ig.  24. 

Rule. — Multiply  height  by  the  circumference  of  sphere,  and  add  product 
to  the  area  of  base. 

Or,  2 prh  — convex  surface  alone. 

Example.  — Height,  bo,  of  a segment,  a b c , Fig.  24,  is  36  ins., 
and  diameter,  b e , of  sphere  100 ; what  is  convex  surface,  and 
what  whole  surface? 

36  X 100  X 3- 1416  mu  309. 76  = height  of  segment  multiplied  by 
circumference  of  sphere. 

To  ascertain  area  of  base  ; diameter  and  versed  sine  being 
given,  diameter  of  base  of  segment,  being  equal  to  chord  of  arc, 
is,  by  Rule,  page  347, 

100  — 36  X 2 = 28 ; V1002  — 28s  = 96- 
96s  X • 7854  = 7238. 2464  = convex  surface , and  7238. 2464  -f  1 1 309. 76  = 18  548. 0064 
= convex  surface  added  to  area  of  base  — square  ins. 

Note. — When  convex  surface  of  a figure  alone  is  required,  area  or  areas  of  base 
or  ends  must  be  omitted. 


* j>  or  k represents  in  this,  and  in  all  cases  where  it  is  used,  ratio  of  circumference  of  a circle  to  its 
diameter,  or  3.1416. 


348  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


When  the  Diameter  of  Base  of  Segment  and  Height  of  it  are  alone  given. 
Rule. — Add  square  of  half  diameter  of  base  to  the  square  of  height ; divide 
this  sum  by  height,  and  result  will  give  diameter  of  sphere. 

2 

Or,  d -T-  2 -f  h2  -4-  h = diameter. 

Spherical  Zone  (or  Frustum  oF  a Sphere). 
Definition.— The  part  of  a sphere  included  between  twa  parallel  chords. 


To  Compute  Surface  of  a Spherical  Zone.— Fig.  25. 


Fig.  25. 


Rule. — Multiply  height  by  the  circumference  of  sphere, 
and  add  product  to  area  of  the  two  ends. 

Or,  h c -f-  a -\-  a'  = surface. 

Or,  2 p r h = convex  surface  alone. 

Example.  — Diameter  of  a sphere,  a b,  Fig.  25,  from  which  a 
zone,  c g , is  cut,  is  25  inches,  and  height,  eg,  is  8 ; what  is  convex 
surface  ? 


25  X 3.1416  X 8 = 628.32  = height  X circumference  of  sphere  = square  ins. 


When  the  Diameter  of  Sphere  is  not  given.  Rule. — Multiply  mean  length 
of  the  two  chords  by  half  their  difference ; divide  this  product  by  breadth 
of  zone,  and  to  quotient  add  breadth.  To  square  of  this  sum  add  square  of 
lesser  chord,  and  square  root  of  their  sum  will  give  diameter  of  sphere. 


Or, 


= — d. 


Spheroids  or  Ellipsoids. 


Definition. — Figures  generated  by  the  revolution  of  a semi-ellipse  about  one  of 
its  diameters. 

When  revolution  is  about  Transverse  diameter  they  are  Prolate,  and  when  it  is 
about  Conjugate  they  arc  Oblate. 


Fig.  26 


To  Compute  Surface  of  a Spheroid..— Fig.  26. 

When  Spheroid  is  Prolate.  Rule. — Square  diameters,  and  multiply 
square  root  of  half  their  sum  by  3.1416,  and  this  product  by  conjugate 
diameter. 

Or,  X 31416  Xd  — surface,  d and  d ' represent- 

ing conjugate  and  transverse  diameters. 

Example. — A prolate  spheroid,  Fig.  26,  has  diameters,  cd 
and  a b,  of  10  and  14  inches;  what  is  its  surface? 

io2  -j-  142  — 296  = sum  of  squares  of  diameters. 
296-4-2  = 148.  and  Vi 48  = 12.1655  = square  root  of  half 
sum  of  squares  of  diameters. 


12.1655  X 3.1416  X 10  = 382.191  ins.  = product  of  root  above  obtained  X 3-I4I6,  : 
and  by  conjugate  diameter. 

When  Spheroid  is  Oblate.  Rule.— Square  diameters,  and  multiply  square  j 
root  of  half  their  sum  by  3.1416,  and  this  product  by  transverse  diameter.  ) 


/d2A-d'2  1 

Or,  / — ~ — X 3. 1416  Xd'  = surface. 

Example.— An  oblate  spheroid  has  diameters  of  14  and  10  inches;  what  is  its 

surface?  , 

I22_ j_  io2  = 296  = su?n  of  squares  of  diameters. 

296  -4-  2 = 148,  and  Vm8  = 12. 1655  = square  root  of  half  sum  of  squares  of  di- 
ameter. 

12.1655  X 3.1416  X 14  = 535-0679  ins.= product  of  root  above  obtained  X 3-j4i6» 
and  by  transverse  diameter. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  349 


To  Compute  Convex  Surface  of  a Segment  of*  a Sphe- 
roid.— Figs.  ST  and.  S8. 

Rule— Square  diameters,  and  take  square  root  of  half  tlieir  sum;  then, 
as  diameter  from  which  the  segment  is  cut  is  to  this  root,  so  is  the  height 
of  segment  to  proportionate  height  required.  Multiply  product  of  other  di- 
ameter and  3.1416  by  proportionate  height  of  segment,  and  this  last  product 
will  give  surface. 


Or, 


Vd2-M'2-^- 2 


XhXd'  or  d X 3. 1416  — surface. 


d or  d ' 

Example.  — Height,  a o,  of  a seg- 
ment, efoi  a prolate  spheroid;  Fig. 
27,  is  4 inches,  diameters  being  10  and 


Fig.  28. 


what  is  convex  surface  of  it  ? 


. m\: 

— ~i&  Square  root  of  half  sum  of  squares 

of  diameters,  12.1655. 

Then- 14: 12.1655  1 : 4 : 3.4758  — height 
cl  of  segment,  proportionate  to  mean  of 

diameters , and  10  X 3- 14^  X 3-4758  = 109. 1957  ins. 

2 Height  co  of  a segment  of  an  oblate  spheroid,  Fig.  28,  is  4 inches,  the  diam- 
eters being  14  and  10;  what  is  convex  surface  of  it?  214.0272  square  ms. 

To  Compute  Convex  Surface  of  a,  Frustum  or  Zone 
of  a Spheroid.  — Figs.  29  and  30. 

Rule.— Proceed  as  by  previous  rule  for  surface  of  a segment,  and  obtain 
proportionate  height  of*' frustum ; then  multiply  product  of  diameter  par- 
allel to  base  of  frustum  and  3.1416  by  proportionate  height  of  frustum,  and 
it  will  give  surface. 

Fig.  2Q.  Example.— Middle  frustum,  0 e,  of  Fig.  30. 

c a prolate  spheroid,  Fig.  29.  is  6 inch-  * 

es,  diameters  of  spheroid  being  10 
and  14;  what  is  its  convex  surface? 

Mean  diameter,  as  per  preceding 
example,  is  12.1655. 

Diameter  parallel  to  base  of  frus- 
tum is  10.  ~d~ 

Then  14  : 12.1655  ::  6 : 5.2138,  and  10  X 3.1416  X 5-2138  = 163.7967  square  ins. 

2.— Middle  frustum  of  an  oblate  spheroid,  as  0 e , Fig.  30,  is  2 inches  in  height, 
diameters  of  spheroid,  as  in  preceding  examples,  being  10  and  14;  what  is  its  con- 
vex surface  ? io7-  01 36  square  ins. 

CircmlaT?  Zone. 

Definition.— A part  of  a circle  included  between  two  parallel  chords. 

To  Compute  Area  of  a Circular  Zone. 

Rule. — From  area  of  circle  subtract  areas  of  segments. 

Or,  see  Table  of  Areas  of  Zones,  page  269. 

When  Diameter  of  Circle  is  not  given—  Multiply*  mean  length  of  the  two 
chords  by  half  their  difference ; divide  this  product  by  breadth  of  zone,  and 
to  quotient  add  the  breadth. 

To  square  of  this  sum  add  square  of  lesser  chord,  and  square  root  of  their 
sum  will  give  diameter  of  circle. 

Example.— Greater  chord,  h g,  is  90  inches;  lesser,  a c,  is  80;  and  breadth  of  zone. 
a 0,  is  72.526;  what  is  its  diameter? 


80  + 9°  x 


— - = 85  X 5 = 425,  and  425---f  72.526  = 78.385. 
72.520 


Then  V 78.38s2  -+-  802  —.f  12  544.2  ==  112  = diameter. 

Gg 


350  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


Fig.  31. 


Cylinder. 

Definition.— A figure  formed  by  revolution  of  a right-angled  parallelogram  around 
one  of  its  sides. 

To  Compute  Surface  of  a Cylinder.— Fig.  31. 

Rule. — Multiply  length  by  circumference,,  find  acid  product  to  area  of 
the  two  ends. 

Or,  l c -j-  2 a = s,  a representing  area  of  end. 

Note. — When  internal  or  convex  surface  alone  is  wanted,  areas  of 
ends  are  omitted. 

Example. — Diameter  of  a cylinder,  ho,  Fig.  31,  is  30  inches,  and  its 
length,  a b , 50;  what  is  its  surface? 

30  X 3.1416  = 94.248,  and  94.248  X 50  = 4712.4. 

Then  302  X .7854  = 706.86  = area  of  one  end;  706.86  X 2 = 1413.72 
= area  of  both  ends , and  4712.4-}-  1413.72  =6125.12  square  ins. 

Prisms. 

Definition.  — Figures,  sides  of  which  are  parallelograms,  and  ends  equal  and 
parallel. 

Note.-— When  ends  are  triangles,  they  are  termed  triangular  prisms  ; when  they 
are  square,  square  or  right  prisms  ; and  when  they  are  a pentagon,  pentagonal 
prisms , etc. 

To  Compute  Surface  of*  a iRiglut  Frism.— Figs.  32  and  33. 


Fig.  32. 


Fig.  34- 


Rule. — Ascertain  areas  of  ends  and  sides,  and  Fig.  33. 
add  them  together.  ® 

Or,  2 a-j-na'  =:  s,  a representing  area  of  ends , a'  area 
of  sides,  and  n their  number. 

Example.  — Side,  a b,  Fig.  32,  of  a square  prism  is  12 
inches,  and  length,  b c,  30;  what  is  surface? 

12X12  = 144  = area  of  one  end  ; 144  X 2 = 288  = area 
of  both  ends ; 12X30  = 30q  = a rea  of  one  side  ; 360  X 4 = 

1440  = area  of  four  sides , and  288  -f- 1440  = 1728  sq.  ins. 

To  Compute  Surface  of  an  ODliqne  or  Irregular  Prism.— 
Fig.  34. 

Rule. — Multiply  circumference  of  one  end,  by  perpendic- 
% a ular  height,  a 0.  Or,  multiply  circumference,  c,  at  a right 
angle  to  sides  by  actual  length  of  figure,  and  add  area  of  ends. 

Example. — Sides,  q c,  of  an  oblique  hexagonal  prism,  Fig.  34,  are 
10  inches,  and  perpendicular  height,  a 0,  is  5 feet;  what  is  its  sur- 
face ? 

To  X 6 = 60  ins.  — length  of  sides. 

60  X 5 X 12  ±=  3600  square  ins.  — area  of  sides,  and  by  table,  page 
342,  102  X 2.59808  X 2 = 519.616  square  ins.,  which  added  to  3600  = 
4x19.616  square  ins. 

Wedge. 

is  a prolate  triangular  prism,  aiid  its  surface  is  computed 
by  rule  for  that  of  a right  prism. 

To  Compute  Surface  of*  a AV edge.— Fig.  35. 


Definition. — A we 


Fig.  35- 


Example. — Back  of  a wedge,  abed,  Fig.  35,  is  20  by  2 inches, 
and  its  end,  ef,  20  by  2;  what  is  its  surface  ? 


202  -f-  2 -=  1 = 401  = sum  of  squares  of  half  base , af  and 
height,  ef,  of  triangle,  efa. 

V401  = 20.025  = square  root  of  above  sum — length  of  e a. 
Then  20.025  X 20X2  = 801  = area  of  sides. 

And  20X2  = 40  = area  of  back ; and  20X2  = 2X2  = 40  = 
area  of  ends.  Hence  801  -}-  40  -f-  40  = 881  square  ins. 


MENSU 


iration  of  areas,  lines,  and  subfaces.  351 


Prismoids. 

Definition— Figures  alike  to  a prism,  having  only  one  pair  of  sides  parallel. 
To  Compete  Surface  of  a Prismoid.-Fig.  30. 

a 1 • -C  Art  n n n OllfiQ  Q 


Fig.  36. 


Fig.  38. 


Fig- 37- 


iT>u.te  ^ ^ — ~ „ 

Rule  - Ascertain  area  of  sides  and  ends  as  by  rules  for 
squares,  triangles,  etc.,  and  add  them  together. 

Example.  -Ends  of  a prismoid,  efg  h and  a ^^U^ts’s^rfa^^ 

8 inches  square,  and  its  slant  height,  d h,  25,  "“at  is  its  suriace . 
jo  X 10  = 100  = area  of  base ; 8 X 8 = 64  — area  of  top. 

i°  + 8 x 25  = 225,  and  225X4  = 9®- area  °fsides- 
„ Then  100  + 64  4"  900  = 1064  = square  ins. 

To  compute  Surface  of  aa  Otolictue  or  Irregular  Prismoid. 
Proceed  undirected  for  an  Oblique  or  Irregular  Prism,  page  350. 

TJrigi^las. 

tbe  base> 

to  compute  Curved^  oPau  ^ula.-Figs.  3., 

When  Section  is  parallel  to  Axis  of  the  Cylinder , Fig.  37-  *»«  *•  Mul‘ 
tinlv  len°th  of  arc  of  one  end  by  height. 

1 ‘ ° Example. Diameter  of  a cylinder,  a c,  from  which  an 

inmilV  Eicr  -27  is  cut  is  io  inches,  its  length,  b d,  50,  and 
verSsed%fnf  o3r?depth  of  ungula  is  5 inches;  what  Is  curved 
surface  ? - ^ rad{u$  0fCyUnder. 

Hence  radius  and  versed  sine  are, equal;  the  arc,  there- 
fore of  ungula  is  one  half  circumference  of  the  cylinder, 

L whtcA  is  31.416  15-708,  and  15.708X50  = 785-4 

" square  ins. 

When  Section  passes  obliquely  through 
inder  Fig.  38.  Rule  2,-Multiply  circumference  of  base  of  cylinder  by 

half  sum  of  greatest  and  1 ^mdrica"^ 2L  Fig.  38,  is  10  inches,  and  great- 
35  and  15  inches;  what  is  its  --ed  surface? 

10  diameter  =31. 416  circumference;  25  + 15  = 4°)  an<*  40—2  = 20.  Hence  3M.6 
X 20  = 628.32  square  ijis. 

When  See, in,  pM  *»«.  If  Sit 

WK^i&T4SWS5  s,  3 i«ii  .w  -i ».  «i  I--,  i,/. 

cn  cle,  Fig.  39.  ,.^  t q from  this  product  subtract  product  of  aic 

and“rHeof’  fi;S?di«  thus  found  by  quotient  of  height,  y c, 

the  lpngest  chord  that  can  he  drawn  in  basa 
what  is  curyed  surface?  ■ 7.  • . 

5X10  = 5°  = sine  of  half  arc  by  diameter. 

, 5^08, ^d'^ver^tTsine  ami  radiu^are  equalj  cosine  is^' 

SO  x I^Fs  = 5°  X 2 = 100  square  mS'  - 


* When  the  cosine  is  o,  this  product  is  o. 


352  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


ceeds 
Fig.  40. 


When  Section  passes  through  Base  of  Cylinder , and  Versed  Sine , a g,  ex - 
ids  Sine , <??•  when  Base  exceeds  a Semicircle , Fig.  40.  Rule  4. — Multiply 
sine  of  half  the  are  of  base  by  diameter  of  cylinder,  and  to  this 
product  add  product  of  arc  and  the  excess  of  versed  sine  over 
the  sine  of  base.  Multiply  sum  thus  found  by  quotient  of 
height  divided  by  versed  sine. 

Example.— Sine,  a d,  of  half  arc  of  an  ungula,  Fig.  40,  is  12  inches; 
versed  sine,  a g,  is  16;  height,  c g,  16;  and  diameter  of  cylinder,  h g , 
25  inches;  what  is  curved  surface? 

12  X 25  = 300  = sine  of  half  arc  by  diameter  of  cylinder,  and  length 
of  arc  of  base,  Rule,  page  344  = arc  of  d h f — circumference  of  base  = 
46-  392- 

Then  46.392X16  — 12.5  = 162.372,  and  300-F  162.272  = 462.372;  16 -4- 16  = i,  and 
462.372  X 1 =462.372  square  ins. 

Fig.  41.  Note.— When  sine  of  an  arc  is  o,  the  versed  sine  is  equal  to  diameter. 

When  Section  passes  obliquely  through  both  Ends  of  Cylinder , 
Fig.  41.  Rule  5. — Conceive  section  to  be  continued  to  m , till  it 
meets  side  of  cylinder  produced ; then,  as  difference  of  versed 
sines,  a e and  d o,  of  arcs  of  two  ends  of  ungula  is  to  versed  sine, 
a e,  of  arc  of  the  less  end,  so  is  height  of  cylinder,  a cZ,  to  the 
part  of  side  produced. 

Ascertain  surface  of  each  of  ungulas  thus  found  by  Rules  3 
and  4,  and  their  difference  will  give  curved  surface. 

Lune. 

Definition. — Space  between  intersecting  arcs  of  two  eccentric  circles. 

To  Compute  Area,  of  a Lune.— Fig.  42 . 

Rule. — Ascertain  areas  of  the  two  segments  from  which  lune  is  formed, 
and  their  difference  will  give  area. 


Fig.  42.  d 


Example.— Length  of  chord  ac,  Fig.  42,  is  20  inches,  height 
ed  is  3,  and  e b 2 ; what  is  area  of  lune  ? 

By  Rule  2,  page  345,  diameters  of  circles  of  which  lune  is 
formed  are  thus  ascertained: 


For  a d c , 


io2  + (3  + 2)2 


-t=  25.  For  aec , 


IO2— {—  22 


Then,  by  Rule  for  Areas  of  Segments 
of  a Circle,  page  267, 


Fig.  43- 


Area  of  ad c is  70.5577  sq.  ins. 

11  aec  “ 27.1638  “ 

Their  difference  43.3939  sq.  ins. 

Note.  — If  semicircles  be  described  on  the  three  sides  of  a right-angled  triangle 
as  diameters,  two  Junes  will  be  formed,  and  their  united  areas  will  be  equal  to  that 
of  triangle. 

Cycloid.. 

Definition. — A curve  generated  by  revolution  of  a circle  on  a plane. 

To  Compute  Area  of  a Cycloid.— Trig.  43. 

Rule. — Multiply  area  of  generating  circle  by  3. 
Example.— Generating  circle  of  a cycloid,  abc.  Fig.  43, 
has  an  area  of  1 15s. 45  sq.  inches;  what  is  area  of  cycloid  ? 
i*5-45  X 3 = 346.$5  *<luare  ins. 

To  Compute  Length,  of  a Cycloidal  Carve. 

Rule. — Multiply  diameter  of  generating  circle  by  4. 

Example. — Diameter  of  generating  circle  of  a cycloid,  Fig.  43,  is  8 inches;  what 
is  length  of  curve  d s c ? 

8 X 4 = 32  = product  of  diameter  and  4 = ins. 

Note. — The  curve  of  a cycloid  is  line  of, swiftest  descent;  that  is,  a body  will  fall 
through  arc  of  this  curve,  from  one  point  to  another,  in  less  time  than  through  any 
other  path. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  353 


Circular  Rings. 

Definition.— Space  between  two  concentric  circles. 

To  Compute  Sectional  Area  of  a Circular  Riiig.-Fig.  4*4,. 
Rule.— From  area  of  greater  circle  subtract  that  of  less. 

Cylindrical  Rings. 

Definition.  —A  ring  formed  by  curvature  of  a cylinder. 

To  Compute  Surface  of  a Cylindrical  Ring.-Fig.  4:4:. 
£ULE  _To  diameter  of  body  of  the  ring  add  inner  diameter  of  the  ring; 
multiply  this  sum  by  diameter  of  the  body,  and  product  by  9.8696. 

* Or,  c X l = surface. 

Example. -Diameter  of  body  of  a cylindrical  ring  a b,  Fig  44, 
innor  riinmpffir  hr.  is  t8:  what  is  surface  01  it: 


is  inches,  and  innerdiameter^V  c,  is  x8;  what  is  surface  of  it? 

2 + 18  = 20=  thickness  of  ring  added  to  inner  diameter. 

2Q  X 2 X 9.8696  = sum  above  obtained  X thickness  of  ring,  and 
that  product  by  9.8696  = 394- 784  ins- 


Link. 


Definition. — An  elongated  ring. 


To  Compute  Surface  of  a Link.  Uigs.  4o  and  4:6. 
Rule.— Multiply  length  of  axis  of  link  by  circumference  of  a section  of 

bod>'>ai-  , ’ Or,lxc  = mrfm. 


To  Compute  Length  of  Axis  and  Circumference. 
When  Rina  is  Elongated.  Rule.-To  less  diameter  add  the  diameter  of 
the  body  of 'the  link,  and  multiply  sum  by  3.1416 ; subtract  less  diameter 
from  greater,  multiply  remainder  by  2,  and  sum  of  these  products  is  length 


Fig.  45- 


of  axis. 


L liAlS. 

Example. -Link  of  a chain,  Fig.  45.  is  1 inch  m diameter 
of  body,  a 6,  and  its  inner  diameters,  b c and  ef  are  12.5 


UI  UUUV  , LO  t/,  wau.  n Cl 

and  2. 5 inches ; what  is  its  circumference  ? 

2.5-j-x  x 3.1416  — 10. 9956  = length  of  axis  of  ends. 

J 2. 3 __  0.  5,  X 2 = 20  = length  of  sides  of  body. 

Then  10.9956  + 20  = 30.9956  = length  of  axis  of  link , and 
— 30.9956  X 3- 1416,  (cir.  of  1 inch)  = 97. 375s  square  ins. 

When  Ring  is  Elliptical,  Fig.  46-  Rule.— Square  diameters  of  axes  of 
ring,  multiply  square  root  of  half  their  sum  by  3.1416,  and  product  is  length 

of  axis.  _ 

Cones. 

Definition.  - A figure  described  by  revolution  of  a right-angled  triangle  about 
one  of  its  legs. 

For  Sections  of  a Cone,  see  Conic  Sections,  page  379. 


To  Compute  Snrface  of  a Cone.-Fig.  . 

Rule —Multiply  perimeter  or  circumference  of  base  by  slant  height,  or 
side  of  cone ; divide  product  by  2,  and  add  the  quotient  to  area  of  the  base. 

0r  c X A +2  + d'  ==  surface , c representing  perimeter. 

FiS'  47'  A Example. -Diameter,  a ft,  Fig.  47,  of  base  of  a cone  is  3 feet, 

^ and  slant  height,  a c,  15 ; what  is  surface  of  cone . 


Circum.  of  3 feet  = 9. 4248,  and  9-  4248  * Jl  = JQ.  68 6 - sur- 


^ face  of  side;  area  of  base  3=7.068,  and  70.686+7.068-77-754 
square  feet. 

ft  a* 


354  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


To  Compute  Surface  of  the  Frustum  of*  a Cone.— 
Fig.  48. 

Rule. — Multiply  sum  of  perimeters  of  two  ends  by  slant  height  of  frus- 
tum ; divide  product  by  2,  and  add  it  to  areas  of  two  ends. 

- c -b  c'  X h 

Or, f -a-j-a  = surface. 

2 

Example.— Frustum,  abed , Fig.  48,  has  a slant  height,  cd,  of  26  inches,  and 
« circumferences  of  its  ends  are  15.71  and  22  inches  respectively; 

g'  4 ' A what  is  its  surface? 

15.71 + 22  X 26 


- = 490. 23  = surface  of  sides  ; X .7854 


d + 


/ 22 
\3M 


7)  X .7854  = 58.119  = areas  of  ends.  Then  496.23 -f- 
4It>/ 

58. 1 19  =548. 349  square  ins. 


Fig.  49. 


Pyramids. 

Definition. — A figure,  base  of  which  has  three  or  more  sides,  and  sides  of  which 
are  plane  triangles. 

To  Compute  Surface  of  a Pyramid.  — Figs.  4r0  and.  £30. 

Rule. — Multiply  perimeter  of  base  by  slant  height;  divide  product  by  2, 
and  add  it  to  area  of  base. 

_ ch  . . Fig.  50. 

Or, (-  a = surface. 

2 

Example.— Side  of  a quadrangular  pyramid,  a b, 

Fig..  49,  is  12  inches,  and  its  slant  height,  a c,  40; 
what  is  its  surface? 

12  X 4 = 48  = perimeter  of  base.  48  * — =.  960  = 

2 !' 

area  of  sides,  and  12  X 12  -}-  960  = 1104  square  ins. 

To  Compute  Surface  of  Frustum  of  a Pyramid.- 
Fig.  SI. 

Rule. — Multiply  sum  of  perimeters  of  two  ends  by  slant  height;  divide 
product  by  2,  and  add  it  to  areas  of  ends. 

~ c.tf  c'  xh  , . , 

Or, \- a -\- a — surface. 

2 !•  . , ...  . 

Example. — Sides  a b,cd,  Fig.  5*,  of  frustum  of  a quadrangular 
pyramid  are  10  and  9 inches,  and  its  slant  height,  a c,  20;  what 
is  its  surface? 

10X4  = 40,  and  9 x 4 = 36 ; 40  -f  36  = 76  = sum  of  perimeters. 

76  X 20  = 1520,  and  I^2-  = 760  = area  of  sides  ; 10  X 10  = 100, 
2 

and  9X9  = 81.  Then  100  -f-  81  -f-  760  = 941  = square  ins. 

When  Pyramid  is  Irregular  sided  or  Oblique.  Rule.  — The  surfaces  of 
each  of  the  sides  and  ends  must  be  computed  and  added  together. 

Helix  (Screw). 

Definition..— A line  generated  by  progressive  rotation  of  a point  around  an  axis 
and  equidistant  from  its  centre. 

To  Compute  Length,  of  a Helix.— Fig.  £5Q. 

Rule. — To  square  of  circumference  described  by  generating  point,  add 
square  of  distance  advanced  in  one  revolution,  extract  square  root  of  their 
sum,  and  multiply  it  by  number  of  revolutions  of  generating  point. 


mensuration  of  areas,  lines,  and  surfaces.  355 

Or,  V(P2  + Z2) n = length,  n representing  number  of  revolutions. 

Fyampif What  is  lepgth  of  a helical  line,  Fig.  52,  running  3.5 

times  around  a cylinder  of  22  inches  in  circumference,  and  advancing 
16  inches  in  each  revolution? 


-big.  52 


Fig.  53 


222  _i_  j(52  — 740  = sum  of  squares  of  circumference  and  of  distance 
advawed.  * Then  V74°  x 3-  5 = 95- 21  ins. 

To  Compute  Length  of  a Revolution  of  Thread  of  a 
Screw. 

Rule.— Proceed  as  above  for  length  and  omit  number  of  revolutions. 
Spirals. 

Definition.- Lines  generated  by  the  progressive  rotation  of  a point  around  a 

fixed  axis.  , . . . . 

A Plane  Spiral  is  when  the  point  rotates  around  a central  point. 

\ Conical  Spiral  is  when  the  point  rotates  around  an  axis  at  a progressing  dis- 
tance from  its  centre,  as  around  a cone. 

To  Compute , Ijengtli  of*  a Plane  Spiral  Line.-Fig.  SO. 

rule  —Add  together  greater  and  less  diameters ; divide  their  sum  by  2 ; 
mtiltinlv  Quotient  by  3.1416.  and  again  by  number  of  revolutions. 

oSh ef  circun/erlnces  are  given,  take  their  mean  length,  and  multiply 

it  by  number  of  revolutions.  . . 

: Qr  _j_  (p  —A—  2 X 3.1416  n — length  of  line;  PXw  = radius,  and 

p r 2 V 1—  pitch.  P representing  the  pitch. 

Fxample  — Less  and  greater  diameters  of  a plane  spiral  Spring, 
as  a b e d,  Fig.  53,  are  2 and  20  inches,  and  number  of  revolutions 
d 10;  what  ’is  length  of  it?  f 

^4^20-4- 2 = n ==  sum  of  diameters  -4-  2 ; n X 3. 1416  = 34.5576 
and  34.5576  X 3- i4i6- 

Then  34.5576  X 10=  345-576  inches. 
vOTE  -Above  rule  is  applicable  to  winding  engines,  see  page  662,  where s it  is ^re- 
qnhed  to  ascertain  length  of  a rope,  its  thickness,  number  of  revoluttons,  dtameter 
of  drum,  etc. 

To  Compute  Length  of  a Conical  Spiral  Line.-Fig.  54. 

— Add  together  greater  and  less  diameters ; divide  their  sum  by 

2’  To^square'o?  produc”1  onhis  circumference  and  number  of  revolutions  of 
spiral,  add  square  of  height  of  its  axis,  and  take  square  root  of  the  sum. 

Fig.  54.  Or,  V(d  + d'±-2  X 3.1416  ^ + h2)  ==  length  of  line. 

Example.— Greater  and  less  diameters  of  a conical  spiral,  Fig.  54,  are 
20  and  2 inches;  its  height,  cd,  10;  and  number  of  revolutions  10;  what 
is  length  of  it? 

20  + 2 -f-  2 ==  11  X 3.14*6  = 34-5576  = sum  of  diameters  -4-  2,  and  X 

3.1416;  35.5576  X io  = 345-576- 

Then  V 345. 5762  102  ==  345- 72  inches. 

Spindles. 

Definition.  -Figures  generated  by  revolution  of  a plane  area,  when  the  curve  is 
revolved  about  a chord  perpendicular  to  its  axis,  or  about  its  double  ordinate,  and 
they  are  designated  by  the  name  of  the  arc  or  curve  from  which  they  are  generated, 
as  Circular,  Elliptic,  Parabolic,  etc. 

* When  the  »piral  is  other  than  a line,  measure  diameters  of  it  from  middle  of  body  composing  it. 


356  MENSURATION  OF  AREAS,  LINES,  AND  SURFACES. 


Circular  Spindle. 

To  Compute  Convex  Surface  of  a Circular  Spindle,  Zone, 
or  Segment  of  it.— L'igs.  55,  50,  and  5*7. 

Rule. — Multiply  length  by  radius  of  revolving  arc ; multiply  this  arc  by 
central  distance,  or  distance  between  centre  of  spindle  and  centre  of  revolv- 
ing arc  ; subtract  this  product  from  former,  double  remain- 
der, and  multiply  it  by  3.1416. 


Or,  l r — ( a 
of  arc , and  c the  spindle 


le  chord. 


) 2 p = surface , a representing  length 


14.142  X 10  = 
= 15.708. 


Example. — What  is  surface- of  a circular  spindle,  Fig.  55,  length 
of  it,/c,  being  14.142  inches,  radius  of  its  arc,  oc,  10,  and  central 
distance,  0 e,  7.071  ? 

141.42  = length  x radius.  Length  of  arc,  fa  c,  by  Rules,  page  344 


15-708  X 7.071  = 111.071:3  — length  ofarcx  central  distance;  141.42  — 111.0713  = 
30. 3487  = difference  of  products.  Then  30. 3487  X 2 X 3. 1416  = 190. 687  square  ins. 


Fig>  56.  Zone. 

Example.— What  is  convex  surface  of  zone  of  a circular 
e spindle,  Fig.  56,  length  of  it,  i c,  being  7.653  inches,  radius  of 

its  arc,  o g,  10,  central  distance,  o e,  7.071,  and  length  of  its 
side  or  arc,  d b,  7.854  inches? 

\ | / 7.653X10=76.53  — length X radius ; 7.854X 7.071=55.5356 

V /'  = length  of  arc  x central  distance  ; 76. 53  — 55. 5356  = 20. 9944 

o — difference  of  products. 

Then  20.9944  X 2 X 3- 1416  = 131.912  square  ins.  dL— — ”"“**>1^ 

Segment. 

Example. —What  is  convex  surface  of  a segment  of  a cir- 
cular  spindle,  Fig.  57,  length  of  it,  ic,  being  3.2495  inches,  \.  j / 

radius  of  its  arc,  0 g,  10,  central  distance,  0 e,  7.071,  and  length 
of  its  side,  id,  3.927  inches?  0 

3.2495  X 10  = 32.495  — lengthx  radius ; 3.927  X 7-071  = 27.7678  = length  of  arc 
X central  distance  ; 32. 495 — > 2 7-  7678  = 4. 7272  = difference  of  products. 

Then  4.7272  X 2 X 3.1416  = 29.702  square  ins. 


General  Formula. — S = 2 (lr  — ac)p  = surface , l representing  length  of  spindle, 
segment , or  zone,  a length  of  its  revolving  arc,  r radius  of  generating  circle , and  c 
ceiitral  distance. 

Illustration. — Length  of  a circular  spindle  is  14.142  inches,  length  of  its  revolv- 
ing arc  is  15.708,  radius  of  its  generating  circle  is  10,  and  distance  of  its  centre  from 
centre  of  the  circle  from  which  it  is  generated  is  7.071 ; what  is  its  surface? 

2 X (14.142  X icr — 15.708  X 7-071)  X 3.1416  = 190.687  square  inches. 

Note. — Surface  of  a frustum  of  a spindle  may  be  obtained  by  division  of  the 
surface  of  a zone. 

Cycloidal  Spindle. 

To  Compute  Convex  Surface  of*  a Cycloidal  Spindle.— 
IX  g.  58. 

Rule. — Multiply  area  of  generating  circle  by  64,  and  divide  it  by  3. 


Or,  - = surface. 

Example.— Area  of  generating  circle,  a be,  of  a cycloidal 
spindle,  de,  is  32  inches;  what  is  surface  of  spindle? 

32  X 64  = 2048  = area  of  circle  x 64 , and  2048  ~ 3 = 
682. 667  square  ins. 


Note.— Area  of  groatest  or  centre  section  of  a oycloidal  spindle  is  twice  area  of 
the  cycloid. 


MENSURATION  OF  AREAS,  LINES,  AND  SURFACES.  357 


Ellipsoid,  IParaDoloid,  or  Hyperboloid  of  Bev- 
olntion. 


Definition. Figures  alike  to  a cone,  generated  by  revolution  of  a conic  section 

around  its  axis. 

Note.— These  figures  are  usually  known  as  Conoids. 

When  they  are  generated  by  revolution  of  an  ellipse,  they  are  termed  Ellipsoids, 

and  when  by  a parabola,  Paraboloids,  etc. 

Revolution  of  an  arc  of  a conic  section  around  the  axis  of  the  curve  will  gne  a 
segment  of  a conoid. 

Ellipsoid. 

To  Compute  Convex  Surface  of  an  Ellipsoid.— Fig.  GO. 

Pm,  — Add  together  square  of  base  and  four  times  square  of  height; 
multiply  square  root  of  half  their  sum  by  3.1416,  *"d  ^ Product  radlus 
of  the  base. 


Fig-  59- 


Or, 


'b2-\-  4/^ 


3. 141 6r  — surface. 


Example.— Base,  a b,  of  an  ellipsoid,  Fig.  59,  is  10  inches,  and 
vertical  height,  cd,  7 ; what  is  its  surface? 


to2_l72  x . — 2n6  = sum  of  square  of  base  and  4 times  square 
of  height;  296 %■  2 2 *48,  and  = ^SS  square  root  of  half 


above  sum.  Then  12.1655  X 3- *4*6  X ™ square  ms. 


To  Compute  Convex  Snrfaoe  of  a Segment,  Frustum, 
or  Zone  of  an  Ellipsoid.— Enr.  59. 


See  Holes  for  Convex  Surface  of  a Segment,  Frustum,  or  Zone  of  a 
Spheroid  or  Ellipsoid,  pages  348-9. 

d or  d'  X 3- 1416  Y.K  — surface , 


and 


mean.  diam.  X h 


li—  h ; then  d X 3-  4^  X h = surface. 


d or  d' 

3?araL>oloid 

To  Compute  Convex  Surface  of*  a Paraboloid.-Fi 


. 60. 


Rule.— From  cube  of  square  root  of  sum  of  four  times  square  of  height, 
and  square  of  radius  of  base,  subtract  cube  of  radius  of  base ; multiply  re- 
mainder by  quotient  of  3.1416  times  radius  of  base  divided  by  six  times 
square  of  height. 


Fig.  60. 


Or,  (4/4/1*  + r*)3  - r3  x = surface. 


Example.— Axis,  6 d,  of  a paraboloid,  Fig.  60,  is  40  inches;  ra- 
dius,  a d , of  its  base  is  18  inches;  what  is  its  convex  surface . 


4o2  X 4 = 6400  = 4 times  square  of  height.;  6400+ i82_  6724  — 
1 sum  of  above  product  and  square  of  radius  of  base  ; (V6724) 3 — 18 
1 y6  _ renminder  of  cube  of  radius  of  base  subtracted  from  cube 
w of  square  root  of  preceding  sum  ; 3. 1416  X (6  X 40  ) _ .005  890  5 

— quotient  of  3.1416  times  radius  of  basest)  times  square  of  height. 

Then  545  536  X -005  890  5 = 3213.48  square  ins. 


Fig.  61. 


Cylinder  Sections. 

To  Compute  Surface  of*  a Cylinder  Section. 

— Eig.  61. 

Rule.  — From  entire  surface  of  cylinder  a o subtract 
surface  of  the  two  lingulas;  r o,  0 c,  as  per  rule,  page  351, 
and  multiply  result  by  4. 


358  MENSURATION  OP  AREAS,  LINES,  AND  SURFACES. 


Any  Riguxe  of  Revolution. 

To  Ascertain  Convex  Surface  of  any  Figure  of  Revolu- 
tion.—Figs.  62,  63,  and.  64. 

Rule.— Multiply  length  of  generating  line  by  circumference  described 
by  its  centre  of  gravity. 

Or,  l 2 r p = surface,  r representing  radius  of  centre  of  gravity. 

Example  i.— If  generating  line,  a c,  of  cylinder,  a cdf  io  inches 
in  diameter,  Fig.  62,  is  10,  then  centre  of  gravity  of  it  will  be  in  b 
radius  of  which  is  b r = 5.  ’ 

Hence  10  X 5 X 2 X 3.1416  = 314.16  ins. 

Again,  if  generating  line  is  eacg , and  it;  is  (ea=z  5,  a c=  10 
and  c g — 5)  — 20,  then  centre  of  gravity,  0,  will  be  in  middle  of 
line  joining  centres  of  gravity  of  triangles  € a c and  ac  q- 
from  r. 


Fig.  62. 


r 3-75 


Hence  20  X 3-75  X 2 X 3.1416  = 471.24  square  ins.— entire  surface. 

Verification,  j ^®“vexf  sur[ac®  as  aboVD-- 31416 

l Area  of  each  end,  io2  X . 7854  X 2 = i57. 08 


Fig.  63. 


471.24  inches. 

2.— if  generating  elements  of  a cone.  Fig.  63,  are  a d = io 
dc  — 10,  and  a c,  generating  line,  = 14. 142,  centre  of  gravity  of 
which  is  in  0,  and  or  ==5, 

Then  14.142  X 5 X 2 X 3.1416=444.285,  con - ^4- 

vex  surface , and  10  X 2 x .7854  = 314-16,  area 
of  base. 

Hence  444.285  '-J-  314. 16  = 758.445,  entire  surface. 

3-— generating  elements  of  a sphere,  Fig.  64,  are  a c = 10,  a b c 
will  be  15.708,  centre  of  gravity  of  which  is  in  0,  and  by  Rule,  page 
606,0  r = 3.183.  516 

Hence  15.708  x 3-183X2  x 3-1416  = 314. 16  square  ins. 


Capillary  Tribe. 


To  Compute  Diameter  of  a Capillary  Tube. 
Rule.— Weigh  tube  when  empty,  and  again  when  tilled  with  mercury; 
subtract  one  weight  from  the  other ; reduce  difference  to  grains,  and  divide 
it  by  length  of  tube  in  inches.  Extract  square  root  of  this  quotient,  multi- 
ply it  by  .0192245,  and  product  will  give  diameter  of  tube  in  inches. 


I iv 

0ri  J jX  .019 224 5 = diameter,  w representing  difference  in  weights  in  grains 
and  l length  of  tube. 


Example.— Difference  in  weights  of  a capillary  tube  when  empty  and  when  filled 
with  mercury  is  90  grains,  and  length  of  tube  is  10  inches;  what  is  diameter  of  it? 

90  = 10  = 9 = weight  of  mercury  -4-  length  of  tube  ; V9  and  3 X .019  224  5 = 
•057  673  5 —square  root  of  above  quotient  X .019  224  5 inches  = diameter  of  tube. 

Proof.— Weight  of  a cube  inch  of  mercury  is  3442.75  grains,  and  diameter  of  a 
circular  inch  of  equal  area  to  a square  inch  is  1.128  (page  342). 

• *£’  th?n>  3442-75  grains  occupy  1 cube  inch,  90  grains  will  require  .026  141  9 cube 
inch,  which,  -f- 10  for  height  of  tube  ^ .002  614  19  inch  for  area  of  section  of  tube. 
Then  .002  614  19  = .051 129  = side  qf  square  of  a column  of  mercury  of  this  area 
Hence  .051 129  X 1.128  (which  is  ratio  between  side  of  a square  and  diameter  of  a 
circle  of  equal  area)  = .057  673  5 ins.  , 


To  ^Ascertain.  Area  of  an.  Irregular  Figure. 
Rule.— Take  a uniform  piece,  of  board  or  pasteboard,  weigh  it,  cut  out 
figure  of  which  area  is  required,  and  w^eigh  it;  then,  as  weight  of  board  or 
pasteboard  is  to  entire  surface,  so  is  weight  of  figure  as  cut  out  to  its  surface. 
Or,  see  rule  page  341,  or  Simpson’s  rule,  page  342. 


MENSURATION  OF  AREAS,  LINES,  SURFACES,  ETC.  359 


To  Ascertain.  Area  of*  any-  Plane  Figure. 

Rule.  — Divide  surfaces  into  squares,  triangles,  prisms,  etc. ; ascertain 
their  areas  and  add  them  together. 


Reduction  of*  an  Ascending  or  Descendinj 
izontal  Measurement. 

In  Link  and  Foot. 


; Line  to  Hor- 


Degrees. 


Link. 


Foot. 


Degrees. 


Link; 


13 

14 

15 

16 


.000099  .00015  7 .004917  .00745 

.000403  .00061  8 .006421  .00973 

.000904  .00137  9 .008125’  .01231 

.00161  .00244  10  .010025  .01519 

.002515  .00381  11  .012124  .01837 

.003617  .00548  12  .014421  .02285 

Illustration  i.— In  an  ascending  grade  of  140,  what  is  reduction  in  500  feet? 

14°  = 500  X .0297  = 14.85  feet  — 14  feet  10.2  ins. 

2. — What  is  reduction  in  500  links? 

140  — 500  X -019  602  = 9.801  feetkkqfeet  9.6  ins. 


Foot. 


Degrees. 


W 

18 


Link. 


.016915 
.019602 
.022  486 
.025  569 
.028  925 
.0323 


Foot. 


.025  63 
.029  7 
.03407 
.03874 
•043  7 
.048  94 


Redaction  ofGrrade  ofan  Ascending  or  Descending  Dine 
to  Degrees. 


Per  100  Links , Feetx  etc. 


Grade. 

Degrees. 

Grade. 

Degrees. 

Grade. 

Degrees.  j 

Grade. 

Degrees. 

•25 

8 

35-2 

i-75’ 

1 

6 10.3 

4-5 

2 34  45-5' 

10 

5 44  20.  7 

•5 

*7 

10.3 

2 

1 

8 45-5 

5 

2 51  5 7-6 

11 

6 18  55.8 

•75 

25 

47-6 

2-5 

1 

25  57-6 

6, 

3 26  22.7 

12 

6 53  3i 

1 

34 

22.7 

3 

1 

43  8,3 

7 

4 0 49.6 

13 

7 28  10.3 

1.25 

42 

57-9 

3-5 

2 

0 20.7 

8 

4 35  18.6 

14 

8 251.7 

i-5 

5i 

35-2 

4 

2 

W 33- 1 

9 

5 9 49-6  1 

15 

8 37  37-2 

To  Plot  Angles  witlioat  a Protractor. 

On  a given  line  prick  off  100  with  any  convenient  scale,  and  from  the 
point  so  pricked  off  lay  off  at  right  angle  with  the  same  scale  the  natural 
tangent  due  to  the  angle,  (see  table  of  Natural  Tangents  and  Sines);  or 
strike  out  a portion  of  a circle  with  radius  100  and  lay  off  a chord  = 2 sin. 
of  half  the  angle  required. 

- " . v.  ' ■ ■ 

To  Compute  Chord.  of  an  Angle. 

Double  sine  of  half  angle. 

Illustration. — What  is  the  chord  of  210  30'? 

Sine  of  21  30  = io°  45',  and  sine  of  io°  45'  == . 186  52,  which,  X 2 = .373  04  chord. 
2 


To  Ascertain  Value  of  a Dower  of  a Qnantity. 

Rule. — Multiply  logarithm  of  quantity  by  fractional  exponent,  and  prod- 
uct is  logarithm  of  required  number. 

Example.— What  is  the  value  of  16^  ? 

% X log.  ‘it  = \ x 1. 204 12  = . 903  09.  Number  for  which  = 8. 


360 


MENSURATION  OF  VOLUMES. 


MENSURATION  OF  VOLUMES. 
Cubes  and.  3?arallelopipedons. 
Cube. 

Definition. — A volume  contained  by  six  equal  square  sides. 


Fig.  1. 


To  Compute  “V olume  of  a Cube.— Fig.  1. 
Rule. — Multiply  a side  of  cube  by  itself,  and  that  product 
again  by  a side. 

Or,  s3  — V,  s representing  length  of  a side , and  V volume. 
Example.— Side,  a 6,  Fig.  r,  is  12  inches;  what  13  volume  of  it? 
12X12X12  = 1728  cube  ins. 

Farallelopipedon. 

Definition.— A volume  contained  by  six  quadrilateral  sides,  every  opposite  two 
of  which  are  equal  and  parallel. 

To  Compute  Volume  of  a IParallelopipedon. 

—Fig.  2. 

Rule. — Multiply  length  by  breadth,  and  that  product 
again  by  depth. 

Or,  l bd  = V. 

Prisms,  Prismoids,  and  Wedges. 

Prisms. 

Definition. — Volumes,  ends  of  which  are  equal,  similar,  and  parallel  planes,  and 
sides  of  which  are  parallelograms. 

Note. — When  ends  of  a prism  or  prismoid  are  triangles,  it  is  termed  a triangular 
prism  or  prismoid;  when  rhomboids,  a rhomboidal  prism , and  when  squares,  a 
square  prism , etc. 


Fig.  3- 


To  Compute  Volume  of  a Prism.— 
IVigs.  3 and  4. 

Rule. — Multiply  area  of*base  by  height. 

Or,  a h = V. 

Example.— A triangular  prism,  a b c,  Fig.  4,  has  sides 
of  2.5  feet,  and  a length,  c b,  of  10;  what  is  its  volume? 


Fig.  4. 
a 

i4v 


By  Rule,  page  339,  2.5  s X -433  = 2.70625  = 
end  a b , and  2.706  25  X 10  = 27.0625  cube  feet. 


- area  of 


Fig.  6. 


When  a Prism  is  Oblique  or  hn'egular. 

Rule.  — Multiply  area  of  an  end  by  height,  as  ao\  or, 
multiply  area  taken  at  a right  angle' to  sides,  as  at  c,  by 
actual  length. 

To  Compute  Volume  of  any  Frustum  of  a 
Prism,  whether  Ltiglit  or  OUliqne.  — IVigs. 
0 and  7. 

Rule. — Multiply  area  of  base  by  perpendicular 
distances  between  it  and  centre  of  gravity  of  upper 
or  other  end. 

Or,  area  at  right  angle  to  side  as  at  e by  actual  length. 

Example. — Area  of  base,  a o,  of  frustum  of  a rectan- 
gular or  cylindrical  prism,  Fig.  6,  is  15  inches,  and 
height  to  centre  of  gravity,  c,  is  12;  what  is  its  volume? 
10X  12  = 1 20  cube  ins. 


MENSURATION  OP  VOLUMES. 


36l 


Prismoids.* 

To  Compute  Volume  of  a,  I^ismoid.— Fig.  8. 
Rule.— To  sum  of  areas  of  the  two  ends  add  four  times  area  of  middle 
section,  parallel  to  them,  and  multiply  this  sum  bv  one  sixth  of  perpendicu- 
lar height. 

Note.— This  is  the  general  rule,  and  known  as  the  Prismoidal  Formula  and  it 
applies  equally  to  all  figures  of  proportionate  or  dissimilar  ends. 

F i g 8 Or,  a + a'  + 4 m X h = 6 = V,  a and  a'  representing  areas  of  ends, 

- and  m area  of  middle  section.  1 

Example.  — What  is  volume  of  a rectangular  prismoid  Fig  8 
\d  lengths  and  breadths,  eg  and  g h,  a b and  b d,  of  two  ends  being 
1 7X6  and  3X2  inches,  and  height  15  feet  ? 

7X6  + 3X2  = 42  + 6 = 48  = sum  of  areas  of  two  ends  ; 7 + 3 — 
2 = 5 = length  of  middle  section  ; 6 + 2 = 2 = 4 = breadth  of  middle 
section  ; 5X4X4=80  =four  times  area  of  middle  section. 

Then  48  + 80  X ^ — ==  128  X 30  = 3840  cube  ins. 

Note  1. — Length  and  breadth  of  middle  section  are  respectively  eaual  to  half 
sum  of  lengths  and  breadths  of  the  two  ends. 

2.— Prismoids,  alike  to  prisms,  derive  their  designation  from  figure  of  their  ends 
as  triangular,  square,  rectangular,  pentagonal,  etc.  ’ 

When  it  is  Irregular  or  Oblique  and  their  ends  are  united  by  plane  or 
curved  surfaces , through  which  and  every  point  of  them , a right  line  may  be 
drawn  from  one  of  the  ends  or  parallel  faces  to  the  other.— Figs  9 10  and  n 


Fig.  9. 


Fig.  11. 


= 120  = sum  of  areas  of  ends  + 4 times  middle  section. 


And 


„ h^It:f~AreaS  ?-f  cnds>  ° c a'ld  0 r s'  F'S-  IQ.  « & c d,  and  i m n u,  Fig.  u and 
ab  ce  and  vxi 0 z,  l ig.  9,  are  each  10  and  30  inches,  that  of  their  middle  section 
20,  and  their  perpendicular  heights  185  what  is  their  volume? 

10 + 30  + 20  x 4 
18 

120  X = 360  cube  ins. 

W edge. 

To  Compute  Volume  of  a Wedge.— Fig.  is. 
Rule.-To  length  of  edge  add  twice  length  of  back;  multiply  this  sum 

of  p?oductdlCUlar  helgh  ’ and  then  by  breadth  0f  back’  a,Kl  tak<5  one  sixtl 
Or,  (l  + V X 2 X h b)  = 6 = V. 

Example.  —Length  of  edge  of  a wedge,  eg,  is  20  inches,  back 
abed,  is  20  by  2,  and  its  height,  ef  20;  what  is  its  volume? 

20  + 20  X 2 = 60  = length  of  edge  added  to  twice  length  of 
oack ; 60  X 20  X 2 = 2400  = above  sum  multiplied  by  height , and 
that  product  by  breadth  of  back.  y ’ 

Then  2400  + 6 = 400  cube  ins. 

Note.  — When  a wedge  is  a true  prism,  as  represented  by 

eaual  to  ATPa  of  an  and  l'  * CU  UJ 


Fig.  12. 


Fig.  volume  of  it  is 


r.*.aprto"d“,i0D  °r  embankme“t  «'  * terminated  by  parallel  cros.  section.,  1.  a r sc  tan- 

Hh 


362 


MENSURATION  OF  VOLUMES. 


To  Compute  F ms  trim  of  a Wedge.-Fig.  13. 

Rule. — To  sum  of  areas  of  both  ends,  add  4 times  area 
of  section  parallel  to  and  equally  distant  from  both  ends, 
and  multiply  sum  by  one  sixth  of  length. 

Or,  A 4-^  + 4 a ^5"  *"**^’ 

Eva-mpt  v Lengths  of  edge  and  back  of  a frustum  of  a wedge 

a^iYcd\rl2T  and  1 X a ms.,  and  height  or  is  2o  ms. ; 

, d what  is  its  volume  ? 

2 + 4 X (20  X 4r)  X ^ = 60+  >20  X “ = 600  cube  ins. 

Note  -When  frustum  is  a true  prism,  as  represented  Fig.  13,  volume  of  it  is  equal 
to  mean  area  of  ends  multiplied  by  its  length. 

Regular  Bodies  (Polyhedrons). 

Definition  —A  regular  body  is  a solid  contained  under  a certain  number  of  simi- 
lar^and  equal  plane  faces,*  all  of  which  are  equal  regular  polygons. 

Note  1 -Whole  number  of  regular  bodies  which  can  possibly  be  formed  is  five. 

* A snhere  may  always  be  inscribed  within,  and  may  always  be  circumscribe 
about  a regular  body  or  polyhedron,  which  will  have  a common  centre. 

Fig.  >4.  Fig.;5.  Fig.  >6.  Fjg.  17. > 


To  Compute  41 V. 

To  Compute  Radius  of  a Sphere  that  will  Circumscribe  a given  Regular 
1 Body,  or  that  may  be  Inscribed  within  it. 

’mol.-L,....  or  . h.„. .droo  U I*  '.  » • - 

radii  of  circumscribing  and  inscribed  spheres . inch  — 

2 X .86602  = 1.73204  inches  — radius  of  circumscribing  sphere,  2 X .5-1  >«- 
radius  of  inscribed  sphere. 

element  required. 

When  Volume  is  given. 

opposite  to  body  in  columns  E and  F in  following  iaDie,  miner  uea 
ment  required.  ; — 

* Angle  of  adjacent  faces  of  a polygon  is  termed  diedral  angle. 


MENSURATION  OF  VOLUMES. 


363 


When  one  of  the  Radii  of  Circumscribing  or  Inscribed  Sphere  alone  is  re- 
quired, the  other  being  given.  Rule. — Multiply  given  radius  by  multiplier 
opposite  to  body  in  columns  G and  H in  Table,  page  364,  under  head  of 
other  radius. 

To  Compute  Linear  Edge. 

When  Radius  of  Circumscribing  or  Inscribed  Sphere  is  given.  Rule. — 
Multiply  radius  given  by  multiplier  opposite  to  body  in  columns  I and  Iv  in 
Table,  page  364. 

When  Surface  is  given.  Rule. — Multiply  square  root  of  it  by  multiplier 
opposite  to  body  in  column  L in  Table,  page  364. 

When  Volume  is  given.  Rule.  — Multiply  cube  root  of  it  by  multiplier 
opposite  to  body  in  column  M in  Table,  page  364. 


To  Compute  Surface. 

When  Radius  of  Circumscribing  Sphere  is  given.  Rule. — Multiply  square 
of  radius  by  multiplier  opposite  to  body  in  column  N in  Table,  page  364. 

When  Radius  of  Inscribed  Sphere  is  given.  Rule. — Multiply  square  of 
radius  by  multiplier  opposite  to  body  in  column  O in  Table,  page  364. 

When  Linear  Edge  is  given.  Rule. — Multiply  square  of  edge  by  multi- 
plier opposite  to  body  in  column  P in  Table,  page  364. 

When  Volume  is  given.  Rule. — Extract  cube  root  of  volume,  and  multi- 
ply square  of  root  by  multiplier  opposite  to  body  in  column  Q in  Table, 
page  364. 

To  Compute  'Volnine. 

When  Linear  Edge  is  given.  Rule. — Cube  linear  edge,  and  multiply  it 
by  multiplier  opposite  to  body  in  column  R in  Table,  page  364. 

When  Radius  of  Circumscribing  Sphere  is  given.  Rule. — Multiply  cube 
of  radius  given  by  multiplier  opposite  to  body  in  column  S in  Table, 
page  364. 

When  Radius  of  Inscribed  Sphere  is  given.  Rule.  — Multiply  cube  of 
radius  given  by  multiplier  opposite  to  body  in  column  T in  Table,  page  364. 

When  Surface  is  given.  Rule. — Cube  surface  given,  extract  square  root, 
and  multiply  the  root  by  multiplier  opposite  to  body  in  column  U in  Table, 
page  364. 

Fig.  18.  Cylinder. 


To  Compute  Volume  of  a Solid.  Cylinder.— 
Fi  g.  18. 

Rule. — Multiply  area  of  base  by  height. 

Example.— Diameter  of  a cylinder,  be,  is  3 feet,  and  its  length,  a b, 
7 feet;  what  is  its  volume? 

Area  of  3 feet  = 7.068.  Then  7.068  x 7 = 49. 176  cube  feet. 


To  Compute  "Volnme  of  a Hollow  Cylinder. 
Rule. — Subtract  volume  of  internal  cylinder  from  that  of  cylinder. 


Fig.  19. 


Cone. 

To  Compute  Volume  of  a Cone.— Fig.  19. 

Rule. — Multiply  area  of  base  by  perpendicular  height, 
and  take  one  third  of  product. 

Example. — Diameter,  a b , of  base  of  a cone  is  15  inches,  and 
height,  c e,  32.5  inches;  what  is  its  volume? 


Area  of  15  inches  = 176.7146.  Then  U6-715X32._5 ; _ 


: 1914. 41 25  cube  ins. 


TJnits  for  Elements  of  tlie  Regular  Bodies. 


364 


MENSURATION  OF  VOLUMES. 


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•339  8 


MENSURATION  OF  VOLUMES. 


365 

To  Compute  Volume  of  Ifrustiixn  of  a,  Cone.-Fig.  SO. 

Rule.— Add  together  squares  of  the  diameters  or  circumferences  of  greater 
and  lesser  ends  and  product  of  the  two  diameters  or  circumferences ; mul- 
tiply their  sum  respectively  by  .7854  and  .07958,  and  this  product  by 
height ; then  divide  this  last  product  by  3. 

Or,  d2-j-d'2-j-d  x d'  X -7854  h-r-  3 = V. 

Or,  c2-l -c'2  + c Xo'  X .079  58  Ah-  3 = V. 

Example. — What  is  volume  of  frustum  of  a cone,  diameters 
of  greater  and  lesser  ends,  bd,ac , being  5 and  3 feet,  and  height 
e o,  9 ? f 

52  -f“32  + 5 X 3 — 49;  and  49  x .7854  = 38.4846  = above  sum 
by  -7854;  and  X ^ =t  115.4538  cube  feet. 


Fig.  20. 


IPyraixLid. 

Note.— Volume  of  a pyramid  is  equal  to  one  third  of  that  of  a prism  having  equal 
bases  and  altitude. 


Fig.  21. 


To  Compute  Volume  of  a Pyramid.-Fig.  SI. 

Rule.— Multiply  area  of  base  by  perpendicular  height,  and 
take  one  third  of  product. 

Example.— What  is  the  volume  of  a hexagonal  pyramid,  Fig.  21 
a side,  a b , being  40  feet,  and  its  height,  e c,  60?  ’ 

402  X 2.5981  (tabular  multiplier,  page  341)  = 4156.96  = area  of  base. 
4156  96  X 60 
3 


= 83  139.2  cube  feet. 


To  Compute  Volume  of  Ifriistyirn  of  a Pyramid . Wig.  SS. 

Rule.— Add  together  squares  of  sides  of  greater  and  lesser  ends,  and 
product  of  these  two  sides ; multiply  sum  by  tabular  multiplier  for  areas  in 
Table,  page  341,  and  this  product  by  height ; then  divide  last  product  by  3. 

Or,  s2  + s'2  + s x s'  X tab.  mult,  x h -4-  3 = V. 

When  A reas  of  Ends  are  known,  or  can  be  obtained  without  reference  to 
a tabular  multiplier , use  following. 

FiS-  22-  .7  Or,  a + a'  + fax  a'  X h +'  3 = V. 

Example. —What  is  the  volume  of  the  frustum  of  a hexagonal 
pyramid,  Fig.  22,  the  lengths  of  the  sides  of  the  greater  and  lesser 
ends  ab  cd  being  respectively  3.75  and  2.5  feet,  and  its  perpen- 
dicular height,  e o,  7.5  ? * 

3-752~i~  2-52  = 20.3125  = sum  of  squares  of  sides  of  greater  and 
lesser  ends;  20.3125  + 3.75  x 2.5  = 29.6875  = above  sum  added  to 
product  of  the  two  sides;  29.6875  X 2.5081  X 7. s = ^S.aS  V tab 
mult. , and  again  by  the  height , which,  + 3 — 192. 83  cube  feet. 

When  Ends  of  a Pyramid  are  not  those  of  a Regular  Polygon , or  when 
Areas  of  Ends  are  given 

Rule.— Add  together  areas  of  the  two  ends  and  square  root  of  their  prod- 
uct ; multiply  sum  by  height,  and  take  one  third  of  product. 

Or,  a + a'+  fa  a'  X 7t  = 3 = V. 

nrSA^f‘TWhat^iS^e  volurae  of  an  ^regular-sided  frustum  of  a pyramid,  the 
areas  of  the  two  ends  being  22  and  88  inches,  and  the  length  20? 

22  + 88  = 110  -sum  of  areas  of  ends  ; 22  x 88  = 1936,  and  fi936  = 44  = square 


root  of  product  of  areas.  Then 


110  + 44  X 20 
3 

IT  H* 


= 1026. 66  cube  ins. 


366 


mensuration  of  volumes. 


Spherical  Pyramid. 

a ^nherical  Pyramid  is  that  part  of  a sphere  included  within  three  or  more  ad- 
• tnff  nhw!  surfaces  meeting  at  centre  of  sphere.  The  spherical  polygon  defined 
by  Tlmse^pSpe^UKaces  of  pyramid  is  termed  the  base,  and  the  lateral  faces  are 

^NotL— ' To  compute  the  Elements  of  Spherical  Pyramids,  see  Docharty  and  Hack- 
ley  s Geometry.  Cylindrical  TTngnlas. 

Definition.—  Cylindrical  Ungulas  are  frusta  of  cylinders.  Conical  Ungulas  are 
frusta  of  cones. 


To  Compute  -Volume  of  a Cylindrical  XJiugula.-Fig.  S3. 

i When  Section  is  parallel  to  Axis  of  Cylinder.  KuLE.-Multiply  area 
Fi  ‘ , of  base  by  height  of  the  cylinder. 

V3'  Or,  a fc= V. 

EXAMFLE—Area  of  base,  d ef.  Fig.  23,  of  a cylindrical  ungula  is  15.5 
inches,  and  its  height,  a e,  20;  what  is  its  volume . 

15.5  x 20=  310  cube  ins. 

2 When  Section  passes  Obliquely  through  opposite  sides  of 
Cylinder , Fig.  24.  Kui.E.-Multiply  area  of  base  of  cylinder  by 
half  sum  of  greatest  and  least  lengths  of  ungula.  Fig.  24. 

Or,aXl  + i'-i-«='r- 

Example  -Area  of  base,  c d,  of  a cylindrical  ungula,  Fig.  24,  is  25 
inches  and  the  greater  and  less  heights  of  it,  a c,  6 d,  are  .5  and  .7, 
what  is  its  volume? 


\ 


Id 


f 


25  X 


15  + 17  . 


: 400  cube  ins. 


3.  When  Section  passes  through  Base  of  Cylinder  °A*t8  ff^a 

SKTStaS  ri**  1 »'  7>~;  j;vS  r v,SS£ 

difference  thus  found  by  quotient  arising  from  height  div  ided  b>  v ersea 
Fig.  25.  0r  2 _ — x . - — V,  v.  sin.  representing  versed  sine. 

( y F„  E3_Sine  a d of  half  arc,  d ef,  of  base  of  an  ungula,  Fig.  25, 

i ' "Ac  is®  inches,  diameter  ot  cylinder  is  10,  and  height,  « g,  of  ungula  10, 

! / J what  is  its  volume? 

i H-A  Two  thirds  of  r3  = 83. 333  = two  tMrds  of  cube  of  sine.  As  versed 
e( :-4Ml  sine  and  radius  of  base  are  equal,  °f  X 

cosine  = 0,  and  83.333  X iod-5  = 166.666  cube  ins. 


4.  When  Section  passes  through  Base , of  Cylinder  and 

Srf&ASS  haT/arc  of  base  all  protlTmt  of  area .of bas< 1 and 
cosine.  Multiply  sum  thus  found  by  quotient  arising  from  height,  div 
by  versed  sine.  _ h 


Fig.  26. 


Or, 


--facX; 


— V. 

^ • v.  sin. 

Sine,  a d,  of  half  arc  of  an  ungnliyFig.  26,  is .'“'hes. 


Fvamptf  —Sine  a d,  ot  hall  arc  oi  au  uiiguw,  a.  * 

ver^ine,  * £ to  »«!  height , pc,  .0,  and  diameter  of  cylinder  25, 
what  is  its  volume?  __  f 

Two  thirds  of  = 1152  = two  thirds  of  cube  of  sine  of  half  arc  of 

^umof^w^thirds'of  mbeof^^ofhaifdM^cfbasP<md  product  bf 
9 area  of  base  and  cosine.  Then  23.3.23  X 20- .6  = 2891.5376  cube  ins. 


MENSURATION  OF  VOLUMES. 


367 


5.  When  Section  passes  Obliquely  through  both  Ends  of  Cylinder , Fig.  27. 
Rule.  — Conceive  section  to  be  continued  till  it  meets  side  of  cylinder 
produced ; then,  as  the  difference  of  versed  sines  of  the  arcs  of  the  two  ends 
of  ungula  is  to  the  versed  sine  of  arc  of  less  end,  so  is  the  height  of  cylinder 
to  the  part  of  side  produced. 

. Ascertain  volume  of  each  of  the  ungulas  by  Rules  3 and  4,  and  take  their 
difference. 


Fig.  27. 


v.  sin/  h 

0r>  v sin.  — v.  sin/~h  ’ U sin'  and  v’  sin/  rePresentin9  versed  sines 

of  arcs  of  the  tico  ends,  h height  of  cylinder,  and  h'  height  of  part  pro- 
duced. 

Example.— Versed  sines,  ae,do,  and  sines,  e and  o,  of  arcs  of  two 
ends  of  an  ungula,  Fig.  27,  are  assumed  to  be  respectively  8.5  and  25 
and  n. 5 and  o inches,  length  of  ungula,  ho , within  cylinder,  cut  from 
one  having  25  inches  diameter,  d o,  is  20  inches;  what  is  height  of  un- 
gula produced  beyond  cylinder,  and  what  is  volume  of  it? 

25  oj  8.5  : 8.5  ::  20  : 10.303  — height  of  ungula  produced  beyond  cyl- 


Greater  ungula,  sine  o being  o,  versed  sine  ==  the  diameter.  Base  of  ungula  being 
a circle  of  25  inches  diameter,  area  = 490. 875.  Versed  sine  and  diameter  of  base 

being  equal  (25),  sine  = 0.  490.875  X 25  a*  ^ = 6135.9375  —product  of  area  of  base 

and  cosine , or  excess  of  versed  sine  over  sine  of  base.  30. 303  -=-25  = 1. 2 12  12  = Quo- 
tient of  height  — versed  sine.  1 

Then  6135.9375  x 1.212  12  = 7437.4926  cube  inches;  and  by  Rules  3 and  4,  volumes 
of  less  and  greater  ungulas  = 515.444,  and  6 922.0486  ==  7437.4926  cube  inches. 

Sphere. 

Definition. — A solid,  surface  of  which  is  at  a uniform  distance  from  the  centre. 

To  Compute  Volume  of  a Spliere.—  Fig.  28. 
Kule. — Multiply  cube  of  diameter  by  .5236. 

Or,  <P  x .5236  — V,  d representing  diameter. 

Example. — What  is  volume  of  a sphere,  Fig.  28,  its  diameter, 
a b,  being  10  inches  ? 

io3  = 1000,  and  1000  X 5236  = 523.6  cube  ins. 

To  Compute  Volume  of  a Hollow  Spliere. 

Rule. — Subtract  volume  of  internal  space  from  that  of  sphere. 

Or,  V — v = volume. 


Fig.  28. 


Segment  of  a Spliere. 

Definition.— A section  of  a sphere. 

rF°  Compute  Volume  of  a Segment  of  a Spliere.— Fig.  29. 

Rule  i.  — To  three  times  square  of  radius  of  its  base  add  square  of  its 
height ; multiply  this  sum  by  height,  and  product  by  .5236. 

Flg*  29'  - Or,  3 r2  -f-  h2  h X . 5236  = V. 

2. — From  three  times  diameter  of  sphere  subtract  twice 
height  of  segment ; multiply  this  remainder  by  square  of 
height,  and  product  by  .5236. 

Or,  3 d — 2 h h2  x . 5236  = V. 

Example. — Segment  of  a sphere,  Fig.  29,  lias  a radius,  a e,  of  7 
inches  for  its  base,  and  a height,  b 0,  of  4 ; what  is  its  volume  ? 

..  . , , 7 2 x 3 -f-  42  = 163  = the  sum  of  three  times  square  of  radius  and 

square  of  height ; 163  X 4 X -5236  = 331.3872  cube  ins. 


368 


MENSURATION  OF  VOLUMES. 


Spherical  Zone  (or  Frustum  of  a Sphere). 
Definition  — Part  of  a sphere  included  between  two  parallel  chords. 

To  Compute  Volume  of  a Spherical  Zone. -Fig.  30. 
Definition.— Part  of  a sphere  included  between  two  parallel  planes. 

Rule.— To  sum  of  squares  of  the  radii  of  the  two  ends  add  one  third  of 
square  of  height  of  zone ; multiply  this  sum  by  height,  and  again  by  1.5708. 


Fig.  30. 


Or,  r 2 -f  r‘ '2  -f  h2  -4-  3 h X 1 • 57°8  — v- 
Example.— What  is  the  volume  of  a spherical  zone,  Fig.  30, 
greater  and  less  diameters, /ft  and  de,  being  20  and  15  inches, 
and  distance  between  them,  or  height  of  zone,  eg,  being  10  ins.? 

io2  _j_  7. 52  = 156.25  ==  sum  of  squares  of  radii  of  the  two  ends  ; 
l56.25q_  io2-t-  3 =3 189.58  = above  sum  added  to  one  third  of 
square  of  the  height. 

Then  189.58  X 10  X 1.5708  = 2977.9226  cube  ins. 
Cylindrical  IR-ing. 

Definition. -A  ring  formed  by  the  curvature  of  a cylinder. 

To  Compute  Volume  of  a Cylindrical  Ring— Fig.  31. 

Rule. — To  diameter  of  body  of  ring  add  inner  diameter  of  ring;  multi- 
ply sum  by  square  of  diameter  of  body,  and  product  by  2.4674- 
Fig.  Or,  d + d’d*  2. 4674  = V. 

Qv  al  =z  y?  a representing  area  of  section  of  body,  and  l length 
of  axis  of  body.. 

Example.— What  is  volume  of  an  anchor  ring,  Fig.  31,  diameter 
of  metal,  a b , being  3 inches,  and  inner  diameter  of  ring,  b c,  8 ? 

3 8 x 32  — 99  = product  of  sum  of  diameters  and  square  of  di- 

ameter of  body  of  ring. 

Then  99  X 2.4674  =.  244.2726  cube  ins. 

Spheroids  (Ellipsoids). 

T!ftvtvttto\  -Solids  generated  by  the  revolution  of  a semi-ellipse  about  one  of  its 
difmeteS  3 When  ^revolution  is  about  the  transverse  diameter  they  are  termed 
Prolate,  and  when  about  the  conjugate  they  are  Oblate. 

To  Compute  Volume  of  a Spheroid.—  Fig.  32. 
Rule.— Multiply  square  of  revolving  axis  by  fixed  axis,  and  this  product 

by  -5236-  0r^  a2  a,  x 5236  _-y  a and  a'  representing  revolving  and 

fixed  axes. 

0rj  4=3X3. 1416  r2r'=V,Y  and  r'  representing  semi- axes. 

b Example.— In  a prolate  spheroid,  Fig.  32,  fixed  axis,  a b is 
14  inches,  and  revolving  axis,  cd,  10;  what  is  its  volume ? 
102  x I4—  1400  ^product  of  square  of  revolving  axis  and 
d fixed  axis.  Then  1400  X 5236  = 733.04  ins- 

Note.— Volume  of  a spheroid  is  equal  to  % of  a cylinder  that  will  circumscribe  it. 

Segments  of  Spheroids. 

ro  Compute  Volume  of  Segment  of  a Spheroid— Fig.  33. 
When  Baie  e f,  is  Circular,  or  parallel  to  revolving  Axis,  as  c d,  Fig-33- 
as  e f to  lids  a b,  Fig.  34.  RuLE.-Multiply  fixed  axis  by  3,  he.ghtof 
icgmenfbv  2,  and  subtract  one  product  from  the  other ; multiply sr“d^ 
w snuare  of  height  of  segment,  and  product  by  .5236.  1 hen  as  squaie  01 

S^ed  axis  Ts  to  square  of  revolving  axis,  so  is  last  product  to  volume  of 
segment. 


Fig.  32. 


MENSURATION  OF  VOLUMES. 


369 


Qr  3 a — 2 A ft2  x. 5236  X a'2 
N “ a?  ~ v- 

\ Example.— In  a prolate  spheroid,  Fig.  33,  fixed  or  trans- 

— ax's;  a A 1S  100  inches,  revolving  or  conjugate,  c ± 60 

J and  height  of  segment,  a o,  10;  what  is  its  volume?  ’ 

. IO?  X 3 — 10X  2 = 280  — twice  the  height  of  segment  sub- 
tiacted  from  three  times  fixed  axis;  280  X 10 2 X . 5236  = 
14600.8  inches --product  of  above  remainder , square  of  height,  and  .5236.  5 Then 
100  : 602  . . 14  669. 8 : 5277. 888  cube  ms.  ^ 

When  Base  ef  is  Elliptical , or  perpendicular  to  revolving  Axis  a b Fi«- 

33,iT  .“!♦*•£  t0  An\  .C  rf’  Fis;  3t  Rule. -Multiply  fixed  aiis  bv  t 
and  height  of  segment  by  2,  and  subtract  one  from  the  other:  multinlv  re- 
mainder by  square  of  height  of  segment,  and  product  by  .52-16.  Then  as 
fixed  axis  is  to  revolving  axis,  so  is  last  product  to  volume  of  segment.  ’ 
Eig.  34. 


Or 


3 a'^-rzh  h2  x .5236  X a 


: V. 


Example.— Diameters  of  an  oblate  spheroid,  Fig.  34  are 
whktis ^its°voTume?and  height  °f  “ Segment  thereof  is  12 i 

‘ , * ~ 12  x 2fr  276,  ~ twice  the  height  of  the  segment  sub- 

q a 7 tracjed  Jrom  three  times  the  revolving  axis  ; 276  X 122  x <2^6 

= 20809.9584=4?™^  of  above  remainder , the  square  of  height,  and  .5236.  "5  ° 

Then  100  : 60  ::  20809.9584  : 12485.975  cube  ins. 


Frasta  of*  Spheroids. 

To  Compute  Volume  of  Middl^Frustum  of  a Spheroid.- 

When  Ends,  ef  and  g h,  are  Circular , or  parallel  to  revolving,  Axis  as  c d 
Fig.  35,  or  a b,  Fig.  36.  Rule.-To  twice'  square  ol  revolving  axi?  add 

rd”Uuct“i°8.  * mUltl'Ply  tMS  SUm  by  len«th  °f 

Or,  2 a/2  + d2x  L26i8  = V. 

Example. -Middle  frustum  of  a prolate  spheroid,  i o. 
Fl§  ,35,  is  36  inches  in  length,  diameter  of  it  being  in 

illume  f d ’ 5°’  and  at  ltS  eDdS’  aUd  9 h'  4°;  what  ^ its 

502  X 2 -J-  402  = 6600  = sum  of  twice  square  of  middle  di 
amder  added  to  square  of  diameter  of  ends,  then  6600  x~ 
30  x 2018  = 62  203.68  cube  ins. 

When  Ends,  ef and g h,are  Elliptical , or  perpendicular  to  revolving  Axis 
ab  Fig.  35,  or  ef  and  g h to  Axis,  c d,  Fig. V Rule.-To  twice  product 
of  transverse  and  conjugate  diameters  of  middle  section,  add  product  of 

aK5sssarA’“‘  "d;  “hVy  “ ” 2 

Flg<  36*  Or,  d d'  x 2 -j -~d  d'  l x .2618  = V. 

Example.— In  middle  frustum  of  a prolate  spheroid  Fig 
36,  diameters  of  its  middle  section  are  50  and  30  inches,  its 
ends  40  and  24,  and  its  length,  0 i,  18;  what  is  its  volume? 

50  X 30  X 2 = 3000  = twice  product  of  transverse  and  con- 
jugate diameters  ; 3000  + 40  X 24  = 3960  = sum  of  above 
of  end  f aUd  product  °f fransverse  and  conjugate  diameters 
Then  3960  X 18  x .2618  = 18  661. 104  cube  ins. 


37° 


mensuration  op  volumes. 


Links. 

Definition.— Elongated  or  Elliptical  rings. 

Elongated,  or  Elliptical  Links. 

T.  Comp.t.  3?'= ”1  " mUPU°‘1 

Eulg- — Multiply  ....  of  . .-.a."  of  a.  1-iy  u itot  by  »*  « 

circumference  of  its  axis.  ^ al  or  c = V. 

NOTE.-B.  R.*.,  ««Wr“id«S  tolh'i.^'^tos,  ‘JSS 

SHgScrs**-- 

jsfflscsj  •«  r.s"r„,  »v£ » «... .. 

W 5 &SASB  a a » » 

— what  is  its  volume? 


nat,  IS  HS  VUiumo  : „ , . 

u o -ot-J-tV-j  tai6  = 10.0056  = 3. 1416  times  sum 


^.S^+^s^Vwse = °faxU  °-rien9tK 

^ffuni^rc^F^rte  “f  the  same  dimenSi0nS  83 

preceding;  what  is  its  volume . 


Fig.  38. 


V133. 

2 


3.1416 


'*<f  sum  of  diamders  squared  X 3-  «4i6  - 
cumference  of  axis  of  ring.  Area  of  1 me  .7  54- 
Then  25.643  X .7854  = *>->4  cube  in& 


Spkerical  Sector. 

fcspsi* sector  of  a ; 

with  the  sector  at  the  centre  of  the  sphere. 


To  Compute  Volume  of  a-  Spherical  Seotor.-FigV  39. 
Rule.— Multiply  external  surface  of  zone,  which  is  base  of  sec  or,  \ 
third  of  the  radius*  of  sphere. 

Or  ar-r-3  = V,  a representing  area  of  base. 

Note.  —Surface  of  a spherical  sector  = sum  of  surface  of  zone  and  sur  aces  o 
two  cones. 


py ample  —What  is  volume  of  a spherical  sec*or’ 
39EgXeAneraIed  b>  sector,  0 a k,  heightofzoneu  6 e *,  be- 
ing  ao,  .2  inches,  and  radius,  g h,  of  sphere  15? 

2 XQ  , 248  = 1130.976  = height  of  zone  X circumference 
Id  of  sphere = external  surface  of  zone  (see  page  350). 

1.30.976  X 1 7^1  = efface  X one  third  of  radius  _ 
5654.88  cube  ins. 


Spindles. 

Definition. —Figures  fn«rat^'1^ 

SetdS  and  they  ^e^esigna^d^  ^h?name  of  arc  from  which  they  are 
generated,  as  Circular,  Elliptic,  Parabolic,  etc. 


MENSURATION  OF  VOLUMES. 


371 


Circular  Spindle. 

To  Compute  Volume  of  a Circular  Spindle.— Fig.  40. 

Rule. — Multiply  central  distance  by  half  area  of  revolving  segment; 
subtract  product  from  one  third  of  cube  of  half  length,  and  multiply  re- 
mainder by  12.5664. 

Or,  — — (c  x ')  x 12.5664  = V,  a representing  area  of  revolving  segment. 


Fig.  40. 


a 


Example.— What  is  volume  of  a circular  spindle,  Fig.  40  when 
central  distance,  oe,  is  7.071067  inches,  length, /c,  14.142  U and 
radius,  oc,  10?  * ■ 5 

Note.— Area  of  revolving  segment;  fe,  being  = side  of  square 
that  can  be  inscribed  in  a circle  of  20,  is  202  X .7854  — 14.142  n* 
4 4 = 28. 54  area.  0 


\ / 7.071 067  X 28. 54-^-2=100.9041  ==  central  distance xhalf  area  of 

revolving  segment ; - — 7 100. 9041  = 16. 947  = remainder  of 

above  product  and  one  third  of  cube  of  half  length. 

Then  16.497  x 12.5664  = 212.9628  cube  ins. 


ITrnstnm  or  Zone  of  a Circnlar  Spindle.* 

To  Compute  Volume  of'  a Frustum  or  Zone  of  a Circnlar 
Spindle.— Fig.  41. 

Rule.— From  square  of  half  length  of  whole  spindle  take  one  third  of 
square  of  half  length  of  frustum,  and  multiply  remainder  by  said  half  length 
of  frustum ; multiply  central  distance  by  revolving  area  which  generates 
the  frustum ; subtract  this  product  from  former,  and  multiply  remainder  by 
6.2832.  J 

Or,  1 4 2 — - ' Xr-  — (c  x a)  x 6.2832  = V,  l and  V representing  lengths  of 

spindle  and  of  frustum,  and  a area  of  revolving  section  of  frustum. 

Note.  — Revolving  area  of  frustum  can  be  obtained  by  dividing  its  plane  into  a 
segment  of  a circle  and  a parallelogram. 


Example.— Length  of  middle  frustum  of  a circular  spindle 
ic,  Fig.  41,  is  6 inches;  length  of  spindle,/#,  is  8;  central  dis- 
tance, 0 e , is  3 ; and  area  of  revolving  or  generating  segment 
is  10;  what  is  volume  of  frustum  ? 


(8  4 2)3- 


(6  4 2)3 


= *3>  and  13  X 3 =39  — product  of  - 


length  of  frustum,  and  remainder  of  one  third  square  of  half 
_ length  of  frustum  subtracted  from  square  of  half  length  of 

spindle ; 39  — 3X10  = 9=  product  of  central  distance  and  area  of  segment  subtracted 
from  preceding  product. 


Then  9X6. 2832  ±=  56. 5488  cube  ins. 


Segment  of  a Circular  Spindle. 

To  Compute  Volume  of  a Segment  of  a Circnlar 
Spindle. — Fig.  42. 

Rule.— Subtract  length  of  segment  from  half  length  of  spindle • double 
remainder,  and  ascertain  volume  of  a middle  frustum  of  this  length.  Sub- 
tract result  from  volume  of  whole  spindle,  and  halve  remainder,  f 

Or,  C — c42  = V,  C and  c representing  volume  of  spindle  and  middle  frustum. 

* Middle  frustum  of  a Circular  Spindle  is  one  of  the  various  forms  of  casks 
teingfirst  obtainedPliCable  t0  8egment  of  any  SPlnd,e  or  anJ  Conoid,  volume  of  the  figure  and  frustum 


372 

Fig.  42- 


MENSURATION  OF  VOLUMES. 

Example.  — Length  of  a circular  spindle,  i a,  Fig.  42,  is 
14.142  13  inches;  central  distance,  o e,  is  7.07107;  radius  of 
a arc,  o a,  is  10;  and  length  of  segment,  i c,  is  3.535  53;  what  is 


its  volume? 

14. 142 13 x 2 = 7.071 07  = double  remainder  of 

2 

Zenpi/i  0/  segment  subtracted  from  half  length  of  spindle  = 
length  of  middle  frustum. 

ft0TE._Area  of  revolving  or  generating  segment  of  whole  spindle  is  28.54  inches, 
and  that  of  middle  frustum  is  19.25. 

The  volume  of  whole  spindle  is 212.9628  cube  ins. 

“ “ middle  frustum  is 162.8982  “ “ 

Hence 50.0646-^2  = 25.0323  cube  ins. 

Cycloidal  Spindle.* 

To  Compute  Volume  of  a Cycloidal  Spindle.— Fig.  43. 

Rule.— Multiply  product  of  square  of  twice  diameter  of  generating  circle 
and  3.927  by  its  circumference,  and  divide  this  product  by  8. 

Fig.  43- 


Or,  2 ^ X 3-927  X d X 3-141^  _ y^  ^ representing  diameter 

of  circle , or  half  width  of  spindle. 

Example.— Diameter  of  generating  circle,  a b e,  of  a cy- 
cloid, Fig.  43,  is  10  inches;  what  is  volume  of  spindle,  del 

jo  xtx  3.927  = 1570.8  =:  product  of  twice  diameter  squared  and  3.927. 

Then  1570.8  X 10  X 3.1416-7-8  = 6168.5316  cube  ins. 

Elliptic  Spindle. 

To  Compute  Volume  of  an  Elliptic  Spindle.— Eig.  44. 

Rule.  — To  square  of  its  diameter  add  square  of  twice  diameter  at  one 
fourth  of  its  length ; multiply  sum  by  length,  and  product  by  .13094 

Or,  d2  -f  7d'  l,  1309  = V,  d and  d'  representing  diameters  as  above. 

Example.  — Length  of  an  elliptic  spindle,  a b,  Fig.  44,  is 
75  inches,  its  diameter,  cd,  35,  and  diameter,  ef  at  .25  of  its 
length,  25;  what  is  its  volume? 

352  -f-  25  X 2 = 3725  = sum  of  squares  of  diameter  0/ 
spindle  and  of  twice  its  diameter  at  one  fourth  of  its  length  ; 
3725  x 75  = 279  375  = above  sum  X length  of  spindle. 

\ Then  279  375  X • 1309  = 36  570. 1875  cube  ins. 

Note— For  all  such  solid  bodies  this  rule  is  exact  when  body  is  fonned  by  a 
conic  section,  or  a part  of  it,  revolving  about  axis  of  section,  and  will  always  ne 
very  near  when  figure  revolves  about  another  line. 

To  Compute  Volume  of  Middle  Frustum  or  Zone  of 
an  Elliptic  Spindle.— Fig.  45. 

Rule. — Add  together  squares  of  greatest  and  least  diameters,  and  square 
of  double  diameter  in  middle  between  the  two ; multiply  the  sum  by  length, 
and  product  by  .13094 

Or,  d2  -f  d'2  -f  Td'W.  1309  = V,  d , d ',  and  d"  representing  different  diameters. 


Fig-  44- 


* Volume  of  a Cvcloidal  Spindle  ia  equal  to  .625  of  ita  circumscribing  cylinder, 
t See  preceding  Note.  * See  Note  above. 


MENSURATION  OF  VOLUMES. 


373 


Fig.  45- 


Example. — Greatest  and  least  diameters,  ab  and  cd  of 
tho  frustum  of  an  elliptic  spindle,  Fig.  45,  are  68  and  so 
inches,  its  middle  diameter,  gh,  60,  and  its  length  e f 7c  • 
what  is  its  volume?  6 ’ Ji75’ 

2 

68  2 + 502  -j-  60  x 2 = 21  524  = sum  of  squares  of  greatest 
and  least  diameters  and  of  double  middle  diameter. 

Then  21  524  X 75  X 1309  = 211  311.87  cube  ins. 

To  Compute  Volume  of  a Segment  ofan  Elliptic  Spin- 
Ole.— Fig.  46. 

Rule  —Add  together  square  of  diameter  of  base  of  segment  and  square 
ot  double  diameter  in  middle  between  base  and  vertex ; multiply  sum  by 
length  of  segment,  and  product  by  .1309.*  \ 

Or,  d2-f-  2 d"  l x • 1309  — V,d  and  d"  representing  diameters. 

Example. — Diameters,  cd  and  g h,  of  the  segment  ofan 
elliptic  spindle,  Fig.  46,  are  20  and  12  inches,  and  length 
oe,  is  16;  what  is  its  volume? 

, 2 

2°2.+  12  x, 2 = 976  — sum  of  squares  of  diameter  at  base 
and  m middle. 

Then  976  X 116  X -1309  = 2044. 134  cube  ins. 
Parabolic  Spindle. 

To  Compute  Volume  of  a Parabolic  Spindle.— Fi~.  4.7-. 
41888V  _MUltiply  S<1Uare  °f  diameter  by  length’  and  the  Product  by 
Or,  d2  l x .41888  = V. 

Rule:  2.— To  square  of  its  diameter  add  square  of  twice  diameter  at  one 
fourth  of  its  length ; multiply  sum  by  length,  and  product  by  .13094 


Fig.  47- 


Or,  d2-|~2<f4x  .1309  = V. 

Example.— Diameter  of  a parabolic  spindle,  a b Fig 
47;  is  40  ms.,  and  its  length,  cd,  10;  what  is  its  volume? 
402  X 10  = 16  000  = square  of  diameter  X length. 
Then  16000  X .418  88  = 6702.08  cube  ins. 

Again,  If  middle  diam.  at  .25  of  its  length  is  30,  Then, 

by  Rule  2,  4o2  + 3°X  2X  40  X .1309  = 6806.8  cube  ins. 

To  Compute  Volume  of  Middle  Frustum  of  a Parabolic 
Spindle.— Fig.  48. 

J™/;- .Add  together  8 times  square  of  greatest  diameter,  3 times 
square  of  least  diameter,  and  4 times  product  of  these  two  diameters ; mill- 
tiply  sum  by  length,  and  product  by  .052  36. 

Or,  d2  8 -f  d'2  3 -f  d d'  x 4 l X .052  36  = V. 

.e,^E/Vmddi-t0ge?*er-'8qu^!f  of  sreatest  and  least  diameters  and 
square  of  double  diameter  in  middle  between  the  two;  multiply  the  sum 
by  length,  and  product  by  .1309.  r J 

Or,  d2  + d'2  -f  2 d"2  l x . 1309  = V,  d"  representing  diameter  between  the  two. 

Fig.  48.  Example.— Middle  frustum  of  a parabolic  spindle,  Fig. 

f 4®i  has  diameters,  a b and  ef  of  40  and  30  inches,  and  its 
— length,  c d,  is  10;  what  is  its  volume  ? 

, 4°“  X 8 + 30  2 X 3 + 40  X 30  X 4 = 20  300  = sum  of  8 

_ times  square  of  greatest  diameter , 3 times  square  of  least 

'''  diameter,  and  4 times  product  of  these. 

Then  20300  X 10  X .052  36  = 10629.08  cube  ins. 


See  Note,  page  372. 


t 8-15  of  .7854. 

I I 


t See  Note,  page  372. 


MENSURATION  OF  VOLUMES. 


374 

To  Compute  Volume  of  a Segment  of  a [Parabolic 
Spindle.— Fig.  49. 

RULE>__Add  together  square  of  diameter  of  base  of  segment  and  square 
of  double  diameter  in  middle  between  base  and  vertex ; multiply  sum  by 
height  of  segment,  and  product  by  .1309. 

Or,  d2  -J-  d"2  l X -1309  = V. 

Example.— Segment  of  a parabolic  spindle,  Fig.  49,  has 
diameters,  ef  and  g h , of  15  and  8.75  inches,  and  height, 
c d,  is  2.5;  what  is  its  volume? 

2e.2-j-8.75  X 2 = 531.25  = sum  of  square  of  base  and  of 
double  diameter  in  middle  of  segment.  Then  531.25  X 2.5 
X .1309  = 173- ?52  oube  ins. 

Hyperbolic  Spinclle. 

To  Compute  Volume  of  a Hyperbolic  Spindle.— Fig.  SO. 

Rule.— To  square  of  diameter  add  square  of  double  diameter  at  one 
fourth  of  its  length  ; multiply  sum  by  length,  and  product  by  .1309. 

2 


Fig.  50. 


Or,  d2-j-2  d'  l X .1309  = V. 

Example.— Length,  a b,  Fig.  50,  of  a hyperbolic  spindle 
is  106  inches,  and  its  diameters,  cd  and  ef  are  150  and 
no,;  what  is  its  volume? 

1502  4-1  xo  x 2 x 109  = 7 090000  = 'product  of  sum  of 
squares  of  greatest  diameter  and  of  twice  diameter  at  one 
fourth  of  length  of  spindle  and  length.  Then  7090000  X 
. 1309  ~r~  928  081  cube  inches. 

To  Compute  Volume  of  Middle  Frustum  of  a.  Hyper- 
bolic  Spindle.— Fig.  ol. 

Rule  —Add  together  squares  of  greatest  and  least  diameters  and  square 
of  double  diameter  in  middle  between  the  two ; multiply  this  sum  by  length, 
and  product  by  .i3°9-t 

Or,  d2-f  d'2-f  (2  d")2  IX  .i3o9  = V. 

51-  Example. — Diameters,  a b and  c d , of  middle  frustum  of  a 

hyperbolic  spindle,  Fig.  51,  are  150  and  no  inches;  diam- 
eter, g h,  140;  and  length,  ef  50;  what  is  its  volume? 

x e;02  _i_  j io2  lfi4o  x.2  = 1 13  000  — sum  of  squares  of  great- 
est and  least  diameters  and  of  double  middle  diameter.  Then 
113000  X 50  X .1309  = 739585  cubeins. 

To  Compute  Volume  of  a Segment  of  a Hyperbolic  Spin- 
die.— Fig.  52. 

Rule. — Add  together  square  of  diameter  of  base  of  segment  and  square 
of  double  diameter  in  middle  between  base  and  vertex ; multiply  sum  by 
length  of  segment,  and  product  by  .1309. 

Or,  d2-\-d"2  l X .1309  = V. 

Example. —Segment  of  a hyperbolic  spindle,  Fig.  52,  has 
diameters,  e f and  gh,  of  no  and  65  inches,  and  its  length,  a b, 
25;  what  is ‘its  volume? 

no2  + = 29  000  = sum  of  squares  of  diameter  of  base 

and  of  double  middle  diameter. 

Then  29000  X 25  X .1309  = 94902.5  cube  ins. 


* See  Note,  page  372. 


t Ibid. 


MENSURATION  OF  VOLUMES. 


375 

Ellipsoid,  Paraboloid,  and  Hyperboloid  of  Revo- 
lution* (Conoids). 

&r—  - * — — « 

Ellipsoid  of  Revolution  (Spheroid). 

Definition.  An  ellipsoid  of  revolution  is  a semi-spheroid.  (See  page  368.) 

Paraboloid  of  Re volution. t 

To  Compute  Volume  of  a.  Paraboloid,  of  devolution  - 
Pig.  53. 


Rule.— Multiply  area  of  base  by  half  height. 


Fig-  53-  c Or,  a h ~ 2 = V. 

Note., -p  This  rule  will  hold  for  any  segment  of  paraboloid 
whether  base  be  perpendicular  or  oblique  to  axis  of  solid.  ’ 

Example. -Diameter,  a b , of  base  of  a paraboloid  of  revolution 
Fl&-  53>  20  inches,  and  its  height,  d c,  20;  what  is  it&  volume?  ’ 

Area  of  20  inches  diameter  of  base  - 3I4.  l6.  Then  314  l6  v 
20  — 3141.6  cube  ms. 

Prustum.  of  a Paraboloid  of  Revolution. 

To  Compute  Volume,  of  a.  Frustum  of  a;  Paraboloid  of 
Revolution.-Fig.  04= . 

Flg  54  MultlPly  Slln}  of  squares  of  diameters  bv 

*’  ' ' height  of  frustum,  and  this  product  by  .3927. 

Or,  d2-\-d'2  h X .3927  = V. 

Example. -Diameters,  a b and  d c,  of  the  base  and  vertex 
of  frustum  of  a paraboloid  of  revolution,  Fig.  54  are  20  and 
11. 5 mches,  and  its  height,  ef  12.6;  what  is  its  volume  ? 

15  v So  25  ~?Um  of  diameters.  Then 

532.25  x 12.6  X -3927  = 2633.5837  cube  ms. 

Segment  of  a Paraboloid,  of  Revolution. 

To  Compute  Volume  of  Segment  of  a Paraboloid  of  Revo- 
lution.— - Wig.  55. 

Rule.  Multiply  area  of  base  by  half  height. 

Or,  a X h4-2  = V 

X 7.4-^-2  = 384.315  ^Tins.  ”-5  iDCheS  diameter  of  base  = Io3.869.  Then  ,03.869 

Hyperboloid  of  Revolution. 

To  Compute  Volume  of  a Hyperboloid  of  Revolution. 

— Pig.  56. 

tm °f  middle  diameter;  mul' 


F'g-  55-  / 


tn0Wn  - C0n0id8-  For  **■“«  »f  » Conoid,  see  KwellOu**. 
.5  of  its  circumference. 


turation,  page  233. 
t Volume  of  a Paraboloid  of  Revolution  is 


MENSURATION  OF  VOLUMES. 

Or,  r2+d2  & X .5236=  V,  d representing  middle  diameter 
Example.  — Base,  a 6,  of  a hyperboloid  of  revolution, 
Fig.  56,  is  80  inches;  middle  diameter,  c d,  66;  and  height, 
ef  60;  what  is  its  volume? 

g0  _i_  2\  66 2 = 5956 =sum  of  square  of  radius  of  base  and 
middie  diam.  Then  5956  X 60  X • 5236  = 87  113.7  (tube  ins. 

Segment  of*  a Hyperboloid  of  Revolution.. 
rro  Compute  Volume  of  Segment  of  a Hyperboloid  of 
He  volution,  as  ITig.  56. 

rule> To  square  of  radius  of  base  add  square  of  middle  diameter ; mul- 

tiply this  sum  by  height,  and  product  by  .5236. 

Or  r2  -j-  d"2  h X • 5236  = V,  r representing  radius  of  base. 

Example.  — Radius,  a e,  of  base  of  a segment  of  a hyperboloid  of  revolution  as 
Fig.  56,  is  21  inches;  its  middle  diameter,  c d,  is  30;  and  its  height,  ef  15;  what  is 
its  volume? 

ot  2 4-  ™2  X 1 s = 20  ns  = product  of  sum.  of  squares  of  radius  of  base  and  middle 
diameter  multiplied  by  height.  Then  20x15  X .5236  = 10532.214  cube  ins. 

Frustum  of  a Hyperboloid  of  Revolution. 
rp0  Compute  Volume  of  Frustum  of  a Hyperboloid  of 
He  volution. — Li  S'-  57 . 

T?UIE__Add  together  squares  of  greatest  and  least  semi-diameters  and 
square  of  diameter  in  middle  of  the  two;  multiply  this  sum  by  height,  and 
product  by  .5236. 

0r?  + cT  2 h X • 5236  — V,  d.  d\  and  d"  representing  several  diameters. 

. 2 . Example.— Frustum  of  a hyperboloid  of  revolution.  Fig. 

tig-  57-  J ig  in  d,  50  inches;  diameters  of  greater  and 

lesser  ends,  a b and  c d,  are  110  and  42 ; and  that  of  middle 
."^§1^  diameter,  g /i,  is  80;  what  is  volume? 

g/.. -i— 110-7-2  = 55,  and  42-^9  = 21.  Hence  55^  + 2i2  + 8o2 

= q866  = $mwi  of  squares  of  semi- diameters  of  ends  ana  of 
^0^  b middle  diam.  Then  9866  X 5°  X • 5236  = 258  291. 88  cube  ins. 

.A-ray  Figure  of  Revolution. 

To  Compute  Volume  of  any  Figure  of  Revolution.— 
Fig.  58. 

Eui.b. — Multiply  area  of  generating  surface  by  circumference  described 
by  its  centre  of  gravity. 

Or  a 2 rp  = V,r  representing  radius  of  centre  of  gravity. 

Fig.  58.  ’ Illustration  i.  - If  generating  surface,  a.  be  d of  cylinder 

/>  b ed  f Fig.  =;8,  is  5 inches  in  width  and  10  in  height,  then  will 
a b — 5 and  b d = 10,  and  centre  of  gravity  will  be  in  0,  the  ra- 
dius of  which  is  r 0 = 5 -r-  2 = 2. 5.  Hence  xo  X 5 = 50  - area 
of  generating  surface. 


\ Then  50X  2.5  X2X3-I4i6  = 785-4  = c”‘fa  a 

i _ of  generating  surface  X circumference  of  its 

_ / centre  of  gravity  = volume  of  cylinder. 

Proof.— Volume  of  a cylinder  10  inches  in  diameter  and  10 
inches  in  height,  io2  X .7854  = 78.54,  and  78.54  X 10  = 785.4. 

2 _lf  generating  surface  of  a cone,  Fig.  59,  is  a e = 10,  d e = 
c then  will  «d  = ii.i8.  and  area  of  triangle.  = 10  X 5^2  = 25, 
centre  of  gravity  of  which  is  in  0,  and  0 r,  by  Rule,  page  607, 

= 1.666. 

Hence,  25  X 1.666  X 2 X 3- 1416  = 261.8  = area  of  generating  surface  X dream- 
ference  of  its  centre  of  gravity  = volume  of  cone. 


MENSURATION  OF  VOLUMES. 


377 

3.— If  generating  surface  of  a sphere,  Fig.  60,  is  a b c,  and  a c 
-.10,  abc  will  be  (1P~  * ~7854)  = 39.27,  cent 
\ which  is  in  0,  and  by  Rule,  page  607,  or  = 2. 122 

Hen< 
ing  sw 
sphere. 


— io}  ab  c will  be  ^ 7 54 j — 39.27,  centre  of  gravity  of 
y Rule,  page 607,  or  = 2. 122. 

/ Hence,  39. 27  x 2.122  x 2 x 3. 1416  = 523. 6 = area  of  general- 
/ tn9  surface  X circumference  of  its  centre  of  gravity  = volume  oj 


Irregular  Bodies. 

To  Compute  Volume  of*  an  Irregular  Body. 

lhJ:UIh;7^S'1it-b?t'U:n  out  of  fre?h  water-  and  note  difference  in 
in  body  ’ 6 5 1S  th‘S  dlfference’ s0  ls  to  number  of  cube  inches 

difference  in  lbs.  by  62.5,  and  quotient  will  give  volume  in 

tor^elIr6irer  'S  t0  be  USed’ aScertained  wei®ht  of  a cube  foot  of  it,  or  64,  is 

b°dy  WCighS  15  IbS' in  water-  and  3o  out;  what 

3°  15  = 15  = difference  of  weights  in  and  out  of  water. 

62.5  : 15  : : 1728  : 414.72 ^volume  in  cube  ins. 

Or,  15 -r- 62.5  — .24,  and  .24  x 1728  = 414.72  ==  volume  in  cube  ins. 

CASK  GAUGING. 

"Varieties  of  Casks. 

To  Compute  Volume  of  a Cask. 
lslJar]efy-  Ordinary  form  of  middle  frustum  of  a Prolate  Spheroid 

a SphericaI  outliae  of  staves,  as  Rum 

^Tf*“To  twic®  S(luare  °f  bung  diameter  add  square  of  head  diameter  • 
multiply  this  sum  by  length  of  the  cask,  and  product  by  .2618,  and  it  will 

fahons  111  C inCheS’  WhiGhj  bdllg  divided  23i,  will  give  resulUn 

2d  Variety.  Middle  frustum  of  a Parabolic  Spindle. 

Thas  Brandy  °f  StaVeS  q“s  at  ^Ime; 

J?257Tq  square  of  a head  diameter  add  double  square  of  bung  diam- 
eter, and  from  sum  subtract  .4  of  square  of  difference  of  diameters  • multinlv 
remainder  by  length,  and  product  by  .2618,  which,  being  divided 
will  give  volume  111  gallons.  ’ ® 1 23L 

3d  Variety.  Middle  frustum  of  a Paraboloid 

Thhf.gtasimfcastsaU  °aSkS  Wh‘Ch  CU™  ^ckens  slightly  at 

Rule.— To  square  of  bung  diameter  add  square  of  head  diameter  • mul- 

wUiyghTvXmTfn  ganonSPr0dUCt  ^ '3927’  WWch’  being  dMded  V 

4 th  Variety.  Two  equal  frustums  of  Cones. 

“S  CrnTpes3  a"  iD  WUich  Cu™  “f  staves  quickens  sharply  at 

Rule.— Add  square  of  difference  of  diameters  to  three  times  sauare  of 
olLT?;  “f'P'y  sum  >-*>;.  length,  and  product  by  .065 % audit wm  give 
gallons.  °Ube  ,nCheS>  WhlCh’  being  divided  V =3i,  will  give  Sf  L 


*•  Weight  of  a cube  foot  of  fresh  water. 


I I* 


t Number  of  inches  in  a cube  foot. 


378 


mensuration  of  volumes. 


Example.— Bung  and  head  diameters  of  a cask  are  24  and  16  inches,  and  length 
36;  what  is  its  volume  in  gallons? 


t: rftf+  (24  + x6)°xT  = 4864,  which  X 36  = >75 104,  and  ,75  104  X .065  66  = 
ii  497.329,  which  -r-  231  = 49-77  gallons. 

Generally. 

Bd+M5  .001 692  L = V.  S.  gallons,  and  .001 416  2 = Imperial  gallons. 

I),  d,  and  M representing  interior , head  and  bung  diameters,  and  L length  of  cask 
in  inches. 


To  Ascertain.  Mean  Diameter  of  a Cask. 

Rine— Subtract  head  diameter  from  bung  diameter  in  inches,  and  mul- 
tinlv  difference  by  following  units  for  the  four  varieties;  add  product  to 
head  diameter,  and  sum  will  give  mean  diameter  of  varieties  required. 

SW;::::::::::::::  i I t 

Example.— Bung  and  head  diameters  of  a cask  of  ist  variety  are  24  and  20  inc  - 
es-  what  is  its  mean  diameter? 

24  — 20  = 4,  and  4 X -7  = 2. 8,  which,  added  to  20,  = 22. 8 ins. 


ULLAGE  CASKS. 

To  Compute  Volume  of  Tillage  Casks. 

When  a cask  is  only  partly  filled,  it  is  termed  an  ullage  cask,  and  is  con- 
sidered in  two  positions,  viz.,  as  lying  on  its  side,  when  it  is  termed  a 
ment  Lying,  or  as  standing  on  its  end,  when  it  is  termed  a Segment  Standing. 


To  XJUage  a Dying  Cask. 

Rule.— Divide  wet  inches  (depth  of  liquid)  by  bung  diameter  ; find  quo- 
tient in  column  of  versed  sines  in  table  of  circular  segments,  page  267,  and 
take  i?s  corresponding  segment;  multiply  this  segment  by  capacity  of  cask 
in  gallons,  and  product  by  1.25  for  ullage  reqtuied. 

Example  -Capacity  of  a cask  is  90  gallons,  bung  diameter  being  32  inches ; what 
is  its  volume  at  8 inches  depth? 

8 -4-  32  = .25,  tab.  seg.  of  which  is  .153  55,  which  X 90  = 13- 8195,  and  again  X 125  _ 

17.2744  gallons.  _ 

To  Ullage  a Standing  Cask. 

rux  F —Add  together  square  of  diameter  at  surface  of  liquor,  square  of 
head  diameter,  and  square  of  double  diameter  taken  m middle  between  the 
two;  multiply  sum  by  wet  inches,  and  product  by  .1309,  and  divide  by  3 
for  result  in  gallons. 


Cask  by  Four  Dimensions. 


To  Compute  Volume  of 

Rule.- Add  together  squares  of  hung  and  head  diameters  and  square  of 
double  diameter  taken  in  middle  between  bung  and  head;  multip  y 

and  product  by  .1309,  and  divide  tins  product  by  231  for 

result  in  gallons. 


To  Compute  Volume  of  any  Cask  from  Three  Dime..- 
sions  only. 


nri  F __\dd  into  one  sum  39  times  square  of  bung  diameter,  25  times 
square  of  head  diameter,  and  26  times  product  of  the  two  diameters ; mul- 
tiply sum  by  length,  and  product  by  .008726;  and  divide  quotient  by  -31 
for  result  iu  gallons.  . 

For  Rules  in  Gauging  in  all  its  conditions  and  for  descr.ptton  and  use  of 
instruments,  see  HaswelCs  Mensuration , pages  307-23. 


CONIC  SECTIONS. 


379 


Fig.  x. 


A 


CONIC  SECTIONS. 

A Cone  is  a figure  described  by  revolution  of  a right-angled  triangle 
about  one  of  its  legs,  or  it  is  a solid  having  a circle  for  its  base  and 
terminated  in  a vertex.  • 

Conic  Sections  are  figures  made  by  a plane  cutting  a cone. 

If  a cone  is  cut  by  a plane  through  vertex  and  base,  section  will  be  a triangle, 
and  if  cut  by  a plane  parallel  to  its  base,  section  will  be  a circle 

revolving 8SS$ftSSi? triaDS’e  reV°1VeS'  ***  h Cirde  Which  is  described 

An  Ellipse  is  a figure  generated  by  an  oblique  plane  cut- 
tmg  a cone  above  its  base. 

Transverse  axis  or  diameter  is  longest  right  line  that  can  be 
drawn  in  it,  as  a b , Fig.  i. 

Conjugate  axis  or  diameter  is  a line  drawn 
through  centre  of  ellipse  perpendicular  to  trans- 
verse axis,  as  c d. 

A Parabola  is  a figure  generated  by  a 
plane  cutting  a cone  parallel  to  its  side,  as  a be,  Fig.  2. 

Axis  is  a right  line  drawn  from  vertex  to  middle  of  base,  as  b 0. 

Note. — A parabola  has  not  a conjugate  diameter. 

A Hyperbola  is  a figure  generated  by  a plane 
cutting  a cone  at  any  angle  with  base  greater  than  that  of 
side  of  cone,  as  a b c,  Fig.  3. 

Transverse  axis  or  diameter,  o 6,  is  that  part  of  axis  e b which 
if  continued,  as  at  0,  would  join  an  opposite  cone,  o fr.  ’ ’ ’ 

Conjugate  axis  or  diameter  is  a right  line  drawn  through  centre 
g > °f  transverse  axis,  and  perpendicular  to  it. 

Straight  line  through  foci  is  indefinite  transverse  axis*  that  part 
pf  it  between  vertices  of  curves,  as  o b , is  definite  transverse  axis. 

~ Its  middle  point,  <7,  is  centre  of  curve. 

Eccentricity  of  a hyperbola  is  ratio  obtained  by  dividing  distance  from  centre  to 
e 1 th er  focus  by  semi -transverse  axis. 

Parameter  is  cord  of  curve  drawn  through  focus  at  right  angles  to  axis. 

Asymptotes  of  a hyperbola  are  two  right  lines  to  which  the  curve  continually  arn 
proaches,  touches  at  an  infinite  distance  but  does  not  pass;  they  are  prolongations 
of  diagonals  of  rectangle  constructed  on  extremes  of  the  axes.  b 

Two  hyperbolas  are  conjugate  when  transverse  axis  of  one  is  conjugate  of  the 
other,  and  contrariwise.  ^uujugaie  01  me 

General  Definitions. 

An  Ordinate  is  a right  line  from  any  point  of  a curve  to  either  of  diameters  as 
and  a 6 an  abscissa*"111  “ & a“d  df'  "e  d°Uble  °rdinates;  c 6.  riK-  5,  is  an  ordinate, 

An  Abscissa  is  that  part  of  diameter  which  is  contained  between 
vertex  and  an  ordinate,  as  ce,  go.  Fig. 4,  and  a b 
Fig.  5. 

Ij  Parameter  of  any  diameter  is  equal  to  four  times 
\(J  ^’stance  from  focus  to  vertex  of  curve;  parameter 
1 of  axis  is  least  possible,  and  is  termed  parameter 
of  curve. 

Parameter  of  curve  of  a conic  section  is  equal 
to  chord  of  curve  drawn  through  focus  perpendic- 
ular to  axis. 

I ar  a meter  of  transverse  axis  is  least,  and  is  termed  parameter  of  curve 
ofcu™em<!to’  °f  * C°DiC  SeCtU>“  a“d  f0Ci  are  sufflcieBt  elements  for  construction 


Fig.  4 cd 


Fig.  5. 


CONIC  SECTIONS. 


380 

A Focus  is  a point  on  principal  axis  where  double  ordinate  to  axis,  through  point, 

18  It^ may° thus:  Divide  scluare  of  ordinate  by  four 
times  abscissa,  and  quotient  will  give  focal  distances,  a 5 and  5,  in  preceding  figures. 

Directrix  of  a conic  section  is  a right  line  at  right  angles  to 
major  axis,  and  it  is  in  such  a position  that 
/:  g \\  u : o. 

Here  a d,  Fig.  6,  is  directrix,  and  o is  offset  to  directrix. 

Latus  Rectum , or  principal  parameter,  passes  through  a focus; 
it  is  a double  ordinate,  which  is  a third  proportion  to  the  axis. 

Or,  A : a::  a:  L. 

A and  a representing  major  and  minor  axes.  (See  Haswell's 
Mensuration , page  232. ) 

A Conoid  is  a warped  surface  generated  by  a right 
line  being  moved  in  such  a manner  that  it  will  touch 
a straight  line  and  curve,  and  continue  parallel  to  a 
given  plane.  Straight  line  and  curve  are  called  di- 
rectrices, plane  a plane  directrix , and  moving  line  the 
generatrix . 

Thus,  let  a b a\  Fig.  7,  be  a circle  in  a horizontal  plane, 
and  d d'  projection  of  right  lines  perpendicular  to  a ver- 
tical plane,  r'  b e;  if  right  lines,  da,rs , r'  b,  r"  s,  and  d'  a, 
be  moved  so  as  to  touch  circle  and  right  line  d d and  bo 
constantly  parallel  to  plane  r*  b e,  it  will  generate  conoid 
w w dab  a' d'. 

Iiadii  restores  arc  lines  drawn  from  the  foci  to  any  point  in  the  curve;  hence  a 
radius  vector  is  one  of  these  lines. 

Traced  angle  is  angle  formed  by  the  radii  vectores  and  the  transverse  diameter. 
Ellipsoid , Paraboloid , and  Hyperboloid  of  Revolution — 1 igures  generated 
by  the  revolution  of  an  ellipse,  parabola,  etc.,  around  their  axes.  (See  Men- 
suration of  Surfaces  and  Solids , pages  357"75*) 

Note  1.— All  figures  which  can  possibly  be  formed  by  cutting  of  a cone  a™  men- 
tioned in  these  definitions,  and  are  five  following — viz.,  a £)  tangle,  a Circle , an  El 
a Hyperbola;  hut  last  three  only  are  termed  Como  Sections. 

2 in  Parabola  parameter  of  any  diameter  is  a third trbportional Jo .abscissa 

and  ordinate  of  any  point  of  curve,  abscissa  and  ordinate  being  referred  to  th-t 
diameter  and  tangent  at  its  vertex. 

3.— In  Ellipse  and  Hyperbola  parameter  of  any  diameter  is  a third  proportional 
to  diameter  and  its  conjugate. 

To  Determine  Parameter  of  an.  Ellipse  or  Hyperbola 


Z7 


Fig.  8. 


Rule.  — Divide  product  of  copjugate 
diameter,  multiplied  by  itself,  by  trans- 
verse, and  quotient  is  equal  to  para- 
meter. 


Fig.  9. 


► In  annexed  Figs.  8 and  9,  of  an  Ellipse  1 ^ 

and  Hyperbola , transverse  and  conjugate 
diameters,  ab,cd , are  each  30  and  20. 

Then  30  : 20  ::  20  : 13- 333  —parameter. 

Parameter  of  curve  — elf  a double  ordinate  passing  through 
focus,  s. 

Ellipse. 

To  Describe  Ellipses.  (See  Geomotry,  page  226.) 

To  Compute  Terms  of  an  Ellipse. 

When  ami  three  of  four  Terms  of  an  Ellipse  are  given , viz.,  Transverse 
and  Conjugate  Diameters,  an  Ordinate,  and  its  Abscissa,  to  ascertain  remain - 
ing  Terms. 


CONIC  SECTIONS. 


381 


To  Compute  Orclinate. 

Transverse  and  Conjugate  Diameters  and  Abscissa  being  given.  Rule. As  trans- 

whTch^d^des^hem  co^ugate’  so  is  square  ro°t  of  product  of  abscissae  to  ordinate 

Example.  —Transverse  diameter,  a b , of  an  ellipse  Fig. 
10,  is  25;  conjugate,  c d,  16;  and  abscissa,  at,  7;  what  is 
length  of  ordinate,  t'e? 


Fig.  ia 


25  — 7 = 18  less  abscissa  ; V7  X 18  = 11.225. 
Hence  25  : 16  ; : 11.225  • 7*184  ordinate. 


0r>  \J c*  “ (fj~j  ~ any  ordinate,  c and  t representing 
Tgtr7nJU9CLte  aUd  tranSverse  diameters,  and  x distance  of  ordinate  from  centre  of 


figure. 

To  Compute  -Abscissae. 

Transverse  and  Conjugate  Diameters  and  Ordinate  being  given.  Rule  —4s  coniu 
S?i«“eter- 18  t:anJsverse>  80  is  square  root  of  difference  of  squares  of  ordinate 
fn<?oaa11iC<0DJUgatKt°  d,stance  between  ordinate  and  centre;  and  this  distance  be- 
ing added  to,  or  subtracted  from,  semi-transverse,  will  give  abscissae  required. 

Example  —Transverse  diameter,  a b,  of  an  ellipse,  Fig.  IO  is  2=;  • coniu^ate  r d 
16 ; and  ordinate,  i e,  7. 184 ; what  is  abscissa,  ibl  7 5 7 J 0 te’  c «> 

Hence,  as  16  : 


: 25 ::  3.52 
| abscissce. 


5*5* 


—7*184  =3.519943.  

Then  25^2  = 12.5,  and  12.5-f- 5.5  = 18  = 6 
25  = 2 = 12.5,  and  12.5  — 5.5  = 7 = at,. 

To  Compute  Transverse  Diameter. 

Conjugate,  Ordinate , and  Abscissa  being  given.  Rule.— To  or  from  semi  coniu 
fprpApapCnfrdm?  M g/eat,  or  less  abscissa  is  used,  add  or  subtract  square  root  of  dif- 
f-  square?  of  01!dinate  and  semi-conjugate.  Then,  as  this  sum  or  difference 
is  to  abscissa,  so  is  conjugate  to  transverse.  uinerence 

Example. — Conjugate^  dimeter,  c d,  of  an  ellipse.  Fig.  10,  is  16-  ordinate  ie 
7. 184 , and  abscissae,  b 1,  i a,  18  and  7 ; what  is  length  of  transverse  diameter  ? ’ ’ 


. . (l6  = 2)2  — 7.1842=3.52. 

1 • 2"r3*52  . 18  ..  16  : 25;  16  = 2 — *3.52  : 7 ::  16  : 25  transverse  diameter. 

To  Compute  Conjugate  Diameter. 

being  given ■ Rvle— As  square  root  of  prod- 
uct  01  abscissae  is  to  ordinate,  so  is  transverse  diameter  to  conjugate.  P 

, EoXA^lTTrknSXYS.e  diameter ,06,  of  an  ellipse,  Fig.  IO  is  2<-  ordinate  ie 
7.1  4,  and^abscissae,  b 1 and  t a,  18  and  7 ; what  is  length  of  conjugate  diameter?  ’ 

18  x 7 = 11225.  Hence  11.225  : 7- 184  ::  25  : 16  conjugate  diameter. 

Rrtp  TJ?  ,C?mpUte  Circumference  of  an  Ellipse. 

3 Rule.  - Multiply  square  root  of  half  sum  of  the  squares  of  two  diameters  by 


« 6 *<*.  of  an  ellipse,  Fig. , 


242-f-2Q2 

2 = 488’  and  ^ 488  = 22  °9-  Hence  22.09  X 3. 1416  = 69.398  circumference. 

p T°  Corupilte  Area  of  an  Ellipse 

on Or,  multiply 

jugate,  cd%;  ^hat'is^ts  area?'ameter  °f  a“  ellipse’  a b’  Fi&  IO>  ls  I2>  aud  its  con- 
12  X 9 X .7854=184.8232  area. 


' ' ? ' ' ' / ~ U//  CU/. 

diam°ItEer-„fAo^  ^ «■*>* 


o uiuor  minor  axis. 

areaLof  drcTe°of  °f  £ircle  °f  40  “ I256  64  5 area  of  ellipse  40X20- 628  32  • 


382 


CONIC  SECTIONS. 


Segment  of  an  Ellipse. 

To  Compute  Area  of  a Segment  of  an  Ellipse. 

When  its  Base  is  parallel  to  either  Axis , as  e if.  RimE  -Divide  heigM  of  seg- 
ment b i.  by  diameter  or  axis,  a b,  of  which  it  is  a part,  and  find  in  Table  of  Areas 
of  Segments  of  a Circle,  page  267,  a segment  having  same  versed  sine  as  this  quo- 
® tient;  then  multiply  area  of  segment  thus  found  and  the 

F'g  11  — e axes  of  ellipse  together. 

Example.  — Height,  b t,  Fig.  u,  is  5,  and  axes  of  ellipse  are 
30  and  20;  what  is  area  of  segment? 

5 -f- 30  = .1666  tabular  versed  sine , the  area  of  which 
(page  267)  is  .085  54. 

Hence  .085  54  X 3°  X 20  = 51.324  area. 

To  Ascertain  Length  of  an  Elliptic  Curve  which  is  less 
than  lialf  of  entire  Figure. 


Fig.  12. 


Let  curve  of  which  length  is  required  be  A b C, 
Fig.  12. 

Extend  versed  sine  b d to  meet  centre  of  curve  in  e. 
n Draw  line  e C,  and  from  e,  with  distance  e b , describe 
b ft,;  bisect  ft,  C in  i,  and  from  e,  with  radius  e i , de- 
■ * scribe  1c  i,  and  it  is  equal  to  half  arc  A b C. 

To  Ascertain  Length,  when  Curve  is  greater  than  half 
entire  Figure. 

Ascertain  by  above  problem  curve  of  less  portion  of  figure;  subtract  it  from  cir- 
cumference of  ellipse,  and  remainder  will  be  length  of  curve  required. 

JParatoola. 

To  Describe  a Parabola.  (See  Geometry,  page  229.) 

To  Compute  either  Ordinate  or  Abscissa  of  a Parabola. 

When  the  other  Ordinate  and  Abscissa,  or  other  Abscissa  and^Ordina^s  are 
given.  Rule.  —As  either  abscissa  is  to  square  of  its  ordinate,  so  is  othei  abscissa  to 
square  of  its  ordinate.  . 

Or,  as  square  of  any  ordinate  is  to  its  abscissa,  so  is  square  of  other  ordinate  to 
its  abscissa. 

Example  i.— Abscissa,  a b , of  parabola,  Fig.  13,  is  9;  its  ordi- 
nate, be,  6 ; what  is  ordinate,  d e,  abscissa  of  which,  a d,  is  16  . 
Hence  9 : 62  I*.  16  : 64,  and  V64  tt  8 length. 

2.— Abscissae  of  a parabola  are  9 and  16,  and  their  correspond- 
ing ordinates  6 and  8;  any  three  of  these  being  taken,  it  is  re- 
quired to  compute  the  fourth. 

62  x_l6  = 8 ordinate.  2.  = 6 ordinate. 

3,  16  ^ — — 9 less  abscissa.  4-  9 = 16  abscissa. 

Laraholic  Curve. 

To  Compute  Length.  of  Curve  of  a Para’lola  cut  off 
a.  Lovable  Ordinate.— F ig.  13. 

Rule. -To  square  of  ordinate  add  ± of  square  of  abscissa,  and  square  root  of 

this  sum,  multiplied  by  two,  will  give  length  of  curve  nearly. 

Example. -Ordinate,  d «,  Fig.  13,  is  8,  and  its  abscissa,  a d,  16;  what  is  length  of 
curve,  fa  e ? 

ga  4 * — — 405. 333,  and  V4°5-  333  X 2 ^ 4°-  2^7  lenffik* 


Fig.  13. 


3 


CONIC  SECTIONS. 


383 


Fig.  14. 


To  Compute  Area  of  a Parabola. 

Rule  -Multiply  base  by  height,  and  take  two  thirds  of  product 


- vuuus  ui  pruauct 

alMogmm7  - Parab°'a  iS  two  thirds  of  its  circumscribing  par- 


Example. —What  is  area  of  parabola,  a b c,  Fhr  ™ heiVht  h, 
being  16,  and  base,  or  double  ordinate,  a c,  16  ? g'  4’  n ghtj  b e> 


16  X 16  — 256,  and  of  25 6 = 170.667  area. 


To  Compute  Area  of  a Segment  of  a Parabola. 


heX  *MatPd^ 


hefghA“eP“is  £?■ Wtaf toS* SSL" ‘ Parab°'a’  ° c and  ^ Fi&  >4,  are  xo  and  6,  and 


io  ^6^10X2  = 15  680,  and  4- 


- __2/v  62  x 3 = 81 .66j  area. 

of  whlchls Same  height’ the  base 
the  two  ends  and  lesser  end.  g 5 creased  by  a third  proportional  to  sum  of 


Hyperbola. 

To  Describe  a Hyperbola.  (See  Geometry,  page  230.) 
To  Compute  Ordinate  of  a Hyperbola, 


Fig.  15. 


6 dia^efa  <-?yperboIa-  ? b Fig.  15,  has  a transverse 

40+ 120  = 160  greater  abscissa , and 
120  : 72  : . -^/ (40  x 160)  : 48  ordinate. 

ic  ^perbolas  lesser  abscissa,  added  to  axis 

(the  transverse  diameter),  gives  greater. 


n o*  * ^ & *•  vutti. 

is  : hyperbola  to  any  point  in  curve 


To  Compute  ^Vbscissee, 


J ugat^dlame ter fs^o* u^veree*™o^s  sma!ue°rMr<lfS  ***,**»■  Rcle.-As  con- 

ThenirUgate  t0  distance  betw’een  ordinate  and  lentre  orXffsum  Iff ' “d 
??  ihne  sum  of  this  distance  and  sem  i transverse  wm  J l ?f  absciss9s- 
their  difference  the  lesser  abscissa  v s will  give  greater  abscissa,  and 


Example  .—Transverse  diameter,  a t,  of  a hvnerbola  Fio>  Tr.  ;0 
72  ’ and  Qrdinate,  a c,  48;  what  are’len’gths  all''  Conjugate’  ** 


To  Compute  Conjugate  Diameter, 


tu  CUUJU^cUt*. 

4s;^s,ra^ 


V40  x 160  — 80  : 48  ::  120  : 72  conjugate. 


CONIC  SECTIONS. 


To  Compute  Transverse  Diameter, 

Conjugate , Ordinate , and  an  Abscissa  being  given.  Rule.— Add  square  of  ordinate 
. enn  iro  of  se mi -conjugate,  and  extract  square  root  of  their  sum. 

Take  sum  or  difference  of  semi-conjugate  and  this  root,  accord mg  as  greater  or 
lpocpr  abscissa  is  used.  Then,  as  square  of  ordinate  is  to  product  of  abscis&a  and 
coni?a»te  sum  or  difference  above  ascertained  to  transverse  diameter  required. 

Note. —When  the  greater  abscissa  is  used,  the  difference  is  taken,  and  con- 

^Example  -Conjugate  diameter,  df. \ of  a hyperbola,  Fig.  15,  is  72;  ordinate,  e c, 

48;  and  lesser  abscissa,  a e,  40;  what  is  length  of  transverse  diameter,  a t? 

-/To 2 1 — 60,  and  60 + 72  -r-  2 = 96  lesser  abscissa , and  40  X 7 2 — 2880. 

v 40  T-  v7  • Hencej  ^g2  . 288o  ::  96  : 120  transverse  diameter. 

To  Compute  Length,  of  any  Arc  of  a Hyperbola,  com- 
mencing at  Vertex. 

RnTE  —To  iq  times  transverse  diameter  add  21  times  parameter  of 
Tn  rf'times  transverse  diameter  add  21  times  parameter,  and  multiply  each  of 
these9sCs  res“ely  by  quotient  of  lesser  abscissa  divided  by  transverse  do 

aToteeacb  of  products  thus  ascertained  add  r5  times  parameter,  and  divide  former 
bv^latter;  then  this  quotient,  multiplied  by  ordinate,  will  give  length  of  arc,  nearly. 

' Xote.  —To  Compute  Parameter,  divide  square  of  conjugate  by  transverse  diam- 

fTv',6  r.  Example. -In  hyperbola,  abc , Fig.  16,  transverse  diameter  is  jeo 
“■  ’ conjugate.  72,  ordinate,  e c,  48,  and  lesser  abscissa,  a e,  40,  wba 

length  of  arc,  a b ? 

Zl_  = 43.2  parameter.  120  X 19  + 43-2  X 21  X — = 1062.4. 

120  - 

^^  + 43.2X  21  X ^ = 662.4.  Then  1062.4  + 43.2  X 15^662.4 

v]c  +43.2X  15  = 1.305,  which  X 48  = 62.64  length. 

Note. -As  transverse  diameter  is  to  conjugate,  so  is  conjugate  to  parameter. 
(See  Rule,  page  380.) 

To  Compute  Area  of  a Hyperbola, 

transverse  diameter  and  lesser  abscissa  10 

‘''^vUieTtim^prodac^of  conjugate^iameter  and  lesser  abscissa  by  transverse 
diameter, +andmt^ hU^tqwtient,  JmBultip.ied  by  former,  will  give  area,  nearly. 

Example.  -Transverse  diameter  of  a hyperbola,  Fig  .6,  is  60,  conjugate  36,  and 
lesser  abscissa  or  height,  a e,  20;  what  is  area  ot  figure . 

60  X no  + !■  of  20=  = 1485.7143,  a«d  VH85-7I43  X 21  = 809.43,  V^X  2°  X 
4 + 809. 43  = 901.02,  which -4- 75  = 12-0136  and ”6  X4  X I2-OI86  = 576  653  a> ea- 
Nora. — For  ordinate,  of  » pnrnbol.  in  divUion.  of  eighths  nnd  tenth.,  see  page  229- 
Delta  Nletal. 

Delta  Metal  is  an  improved  composition  of  Aluminimn  a, ,d  its  alloys , it  is 
non-corrosive,  capable  of  being  cast,  forged,  and  hot  rolled. 

Tensile  Strength  per  Sq.  Inch. 

Cast  in  green  sand 48380  lbf.  I KoUe, M^ahA ; ; ; ; ; ; ; ; ; ; 

Boiled,  hard 75  260  | Wire,  no.  22  tt  vx. 


PLANE  TRIGONOMETRY. 


385 


PLANE  TRIGONOMETRY. 

By  Plane  Trigonometry  is  ascertained  how  to  compute  or  determine 
four  of  the  seven  elements  of  a plane  or  rectilinear  triangle  from  the 
other  three,  for  when  any  three  of  them  are  given,  one  of  which  bein«- 
a side  or  the  area,  the  remaining  elements  may  be  determined ; and 
this  operation  is  termed  Solving  the  Triangle. 

The  determination  of  the  mutual  relation  of  the  Sines,  Tangents,  Secants 
etc.,  of  the  sums,  differences,  multiples,  etc.,  of  arcs  or  angles  is  also  classed 
under  this  head. 

For  Diagram  and  Explanation  of  Terms , see  Geometry,  pp.  219-21. 


Right-angled.  Triangles. 

For  Solution  by  Lines  and  Areas , see  Mensuration  of  Areas,  Lines 
and  Surfaces, pp.  335-39. 

To  Compute  a Side. 

When  a Side  and  its  Opposite  Angle  is  given.  Rule. — As  sine  of  an<de 
opposite  given  side  is  to  sine  of  angle  opposite  required  side,  so  is  given  side 
to  required  side. 

To  Compute  an  Angle. 

Rule.— As  side  opposite  to  given  angle  is  to  side  opposite  to  required 
angle,  so  is  sine  of  given  angle  to  sine  of  required  angle. 

To  Compute  Base  or  Perpendicular  in  a Right-angled 
Triangle. 

When  Angles  and  One  Side  next  Right  Angle  are  given.  Rule  —As  ra- 
dius is  to  tangent  of  angle  adjacent  to  given  side,  so  is  this  side  to  other  side. 


To  Compute  the  other  Side. 

. When  Two  Sides  and  Included  Angle  are  given.  Rule. — As  sum  of  two 
given  sides  is  to  their  difference,  so  is  tangent  of  half  sum  of  their  opposite 
angles  to  tangent  of  half  their  difference;  add  this  half  difference  to  half 
sum,  to  ascertain  greater  angle ; and  subtract  half  difference  from  half  sum 
above6rtam  ^SS  angle*  The  otlier  side  may  then  be  ascertained  by  Rule 
To  Compute  Angles. 

When  Sides  are  given.  Rule.— As  one  side  is  to  other  side,  so  is  radius 
to  tangent  of  angle  adjacent  to  first  side. 


To  Compute  an  Angle. 

r,Fhm,/tee  S}^  are  Pivfn-  Eule  r- — Subtract  sum  of  logarithms  of 
o/hnlf hlCh  7;  .!m  r?«luired  angle,  from  20;  to  remainder  add  logarithm 
of  half  sum  of  three  sides,  and  that  of  difference  between  this  half  sum  and 
Side  opposite  to  required  angle.  Half  the  sum  of  these  three  logarithms”  s 

lalmf  byRule  above  req"ired  The  °ther  anSles  be  ascer- 

,tr« »•»  -“tts 

resaU^  ~RiU<?11 Case?  eLther  of  these  ru,es  wil1  Sive  sufficiently  accurate 
it  is  less  than  9o°  US  squired  angle  exceeds  90O;  and  Rule  2 when 

K Iv 


PLANE  TRIGONOMETRY. 


386 

Example— The  sides  of  a triangle  are  3>  4,  and  5;  what  are  the  angles  of  the 
hypothenuse  ? 

20  __  (Log.  4 = .602  06  -f  Log.  5 = ^698  97)  = 18.698  97 ; Log.  3 + 4 + 5 • 2 4 

■50103;  and  Log.  3 + 4 + 5^~2  — 5 — °-  . QO  ,, 

3 Then  18.69897  + . 30103  = 19, which^-2  = 9.5  = log.  sin.  of  half  angle  _ x8  2 , 

which  X 2 = 36°  52'  angle. 

Hence  9o°  — 36°  52'  = 53°  8'  remaining  angle. 

=B?y ^Sin^TaiL,1  Sec.',  etc.,  A B^  etc.,. is  expressed  Sine,  Tangent,  Secant,  etc.,  of 
angles,  A,  B,  etc. 

To  Compute  Sides  A C and  BE. -Figs.  1 and  3. 

When  Hyp.,  Side  B A,  and  An 
Sin.  B X B A 


Fig.  i, 


Cosine.  A Vers. 


- = AC. 

Sin.  C 

BAX  Cot.  C = A C. 
Hyp.  X Cos.  C = AC. 
Hyp.  X Sin.  B = A C. 
BA 
Sin.  C ’ 

AC 
Sin  B 


- = BC. 
= B C. 


To  Compute  Side  A C and.  Angles. 
When  Hyp . and  Side  B A are  given.— Fig.  i and  2. 


AC 

Hyp.' 


A C 


I Sin. B. 


-®A  = Sin.C.  BAJlS!j—  = AC- 
Hyp.  fein-  C 


B C X Sin.  B = AC. 


BA 


Fig.  3- 


To  Compute  Side  B € and  Hyp.  or  Angles 
When  both  Sides  are  given.— Fig.  2. 

= Tan.  B.  ~ = BC.  VaC*  + B A^BC. 


B A 
AC 


= Tan.  C. 


?4=sin' c- 

B C 


^ = Sin.B. 


To  Compute  Sides.— Figs.  3 and  4. 
When  a Side  and  an  Angle  are 
J'  given, 


Fig.  4- 


| \';h  A C X Tan.C 
— Rad‘ 


Rad. 


BCX  Cos. 

B = BA. 

B C X Sin. 

B = A C. 

A B X Sec. 

b=bc. 

lC  BA 

AC  X Sin.C 

A. 

Sin.  B 

;C-BC. 

A C X Bad. 
Sin.  B 

Tangent. 


In  B A C Fig.  5,  a right-angled  triangle,  C A,  is  assumed  to  be  radius ; 
B A tangent  MC, and  BC  secant  to  that  radius ; Or, dividing  each  of  tttese 
by  base,  there  is  obtained  the  tangent  and  secant  of  C respectively  to  radius  . 


Radius. 


PLANE  TRIGONOMETRY. 


387 


O Radius,  (f 


Radius  C A = 1 

Secant  CB  = i.4i42 

Tangent  AB=i 
Co  secant  CB  = 1.4142 
Co  tangent  e B = 1 

a/  A C2  + B A2  — hyp.  B C. 

AC-r Cos.  C = hyp.  B C. 

A Area  Cos.  C 

V tSTc  =Rad-  sn=  = Cot.C, 


Sine  d g=  .7071 

Cosine  C g or  od  = .7071 
Versed  sine  g A=  .2929 
Co- versed  sine  oe=  .2929 
Angle  CAB  = 9o° 


BA-f- Sin.  C = hyp.  B C. 
1 -f-  Tan.  C = Cot.  C. 

B C2  x Sin.  2 C 


- —Area. 


BC  X Cos.  C = Rad. 
B A x Tan.  B = Rad. 
BC4-B  A = Sec.  B. 
B C X Cos.  B = B A. 


Sin.  C ' 

BAxSec.  B = RC. 

B A x Cot.  C — Rad.  B C x Sin.  B = Rad. 

B C x Sin.  C = B A.  A C X Tan.  0 = B A 

1 4-  Sin.  C = Cosec.  C.  1 - Sin.  C ~ Co-ver.  sin. 

Cos.  C 4-  Sin.  C = Cot.  C.  C B x Sin.  B = A C. 


Perp.  4-  hyp.  = Sin.  C. 
Base  4-  hyp.  = Cos.  C. 
Base  4-  hyp.  =Sin.  B. 
Base  4-  perp.  = Cotan.  C. 


Trigonometrical  Equivalents. 


V (1  — sin.2)  ==  Cos. 
Sin.  -4-  tan.  = Cos. 
Sin.  x cot.  = Cos. 
Sin.  4-  cos.  rz  Tan. 
Cos.  4-  cot.  — Sin. 
Cos.  ‘4-  sin.  — Cot. 


Hyp.  4-  base  = Sec.  C.  Perp.  4-  base  = Tan  C 

Base  4-  perp.  = Tan.  B.  Hyp.  4-  perp.  = Sec.  B. 

Perp.  4-  hyp.  = Cos.  B.  Hyp.  4-  perp.  = Cosec.  C. 

Hyp.  — Base  = Versin.  Hyp.  — Perp.  — Co-ver.  sin.  CT. 
Tan.  4-  sin.  = Sec. 

Tan.  4-  sec.  — Sin. 


Tan.  x cot.  = Rad. 


•cos.2)  t=  Sin. 
cot.  ==  Tan. 
4-  sin.  Cosec. 


1 4-  cos.  = Sec. 

1 4-  cosec.  = Sin. 

1 4-  sec.  = Cos. 

x — cos.  — Versin. 

1 — sin.  = Co-ver.  sin. 
tan.  = Cotan. 


T ' 1 ~ tan.  zzr  ootan. 

A B 0f  a right-aD8,ed  tri“gJ«  ^ *00,  and  angle  C 


Fig.  6. 


B Ofolicpne-anglecL  Triangles. 

To  Compute  Sides  B A and  B C. 
When  Side  A C and  Angles  are  given.— Fig.  6. 


Sin.  C x A C 


Sin.  B 


= BA. 

Sin.  A x A C 


Sin.  0 X B C 


Sin.  A 


= BA. 


Sin  B 


= BC. 


To  Compute  Angles  and  Side  A C 
Wken  8ides  A B,  B C,  and  one  of  the  Angles  are  given.- FtV  6 

/ Sin  n o * 


B C X Sin.  B 


AC 


= Sin.  A. 


Sin.  C x A C 


B A 
Sin.  B x B C 


= Sin.  B. 


A B x Sin.  B 


AC 


: Sin.  C. 


= AC. 


Fig.  7. 


Sin.  A 

To  Compute  Sides  B A and  B C. 
When  Side  A C and  Angles  are  given.— Fio*.  7. 
Sin.  C x B C _ . Sin.  Ax  AC 


Sin.  A 


= BA. 


Sin.  B 


-=BC. 


When  Side  B C and  Angles  are  qiven. — Fig-  7 
B C x Sin.  C ~ 


Sin.  (C  + B) 


= B A. 


Sin.  C x AC 


Sin.  B 


:BA. 


pU™tISim!  "*  C°Sine  °fan  arc  are  each  W ^ sine  and  cosine  of  their  sup. 


Spherical  Triangles , Right  - angled  and  Oblique. 

Molesworth,  Lond.,  1878,  pp.  435-6.  * 


For  full  formulas  see 


388 


PLANE  TRIGONOMETRY. 


To  Compute  Angles  and.  Side  AC. 
When  Sides  A B,  B C,  and  Angle  B are  given.— Fig.  7. 

_ ^ . o,-  T.  RA  v Sin  TC 


B C X Sin.  B 
AC 
BAX  Sin.  A 


BC 


= Sin.  A. 
= Sin.  C. 


BAX  Sin.  B 
AC! 

B C X Sin.  C 


= Sin.  C. 
= Sin.  A. 


BC,  ' AB 

To  Compute  all  tlie  Angles. 

When  all  the  Sides  are  given , Figs.  6 arid  7.  Rule.— Let  fall  a perpen- 
dicular, B d,  opposite  to  required  angle.  Then,  as  A C : s^  °f  A B,  B C 
their  difference  : twice  d g , the  distance  of  perpendicular,  B d,  from  middle 

°f  Hence  SA*  d,  C a are  known,  and  triangle,  A B C,  is  divided  into  two  right- 
angled  triangles!  BCd,B  A <2;  then,  by  rules  for  right-angled  triangles, 
ascertain  angle  A or  0.  . . „ „ r 

Operation. -A  C,  Fig.  6, . 5014  : A B + B C,  1. 1 174  + MH2  = 2-  53*6 . . A B a>  B L, 

1. 4142  — 1. 1174  = .2968  : 2 X d g — 1 4986. 

Hence  A c!  = ^ ^ • 4986,  and  C d = A d + A C = x. 

Consequently,  triangle  B d C,  Fig.  6,  is  divided  into  two  triangles,  B A C and  B d A. 
To  Compute  Side  A B and  Angles. 

When  Two  Sides  and  One  Angle,  or  One  Side  and  Two  Angles,  are  given.- 
Fig.  6. 


AC  X Sin.  C 


B C X Sin.  B 
AC 
A B X Sin.  B 


= Sin.  A. 
= Sin.  C. 


A C X Sin.  A 
AB— (ACxCos.  A) 
AC  X Sin.  C 


= Tah.  B. 
Tan.  B. 


2 Area 
B C,  Sin.  C 


BC— (ACxCos.  C) 

To  Compute  Area  of  a Triangle. -Fig.  S. 
BAxBCxSin.  B AC  X BC  X Sin.  C B A X A C X Sin.  A 

7 T,  I ’ 2 

Sin.  2 C,  B C2  A C-,  Tan.  C an(J  B Aa,  Cot.  C _ 

^ ’ 2 ’ 2 
Note.— For  other  rules,  see  Mensuration  of  Areas,  Lines,  and 
JA  Surfaces,  page  335. 

To  Compute  Sides. 

When  Areas  and  Angles  are  given.— Figs.  6 and  7. 

2 Area 


= AC. 


A C,  Sin.  A 


- = BA. 


J Si 


2 Area,  Sin.  A _ g q 
Sin.  C,  Sin.  (A  -pC) 


To  Ascertain  Distance  of  Inacces- 
sible Objects  on  a Level  Diane.— 
Digs.  9 and  IO 


Fig.  10. 


Fig.  9. 


Operation.— Lay  off  perpendic- 
ulars to  line  A B,  Fig.  9,  as  B c,  d e, 
on  line  A d , terminating  on  line 
e A. 

Then  e d — c B : c B : : B d : BA. 

When  there  are  Two  Inacces- 
sible Objects , as  Fig.  10. 

Operation.  — Measure  a base 
line,  A B,  Fig.  10,  and  angles  cAB,  £ 
dB  A,  Acd,  Bdc,etc.  Then  pro- 
ceed by  formulas,  page  387,  to  deduce  cd. 

Note. If  course  of  cd  is  required,  take  difference  of  ang 

d c A and  cd  B from  course  A B. 


PLANE  TRIGONOMETRY. 


B Fig.  ”• 


When  the  Objects  can  be  aligned. — 
Fig.  ii.  I 

Operation.— Align  c B,  Fig.  n,  at  A, 
measure  a base  line  at  any  angle  there- 
to, as  A o,  and  angles  o A c,  B A c,  and 
Bo  A.  Then  proceed  as  per  formula 
page  386,  to  deduce  c B. 

To  Compute  Distance  from 
a Given  Point  to  an  In- 
accessible Object.  — Die:. 

13. 


389 

Fig.  12. 


Operation.— Measure  a level  line,  Ac,  Fig.  12,  and  ascertain  angles  B Ac  c A B 
SrmTne  A BDg  ’ A ***  tW°  angles’  pr0Ceed  as  formula,  page  38 ^to  de- 


To  Compute  Height  of  an  Elevated.  Doint.— DP 


. 13. 


, . , compute  uisiance  Ac,  Fjc. 

t»4a  ascrep^ta,n  anSle  °f  depression  Aoc  and  of  elevation 

JtS  A o.  I hen  nrnnoorl  o o n ^ . 


Operation.  — Measure 
distance  on  a horizontal 
line,  A c,  Fig.  13;  ascertain 
Angle  B A c.  Then  pro- 
ceed as  per  formulas,  pp. 
386-8,  to  ascertain  B c. 

When  a Horizontal 
Base  is  not  Attainable. 

—Fig.  14. 

Operation. — Measure  or 
compute  distance  Ac,  Fig. 


Fig.  15. 


When  a Full  Base  Line  is  not  A ttain- 
B able.  Fig.  15. 

Operation.  — Measure  a base 
line,  A c,  Fig.  15,  and  ascertain 
angles  A c B,  c A B. 

Then  proceed  as  per  for- 
mula, page  386,  to  ascer- 
tain d B. 


Fig.  16. 


Without  Use  of  an  Instrument. 

—Fig.  16. 

t^^ft^a^UkTeWaUon^ro^^se^ni^rf  d'W’*A*e*  up  a staff  at  each  ex- 

ojr‘Se  “^ight  ofeyjfrom 

Then  D 7 


x—y 


and  D length  of  line  d d. 


+ A + i = height,  s representing  height  of  line  of  sight  from  base  d d, 


K K* 


39° 


O -£ 
2*  « 
SV  c- 

29 


natural  sines  and  cosines. 


ISTatnral  Sines 


3 

3 

4 

4 

5 

5 

6 
6 
7 

7 

8 
8 
9 
9 


0° 


N.  sine. 


13 

13 

14 

14 

15 
15 

15 

16 

16 

17 
17 


23 

23 

24 

24 

25 

25 

26 

26 

27 

27 

28 

28 

29 
29 


34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

8 


.00000 

.00029 

.00058 

.00087 

.00116 

.00145 

.00175 

.00204 

.00233 

.00262 

.00291 

.0032 

.00349 

.00378 

.00407 

.00436 

.00465 

.00495 

00524 

.00553 

.00582 

.00611 

.0064 

.00669 

.00698 

.00727 

.00756 

.00785 

.00814 

.00844 

.00873 

.00902 

.00931 

.0096 

.009'' 


1° 


N.  sine. 


.01018 

.01047 

.01076 

,01105 

.OH34 

.01164 

.01193 

.01222 

.01251 

.0128 

.01309 

.01338 

.01367 

.01396 

.01425 

01454 

.01483 

.01513 

.01542 

.01571 

.016 

.01629 

.01658 

.01687 

.01716 

.01745 


.99999 

.99999 

.99999 

•99999 

.99999 

.99999 

.99999 

.99999 

.99998 

.99998 

.99998 

.99998 

.99998 

.99998 

•99997 

•99997 

•99997 

.99997 

.99996 

.99996 

.99996 

.99996 

•99995 

•99995 

•99995 

•99995 

•99994 

.99994 

•99994 

•99993 

•99993 

•99993 

.99992 

.99992 

.99991 

.99991 

.99991 

•9999 

•9999 

.99989 

.99989 

.99989 

.99988 

.99988 

.99987 

.99987 

.99986 

. 99986 

•99985 

.99985 


N.  cos.  N.  sine.  N.  cos. 


.01745 
•01774 
01803 
.01832 
.01862 
.01891 
.0192 
.01949 
.01978 
,02007 
,02036 
.02065 
. 02094 
,02123 
.02152 
.02181 
.02211 
,0224 
.02269 
.02298 
,02327 
,02356 
.02385 
.02414 
.02443 
.02472 
.02501 
.0253 
.0256 
.02589 

,02618 

,02647 

.02676 

.02705 

.02734 

.02763 

.02792 

.02821 

.0285 

02879 


and.  Cosines. 

2° 


N.  cos, 


N.  sine, 

89° 


.02908 

.02938 

.02967 

02996 

03025 

.03054 

.03083 

.03112 

.03141 

.0317 

.03199 

.03228 

.03257 

.03286 

.03316 

•03345 

•03374 

.03403 

•03432 

.03461 

•0349 


.99985 

.99984 

.99984 

.99983 

.99983 

.99982 

.99982 

.99981 

•999? 

.9998 

•99979 

•99979 

.99978 

.99977 

•99977 

.99976 

.99976 

•99975 

•99974 

•99974 

•99973 

.99972 

.99972 

•9997 

•9997 

.99969 

.99969 

.99968 

.99967 

. 99966 

. 99966 

•99965 

.99964 

,99963 

,99963 

, 99962 

.99961 

.9996 

•99959 

•99959 

.99958 

•99957 

•99956 

•99955 

•99954 

•99953 

•99952 

.99952 

•99951 

•9995 

•99949 

.99948 

•99947 

•99946 

•99945 

.99944 

•99943 

•99942 

.99941 

•9994 

•99939 


3° 


N.  sine. 


•99939 

•99938 

•99937 

•99936 

•99935 

•99934 

•99933 

.99932 

•99931 

•9993  I 

.99929  | 

.99927 

.99926 

.99925 

.99924 

.99923 

.99922 

.99921 

.99919 

•999l8 

.99917 

.99916 

•99915 

•999*3 

•99912 

•99911 

.9991 

.99909 

.99907 

.99906 

.99905 
.99904 
.99902 
• 99901 
•999 

.99898 
•99897 
. 99896 

•99894 

.99893 

.99892 

.9989 


.99886 

.99885 

•99883 

.99882 

.99881 

.99879 

.99878 

.99876 

.99875 

•99873 

.99872 

.9987 

.99869 

.99867 

.99866 

.99864 

.99863 


05234 

.05263 

.05292 

.05321 

•0535 

05379 

>5408 

•05437 

.05466 

05495 

.05524 

•05553 

.05582 

.0561 

,0564 

.05669 

05698 

,05727 

■05756 

.05785 

.05814 

05844 

.05873 

05902 

•05931 

0596 

,05989 

06018 

,06047 

.06076 

.06105 

.06134 

.06163 

.06192 

.0622 

,0625 


99863 

99861 

.9986 

99858 

•99857 

.99855 

.99854 

.99852 

,99851 


.06279 

.06308 

•06337 

.06366 

.06395 

.06424 

.06453 

.06482 

.0651 

.0654 

.06569 

.06598 

.06627 

.06656 

.06685 

.06714 

.06743 

.06773 


g*  5 

P<  t 


.99847 

,99846 

.99844 

.99842 

.9984 


.99838 

99836 

■99834 

.99833 

99831 

.99829 

.99827 

,99826 

99824 


59 

58 

57 

56 

55 

54 

53 

52 

5i 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 


99821 
.99819 
.99817 
,99815 
.99813 
.99812 
■99?1 

. 99808 
. 99806 
.99804 

.99803 

.99801 
•99799 
•99797 
•99795 
•99793 
.99792 

•9979 

.99788 
.99786 
•99784 
.99782 
•9978 

.9977? 

.99776 

•99774 

.99772 

I -9977 

.06802  .99708 
.06831  .99766 

.0686  I .99764 
.06889  .99762 

.06918  .9976 

.06947  | -99758 
.06976  ! .99756  , o 
N.  cos.  I N.  sine.  ! ' 

860  I 


NATURAL  SINES  AND  COSINES. 


391 


Prop. 

parts. 

40 

I 

5° 

6° 

70 

29 

N.  sine 

i.  N.  cos 

1 N.  sine 

. N.  cos 

. N.  sine 

5.  N.  cos 

. N.  sine 

}.  N.  cos. 

0 
.0 

1 

1 

2 
2 
3 

3 

4 

4 

5 

5 

6 
6 
7 

7 

8 
8 
9 
9 

10 

10 

11 

11 

12  2 

12  2 

13  2 

13  2 

14  2 

14  2 

15  3 
15  3 

15  3 

16  3 

16  3. 

17  3^ 

17  3< 

18  3; 

18  3* 

19  3c 

19  4c 

20  41 

20  42 

21  43 

21  44 

22  45 

22  46 

23  47 

23  48 

24  49 

24  50 

25  5i 

25  52 

26  53 

26  54 

27  55 

27  56 

28  57 

28  58 

29  59 

29  60 

0 .0697* 

1 . 0700 

2 0703^ 

3 .0706; 

4 .07095 

5 .07121 

6 .0715 

7 .07175 

0 .0720$ 

9 -07237 

10  .07266 

11  .07295 

12  .07324 
r3  .07353 

14  .07382 

15  .07411 

16  .0744 

17  -07469 
r8  .07498 
[9  .07527 

20  .07556 

21  -07585 

'2  .07614 

3 -07643 

4 .07672 

5 .07701 

6 -0773 

7 -07759 

8 .07788 

9 .07817 

0 .07846 

1 .07875 

2 . 07904 

3 .07933 

\ .07962 
5 .07991 
3 . 0802 

7 .08049 

3 .08078* 

) .08107 

> .08136 
.08165 
! .08194 

1 .08223 

.08252 
.08281 
•0831 
-08339 
.08368 
08397 
.08426 
•08455 
.08484  . 

•08513  . 

.08542  . 

-08571  . 

.086 

.08629  * 
.08658  . 
.08687  • 

.08716  . 

.99756 

•99754 

•99752 

•9975 

•99748 

.99746 

•99744 

•99742 

•9974 

•99738 

.99736 

•99734 

•9973i 

•99729 

.99727 

•99725 

.99723 

• 99721 

.99719 

.93716 

.99714 

.99712 

.9971 

.99708 

•99705 

•99703 

.99701 

.99699 

.99696 

.99694 

•99692 

.99689 

.99687 

.99685 

.99683 

.9968 

.99678 

.99676 

.99673 

.99671 

. 99668 

.99666 

.99664 

.99661 

.99659 

•99657 

.99654 

.99652 

.99649 

.99647 

.99644 

.99642 

.99639 

99637 

99635  • 

99632 

9963  • 

99627  . 

99625  . 

99622 
99619  . 

.o87i 

.0874 

.0877. 

.0880 

.0883' 

.0886 

.0888c 

.0891$ 

.08947 

.08976 

.09005 

•09034 

.09063 

.09092 

.09121 

.0915 

•°9I79 

.09208 

.09237 

.09266 

.09295 

•09324 

•09353 

.09382 

.09411 

•0944 

. 09469 

.09498 

.09527 

•09556 

•09585 

.09614 

.09642 

.09671 

.097 

.09729 

.09758 

.09787 

.09816 

•09845 

.09874 

.09903 

•09932 

.09961 

•°999 
. 10019 
. 10048 
10077 
. 10106 
•10135 
.10164 
. 10192 
.10221 
• 1025 
.10279  ■ 

. 10308  . 
ID337  • 
10366  . 

10395  • 

10424  . 

IQ453  • 

5 .9961c 
•9961; 

.99614 

.99612 

.99605 

.99607 

.99604 

. 99602 

•99599 

•99596 

•99594 

•9959i 

.99588 

.99586 

•99583 

•9958 

•99578 

•99575 

•99572 

•9957 

•99567 

•99564 

•99562 

•99559 

•99556 

•99553 

•99551 

.99548 

•99545 

.99542 

•9954 

•99537 

•99534 

•9953i 

•99528 

•99526 

•99523 

•9952 

•99517 

•99514 

•995II 

.995o8 

.99506 

•99503 

•995 

•99497 

•99494 

•99491 

.99488 

•99485 

.99482 

•99479 

■99476 

•99473 

9947 

99467 

99464 

99461 

99458 

99455  • 

99452  . 

) -io45; 

r . 1048: 
l -1051: 
■ • 1054 

) • 1056c 

' • 10597 

. 1062^ 
! -1065c 

i . 10684 
. IO713 
• IO742 
.IO771 
. 108 
. 10829 
. 10858 
.10887 
.IO916 
• io945 
• io973 
. 1 1002 
.11031 
.1106 
. 1 1089 
.11118 
.H147 
.11176 
.11205 
.11234 
.11263 
.11291 
.1132 

• XI349 
.11378 
.11407 
.11436 
.11465 
.11494 

• ri523 
•11552 

.1158 

.11609 

.11638 

.11667 

.11696 

.11725 

•II754 

• H783 

. 11812 
.1184 
.11869 
.11898 
.11927 
.11956 
.11985  . 

.12014  . 

• I2043  . 

.12071  . 

. 121 

.12129  • 
.12158  . 
.12187  • 

3 -9945= 
2 .9944c 

t .9944^ 
•99443 
) -9944 
r -99437 
> -99434 
; .99431 

■ .99428 

• 99424 
•99421 
.99418 
•99415 
.99412 
•99409 

• 99406 
.99402 
•99399 
•99396 
•99393 
•9939 
.99386 
•99383 
•9938 
•99377 
•99374 
•9937 
•99367 
•99364 
•9936 
•99357 
•99354 
•99351 
•99347 
•99344 
•9934i 
•99337 
•99334 
•9933i 
•99327 
•99324 
•9932 
.99317 
•99314 
•993i 
•99307 
•99303  • 
•993 
.99297 
.99293 
•9929 

. 99286 
.99283 
.99279 
99276 
99272  . 

99269  . 

99265  . 

99262  . 

99258  . 

99255  • 

2 .1218; 

) . I22l( 

) .1224c 

! .12274 

.12302 
•12331 
.1236 
.12389 
. 12418 
.12447 
.12476 
.12504 
•12533 
. 12562 
.12591 
.1262 
.12649 
.12678 
. 12706 
• I2735 
.12764 

• i2793 
.12822 
.12851 
. 1288 
. 12908 
• 12937 
.12966 
•I2995 
• 13024 

•13053 

.13081 
.1311 
• I3I39 
.13168 

• I3I97 
•13226 

•13254 

.13283 

•13312 

• I334I 

•1337 

•13399 

• i3427 

•i3456 

•13485 

•I35I4 
•13543 
•13572 
.136 
.13629 
.13658  , 

•13687  . 

•13716  . 
• 33744  • 
13773  • 
13802  . 

*3831  • 
1386  . 

13889  . 

I39I7  • 

1 -99255  60 

> -99251  59 

> .99248  58 

!•  -99244  57 
! .9924  56 

: -99237  55 

•99233  54 

1 -9923  53 

.99226  52 

.99222  51 

.99219  50 

.99215  49 

.99211  48 

.99208  47 

. 99204  46 

•992  45 

.99197  44 

•99193  43 

.99189  42 

.99186  41 

.99182  40 

.99178  39 

.99175  38 

•99I7I  37 

•99i67  36 

•99163  35 

.9916  34 

.99156  33 

•99I52  32 

•99j48  31 

•99I44  3° 
.99141  29 

.99137  28 

•99J33  27 
.99129  26 

•99I25  25 

.99122  24  : 

.99ji8  23  : 

.99114  22 

.9911  21  j 

.99106  20  ] 

.99102  19  ] 

.99098  18  1 

•99094  17  1 

.99091  16  1 

.99087  l5  1 

•99083  14  1 

.99079  13  1 

.99075  12  1 

.99071  11  1 

.99067  10  1 

.99063  9 1 

.99059  8 1 

99°55  7 0 

99051  6 0 

99047  5 0 

99043  4 0 

99039  3 0 

99°35  2 0 

99031  1 0 

99027  0 0 

N.  COB.  I 

85° 

sine.  N.  cos.  I 

II  84° 

4.  sine. 

N.  cos.  I 
83o 

4.  sine.  ] 

N-.  cos.  P 

82° 

4.  sine.  ' 

Prop. 

parts. 


392 


natural  sines  and  cosines. 


p*2 
0 T. 

8° 

e-  2. 

28 

' 

N . sine.  I 

0 " 

0 

.13917  ■ 

0 

1 

.13946  . 

1 

2 

•13975  • 

1 

3 

.14004  . 

2 

4 

•M033  * 

2 

5 

.14061  . 

3 

6 

.1409  . 

3 

7 

.14119  • 

4 

8 

.14148  . 

4 

9 

.14177  • 

5 

10 

.14205  . 

5 

11 

.14234  . 

6 

12 

.14263  . 

6 

13 

.14292 

7 

14 

.1432 

7 

i5 

•14349 

7 

16 

•14378 

8 

17 

.14407 

8 

18 

.14436 

9 

19 

.14464 

9 

20 

•14493 

10 

21 

.14522 

10 

22 

.14551 

11 

23 

.1458 

11 

24 

. 14608 

12 

25 

•14637 

12 

26 

.14666 

13 

27 

• 14695 

13 

28 

•14723 

14 

29 

-X4752 

1\ 

30 

.14781 

,99019 

•99OI5 
.99011 
.99006 
. 99002 


.98994 

.9899 

.98986 

,98982 

,98978 

•98973 

. 98969 
.98965 
.98961 

•98957 

.98953 

.98948 

.98944 

.9894 

.98936 

.98931 

.98927 

.98923 

.98919 

.98914 

•<*9? 

. 98906 
. 98902 
.98897 
.98893 


•15643 

.15672 

• 15701 
•1573 
•15758 
•15787 

.15816 

•15845 

•15873 

.15902 
• I593I 
•15959 
.15988 
.16017 
. 16046 
.16074 
.16103 
. 16132 
. 1616 
.16189 
16218 
[6246 
16275 
16304 
i6333 
6361 
1639 
16419 
16447 

6476 


.98876 
.98871 
.98867 
.98863 
.98858 
.98854 
.98849 
.98845 
.98841 
.98836 
.98832 
.98827 
.98823 
.98818 
.98814 
. 98809 
.98805 
.988 
.98796 

•98791 

.98787 
.98782 
.98778 

•98773 

•98769 

N.  cos.  N.sirte* 

61° 


6591 

1662 


.1682 


D 

N.  cos.  j j 

10° 

PsT.  sine.  1 

1 

S’,  cos;*  | N 

.98769 

.17365  • 

98481  J .1 

.98764 

•17393  ■ 

.98476  .1 

.9876 

.17422  . 

,98471  .1 

•98755 

.17451  ■ 

,98466  j .: 

•98751 

•17479 

.98461  j .: 

.98746 

.17508 

.98455  . 

.98741 

•17537 

.9845 

•98737 

•17565 

.98445  • 

.98732 

•17594 

.9844 

.98728 

.17623 

.98435  . 

•98723 

.17651 

.9843 

.98718 

.1768 

.98425  . 

.98714 

.17708 

.9842 

.98709 

.17737 

.98414  • 

.98704 

.17766 

.98409 

•987 

•17794 

.98404  • 

.98695 

.17823 

.98399  • 

.9869 

.17852 

.98394  • 

.98686 

. 1788 

.98389  1 . 

.98681 

.17909 

.98383 

.98676 

•17937 

•98378 

.98671 

. 17966 

9o3aq 

.98667 

•17995 

.98368 

.98662 

.18023 

.98362 

.98657 

. 18052 

.98357 

.98652 

.18081 

.98352 

.98648 

.18109 

•98347 

.98643 

.18138 

.98341 

.98638' 

.18166 

•98336 

' -98633 

.18195 

•98331 

; .98629 

.18224 

.98325 

1 .98624 

.18252 

.9832 

- .98619 

.18281 

•98315 

.98614 

.18309 

.9831 

.98609 

.18338 

.98304 

5 .98604 

.18367 

.98299 

r .986 

.18395 

.98294 

> -98595 

.18424 

.98288 

\ -9859 

.18452 

.98283 

3 .98585 

.18481 

.98277 

2 .9858 

.18509 

.98272 

•98575 

.18538 

.98267 

? -9857 

.18567 

.98261 

8 .93565 

.18595 

.98256 

6 .98561 

.18624 

•9825 

5 -98556 

. 18652 

.98245 

4 -98551 

. r868i 

.9824 

2 .98546 

.1871 

.98234 

1 .98541 

.18738 

.98229 

i .98536 

| .18767 

.98223 

’8  .98531 

1 -18795 

, .98218 

>7  .98526 

1 .18824 

. .98212 

>6  .98521 

.18852 

! . 98207 

34  .98516 

> .18881 

.982OI 

)3  -98511 

.1891 

.98196 

22  .98506 

> .1893? 

l .9819 

5 .98501 

t .18965 

r .98185 

79  -9849( 

3 .1899; 

5 -98x79 

08  .98491 

i .19024  .90174 

36  . 9848* 

S .19052  j .98168 

65  .9848 

1 j .19081  j .98163 

11° 

N.  sine.  N.  cos. 


19281 
19309 
^9338 
19366 
*9395 
19423 
19452 
[9481 
19509 
'9538 
19566 
19595 
19623 
.19652 
.1968 
19709 
19737 
.19766 
19794 
.19823 
19851 
1988 
19908 
19937 
• 19965 
19994 
.20022 
.2005 
. 20079 
-20108 
.20136 
.20165 
.20193 
. 20222 
.2025 
. 20279 
. 20307 
.20336 
. 20364 
.20393 
.20421 
.2045 
. 20478 
.20507 
.20535 
.20563 
. 20592 
.2062 
. 20649 
.20677 
.20706 
.20734 
.20763 
.20791 


N.  cos.  | N.  sine,  j 

800 


790 


.98004 

.97988 

.97992 

.97987 

.97981 

•97975 

•97969 

•97963 

•97958 

•97952 

•97946 

•9794 

•97934 

.97928 

.97922 

•979l6 

.9791 

.97905 

.97899 

•97893 

.97887 

.97881 

.97875 

.97869 

.97863 

.97857 

•97851 

•97845 
.97839  ! 
•97833  ! 
.97827 
.97821  j 
.97815 


N.  cos.  ! N.  sine,  j 

780  ] 


Prop. 
Cn  parts. 


NATURAL  SINES  AND  COSINES. 


© t 
£ 

2 7 

I 

. 

' N.  sine 

12° 

13° 

I40 

150 

Prop. 

parts. 

o 

c 

.20791 

•97815 

•22495 

•97437 

.24192 

•9703 

to 

00 

•96593 

60 

9 

o 

1 

.2082 

.97809 

•22523 

•9743 

.2422 

.97023 

.2591 

•96585 

59 

9 

I 

2 

. 20848 

•97803 

•22552 

•97424 

.24249 

•97015 

•25938 

.96578 

58 

9 

I 

0 

.20877 

•97797 

.2258 

•97417 

•24277 

.97008 

.25966 

•9657 

57 

9 

2 

< 

.20005 

.97791 

.22608 

.97411 

•24305 

.97001 

•25994 

.96562 

56 

8 

2 

5 

.20933 

.97784 

.22637 

•97404 

•24333 

•96994 

. 26022 

•96555 

55 

8 

3 

6 

.20962 

•97778 

.22665 

•97398 

•24362 

.96987 

.2605 

•96547 

54 

8 

3 

7 

.2099 

.97772 

.22693 

•97391 

•2439 

.9698 

.26079 

•9654 

53 

8 

4 

8 

.21019 

.97766 

.22722 

•97384 

.24418 

•96973 

.26107 

.96532 

52 

8 

4 

9 

.21047 

.9776 

•2275 

•97378 

.24446 

.96966 

•26135 

•96524 

5i 

8 

5 

IO 

.21076 

•97754 

.22778 

•9737i 

•24474 

•96959 

.26163 

•96517 

50 

8 

5 

11 

.21104 

.97748 

.22807 

•97365 

•24503 

•96952 

. 26191 

•96509 

49 

7 

5 

12 

.21132 

.97742 

.22835 

•97358 

•24531 

•96945 

.26219 

.96502 

48 

7 

6 

13 

.21161 

•97735 

.22863 

•9735i 

•24559 

•96937 

.26247 

.96494 

47 

7 

6 

14 

.21189 

.97729 

.22892 

•97345 

•24587 

•9693 

.26275 

.96486 

46 

7 

7 

IS 

.21218 

•97723 

.2292 

•97338 

•24615 

•96923 

•26303 

.96479 

45 

7 

7 

16 

.21246 

.97717 

.22948 

•97331 

. 24644 

.96916 

.26331 

.96471 

44 

7 

8 

1 7 

.21275 

.97711 

.22977 

•97325 

.24672 

.96909 

•26359 

•96463 

43 

6 

8 

18 

.21303 

•97705 

•23005 

•973i8 

•247 

.96902 

.26387 

.96456 

42 

6 

9 

i9 

•21331 

.97698 

•23033 

•973H 

.24728 

.96894 

•26415 

.96448 

4i 

6 

9 

20 

.2136 

.97692 

.23062 

•97304 

.24756 

.96887 

• 26443 

•9644 

40 

6 

9 

21 

.21388 

.97686 

•2309 

.97298 

•24784 

.9688 

.26471 

•96433 

39 

6 

IO 

22 

.21417 

.9768 

.23118 

.97291 

•24813 

•96873 

•265 

•96425 

38 

6 

IO 

23 

.21445 

•97673 

•23146 

.97284 

.24841 

.96866 

.26528 

.96417 

37 

6 

II 

24 

.21474 

.97667 

•23175 

•97278 

. 24869 

.96858 

•26556 

.9641 

36 

5 

II 

25 

.21502 

.97661 

•23203 

.97271 

.24897 

.96851 

.26584 

.96402 

35 

5 

12 

26 

•2153 

•97655 

•23231 

.97264 

•24925 

.96844 

.26612 

•96394 

34 

5 

12 

27 

•21559 

.97648 

.2326 

•97257 

•24954 

•96837 

.2664 

.96386 

33 

5 

13 

28 

.21587 

.97642 

.23288 

•97251 

. 24982 

.96829 

.26668 

•96379 

32 

5 

13 

29 

.21616 

•97636 

•23316 

.97244 

.2501 

96822 

. 26696 

•96371 

31 

5 

14 

30 

.21644 

•9763 

•23345 

.97237 

•25038 

.96815 

.26724 

•96363 

3° 

5 

14 

3i 

.21672 

.97623 

•23373 

•9723 

. 25066 

. 96807 

■26752 

•96355 

29 

4 

14 

32 

.21701 

.97617 

• 23401 

•97223 

.25094 

.968 

.2678 

•96347 

28 

4 

15 

33 

.21729 

.97611 

•23429 

.97217 

.25122 

•96793 

.26808 

•9634 

27 

4 

15 

34 

.21758 

.97604 

•23458 

.9721 

•25151 

.96786 

. 26836 

•96332 

26 

4 

l6 

35 

.21786 

•97598 

.23486 

.97203 

•25179 

.96778 

.26864 

•96324 

25 

4 

l6 

36 

.21814 

•97592 

•23514 

.97196 

.25207 

.96771 

.26892 

.96316 

24 

4 

*7 

37 

.21843. 

•97585 

•23542 

.97189 

•25235 

.96764 

.2692 

. 96308 

23 

3 

17 

38 

.21871 

•97579 

•23571 

.97182 

•25263 

•96756 

. 26948 

.96301 

22 

3 

18 

39 

.21899 

•97573 

•23599 

.97176 

•25291 

.96749 

.26976 

•96293 

21 

3 

18 

40 

.21928 

.97566 

•23627 

.97169 

•2532 

•96742 

.27004 

.96285 

20 

3 

18 

4i 

.21956 

•9756 

•23656 

.97162 

•25348 

•96734 

.27032 

.96277 

J9 

3 

*9 

42 

.21985 

•97553 

.23684 

•97155 

•25376 

.96727 

.2706 

.96269 

18 

3 

43 

.22013 

•97547 

.23712 

•97!48 

.25404 

.96719 

. 27088 

.96261 

17 

3 

20 

44 

.22041 

•97541 

•2374 

.97141 

•25432 

.96712 

.27116 

•96253 

16 

2 

20 

45 

.2207 

•97534 

•23769 

•97134 

•2546 

.96705 

.27144 

-96246 

15 

2 

21 

46 

.22098 

.97528 

•23797 

.97127 

.25488 

.96697 

.27172 

.96238 

!4 

2 

21 

47 

.22126 

•97521 

.23825 

.9712 

.25516 

.9669 

.272 

•9623 

13 

2 

22 

48 

•22155 

•97515 

•23853 

•97H3 

•25545 

.96682 

.27228 

.96222 

12 

2 

22 

49 

.22183 

.97508 

.23882 

.97106 

•25573 

•96675 

.27256 

.96214 

11 

2 

23 

50 

.22212 

.97502 

.2391 

.971 

.25601 

. 96667 

.27284 

. 96206 

10 

2 

23 

5i 

.2224 

.97496 

•23938 

.97093 

.25629 

.9666 

.27312 

.96198 

9 

1 

23 

52 

.22268 

•97489 

.23966 

.97086 

•25657 

•96653 

•2734 

.9619 

8 

1 

24 

53 

.22297 

•97483 

•23995 

•97079 

.25685 

•96645 

.27368 

.96182 

7 

1 

24 

54 

.22325 

.97476 

. 24023 

.97072 

•25713 

.96638 

.27396 

.96174 

6 

1 

25 

55 

•22353 

•9747 

.24051 

.97065 

•25741 

•9663 

.27424 

.96166 

5 

1 

25  : , 

56 

.22382 

•97463 

.24079 

.97058 

.25769 

.96623 

•27452 

.96158 

4 

1 

26 

57 

.2241 

•97457 

.24108 

•97051 

•25798 

•96615 

.2748 

.9615 

3 

0 

26  1 

58 

.22438 

•9745 

.24136 

.97044 

.25826 

.96608 

.27508 

.96142 

2 

0 

27  ; 

59 

.22467  . 

.97444 

.24164 

•97037 

.25854  , 

.966 

•27536 

.96134 

1 

0 

27  | < 

5o 

.22495  . 

97437 

.24192 

•9703 

.25882 

■96593 

.27564 

.96126 

0 

X.  cos.  ] 

X.  sine. 

N.  cos.  1 j 

X.  sine. 

X.  cos.  ] 

X.  sine. 

N.  cos.  1 1 

S’,  sine. 

✓ 

» 

770 

750 

75°  1 

74°  1 

Prop. 


394 


NATURAL  SINES  AND  COSINES. 


16° 


N.  sine. 


10 

23 

II 

24 

II 

25 

12 

26 

12 

27 

13 

28 

13 

29 

14 

30 

14 

3i 

32 

15 

33 

15 

34 

l6 

35 

l6 

36 

17 

37 

17 

38 

18 

39 

18 

40 

18 

4i 

i9 

42 

*9 

43 

20 

44 

20 

45 

21 

46 

21 

47 

23 


•27564 

.27592 

.2762 

.27648 

.27676 

.27704 

•27731 

•27759 

.27787 

.27815 

.27843 

.27871 

.27899 

.27927 

•27955 

.27983 

.28011 

. 28039 

. 28067 

.28095 

.28123 

.2815 

.28178 

.28206 

.28234 

.28262 

.2829 

.28318 

.28346 

.28374 

.28402 

.28429 

.28457 

.28485 

.28513 

.28541 

.28569 

.28597 

.28625 

.28652 

.2868 

.28708 

.28736 

.28764 

.28792 

.2882 

.28847 

.28875 

. 28903 

•28931 

.28959 

.28987 

.29015 

.29042 

.2907 

. 29098 

.29126 

•29154 

.29182 

.29209 

.29237 


.96126 
.96118 
.9611 
.96102 
. 96094 
. 96086 
.96078 
.9607 
. 96062 
.96054 
. 96046 
.96037 
. 96029 
.96021 
.96013 
. 96005 

•95997 

.95989 

.95981 

•95972 

.95964 

•95956 

•95948 

•9594 

•9593i 

•95923 

•959I5 

•959°7 

.95898 

•9589 

.95882 

•95874 

.95865 

•95857 


.9584] 

•95832 

.958: 

958i 


•9574 


•9569 


N.  cos. 


73° 


17° 

N.  sine,  j 

N.  cos.  N, 

•29237 

•9563  -3 

.29265 

.95622  .3 

.29293 

.95613  -3 

.29321 

•95605  -3 

.29348 

.95596  -3 

•29376 

•95588  .3 

.29404 

•95579  -3 

.29432 

•95571  -3 

.2946 

•95562  .3 

.29487 

•95554  -3 

•295X5 

•95545  -3 

•29543 

•95536  .3 

•29571 

•95528  .3 

•29599 

•95519  -3 

.29626 

•955ii  .3 

.29654 

•95502  .3 

.29682 

•95493  -3 

.2971 

•95485  -3 

•29737 

•95476  •: 

.29765 

•95467  •: 

•29793 

•95459  •: 

.29821 

•9545  •: 

.29849 

•95441  •: 

.29876 

•95433  •: 

.29904 

• 95424  •: 

.29932 

•95415  •: 

.2996 

•95407  •: 

.29987 

•95398  • 

.30015 

•95389  • 

.30043 

•9538 

.30071 

•95372  • 

.30098 

•95363  • 

.30126 

•95354  • 

•30154 

•95345  • 

.30182 

•95337  • 

.30209 

•95328  . 

•30237 

•95319  • 

.30265 

•953i 

> . 30292 

•953oi  • 

r .3032 

•95293  • 

) .30348 

.95284  • 

: . 30376 

•95275 

2 .30403 

.95266 

I-  -30431 

•95257 

3 .30459 

.95248 

7 . 30486 

•9524 

9 -30514 

.95231 

•30542 

.95222 

2 .3057 

.95213 

4 -30597 

•95204 

5 • 30625 

7 -30653 

, .95186 

8 . 3068 

•95!77 

. 30708 

! .95168 

i -3073^ 

> -95I59 

3 -30763 

5 -9515 

4 -30791 

: -95142 

,6  .3081c 

) -95133 

.7  .3o84( 

) .95124 

19  • 3087^ 

t -95115 

1 .309°' 

2 .95106 

ie.  N.  cos.  | N.  sine. 

72° 

18o 


N.  cos. 


•31593 

.3162 

.31648 

.31675 

•3i703 

•3173 


95106 

95097 

95088 

95079 

9507 

95061 

95052 

95043 

95033 

95024 

950X5 

95006 

94997 

94988 

94979 

9497 

94961 

94952 

94943 

94933 

94924 

949 1 5 

94906 

94897 


94878 

9486c 

9486 

94851 


9474 

9473 


,31868 
31896 

31923 

3I95I 
31979 
.32006 

.32034 

.32061 
.32089 
.32116 

•32144 

.3217 
.32199 
.32227 

•32254 

.32282 
•32309 
‘32337 

•32364 

• 32392 
•32419 

• 32447 
•32474 
.32502 

•32529 

•32557  I -94552 


710 


19° 

N.  sine.  | 

N.  cos. 

•32557 

•94552  6c 

•32584 

•94542  59 

.32612 

•94533  5* 

.32639 

•94523  5/ 

. 32667 

•94514  5< 

.32694 

•94504  5; 

.32722 

•94495  5< 

•32749 

.94485  5; 

•32777 

•94476  5i 

.32804 

.94466  5: 

•32832 

•94457  5< 

.32859 

•94447  4< 

•32887 

•94438  4' 

•32914 

• 94428  4' 

•32942 

.94418  4{ 

. 32969 

•94409  4 

.32997 

•94399  4 

•33024 

•9439  4 

•33051 

•9438  4 

•33079 

•9437  4 

. 33106 

.94361  4 

j -33134 

94351  3 

.33i6i 

•94342  3 

•33189 

•94332  3 

.33216 

•94322  3 

•33244 

•94313  3 

1 -33271 

•94303  3 

•33298 

•94293  3 

•33326 

.94284  3 

: ! -33353 

•94274  3 

: -33381 

.94264  3 

i -33408 

•94254  1 

^ -33436 

•94245  2 

5 -33463 

•94235  2 

> -3349. 

.94225  J 

5 .335x8 

.94215  : 

7 -33545 

. 94206  : 

5 -33573 

.94196  : 

3 .336 

.94186 

9 *33627 

.94176 

.33655 

•94i67 

.33682 

•94157 

1 -3371 

•94147 

2 -33737 

•94137 

2 .33764 

•94127 

3 .33792 

.94118 

4 .33819 

.94108 

4 -33846 

.94098 

>5  -33874 

. 94088 

,6  .33901 

.94078 

^6  .33929 

.94068 

\7  -33956 

.94058 

57  -33983 

•94049 

c8  .34011 

•94039 

39  .34038 

.94029 

)9  -34065 

.94019 

9 -34093 

.94009 

3 .3412 

7 1 -34H7 

•93999 

•93989 

5i  .34175 

•93979 

52  .34202 

•93969 

ne.  N.  cos. 

N.  sine. 

70° 

9 

9 

9 

9 

8 

8 

8 

8 

8 

8 

8 

7 

,7 

7 

7 

7 

7 

6 

6 

6 

6 

6 

6 

6 

5 

5 

5 

5 

5 

5 

5 

4 

4 

4 

4 

4 

4 

3 

3 

3 

3 

3 

3 

3 


NATURAL  SINES  AND  COSINES, 


395 


c.® 
C Z 

20° 

210 

220 

27 

' 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

1 

j N.  cos. 

o 

0 

.34202 

•93969 

•35837 

•93358 

•3746i 

.92718 

o 

z 

.34229 

•93959 

•35864 

•93348 

•37488 

. 92707 

I 

2 

•34257 

•93949 

•35891 

•93337 

•37515 

.92697 

I 

3 

.34284 

•93939 

•359l8 

•93327 

•37542 

.92686 

2 

4 

•343** 

.93929 

•35945 

•933x6 

•37569 

•92675 

2 

5 

•34339 

•939*9 

•35973 

•933o6 

•37595 

.92664 

3 

6 

.34366 

•93909 

•36 

•93295 

.37622 

.92653 

3 

7 

•34393 

•93899 

.36027 

•93285 

•37649 

.92642 

4 

8 

.34421 

.93889 

•36054 

•93274 

•37676 

.92631 

4 

9 

1 -34448 

•93879 

.36081 

.93264 

•37703 

.9262 

5 

IO 

•34475 

.93869 

.36108 

•93253 

•3773 

.92609 

5 

11 

•34503 

•93859 

•36135 

•93243 

•37757 

.92598 

5 

12 

•3453 

•93849 

.36162 

•93232 

•37784 

.92587 

6 

13 

•34557 

•93839 

•3619 

.93222 

.37811 

.92576 

6 

14 

•34584 

•93829 

.36217 

.93211 

•37838 

•92565 

7 

i5 

.34612 

.93819 

.36244 

.93201 

•37865 

•92554 

7 

16 

•34639 

•93809 

.36271 

•93x9 

•37892 

•92543 

8 

17 

.34666 

•93799 

.36298 

.9318 

•379*9 

•92532 

8 

18 

.34694 

•93789 

•36325 

.93169 

•37946 

.92521 

9 

i9 

•34721 

•93779 

•36352 

•93159 

•37973 

.9251 

9 

20 

•34748 

•93769 

•36379 

.93148 

•37999 

.92499 

9 

21 

•34775 

•93759 

.36406 

•93x37 

.38026 

.92488 

IO 

22 

•34803 

•93748 

•36434 

•93127 

•38053 

.92477 

IO 

23 

•3483 

•93738 

.36461 

.93116 

.3808 

. 92466 

ii 

24 

34857 

•93728 

.36488 

.93106 

•38107 

•92455 

ii 

25 

.34884 

•937*8 

•36515 

•93095 

•38134 

•92444 

12 

26 

•34912 

.93708 

•36542 

•93084 

.38161 

.92432 

12 

27 

•34939 

.93698 

•36569 

•93074 

.38188 

.92421 

13  1 

28 

.34966 

.93688 

•36596 

.93063 

•38215 

.9241 

*3 

29 

•34993 

•93677 

.36623 

•93052 

.38241 

•92399 

14 

30 

.35021 

•93667 

•3665 

,93042 

.38268 

.92388 

i4 

3i 

•35048 

•93657 

•36677 

•93031 

•38295 

•92377 

14 

32 

•35075 

•93647 

•36704 

•9302 

•38322 

.92366 

15 

33 

•35102 

•93637 

•36731 

.9301 

•38349 

•92355 

15 

34 

•3513 

.93626 

•36758 

.92999 

•38376 

•92343 

16 

3| 

•35157 

.93616 

•36785 

.92988 

•38403 

•92332 

16 

36 

•35184 

.93606 

.36812 

.92978 

•3843 

.92321 

i7 

37 

•352U 

•93596 

•36839 

•92967 

•38456 

.9231 

i7 

38 

•35239 

•93585 

.36867 

.92956 

•38483 

. 92299 

18 

39 

.35266 

•93575 

• 36894 

.92945 

•3851 

.92287 

18 

40 

•35293 

•93565 

.36921 

•92935 

•38537 

.92276 

1 8 

41 

•3532 

•93555 

.36948 

.92924 

•38564 

.92265 

^9 

42 

•35347 

•93544 

•36975 

.92913 

.3859* 

.92254 

19 

43 

•35375 

•93534 

.37002 

. 92902 

•38617 

.92243 

20 

44 

•35402 

•93524 

•37029 

.92892 

.38644 

.92231 

20 

45 

•35429 

•93514 

•37056 

.92881 

.38671 

.9222 

21 

46 

•35456 

•93503 

•37083 

.9287 

.38698 

. 92209 

21 

47 

•35484 

•93493 

•3711 

.92859 

•38725 

.92198 

22 

48 

•355** 

•93483 

•37x37 

.92849 

•38752 

.92186 

22 

49 

•35538 

•93472 

•37164 

.92838 

•38778 

•92X75 

23 

5o 

•35565 

.93462 

•37I9I 

.92827 

.38805 

.92164 

23 

5i 

•35592 

•93452 

.37218 

.92816 

•38832 

.92152 

23 

52 

•35619 

•93441 

•37245 

92805 

•38859 

.92141 

24 

53 

•35647 

•93431 

•37272 

•92794 

.38886 

.9213 

24 

54 

•35674 

•9342 

•37299 

.92784 

.38912 

.92119 

25 

55 

•357oi 

•934i 

•37326 

•92773 

•38939 

.92107 

25 

56 

•35728 

•934 

•37353 

.92762 

.38966 

. 92096 

26 

57 

•35755 

•93389 

•3738 

•92751 

•38993 

.92085 

26 

58 

•35782 

•93379 

•37407 

•9274 

.3902 

• 92073 

27 

•358i 

•93368 

•37434 

.92729 

•39046 

. 92062 

27 

60 

•35837 

•93358 

•3746i 

.92718 

•39073 

.9205 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

65° 

68°  1 

67° 

23° 


N.  sine. 

N.  cos. 

11 

•39073 

.9205 

60 

11 

•39* 

•92039 

59 

11 

•39I27 

.92028 

58 

11 

•39*53 

.92016 

57 

10 

•39*8 

.92005 

56 

10 

•39207 

•9*994 

55 

10 

•39234 

.91982 

54 

10 

• 3926 

9*97* 

53 

10 

.39287 

•9*959 

52 

10 

•393*4 

.91948 

5* 

9 

•3934* 

•9*936 

5o 

9 

•39367 

•9*925 

49 

9 

•39394 

•9*9*4 

48 

9 

.39421 

.91902 

47 

9 

•39448 

.91891 

46 

8 

•39474 

•9*879 

45 

8 

•395oi 

.91868 

44 

8 

•39528 

.91856 

43 

8 

•39555 

•9*845 

42 

8 

•3958i 

•9*833 

4i 

8 

.39608 

.91822 

4o 

7 

•39635 

.9181 

39 

7 

.39661 

•9*799 

38 

7 

.39688 

.91787 

37 

7 

•397*5 

•9*775 

36 

7 

•3974* 

.91764 

35 

6 

•39768 

•9*752 

34 

6 

•39795 

.91741 

33 

6 

.39822 

.9*729 

32 

6 

.39848 

.91718 

3* 

6 

•39875 

.91706 

30 

6 

. 39902 

.91694 

29 

5 

•39928 

.91683 

28 

5 

•39955 

.91671 

27 

5 

.39982 

.9166 

26 

5 

. 40008 

.91648 

25 

5 

•40035 

.91636 

24 

4 

.40062 

•9*625 

23 

4 

. 40088 

.9*613 

22 

4 

•40**5 

.91601 

21 

4 

.40141 

•9*59 

20 

4 

.40168 

•9*578 

*9 

3 

.40195 

.9*566 

18 

3 

.40221 

•9*555 

*7 

3 

.40248 

•9*543 

16 

3 

•40275 

•9*53* 

*5 

3 

.40301 

•9*5*9 

*4 

3 

.40328 

•9*508 

*3 

2 

•40355 

.91496 

12 

2 

.40381 

.91484 

11 

2 

. 40408 

.91472 

10 

2 

•40434 

.91461 

9 

2 

.40461 

•9*449 

8 

1 

.40488 

•9*437 

7 

1 

.40514 

•9*425 

6 

1 

•40541 

•9*4*4 

5 

1 

•40567 

.91402 

4 

1 

•40594 

•9*39 

3 

1 

.40621 

•9*378 

2 

0 

.40647 

.91366 

1 

0 

.40674 

•9*355 

0 

0 

N.  cos. 

N.  sine. 

' 

660 


Prop. 

parts. 


NATURAL  SINES  AND  COSINES, 


396 


Oh  .2 
2 a 

24° 

250 

26° 

270 

Prop. 

parts. 

Ph  p. 

26 

' 

N.  sine. 

N.  cos. 

N.  Bine. 

N,  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

14 

0 

0 

.40674 

•9I355 

.42262 

.90631 

.43837 

.89879 

•45399 

.89101 

60 

14 

0 

1 

•4°7 

•9I343 

.42288 

.90618 

.43863 

.89867 

•45425 

. 89087 

59o 

4 

1 

2 

.40727 

•9^331 

•42315 

. 90606 

.43889 

.89854 

•45451 

.89074 

58 

14 

1 

•4°753 

■9I3*9 

•42341 

•90594 

.43916 

.89841 

•45477 

.89061 

57 

13 

2 

4 

.4078 

•9I3°7 

.42367 

.90582 

•43942 

.89828 

•45503 

.89048 

56 

3 

2 

5 

.40806 

.91295 

•42394 

.90569 

.43968 

.89816 

45529 

•89035 

55 

13 

3 

6 

•4o833 

.01283 

.4242 

•90557 

•43994 

. 89803 

•45554 

.89021 

54 

13 

3 

7 

.4086 

.91272 

.42446 

•90545 

.4402 

.8979 

4558 

.89008 

53 

12 

3 

8 

.40886 

.9126 

•42473 

•90532 

.44046 

•89777 

.45606 

.88995 

52 

12 

4 

9 

•4°9I3 

.91248 

•42499 

•9052 

.44072 

.89764 

•45632 

.88981 

5i 

12 

4 

10 

.40939 

.01236 

•42525 

.90507 

.44098 

•89752 

•45658 

.88968 

50 

12 

5 

11 

. 40966 

.91224 

•42552 

•90495 

.44124 

•89739 

.45684 

•88955 

49 

II 

5 

12 

.40992 

.91212 

•42578 

.90483 

•44i5i 

.89726 

•457i 

.88942 

48 

II 

6 

13 

.41019 

.912 

.42604 

•9047 

•44177 

•89713 

•45736 

.88928 

47 

1 

6 

14 

•4io45 

.91188 

.42631 

.90458 

•44203 

•897 

•45762 

.88915 

46 

II 

7 

15 

.41072 

.91176 

.42657 

.90446 

.44229 

.89687 

•45787 

.88902 

45 

II 

7 

16 

.41098 

.91164 

.42683 

•90433 

•44255 

.89674 

•45813 

.88888 

44 

IO 

7 

17 

•4II25 

.91152 

.42709 

.90421 

.44281 

. 89662 

•45839 

•88875 

43 

IO 

8 

18 

•4II5I 

•9II4 

.42736 

. 90408 

•44307 

.89649 

•45865 

.88862 

42 

IO 

8 

19 

.41178 

.91128 

.42762 

.90396 

•44333 

.89636 

•45891 

. 88848 

41 

IO 

9 

20 

.41204 

.91116 

.42788 

•90383 

•44359 

.89623 

•45917 

.88835 

4° 

9 

9 

21 

.41231 

.91104 

.42815 

•90371 

•44385 

.8961 

•45942 

.88822 

39 

9 

10 

22 

•41257 

. QIOQ2 

.42841 

.90358 

.44411 

•89597 

.45968 

. 88808 

38 

9 

10 

23 

.41284 

.9108 

.42867 

.90346 

•44437 

.89584 

•45994 

.88795 

37 

g 

10 

24 

.4131 

.91068 

.42894 

•90334 

•44464 

-8957X 

4602 

.88782 

36 

8 

8 

11 

25 

•4X337 

.91056 

.4292 

.90321 

•4449 

•89558 

. 46046 

.88768 

35 

11 

26 

.41363 

.91044 

.42946 

•90309 

.44516 

•89545 

.46072 

•88755 

34 

8 

12 

27 

• 4I39 

.9IO32 

•42972 

.90296 

•44542 

•89532 

46097 

.88741 

33 

8 

12 

28 

.41416 

.9IO2 

•42999 

.90284 

.44568 

.89519 

.46123 

.88728 

32 

7 

13 

29 

•4T443 

.91008 

.43025 

90271 

•44594 

. 89506 

.46149 

.88715 

31 

7 

13 

3° 

.41469 

.90996 

•43051 

.90259 

.4462 

.89493 

•46175 

.88701 

3° 

7 

13 

31 

.41496 

. 90984 

•43077 

.90246 

.44646 

. 8948 

46201 

. 88688 

29  ; 

7 

14 

32 

.41522 

.QOQ72 

.43104 

•90233 

•44672 

.89467 

.46226 

.88674 

2S 

7 

14 

33 

•4i549 

.9096 

•4313 

.90221 

.44698 

.89454 

.46252 

.88661 

27 

6 

15 

34 

•4i575 

.90948 

•43156 

. 90208 

•44724 

.89441 

.46278 

.88647 

26 

6 

15 

35 

.41602 

.90936 

.43182 

.90196 

•4475 

.89428 

.46304 

.88634 

25 

6 

16 

36 

.41628 

.90924 

.43209 

.90183 

•44776 

.89415 

•4633 

.8862 

24 

6 

16 

37 

.41655 

.909II 

•43235 

.90171 

. 44802 

. 89402 

•46355 

.88607 

23 

5 

16 

38 

.41681 

.90899 

.43261 

90158 

.44828 

.89389 

.46381 

•88593 

22 

5 

17 

39 

.41707 

.90887 

•43287 

.90146 

•44854 

•89376 

.46407 

.8858 

21 

5 

17 

40 

•4I734 

.90875 

•43313 

•9OI33 

. 4488 

•89363 

•46433 

.88566 

20 

5 

18 

41 

.4176 

.90863 

•4334 

.9012 

.44906 

•8935 

.46458 

•88553 

*9 

4 

18 

42 

.41787 

.90851 

.43366 

.90108 

•44932 

•89337 

.46484 

.88539 

18 

4 

19 

43 

.41813 

.90839 

•43392 

.90095 

•44958 

.89324 

.4651 

.88526 

17 

4 

19 

44 

.4184 

. 90826 

.43418 

90082 

•44984 

•893» 

•46536 

.88512 

10  i 

4 

20 

45 

.41866 

.90814 

•43445 

.9007 

.4501 

.89298 

.46561 

.88499 

15  i 

4 

20 

46 

.41892 

. 90802 

•4347i 

•90057 

•45036 

.89285 

•46587 

. 88485 

14 1 

3 

20 

47 

•4I9I9 

.9079 

•43497 

.90045 

.45062 

.89272 

.46613 

.88472 

13 

1 l 

3 

21 

48 

•41945 

.90778 

•43523 

.90032 

.45088 

.89259 

.46639 

.88458 

12 

3 

21 

49 

.41972 

.90766 

•43549 

.90019 

•45“4 

.89245 

.46664 

•88445 

II  | 

3 

22 

5° 

.41998 

•9°753 

•43575. 

.90007 

.4514 

.89232 

.4669 

.88431 

1°  ] 

2 

22 

51 

.42024 

•9°74I 

. 43602 

• 89994 

.45166 

.89219 

.46716 

.88417 

9 

8 

2 

23 

* 52 

.42051 

.90729 

.43628 

.89981 

.45192 

. 89206 

•46742 

. 88404 

2 

23 

53 

.42077 

.90717 

•43654 

. 89968 

.45218 

•091?3 

•46767 

.8839 

7 

2 

23 

54 

.42104 

.90704 

4368 

.89956 

•45243 

.8918 

•46793 

•88377 

O 

1 

24 

55 

.4213 

. 90692 

.43706 

•89943 

.45269 

.89167 

.46819 

.88363 

1 

24 

56 

.42156 

.9068 

•43733 

•8993 

•45295 

•89153 

.46844 

.88349 

\ 

1 

25 

57 

.42183 

.90668 

•43759 

.89918 

4532i 

.8914 

.4687 

.88336 

3 

1 

25 

26 

58 

59 

.42209 

.42235 

• 90655 
.90643 

•43785 

.43811 

.89905 
. 89892 

•45347 

•45373 

.89127 

.89114 

.46896 

.46921 

.88322 

88308 

2 

1 

0 

0 

26 

60 

.42262 

.90631 

•43837 

.89879 

•45399 

1 .89101 

•46947 

8829; 

1 1 

0 

0 

N.  cos. 

( 

N.  sine. 

55° 

N.  cos.  * N.  siue. 

64° 

N.  cos.  ^ N.  sine. 

1 63° 

N.  cos. 

< 

N.  sine 

52° 

NATURAL  SINES  AND  COSINES. 


397 


& 

25 

: 

1 

N.  sine. 

28° 

. N.  cos. 

0 

0 

.46947 

.88295 

0 

1 

•46973 

.88281 

I 

2 

•46999 

.88267 

I 

3 

.47024 

.88254 

2 

4 

•4705 

.8824 

2 

3 

.47076 

.88226 

3 

6 

.47101 

.88213 

3 

7 

.47127 

.88199 

3 

8 

•47153 

.88185 

4 

9 

.47178 

‘oo172 

4 

10 

.47204 

.88158  1 

5 

11 

.47229 

.88144 

5 

12 

•47255 

.8813 

5 

*3 

.47281 

.88117 

6 

*4 

47306 

. 88103 

6 

i5 

•47332 

.88089 

7 

16 

•47358 

.88075 

7 

17 

•47383 

. 88062 

8 

18 

.47409 

.88048 

8 

*9 

•47434 

. 88034 

8 

20 

.4746 

.8802 

9 

21 

.47486 

.88006 

9 

22 

•47511 

.87993 

10 

23 

•47537 

.87979 

10 

24 

.47562 

.87965 

10 

25 

.47588 

•87951 

11 

26 

.47614 

•87937 

11 

27 

•47639 

.87923 

12 

28 

.47665 

•87909 

12 

29 

.4769 

.87896 

*3 

30 

•477l6 

.87882 

*3 

3i 

•4774i  I 

.87868  . 

*3 

32 

•47767  ! 

•87854  ■ 

14 

33 

•47793  j 

.8784 

J4 

34 

•47818  ! 

.87826  . 

r5 

35 

•47844 

.87812 

15 

36 

.47869  ■ 

.87798  . 

15 

37 

•47895 

.87784  . 

16 

38 

.4792  | 

.8777 

16 

39 

•47946  | 

.87756  . 

*7 

40 

•47971  1 

•87743  • 

*7 

4i 

•47997  ! 

.87729  . 

18 

42 

.48022  j 

•87715  . 

18 

43 

.48048 

.87701 

18 

44 

•48073  ! 

.87687 

*9 

45 

.48099 

.87673  . 

z9 

46 

.48124 

.87659  . 

20 

47 

.4815 

.87645  . 

20 

48 

•48175 

.87631  . 

20 

49 

48201 

.87617 

21 

5o 

.48226 

. 87603 

21 

5i 

.48252 

.87589  . 

22 

52 

.48277 

•87575  • 

22 

53 

•48303 

•87561  . 

23 

54 

.48323 

• 87546  • 

23 

55 

•48354 

.87532  . 

23 

56 

•48379 

•87518  . 

24 

57 

•48405 

.87504  ., 

24 

58 

•4843 

.8749 

25 

i9 

• 48456 

.87476  .. 

25 

60 

.48481 

.87462  .; 

N.  cos. 

N.  sine.  P 

6lo 

.48481 

.48506 

.48532 

•48557 

.48583 

. 48608 

.48634 

.48659 

.48684 

.4871 

•48735 

.48761 

48786 

48811 

48837 


.48913 

,48938 

48964 

48989 

49014 

4904 

49065 

4909 

49116 

49141 

.49166 

.49192 

49217 

.49242 

49268 

49293 

.49318 

•49344 

493% 

49394 

49419 

•49445 

4947 

49495 

49521 

.49546 

4957i 

49596 

49622 

49047 


49748 


290 

N.  sine.  N.  cos. 

.87462 
.87448 
•87434 

.8742 
■ 87406 
•87391 
•87377 
.87363 
•87349 
•87335 
.87321 
.87306 
.87292 
.87278 
.87264 
.8725 

•87235 

.87221 
.87207 
•87193 
87178 
.87164 

•8715 

.87136 
.87121 
.87107 
87093 
.87079 
. 87064 
.8705 
. 87036 
. 87021 
. 87007 
,86993 
.86978 
.86964 
.86949 
•86935 
.86921 
. 86906 
.86892 
.86878 
.85853 
. 86849 
•86834 
.8682 
. 86805 
.86791 
.86777 
. 86762 
.86748 
•86733 
•86719 
. 86704 
.8669 
. 86675 
.86661 
.86646 
.86632 
.86617 
.86603 


300 


•5 

.50025 
.5005 
. 50076 
.50101 
.50126 

•50151 

.50176 
. 50201 
.50227 
•50252 
•50277 
. 50302 

• 50327 
•50352 
•50377 

• 50403 

. 50428 
•50453 
.50478 

• 50503 

•50528 

•50553 

•50578 

•50603 
. 50628 
•50654 
.50679 

• 50704 

• 50729 

• 50754 

•50779 

. 50804 
.50829 

• 50854 
■ 50879 
.50904 
. 50929 

•50954 

•50979 

.51004 

.51029 

•51054 

51079 

51104 

•51129 

•5ii54 

.51179 

.51204 

•51229 

•51254 

.51279 

•51304 

■51329 

•51354 

'51379 

,51404 

51429 

•51454 

•5M79 

•51504 


.8653 


.86471 


.86427 
.86413 
.86398 
.86384 
. 86369 
•86354 
8634 
86325 
.8631 
.86295 
.86281 
.86266 
.86251 
.86237 
.86222 
.86207 
.86192 
.86178 
.86163 
.86148 
.86133 
.86119 
.86104 
. 86089 
. 86074 
.86059 
. 86045 
.8603 
.86015 
.86 

•85985 

•8597 

•85956 

.85941 
.85926 
.85911 
.85896 
.85881 
.85866 
•85851 
.85836 
(. 85821 
.85806 
.85792 
•85777 
.85762 
•85747 
•85732 
•857r7 


310 

I 

I 

3 *51504 

•85717 

60 

3 -51529 

.85702 

59 

3 *51554 

.85687 

58 

? • 5*579 

.85672 

57 

[ -51604 

•85657 

56 

.51628 

.85642 

55 

5 -51653 

.85627 

54 

.51678 

.85612 

53 

> -51703 

•85597 

52 

.51728 

.85582 

5i 

p *51753 

•85567 

5o 

.51778 

•85551 

49 

.51803 

•85536 

48 

•5i828 

•85521 

47 

.51852 

.85506 

46 

•51877 

.85491 

45 

.51902 

.85476 

44 

•5i927 

.85461 

43 

•51952 

.85446 

42 

•51977 

•85431 

4i 

.52002 

.85416 

40 

. 52026 

• 85401 

39 

•52051 

•85385 

38 

.52076 

•8537 

37 

.52101 

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36 

.52126 

•8534 

35 

.52151 

•85325 

34 

•52175 

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33 

. 522 

.85294 

32 

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3i 

.5225 

.85264 

30 

•52275 

.85249 

29 

.52299 

•85234 

28 

.52324 

.85218 

27 

•52349 

•85203 

26 

•52374 

.85188 

25 

•52399 

•85173 

24 

•52423 

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23 

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22 

•52473 

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21 

.52498 

.85112 

20 

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J9 

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18 

•52572 

. 85066 

17 

•52597 

85051 

16 

.52621 

•85035 

15 

. 52646 

• 8502 

14 

•52671 

.85005 

*3 

. 52696 

. 84989 

12 

•5272 

.84974 

11 

•52745 

.84959 

10 

•5277 

.84943 

9 

•52794 

.84928 

8 

•52819 

.84913 

7 

.52844 

.84897 

6 

.52869 

.84882 

5 

•52893 

.84866 

4 

.52918 

.84851 

3 

• 52943 

.84836 

2 

.52967 

. 8482 

1 

.52992 

.84805 

0 

! n.  cos.  : 

N.  sine,  j 

' I 

f 580  | 

1 

Prop. 

parts. 


NATURAL  SINES  AND  COSINES. 


398 


#•  « 
pH  P- 

23 

o 

O 


2 

2 

2 

3 

3 

3 

4 

4 

5 
5 

5 

6 
6 
7 
7 

7 

8 
8 
8 

9 
9 

10 

10 

10 

11 

11 

12 
12 

12 

13 
13 

13 

14 

14 

15 
15 

15 

16 
16 

16 

17 

17 

18 

18 

18 

*9 

*9 

20 

20 

20 

21 
21 

21 

22 

22 

23 
23 


32c 

33° 

34° 

35C 

< 

< 

p 

H 

• e* 
m P. 

II 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

*9 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

£ 

52992 

•53oi7 

•53041 

. 53066 

•53091 

•53**5 

•53i4 

•53164 

•53189 

•53214 

•53238 

•53263 

.53288 

•53312 

•53337 

•5336i 

•53386 

•534** 

•53435 

•5346 

•53484 

•53509 

•53534 

•53558 

•53583 

•53607 

•53632 

•53656 

•53681 

•53705 

•5373 

•53754 

•53779 

•53804 

.53828 

•53853 

•53877 

.53902 

•53926 

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•53975 

•54 

.54024 

.54049 

•54073 

•54097 

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.54146 

•54*7* 

•54195 

.5422 

.54244 

.54269 

•54293 

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•54415 

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• 54464 

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•84759 

•84743 

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.84681 

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.8465 

•84635 

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. 84604 

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•84495 

. 8448 

. 84464 

.84448 

•84433 

•844I7 

. 84402 

.84386 

•8437 

.84355 

•84339 

.84324 

.84308 

.84292 

.84277 

.84261 

.84245 

.8423 

.84214 

.84198 

.84182 

.84167 

.84151 

•84135 

.8412 
.84104 
. 84088 
.84072 
.84057 
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. 84009 
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•83978 
83962 
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.83883 

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•54513 

•54537 

•5456i 

•54586 

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• 54659 

• 54683 

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• 54878 
• 549°2 
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.55024 
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• 55072 
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83867 

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. 83804 
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•8374 

.83724 

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.83692 

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.8366 

.83645 

.83629 

.83613 

•83597 

.83581 

.83565 

•83549 

•83533 

•83517 

.83501 

.83485 

.83469 

•83453 

•83437 

.83421 

.83405 

•83389 

•83373 

.83356 

•8334 

•83324 

.83308 

.83292 

•83276 

.8326 

•83244 

.83228 

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•83195 

•83*79 

•83163 

•83*47 

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.83098 
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.83001 

.82985 

. 82969 

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.8292 

.82904 

*559I9  • 
•55943  • 
•55968 
• 55992 
.56016 
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. 56064 
. 56088 
.56112 

.56136 

.5616 

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. 56208 

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.56256 

.5628 

•56305 

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• 56353 
•56377 
. 56401 

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•56449 

•56473 

•56497 

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•56545 

• 56569 

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.56641 

• 56665 
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•567*3 

1 .56736 
•5676 
.56784 

.56808 

.56832 

.56856 

.5688 

•56904 

.56928 

.56952 

.56976 

•57 

.57024 

•57047 

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•57095 

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•57*9* 

•57215 

•57238 

.57262 

.57286 

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.82904 
.82887 
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.82822 
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.82741 
.82724 
. 82708 
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.82675 
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.82643 
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.82561 
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.82528 
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.82495 
.82478 
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.82396 
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- 82363 
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• 57904 
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• 58047 

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.58141 

.58165 

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. 58236 

.5826 

.58283 

• 58307 
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.58425 

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.58496 

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•58543 

•58567 

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• 58614 

•58637 

.58661 
. 58684 
. 58708 

•5873* 

•58755 

•58779 

.81915 

.81899 

.81882 

.81865 

.81848 

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.81815 

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.81782 

.81765 

.81748 

•81731 

.81714 

.81698 

.81681 

.81664 

.81647 

.81631 

.81614 

.81597 

.8158 

.81563 

•81546 

•8153 

•81513 

.81496 

.81479 

.81462 

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.81428 

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•81395 

.81378 

.81361 

.81344 

.81327 

.8131 

.81293 

.81276 

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.81191 

.81174 

.81157 

.8114 

.81123 

.81106 

.81089 

.81072 

.81055 

.81038 

.81021 

.81004 

.80987 

.8097 

•80953 

.80936 

.80919 

.80902 

60 

59 

58 

57 

56 

55 

54 

53 

52 

5* 

50 

49 

48 

47 

46 

45 
44 
43 
42 
4*  ; 

40 
39 
38 
37  j 

36 

35 

34 

33 

32 

3* 

3° 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

*9 

18 

17 

1 6 

15 

*4 

*3 

12 

11 

10 

l 

7 

6 

5 

4 

3 

2 

1 

0 

l6 

l6 

*5 

i5 

*5 

*5 

*4 

*4 

*4 

*4 

*3 

*3 

*3 

*3 

12 

12 

12 

11 

11 

11 

11 

10 

10 

10 

10 

9 

9 

9 

9 

8 

8 

8 

7 

7 

7 

7 

6 

6 

6 

6 

5 

5 

5 

5 

4 

4 

4 

3 

3 

3 

3 

2 

2 

2 

2 

i 

N.  cos.  N.  sine. 

57° 

N.  cos.  1 N.  sine. 

1 56° 

N.  cos. 

N.  sine. 

>5° 

1 N.  cos.  • N.  sine,  j ' | 

64°  I I 

NATURAL  SINES  AND  COSINES.  399 


Prop. 

parts. 

36° 

3 

70 

3 

8° 

39° 

Prop. 

parts. 

23 

' 

N.  sine. 

18 

O 

0 

•58779 

. 80902 

.60182 

.79864 

.61566 

.78801 

.62932 

•777*5 

60 

18 

0 

1 

. 58802 

.80885 

. 60205 

.79846 

.61589 

•78783 

•62955 

. 77696 

59 

18 

I 

2 

.58826 

. 80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678 

58 

*7 

I 

3 

.58849 

.8085 

.60251 

.79811 

•61635 

•78747 

•63 

.7766 

57 

*7 

2 

4 

•58873 

.80833 

. 60274 

•79793 

.61658 

.78729 

.63022 

.77641 

56 

*7 

2 

S 

.58896 

.80816 

.60298 

•79776 

.61681 

.78711 

•63045 

.77623 

55 

*7 

2 

6 

.5892 

.80799 

.60321 

•79758 

.61704 

.78694 

. 63068 

.77605 

54 

16 

3 

7 

•58943 

.80782 

.60344 

•7974* 

.61726 

.78676' 

.6309 

.77586 

53 

16 

3 

8 

• 58967 

. 80765 

. 60367 

•79723 

.61749 

.78658 

•63113 

•77568 

52 

16 

3 

9 

•5899 

. 80748 

.6039 

.79706 

.61772 

.7864 

•63135 

•7755 

5* 

*5 

4 

10 

•59OI4 

•8073 

.60414 

.79688 

•6i795 

.78622 

•63158 

•77 53* 

50 

*5 

4 

11 

• 59°37 

•80713 

.60437 

.79671 

.61818 

. 78604 

.6318 

•775*3 

49 

*5 

5 

12 

.59061 

. 80696 

.6046 

•79653 

.61841 

78586 

•63203 

•77494 

48 

*4 

5 

13 

• 59o84 

. 80679 

.60483 

•79635 

.61864 

.78568 

.63225 

.77476 

47 

*4 

3 

14 

.59108 

.80662 

. 60506 

.79618 

.61887 

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46 

*4 

6 

13 

•59*3* 

. 80644 

.6052Q 

•796 

.61909 

•78532 

•63271 

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45 

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6 

16 

•59*54 

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•60553 

79583 

61932 

•785*4 

.63293 

.77421 

44 

13 

7 

17 

• 59*78 

.8061 

.60576 

•79565 

•61955 

.78496 

•63316 

.77402 

43 

13 

7 

18 

• 59201 

•80593 

.60599 

•79547 

.61978 

•78478 

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42 

*3 

7 

19 

•59225 

.80576 

. 60622 

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.7846 

63361 

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4* 

12 

8 

20 

• 59248 

.80558 

.60645 

•795*2 

.62024 

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•63383 

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4° 

12 

8 

21 

• 59272 

.80541 

.60668 

•79494 

.62046 

.78424 

. 63406 

•77329 

39 

12 

8 

22 

•59295 

.80524 

.60691 

•79477 

. 62069 

.78405 

.63428 

•773* 

33 

11 

9 

23 

•593*8 

•80507 

.60714 

•79459 

.62092 

.78387 

•63451 

.77292 

37 

11 

9 

24 

•59342 

. 80489 

.60738 

.79441 

.62115 

.78369 

•63473 

•77273 

36 

11 

10 

25 

•59365 

.80472 

.60761 

•79424 

.62138 

•78351 

,63496 

•77255 

35 

11 

10 

26 

•59389 

•80455 

.60784 

.79406 

.6216 

•78333 

.63518 

.77236 

34 

10 

10 

27 

•594*2 

.80438 

. 60807 

•79388 

.62183 

•78315 

•6354 

.77218 

33 

10 

11 

28 

•59436 

.8042 

.6083 

•7937* 

. 62206 

.78297 

•63563 

•77*99 

32 

10 

11 

29 

•59459 

.80403 

.60853 

•79353 

.62229 

.78279 

•63585 

.77181 

3* 

9 

12 

30 

•59482 

. 80386 

.60876 

•79335 

.62251 

.78261 

. 63608 

.77162 

30 

9 

12 

3i 

•59506 

. 80368 

.60899 

•793*8 

.62274 

.78243 

•6363 

•77*44 

29 

9 

12 

32 

•59529 

•80351 

.60922 

•793 

.62297 

.78225 

•63653 

•77*25 

28 

8 

13 

33 

•59552 

•80334 

.60945 

.79282 

.6232 

.78206 

•63675 

•77*07 

27 

8 

13 

34 

•59576 

.80316 

. 60968 

.79264 

.62342 

.78188 

. 63698 

.77088 

26 

8 

13 

35 

•59599 

. 80299 

.60991 

•79247 

•62365 

.7817 

.6372 

.7707 

25 

8 

14 

36 

. 59622 

.80282 

.61015 

.79229 

.62388 

78152 

.63742 

•77051 

24 

7 

14 

37 

.59646 

. 80264 

.61038 

.79211 

.62411 

•78134 

•63765 

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23 

7 

15 

38 

. 59669 

.80247 

.61061 

•79*93 

•62433 

78116 

•63787 

•77014 

22 

7 

15 

39 

•59693 

.8023 

.61084 

•79*76 

.62456 

. 78098 

.6381 

.76996 

21 

6 

15 

40 

•597*6 

.80212 

.61107 

•79*58 

.62479 

.78079 

.63832 

•76977 

20 

6 

16 

4i 

•59739 

•80195 

.6113 

•79*4 

.62502 

.78061 

•63854 

•76959 

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6 

16 

42 

•59763 

. 80178 

.61153 

.79122 

.62524 

.78043 

•63877 

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18 

5 

16 

43 

•59786 

.8016 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921 

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5 

*7  1 

44 

.59809 

.80143 

.61199 

.79087 

.6257 

. 78007 

.63922 

•76903 

16 

5 

*7 

45 

.59832 

. 80125 

.61222 

. 79069 

62592 

.77988 

•63944 

.76884 

*5 

5 

18  : 

46 

.59856 

. 80108 

.61245 

•79°5* 

.62615 

•7797 

.63966 

.76866 

*4 

4 

*g| 

47 

•59879 

.80091 

.61268 

•79°33 

.62638 

•77952 

.63989 

•76847 

*3 

4 

18  ; 

48 

. 59902 

. 80073 

.61291 

.79016 

.6266 

•77934 

.64011 

.76828 

12 

4 

*9 

49 

• 59926 

. 80056 

.61314 

.78998 

.62683 

•779*6 

•64033 

.7681 

11 

3 

*9 

50 

•59949 

.80038 

•61337 

.7898 

.62706 

•77897 

.64056 

7679* 

10 

3 

20 

5i 

•59972 

.80021 

.6136 

. 78962 

.62728 

•77879 

. 64078 

.76772 

9 

3 

20 

52 

•59995 

.80003 

•61383 

.78944 

.62751 

.77861 

641 

•76754 

8 

2 

20 

53 

.60019 

.79986 

61406 

.78926 

.62774 

•77843 

.64123 

•76735 

7 

2 

21 

54 

.60042 

.79968 

.61429 

. 78908 

.62796 

.77824 

64145 

.76717 

6 

2 

21 

55 

.60065 

•7995* 

.61451 

.78891 

.62819 

. 77806 

.64167 

.76698 

5 

2 

21 

56 

.60089 

•79934 

.61474 

•78873 

62842 

.77788 

.6419 

.76679 

4 

1 

22 

57 

.60112 

.79916 

•6i497 

•78855 

.62864 

•77769 

.64212 

.76661 

3 

1 

22 

58 

.60135 

•79899 

• 6152 

•78837 

.62887 

•7775* 

•64234 

. 76642 

2 

1 

23 

59 

.60158 

.79881 

•61543 

78819 

. 62909 

•77733 

.64256 

.76623 

1 j 

0 

23 

60 

.60182 

.79864 

.61566 

.78801 

.62932 

•777*5 

.64279 

. 76604 

0 

0 

N.  cos. 

N.  sine. 

N.  cos.  1 

N.  sine. 

N.  cos. 

N.  sine. 

N.  cos. 

N.  sine. 

' 

1 

53°  II 

52°  1 

51° 

50° 

1 

400 


NATURAL  SINES  AND  COSINES. 


c r 

40° 

410 

42° 

i 43° 

pi  & 

22 

' 

N.  sine. 

N.  eos. 

N.  sine. 

N.  C08. 

N.  sine.  | 

N.  cos. 

j N.  sine. 

N.  COR. 

0 

0 

.64279 

. 76604 

.65606 

•7547z 

• 66913 

•743*4 

.682 

•73*35 

60 

0 

1 

.64301 

• 76586 

.65628 

•75452 

•66935 

•74295 

; .68221 

•73**6 

59 

1 

2 

•64323 

.76567 

•6565 

•75433 

.66956 

•74276 

.68242 

.73096 

58 

1 

3 

.64346 

•76548 

.65672 

•754z4 

.66978 

74256 

| .68264 

.73076 

57 

1 

4 

.64368 

•7653 

.65694 

'75395 

.66999 

•74237 

j .68285 

• 73056 

56 

2 

5 

•6439 

.76511 

.65716 

•75375 

.67021 

•74217 

| .68306 

•73036 

55 

2 

6 

.64412 

.76492 

•65738 

75356 

.67043 

74198 

1 .68327 

.730*6 

54 

3 

7 

•64435 

•76473 

•65759 

•75337 

. 67064 

•74178 

; -68349 

. 72996 

53 

3 

8 

•64457 

76455 

65781 

•753z8 

. 67086 

•74*59 

1 -6837 

.72976 

52 

3 

9 

•64479 

76436 

•65803 

•75299 

.67107 

•74*39 

•68391 

•72957 

5i 

4 

10 

.64501 

.76417 

•65825 

.7528 

.67129 

•74*2 

1 .68412 

•72937 

50 

4 

11 

.64524 

•76398 

•65847 

•75261 

.67151 

.741 

! -68434 

.72917 

49 

4 

12 

.64546 

.7638 

.65869 

75241 

.67172 

.7408 

; 68455 

.72897 

48 

5 

13 

.64568 

.76361 

•65891 

.75222 

.67194 

.74061 

i .68476 

.72877 

47 

5 

14 

•6459 

.76342 

•659*3 

•75203 

.67215 

•74041 

•68497 

72857 

46 

6 

15 

.64612 

•76323 

•65935 

•75184 

•67237 

. 74022 

■ 68518  ; 

.72837 

45 

6 

16 

•64635 

•76304 

65056 

•75165 

.67258 

. 74002 

i -68539 

.72817 

44 

6 

17 

.64657 

.76286 

.65978 

•75146 

.6728 

•73983 

! -68561  ! 

.72797 

43 

7 

18 

. 64670 

.76267 

.66 

•75126 

.67301 

•73963 

j .68582 

•72777 

42 

7 

*9 

.64701 

.76248 

. 66022 

•75107 

.67323 

•73944  1 

i .68603 

•72757 

4* 

7 

20 

.64723 

.76229 

.66044 

.75088 

.67344 

•73924 

.68624 

•72737 

40 

8 

21 

.64746 

.7621 

.66066 

.75069 

•67366 

•739°4 

. 68645 

.72717 

39 

8 

22 

. 64768 

.76192 

.66088 

•7505 

.67387 

• 73885 

.68666 

. 72697 

38 

8 

23 

•6479 

•76i73 

.66109 

•7503 

.67409 

•73865 

.68688 

72677 

37 

9 

24 

.64812 

•76154 

.66131 

.75011 

•6743 

73846 

.68709 

•72657 

36 

9 

25 

•64834 

•76135 

•66153 

.74992 

•67452 

.73826 

•6873 

•7*637 

35 

10 

26 

.64856 

.76116 

•66175 

•74973 

•67473 

.73806 

.68751 

.72617 

34 

10 

27 

.64878 

.76097 

.66197 

•74953 

•67495 

•73787 

.68772 

•72597 

33 

10 

28 

.64901 

.76078 

.66218 

•74934 

.67516 

•73767 

.68793 

•72577 

32 

11 

29 

.64923 

.76059 

.6624 

•749*5 

•67538 

•73747 

.68814 

•72557 

3* 

11 

30 

.64945 

. 76041 

.66262 

.74896 

•67559 

.73728 

.68835 

•72537 

3° 

11 

3i 

.64967 

. 76022 

.66284 

.74876 

•6758 

.73708 

.68857 

•725*7 

29 

12 

32 

.64989 

.76003 

.66306 

•74857 

.67602 

.73688 

.68878 

72497 

28 

12 

33 

.65011 

•75984 

.66327 

.74838 

.67623 

•73669 

.68899 

.72477 

27 

12 

34 

•65033 

•75965 

• 66349 

.74818 

.67645 

• 73649 

.6892 

•72457 

26 

13 

35 

•65055 

•75946 

.66371 

•74799 

.67666 

• 73629 

.68941 

•72437 

25 

13 

; 36 

.65077 

.75927 

•66393 

.7478 

.67688 

•736i 

.68962 

•724*7 

24 

14 

i 37 

.651 

• 759°8 

.66414 

.7476 

.67709 

•7359 

.68983 

72397 

23 

14 

! 38 

.65122 

.75889 

.66436 

7474* 

•6773 

•7357 

.69004 

•72377 

22 

14 

; 39 

.65144 

•7587 

.66458 

•74722 

•67752 

•7355* 

.69025 

•72357 

21 

15 

i 40 

.65166. 

•75851 

.6648 

•747°3 

•67773 

•7353* 

. 69046 

•72337 

20 

15 

; 41 

.65188 

75832 

.66501 

•74683 

•67795 

•735H 

.69067 

•723*7 

*9 

i5 

42 

•6521 

•758i3 

.66523 

.74664 

.67816 

•7349* 

. 69088 

.72297 

18 

16 

i 43 

•65232 

•75794 

•66545 

74644 

•67837 

•73472 

.69109 

.72277 

*7 

16 

i 44 

•65254 

1 -75775 

.66566 

.74625 

.67859 

•73452 

•69*3 

.72257 

16 

*7 

45 

.65276 

! 75756 

66588 

. 74606 

.6788 

•73432 

.69151 

.72236 

*5 

17 

46 

.65298 

i -75738 

.6661 

.74586 

.67901 

•734*3 

.69172 

.72216 

*4 

17 

47 

6532 

I -75719 

.66632 

•74567 

.67923 

•73393 

.69193 

.72196 

*3 

18 

48 

•65342 

| -757 

.66653 

•74548 

•67944 

•73373 

.69214 

. 72176 

12 

18 

49 

•65364 

| 7568 

.66675 

•74528 

.67965 

•73353 

•69235 

72156 

11 

18 

50 

.65386 

S -75661 

. 66697 ^ 

•74509 

.67987 

•73333 

.69256 

.72136 

10 

*9 

! 51 

.65408 

75642 

.66718 

.74489 

.68008 

•733*4 

.69277 

.72116 

9 

*9 

; 52 

•6543 

•75623 

.6674 

7447 

. 68029 

•73294 

.69298 

72095 

8 

19 

! 53 

•65452 

•75604 

.66762 

•7445* 

.68051 

•73274 

•693*9 

72075 

7 

20 

54 

•65474 

•75585 

.66783 

•7443* 

.68072 

•73254 

.6934 

•72055  | 6 

20 

55 

.65496 

•75566 

. 66805 

•744*2 

.68093 

•73234 

.69361 

•72035 

5 

21 

56 

.65518 

•75547 

.66827 

■74392 

.68115 

•73215 

.69382 

.72015 

4 

21 

57 

.6554 

•75528 

.66848 

i -74373 

.68136 

1 -73*95 

69403 

1 -7*995 

3 

21 

58 

• 65562 

•75509 

.6687 

1 -74353 

68157 

•73*75 

.69424 

I -7*974 

2 

22 

59 

65584 

•7549 

.66891 

•74334 

68179 

1 -73*55 

•69445 

| -7*954 

1 

22 

60 

.65606 

•75471 

•66913 

•743*4 

.682 

•73*35 

69466 

•7*934 

0 

I ' 

1 49° 

48° 

470 

11  46° 

l 

£9 

*9 

*9 

18 

18 

18 

*7 

i7 

17 

16 

16 

16 

16 

i5 

i5 

i5 

i4 

14 

14 

13 

13 

13 

12 

12 

12 

iz 


10 

10 

10 

10 

9 

9 

1 
8 
8 
7 
7 
7 
6 
6 
6 
5 
5 
5 
4 
4 
4 
3 
3 
3 
3 

2 
2 
2 


o 

o 


parts. 


NATURAL  SINES  AND  COSINES, 


401 


h Prop. 
M parts. 

, 

4' 

N.  sine. 

1° 

N.  cos. 

h Prop. 
p°  parts. 

u Prop. 
w parts. 

, 

44 

N.  sine. 

LO 

N.  cos. 

H 

p«  p, 

9 , 

0 

0 

. 69466 

•7T934 

60 

19 

11 

3i 

.70112 

•71305 

29 

9 

O 

1 

.69487 

.71914 

59 

19 

12 

32 

.70132 

.71284 

28 

9 

I 

2 

.69508 

•7i894 

58 

18 

12 

33 

•70153 

.71264 

27 

9 

I 

3 

.69529 

•7i873 

57 

18 

12 

34 

•7OI74 

•71243 

26 

8 

I 

4 

.69549 

•7l853 

56 

18 

13 

35 

•70195 

.71223 

25 

8 

2 

5 

•6957 

•7i833 

55 

17 

13 

36 

.70215 

.71203 

24 

8 

2 

6 

.69591 

•7i8i3 

54 

17 

14 

37 

.70236 

.71182 

23 

7 

3 

7 

.69612 

.71792 

53 

17 

14 

38 

.70257 

.71162 

22 

7 

3 

8 

.69633 

.71772 

52 

16 

14 

39 

.70277 

.71141 

21 

7 

3 

9 

.69654 

•7W52 

5i 

16 

15 

40 

.70298 

.71121 

20 

6 

4 

10 

.69675 

•7W32 

5o 

16 

15 

4i 

•70319 

■7« 

x9 

6 

4 

11 

. 69696 

.71711 

49 

16 

15 

42 

•70339 

.7108 

18 

6 

4 

12 

.69717 

.71691 

48 

15 

16 

43 

.7036 

•7io59 

17 

5 

5 

13 

•69737 

.71671 

47 

15 

16 

44 

.70381 

•7IQ39 

16 

5 

5 

14 

.69758 

•7i65 

46 

15 

17 

45 

. 70401 

.71019 

x5 

5 

6 

i5 

.69779 

•7i63 

45 

14 

17 

46 

.70422 

.70998 

14 

4 

6 

16 

.698 

. 7161 

44 

14 

W 

47 

•70443 

.70978 

13 

4 

6 

17 

.69821 

•7*59 

43 

H 

18 

48 

.70463 

•70957 

12 

4 

7 

18 

.69842 

•7i569 

42 

13 

18 

49 

.70484 

•70937 

11 

3 

7 

19 

.69862 

•7i549 

41 

13 

18 

50 

•70505 

.70916 

xo 

3 

7 

20 

.69883 

•71529 

40 

13 

x9 

5i 

•70525 

.70896 

9 

3 

8 

21 

. 69904 

.71508 

39 

12 

1 9 

52 

.70546 

.70875 

8 

3 

8 

22 

.69925 

.71488 

S8 

12 

x9 

53 

.70567 

•70855 

7 

2 

8 

23 

.69946 

.71468 

37 

12 

20 

54 

•70587 

.70834 

6 

2 

9 

24 

. 69966 

•71447 

36 

II 

20 

55 

.70608 

.70813 

5 

2 

9 

25 

.69987 

.71427 

35 

II 

21 

56 

.70628 

•70793 

4 

i 

10 

26 

.70008 

.71407 

34 

II 

21 

57 

.70649 

.70772 

3 

X 

10 

27 

.70029 

.71386 

33 

IO 

21 

S8 

.7067 

.70752 

2 

1 

10 

28 

.70049 

.71366 

32 

IO 

22 

59 

.7069 

•70731 

1 

0 

11 

29 

.7007 

•71345 

3i 

IO 

22 

60 

7071 1 

.70711 

0 

0 

11 

30 

.70091 

•71325 

30 

IO 

N.  cos. 

N.  sine. 

' 

N.cos. 

N.  sine. 

' 

45° 

450 

Preceding  Table  contains  Natural  Sine  and  Cosine  for  every  minute 
of  the  Quadrant  to  Radius  1. 

If  Degrees  are  taken  at  head  of  columns,  Minutes,  Sine,  and  Cosine  must 
be  taken  from  head  also ; and  if  they  are  taken  at  foot  of  column,  Minutes, 
etc.,  must  be  taken  from  foot  also. 

Illustration. — .3173  is  sine  of  180  30',  and  cosine  0171°  30'. 


To  Compute  Sine  or  Cosine  for  Seconds. 

When  Angle  is  less  than  450.  Rule. — Ascertain  sine  or  cosine  of  angle 
for  degrees  and  minutes  from  Table;  take  difference  between  it  and  &ine- 
or  cosine  of  angle  next  below  it.  Look  for  this  difference  or  remainder,* 
if  Sine  is  required,  at  head  of  column  of  Pi'oportional  Parts , on  left  side ; 
and  if  Cosine  is  required,  at  head  of  column  on  right  side ; and  in  these- 
respective  columns,  opposite  to  number  of  seconds  of  angle  in  column,,  is- 
number  or  correction  in  seconds  to  be  added  to  Sine,  or  subtracted  from-  | 
Cosine  of  angle. 

Illustration  i. — What  is  sine  of  8°  9'  10"? 


Sine  of  8°  9',  per  Table  = 
Sine  of  8°  10',  “ = 


•14177; 
.142  05; 


.00028  difference. 


In  left  side  column  of  proportional  parts,  under  28,  and  opposite  to  10',  is  5,  cor- 
rection for  10',  which,  being  added  to  . 141  77  = .141  82  Sine. 


* The  table  in  some  instances  will  give  a unit  too  much,  but  this,  in  general,  is  of  little  importance. . 

L L* 


402 


NATURAL  SINES  AND  COSINES. 


2. — What  is  cosine  of  8°  9'  10"? 

Cosine  bf  8°  9'  per  Table  = -989  9°  1 ) .000 04  difference. 

Cosine  of  8°  10  , “ =.99986;) 

In  right-side  column  of  proportional  x>arts,  under  4,  and  opposite  to  10 , is  1,  the 
correction  for  10', -which,  being  subtracted  from  .989 90  = .989  89  cosine. 

When  Angle  exceeds  450.  Rule— Ascertain  sine  or  cosine  for  angle  in 
degrees  and  minutes  from  Table,  taking  degrees  at  the  foot  of  it ; then  take 
difference  between  it  and  sine  or  cosine  of  angle  next  above  it.  Look  for  re- 
mainder, if  Sine  is  required,  at  head  of  column  of  Proportional  Parts , on  right 
side ; and  if  Cosine  is  required,  at  head  of  column  on  left  side ; and  in  these 
respective  columns,  opposite  to  seconds  of  angle,  is  number  or  correction  in 
seconds  to  be  added  to  Sine,  or  subtracted  from  Cosine  of  angle. 

Illustration.— What  is  the  Sine  and  Cosine  of  8i°  50'  50"? 

Sine  of  8i°  50'  per  Table  = .989  86; ) OOQO  difference. 

Sine  of  8i°  51  , “ =-9^995  ) 

In  right-side  column  of  proportional  parts,  and  opposite  to  50',  is  3,  which,  added 
to  . 989  86  = . 989  89  Sine. 

Cosine  of  8i°  50',  per  Table  = .142 05 ;)  Q difference. 

Cosine  of  8x°  51  , =-i4I77j) 

In  left-side  column  of  proportional  parts , and  opposite  to  50',  is  24,  which,  sub- 
tracted from  .14205  ==.141  81  Cosine. 

T'o  Ascertain  or  Compute  jNTxixn'ber  of  Degrees,  Minutes, 
and  Seconds  of  a given  Sine  or  Cosine. 

When  Sine  is  given.  Rule. — If  given  sine  is  in  Table,  the  degrees  of  it 
will  be  at  top  or  bottom  of  page,  and  minutes  in  marginal  column,  at  left  or 
right  side,  according  as  sine  corresponds  to  an  angle  less  or  greater  than  450. 

If  given  sine  is  not  in  Table,  take  sine  m I able  which  is  next  less  than  the 
one  for  which  degrees,  etc.,  are  required,  and  note  degrees,  etc.,  for  it.  Sub- 
tract this  sine  from  next  greater  tabular  sine,  and  also  from  given  sine. 

Then,  as  tabular  difference  is  to  difference  between  given  sine  and  tabu- 
lar sine,  so  is  60  seconds  to  seconds  for  sine  given. 

Example. — What  are  the  degrees,  minutes,  and  seconds  for  sine  of  .75? 

Next  less  sine  is  .74992,  arc  for  which  is  48°  35'.  Next  greater  sine  is  .75011, 
difference  between  which  and  next  less  is  .75011 — .749 92 ^=. 000 19.  Difference  be- 
tween kss  tabular  sine  and  one  given  is  . 75  — .749  92  = 8. 

TLeai  i-g  : 8 il.fp  ; 25+,  which,  added  to  48°  35'  = 48°  35'  25". 

When  Cosine  is  given.  Rule. — If  given  cosine  is  found  in  Table,  degrees 
of  it  will  be  found  as  in  manner  specified  when  .sine  is  given.  . 

If  given  cosine  is  not  in  Table,  take  cosine  in  T able  which  is  next  greater 
than  one  for  which  degrees,  etc.,  are  required,  and  note  degrees,  etc.,  for  it. 
Subtract  this  cosine  from  next  less  tabular  cosine,  and  also  from  given  cosine. 

Then,  as  tabular  difference  is  to  difference  between  given  cosine  and  tabu- 
lar cosine,  so  is  60  seconds  to  seconds  for  cosine  given. 

Example. — What  are  the  degrees,  minutes,  and  seconds  for  cosine  of  .75? 

Next  greater  cosine  is  .750  n,  arc  for  which  is  410  24'.  Next  less  cosine  is  ^749  92, 
difference  between  which  and  next  greater  is  .75011  — .74992  = .000 19.  Difference 
between  greater  tabular  cosine  and  one  given  is  .750 11  — .75000  — 11. 

Then  19  : n ::  60  : 35 — , which,  added  to  410  24'  = 41°  24'  35". 

To  Compnte  Versed  Sine  of  an  Angle. 

Subtract  cosine  of  angle  from  1. 

Illustration.— What  is  the  versed  sine  of  210  30' ? 

Cosine  of  210  30'  is  .93042,  which,  — 1 = .06958  versed  sine. 

To  Compute  Co-versed  Sine  of  an  Angle. 

Subtract  sine  of  angle  from  1. 

Illustration. — What  is  the  co- versed  sine  of  210  30'? 

The  sine  of  210  30'  is  .3665,  which,  — 1 = .6335  co-versed  sine. 


NATURAL  SECANTS  AND  COSECANTS, 


403 


]N"atnral  Secants  and.  Co -secants. 


0° 

1° 

2°  I 

30 

' 

Secant. 

Co-SKCANT. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

r 

0 

j 

Infinite. 

I. OOOI 

57-299 

1.0006 

28.654 

1. 0014 

19.107 

60 

I 

I 

3437*7 

.OOOI 

6-359 

.0006 

8.417 

.0014 

9.002 

59 

2 

I 

1718.9 

.0002 

5-45 

.0066 

8.184 

.0014 

8.897 

58 

3 

I 

145-9 

.0002 

4-57 

.0006 

7-955 

.0014 

8.794 

57 

4 

I 

859.44 

.0002 

3-7i8 

.0006 

7-73 

.0014 

8.692 

56 

5 

I 

687.55 

1.0002 

52.891 

1.0007 

27. 508 

1. 0014 

18.591 

55 

6 

I 

572.90 

.0002 

2.09 

.0007 

7.29 

•0015 

8.491 

54 

7 

I 

491. 1 1 

.0002 

i-3i3 

.0007 

7-075 

.0015 

8-393 

53 

8 

I 

29.72 

.0002 

o-558 

.0007 

6.864 

•0015 

8.295 

52 

9 

I 

38i-97 

.0002 

49.826 

.0007 

6.655 

.0015 

8. 198 

5* 

10 

I 

343-77 

1.0002 

49.  H4 

1.0007 

26.45 

1. 0015 

18. 103 

50 

11 

I 

12.52 

.0002 

8.422 

.0007 

6. 249 

•0015 

8.008 

49 

12 

I 

286. 48 

.0002 

7-75 

.0007 

6.05 

.0016 

7.914 

48 

13 

I 

64.44 

.0002 

7.096 

.0007 

5-854 

.0016 

7.821 

47 

14 

I 

45-55 

0002 

6.46 

.0008 

5.661 

.0016 

7-73 

46 

15 

I 

229.18 

1.0002 

45-84 

1.0008 

25.471 

1.0016 

17.639 

45 

16 

I 

14.86 

.0002 

5-237 

0008 

5-284 

.0016 

7-549 

44 

17 

I 

02.22 

.0002 

4-65 

.0008 

5-i 

.0016 

7.46 

43 

18 

I 

190.99 

.0002 

4.077 

.0008 

4.918 

.0017 

7-372 

42 

*9 

I 

80.73 

.0003 

3-52 

.0008 

4-739 

.0017 

7.285 

4* 

20 

I 

i7i-89 

I.OOO3 

42.976 

1.0008 

24,562 

1. 0017 

17.198 

40 

21 

X 

63-7 

.OOO3 

2.445 

.0008 

4-358 

.0017 

7-**3 

39 

22 

I 

56.26 

.0003 

1.928 

.0008 

4.216 

.0017 

7.028 

38 

23 

I 

-49-47 

.0003 

1.423 

.0009 

4.047 

.0017 

6.944 

37 

24 

I 

43-24 

.OOO3 

4°-93 

.0009 

3-88 

.0018 

6.861 

36 

25 

I 

I37-5I 

I.OOO3 

40.448 

1.0009 

23.716 

1. 0018 

16.779 

35 

26 

I 

32.22 

.0003 

39.978 

.0009 

3-553 

.0018 

6.698 

34 

27 

I 

.27-32 

.0003 

9-5i8 

.0009 

3-393 

.0018 

6.617 

33 

28 

I 

22.78 

.OOO3 

9.069 

.0009 

3-235 

.0018 

6.538 

32 

29 

I 

18.54 

.0003 

8.631 

.0009 

3-079 

.0018 

6-459 

3* 

30 

I 

1 14- 59 

I.OOO3 

38.201 

1.0009 

22.925 

1. 0019 

16.38 

3° 

3i 

I 

10.9 

.0003 

7.782 

.001 

2.774 

.0019 

6.303 

29 

32 

I 

07-43 

.0003 

7-37* 

.001 

2.624 

.0019 

6.226 

28 

33 

I 

04.17 

.OOO4 

6.969 

.001 

2.476 

.0019 

6.15 

27 

34 

I 

01. 11 

.OOO4 

6.576 

.001 

2-33 

.0019 

6.075 

26 

35 

I 

98.223 

I.OOO4 

36.191 

1. 001 

22. 186 

1. 0019 

16 

25 

36 

I 

5-495 

.OOO4 

5.814 

.001 

2.044 

.002 

5.926 

24 

37 

I 

2.914 

.OOO4 

5-445 

.001 

1.904 

.002 

5-853 

23 

38 

I. OOOI 

2.469 

.OOO4 

5-084 

.001 

1-765 

.002 

5-78 

22 

39 

.OOOI 

88. 149 

.OOO4 

4.729 

.0011 

1.629 

.002 

5.708 

21 

40 

I. OOOI 

85.946 

I.OOO4 

34-382 

1. 001 1 

21.494 

1.002 

*5-637 

20 

4i 

.OOOI 

3-849 

.OOO4 

4.042 

.0011 

1.36 

.0021 

5.566 

*9 

42 

.OOOI 

1-853 

.OOO4 

3.708 

.0011 

1.228 

.0021 

5-496 

18 

43 

.OOOI 

79-95 

.OOO4 

3-38i 

.0011 

1.098 

.0021 

5-427 

*7 

44 

• OOOI 

8-133 

.OOO4 

3.06 

.0011 

20.97 

.0021 

5-358 

16 

45 

I.  OOOI 

76.396 

I.OOO5 

32-745 

1. 001 1 

20.843 

1. 0021 

15.29 

i5 

46 

• OOOI 

4-736 

.0005 

2-437 

.0012 

0.717 

.0022 

5.222 

14 

47 

.OOOI 

3.146 

.0005 

2.134 

.0012 

o-593 

.0022 

5-*55 

*3 

48 

.OOOI 

1.622 

.0005 

1.836 

.0012 

0.471 

.0022 

5-089 

12 

49 

.OOOI 

1. 16 

.0005 

1-544 

.0012 

0-35 

.0022 

5-023 

11 

50 

I. OOOI 

68.757 

I.OOO5 

31-257 

1. 001 2 

20.23 

1.0022 

14.958 

10 

5i 

.OOOI 

7.409 

.0005 

30.976 

.0012 

0. 112 

.0023 

4-893 

9 

52 

.OOOI 

6. 1 13 

.0005 

0.699 

.0012 

19.995 

.0023 

4.829 

8 

53 

.OOOI 

4.866 

.0005 

0.428 

.0013 

9.88 

.0023 

4-765 

7 

54 

.OOOI 

3.664 

.0005 

0. 161 

•0013 

9.766 

.0023 

4.702 

6 

55 

I. OOOI 

62. 507 

I.OOO5 

29. 899 

1. 0013 

19-653 

1.0023 

14.64 

5 

56 

.OOOI 

*•39* 

.0006 

9.641 

.0013 

9-54* 

.0024 

4-578 

4 

57 

.OOOI 

*-3i4 

.0006 

9.388 

•0013 

9-431 

.0024 

4-5*7 

3 

58 

.OOOI 

59-274 

.OO06 

9*  39 

.0013 

9.322 

.0024 

4-456 

2 

I9 

.OOOI 

8.27 

.0006 

8.894 

.0013 

9.214 

.0024 

4-395 

1 

60 

I. OOOI 

57-299 

I.OO06 

28.654 

1. 0014 

19.107 

1.0024 

*4-335 

0 

' 

Co-SEC’T. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

89° 

88° 

| 87° 

86° 

404 


NATURAL  SECANTS  AND  CO-SECANTS, 


40  , 

50 

6° 

70 

9 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant.  [ 

Co-sec’t. 

' 

o 

1.0024 

14-335 

1.0038 

11.474 

1.0055 

9. 5668 

1.0075 

8.2055 

60 

i 

.0025 

4.276 

.0038 

1.436 

.0055 

•5404 

.0075 

. 1861 

59 

2 

.0025 

4-2i7 

.0039 

1.398 

.0056 

•5Mi 

.0076 

.1668 

58 

3 

.0025 

4-  *59 

.0039 

1.36 

.0056 

.488 

.0076 

.1476 

57 

4 

.0025 

4.  IOI 

.0039 

I-323 

.0056 

.462 

.0076 

• 1285 

56 

5 

1.0025 

14.043 

1.0039 

11.286 

1.0057 

9.4362 

1.0077 

8. 1094 

55 

6 

.0026 

3.986 

.004 

1.249 

•0057 

.4105 

•°°77 

• 0905 

54 

7 

.0026 

3-93 

.004 

1-213 

.0057 

.385 

.0078 

.0717 

53 

8 

.0026 

3-874 

.004 

1.176 

.0057 

•3596 

.0078 

.0529 

52 

9 

.0026 

3.818 

.004 

1.14 

.0058 

•3343 

.0078 

.0342 

5i 

IO 

1.0026 

i3-763 

1. 0041 

11. 104 

1.0058 

9. 3092 

1.0079 

8.0156 

50 

ii 

.0027 

3.708 

.0041 

1.069 

.0058 

.2842 

.0079 

7.9971 

49 

12 

.0027 

3-654 

.0041 

1-033 

.0059 

-2593 

.0079 

.9787 

48 

*3 

.0027 

3-6 

.0041 

0.988 

.0059 

.2346 

.008 

.9604 

47 

.0027 

3-547 

.0042 

0.963 

.0059 

.21 

.008 

.9421 

46 

15 

1.0027 

13-494 

1.0042 

10.929 

1.006 

9-i855 

1.008 

7.924 

45 

16 

.0028 

3-44i 

.0042 

0. 894 

.006 

.1612 

.0081 

•9°59 

44 

17 

.0028 

3-389 

.0043 

0.86 

.006 

•137 

.0081 

.8879 

43 

18 

.0028 

3-337 

.0043 

0.826 

.0061 

.1129 

.0082 

.87 

42 

19 

.0028 

3. 286 

.0043 

0.792 

.0061 

.089 

.0082 

.8522 

41 

20 

1.0029 

13-235 

I.0043 

10.758 

1. 0061 

9.0651 

1.0082 

7-8344 

40 

21 

.0029 

3.184 

.0044 

0.725 

.0062 

.0414 

.0083 

.8168 

39 

22 

.0029 

3-134 

.0044 

0. 692 

.0062 

.0179 

.0083 

•7992 

38 

23 

.0029 

3.084 

.0044 

0.659 

.0062 

8.9944 

.0084 

.7817 

37 

24 

.0029 

3-034 

.0044 

0.626 

.0063 

.9711 

.0084 

.7642 

36 

25 

1.003 

12.Q85 

1.0045 

10.593 

1.0063 

8.9479 

1.0084 

7.7469 

35 

26 

.003 

2-937 

.0045 

0. 561 

.0063 

.9248 

.0085 

.7296 

34 

27 

.003 

2.888 

.0045 

0. 529 

.0064 

.9018 

.0085 

.7124 

33 

28 

.003 

2.84 

.0046 

0.497 

.0064 

•879 

.0085 

•6953 

32 

29 

.0031 

2-793 

.0046 

0.465 

.0064 

.8563 

.0086 

.6783 

3i 

30 

1. 0031 

12.745 

1.0046 

10.433 

1.0065 

8-8337 

1.0086 

7.6613 

30 

31 

.0031 

2.698 

.0046 

0. 402 

.0065 

.8112 

.0087 

•6444 

29 

32 

.0031 

2.652 

.0047 

0.371 

.0065 

.7888 

.0087 

.6276 

23 

33 

.0032 

2.606 

.0047 

o-34 

.0066 

.7665 

.0087 

.6108 

27 

34 

.0032 

2. 56 

.0047 

0. 309 

.0066 

•7444 

.0088 

•5942 

26 

35 

1.0032 

12.514 

1.0048 

10.278 

1.0066 

8.7223 

1.0088 

7-5776 

25 

36 

.0032 

2.469 

.0048 

0.248 

.0067 

.7004 

.0089 

.5611 

24 

37 

.0032 

2.424 

.0048 

0.217 

.0067 

.6786 

.0089 

•5440 

23 

38 

• 0033 

2-379 

.0048 

0.187 

.0067 

.6569 

.0089 

.5282 

22 

39 

.0033 

2-335 

.0049 

o.i57 

.0068 

•6353 

.009 

•5119 

21 

40 

1.0033 

12.291 

1.0049 

10.127 

1.0068 

8.6138 

1.009 

7-4957 

20 

4i 

.0033 

2.248 

.0049 

0.098 

.0068 

•5924 

.009 

•4795 

r9 

42 

.0034 

2.204 

.005 

0.068 

.0069 

.5711 

.0091 

•4634 

18 

43 

• oc>34 

2. 161 

.005 

0.039 

.0069 

•5499 

.0091 

•4474 

17 

44 

.0034 

2.118 

.005 

0.01 

.0069 

.5289 

.0092 

•4315 

16 

45 

1.0034 

12.076 

1.005 

9.9812 

1.007 

8.5079 

1.0092 

7-4I56 

15 

46 

•0035 

2.034 

.0051 

.9525 

.007 

.4871 

.0092 

•399s 

14 

47 

•0035 

1.992 

.0051 

•9239 

.007 

.4663 

.0093 

•384 

13 

48 

•0035 

x-95 

.0051 

•8955 

.0071 

•4457 

.0093 

•3683 

12 

49 

•0035 

1.909 

.0052 

.8672 

.0071 

.4251 

.0094 

•3527 

11 

50 

1.0036 

11.868 

1.0052 

9.8391 

1. 007 1 

8. 4046 

1.0094 

7-3372 

10 

5i 

.0036 

1.828 

.0052 

.8112 

.0072 

•3843 

.0094 

•3217 

9 

52 

.0036 

1.787 

•0053 

.7834 

.0072 

.3640 

.0095 

.3063 

8 

53 

.0036 

1-747 

•0053 

•7558 

.0073 

•3439 

.0095 

.2909 

7 

54 

.0037 

1.707 

.0053 

• 7283 

•0073 

•3238 

.0096 

•2757 

6 

55 

1.0037 

11.668 

1-0053 

9.701 

1.0073 

8.3039 

1.0096 

7.2604 

5 

56 

.0037 

1.628 

.0054 

.6739 

.0074 

.2840 

.0097 

•2453 

4 

57 

.0037 

1.589 

.0054 

.6469 

.0074 

.2642 

.0097 

.2302 

3 

58 

.0038 

i-55 

.0054 

.62 

.0074 

.2446 

.0097 

.2152 

2 

59 

.0038 

i-512 

•0055 

•5933 

•0075 

.225 

.0098 

.2002 

1 

60 

1.0038 

n-474 

1.0055 

9. 5668 

1.0075 

8. 2055 

.0098 

00 

u> 

0 

r 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

# 

85° 

84° 

83° 

82° 

NATURAL  SECANTS  AND  CO-SECANTS.  405 


80 

II  9° 

| 10° 

| 11° 

9 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

| Secant. 

Co-sec’t. 

' Secant. 

Co-sec’t. 

' 

0 

1.0098 

7-i853 

1. 0125 

6.3924 

i- 01 54 

5-7588 

1.0187 

5. 2408 

60 

I 

.0099 

.1704 

.0125 

.3807 

•0155 

•7493 

.0188 

•233 

59 

2 

.0099 

•1557 

•0125 

•369 

•oi55 

•7398 

.0188 

.2252 

58 

3 

.0099 

.1409 

.0126 

•3574 

.0156 

•7304 

.0189 

.2174 

57 

4 

.01 

.1263 

.0126 

•3458 

.0156 

.721 

.0189 

.2097 

56 

5 

1. 01 

7.m7 

1. 0127 

6-3343 

1.0157 

5-7“7 

1. 019 

5.2019 

55 

6 

.0101 

.0972 

.0127 

.3228 

.0157 

.7023 

.0191 

.1942 

54 

7 

.0101 

.0827 

.0128 

•3ii3 

.0158 

•693 

.0191 

.1865 

53 

8 

.0102 

.0683 

.0128 

.2999 

.0158 

.6838 

.0192 

.1788 

52 

9 

.0102 

•0539 

.0129 

.2885 

.0159 

•6745 

.0192 

.1712 

5i 

10 

1. 0102 

7.0396 

1. 0129 

6.2772 

1.0159 

5-6653 

1-0193 

5.1636 

50 

11 

.0103 

.0254 

.013 

.2659 

.016 

.6561 

.0193 

• 156 

49 

12 

• 0103 

.0112 

.013 

.2546 

.016 

•647 

.0194 

.1484 

48 

13 

.0104 

6.9971 

.0131 

•2434 

.0161 

•6379 

•0195 

.1409 

47 

14 

.0104 

.983 

.0131 

.2322 

.0162 

.6288 

.0195 

•1333 

46 

15 

1. 0104 

6.969 

1. 0132 

6.2211 

1.0162 

5-6197 

1.0196 

5-1258 

45 

16 

.0105 

•955 

.0132 

.21 

.0163 

.6107 

.0196 

.1183 

44 

17 

.0105 

•9411 

•0133 

.199 

.0163 

.6017 

•0197 

.1109 

43 

18 

.0106 

•9273 

•0133 

.188 

.0164 

.5928 

.0198 

.1034 

42 

*9 

.0106 

•9*35 

.9134 

.177 

.0164 

.5838 

.0198 

.096 

4i 

20 

1. 0107 

6. 8998 

1.0134 

6.- 1661 

1.0165 

5-5749 

I-OI99 

5.0886 

40 

21 

.0107 

.8861 

•0135 

•1552 

.0165 

.566 

.0199 

.0812 

39 

22 

.0107 

.8725 

•0135 

•1443 

.0166 

•5572 

.02 

•0739 

38 

23 

.0108 

.8589 

.0136 

•1335 

.0166 

•5484 

.0201 

.0666 

37 

24 

.0108 

•8454 

.0136 

.1227 

.0167 

•5396 

.0201 

•0593 

36 

25 

1. 0109 

6. 832 

1.0136 

6. 112 

1.0167 

5-53o8 

1.0202 

5-052 

35 

26 

.0109 

.8185 

•QI37 

.1013 

.0168 

.5221 

.0202 

•0447 

34 

27 

.on 

.8052 

’ -0137 

. 0906 

.0169 

•5i34 

.0203 

•0375 

33 

28 

• on 

.7919 

.0138 

.08 

.0169 

•5047 

.0204 

.0302 

32 

29 

.0111 

.7787 

.0138 

•o694 

.017 

.496 

.0204 

.023 

3i 

30 

I.OIII 

6-7655 

1-0139 

6.0588 

1. 017 

5-4874 

1.0205 

5.0158 

30 

3i 

.0111 

•7523 

.0139 

.0483 

.0171 

.4788 

.0205 

.0087 

29 

32 

.0112 

•7392 

.014 

•0379 

.0171 

.4702 

.0206 

•0015 

28 

33 

.0112 

.7262 

.014 

.0274 

.0172 

.4617 

.0207 

4-9944 

27 

34 

•0113 

.7132 

.0141 

.017 

.0172 

•4532 

.0207 

•9873 

26 

35 

1. 0113 

6.7003 

1. 0141 

6.0066 

I-OI73 

5-4447 

1.0208 

4.9802 

25 

36 

.0114 

.6874 

.0142 

5-9963 

.0174 

.4362 

.0208 

•9732 

24 

37 

.0114 

•6745 

.0142 

,986 

•OI74 

.4278 

.0209 

.9661 

23 

38 

.0115 

.6617 

.0143 

•9758 

•0175 

.4194 

.021 

•9591 

22 

39 

.0115 

.649 

.0143 

•9655 

•OI75 

.411 

.021 

.9521 

21 

40 

1. 0115 

6.6363 

1. 0144 

5-9554 

1.0176 

5.4026 

1. 0211 

4.9452 

20 

4i 

.0116 

•6237 

.0144 

•9452 

.0176 

•3943 

.0211 

.9382 

19 

42 

.0116 

.6m 

.0145 

•935i 

•OI77 

.386 

.0212 

•93i3 

18 

43 

.0117 

' -5985 

.0145 

•925 

.0177 

•3777 

•0213 

•9243 

17 

44 

.0117 

.586 

.0146 

•915 

.0178 

•■3695 

.0213 

•9I75 

16 

45 

1. 0118 

6.5736 

1.0146 

5.9049 

1. 0179 

5-3612 

1. 0214 

4.9106 

15 

46 

.0118 

.5612 

.0147 

•895 

.0179 

•353 

.0215 

•9°37 

14 

47 

.0119 

• 5488 

.0147 

.885 

.018 

•3449 

.0215 

.8969 

13 

48 

.0119 

•5365 

.0148 

•8751 

.018 

•3367 

.0216 

.8901 

12 

49 

.0119 

•5243 

.0148 

.8652 

.0181 

. 3286 

.0216 

•8833 

11 

50 

1. 012 

6.5121 

1. 0149 

5-8554 

1. 0181 

5-3205 

1. 0217 

4.8765 

10 

5i 

.012 

•4999 

• 015 

.8456 

.0182 

.3124 

.0218 

.8697 

9 

52 

.0121 

.4878 

.015 

•8358 

.0182 

•3044 

.0218 

.863 

8 

53 

.0121 

•4757 

.0151 

.8261 

.0183 

.2963 

.0219 

.8563 

7 

54 

.0122 

•4637 

.0151 

•8163 

.0184 

.2883 

.022 

.8496 

6 

55 

1 0122 

6.4517 

1. 0152 

5.8067 

1.0184 

5.2803 

1.022 

4.8429 

5 

56 

.0123 

•4398 

.0152 

•797 

.0185 

•2724 

.0221 

.8362 

4 

57 

.0123 

4279 

•0153 

.7874 

.0185 

.2645 

.0221 

.8296 

3 

58 

.0124 

.416 

•0153 

‘777 8 

.0186 

.2566 

.0222 

.8229 

2 

59 

.0124 

.4042 

.0154 

.7683 

.0186 

.2487 

.0223 

.8163 

1 

60 

1-0125 

6.3924 

1.0154 

5-7588 

1.0187 

5. 2408 

1.0223 

4.8097 

0 

' 1 

Co-sec’t. 

Secant. 

Co-sf.c’t. 

Secant. 

Co-sec’t.  | 

Secant. 

O 

0 

4* 

Secant. 

/ 

1 

81° 

80°  | 

790 

780 

406  natural  secants  and  co-secants. 


12° 

13° 

140 

150 

/ 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant.  1 

Co-sec’t. 

' 

o 

1.0223 

4.8097 

1.0263 

4-4454 

1.0306 

4-I336 

1-0353 

3-8637 

60 

I 

.0224 

.8032 

.0264 

•4398 

•0307 

. 1287 

•0353 

.8595 

59 

2 

.0225 

.7966 

.0264 

•4342 

.0308 

.1239 

•0354 

•8553 

58 

3 

.0225 

.7901 

.0265 

.4287 

.0308 

.1191 

•0355 

•8512 

57 

4 

.0226 

•7835 

.0266 

.4231 

.0309 

.1144 

•0356 

•847 

56 

5 

1.0226 

4-777 

1*0266 

4.4176 

1.031 

4.1096 

1-0357 

3. 8428 

55 

6 

.0227 

.7706 

.0267 

.4121 

.0311 

. 1048 

•0358 

-8387 

54 

7 

.0228 

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.0311 

. 1001 

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.8346 

53 

8 

.0228 

•7576 

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• 0312 

•0953 

•0359 

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52 

9 

.0229 

•7512 

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•3956 

•0313 

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.036 

.8263 

5i 

IO 

1.023 

4. 7448 

1.027 

4-39 01 

1.0314 

4.0859 

1.0361 

3.8222 

50 

XI 

.023 

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•3847 

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.8181 

49 

12 

.0231 

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.814 

48 

*3 

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47 

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46 

is 

1.0233 

4-7*3 

1.0273 

4-363 

1-0317 

4.0625 

1.0365 

3.8018 

45 

16 

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•0579 

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•7978 

44 

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•7937 

43 

18 

•0235 

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•3469 

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.0486 

•0367 

•7897 

42 

19 

•0235 

.6879 

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•34i5 

.032 

•044 

.0368 

•7857 

4i 

20 

1.0236 

4.6817 

1.0277 

4-3362 

1. 0321 

4-0394 

1.0369 

3.7816 

40 

21 

.0237 

•6754 

.0278 

•3309 

.0322 

.0348 

.037 

•7776 

39 

22 

.0237 

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•7736 

38 

23 

.0238 

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•0371 

•7697 

37 

24 

•023Q 

.6569 

.028 

•3i5 

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.0211 

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•7657 

36 

25 

I.O239 

4.6507 

1.028 

4.3098 

1.0325 

4.0165 

1-0373 

3-76i7 

35 

26 

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.6446 

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•0374 

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34 

27 

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•6385 

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33 

28 

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32 

29 

.0242 

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3.9984 

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30 

I.O243 

4. 6202 

1.0284 

4.2836 

1.0329 

3-9939 

1-0377 

3-742 

30 

3i 

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6142 

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•033 

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29 

32 

.0244 

608  X 

.0285 

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28 

33 

.0245 

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27 

34 

.0245 

.5961 

.0287 

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.0332 

.976 

.0381 

•7263 

26 

35 

I.O246 

4.5901 

1.0288 

4-2579 

1-0333 

3.9716 

1.0382 

3.7224 

25 

36 

.0247 

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.9672 

.0382 

.7186 

24 

37 

.0247 

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.0289 

.2476 

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•0383 

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23 

38 

.0248 

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.2425 

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.7108 

22 

39 

.0249 

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•2375 

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•9539 

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21 

40 

I.O249 

4. 5604 

x.0291 

4.2324 

i-o337 

3-9495 

1.0386 

3-7°3i 

20 

4i 

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42 

.0251 

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•0338 

.9408 

•0387 

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18 

43 

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•0339 

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17 

44 

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16 

45 

1-02.53 

4- 53i  1 

1-0295 

4.2072 

1. 0341 

3-9277 

1.039 

3.684 

15 

46 

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14 

47 

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13 

48 

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12 

49 

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11 

50 

1.0256 

4.5021 

1.0299 

4.1824 

1-0345 

3.9061 

1-0394 

3-6651 

10 

5i 

.0257 

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.0299 

•1774 

•0345 

.9018 

•0395 

.6614 

9 

52 

.0257 

•49°7 

•03 

•1725 

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.8976 

•0396 

•6576 

8 

53 

.0258 

•485 

• 0301 

. 1676 

•0347 

•*933, 

•0397 

■6539 

7 

54 

.0259 

•4793 

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.1627 

.0348 

•fe9 

.0398 

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6 

55 

1.026 

4-4736 

1.0302 

4-I578 

1.0349 

3.8848 

1.0399 

3.6464 

5 

56 

.026 

•4679 

V0303 

.1529 

•0349 

.8805 

•0399 

•6427 

4 

57 

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.8763 

.04 

•639 

3 

58 

.0262 

.4566 

•0305 

•1432 

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.8721 

.0401 

•6353 

2 

59 

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•45i 

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.0352 

.8679 

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.6316 

X 

60 

1.0263 

4-4454 

1.0306 

4-I336 

1-0353 

3-8637 

1.0403 

3.6279 

0 

/ 

Co-SEC’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

| Secant. 

770 

760 

75° 

740 

NATUKAL  SECANTS  AND  CO-SECANTS. 


407 


/ 

1 

| Secant. 

L6° 

| Co-sec’t 

Secant. 

L7° 

Co-sec’t. 

3 

Secant. 

.8° 

Co-sec’t. 

1 

Secant. 

.90 

Co-sec’t 

/ 

0 

1.0403 

3.6279 

1-0457 

3-4203 

1-0515 

3.2361 

1.0576 

3-0715 

60 

I 

.0404 

.6243 

.0458 

.417 

.0516 

.2332 

•0577 

.069 

59 

2 

.0405 

.6206 

•0459 

.4138 

•0517 

.2303 

.0578 

.0664 

58 

3 

.0406 

.6169 

.046 

.4106 

•0518 

.2274 

•0579 

.0638 

57 

4 

.0406 

•6133 

.0461 

•4073 

.0519 

.2245 

.058 

.0612 

56 

5 

1.0407 

3.6096 

1.0461 

3.4041 

1.052 

3.2216 

1.0581 

3-0586 

55 

6 

.0408 

.606 

.0462 

.4009 

.0521 

.2188 

.0582 

•0561 

54 

7 

.0409 

.6024 

.0463 

•3977 

.0522 

.2159 

.0584 

•0535 

53 

8 

.041 

•5987 

.0464 

•3945 

•0523 

.2131 

•0585 

•0509 

52 

9 

.0411 

•5951 

.0465 

•39i3 

.0524 

.2102 

.0586 

.0484 

51 

10 

1. 0412 

3-59I5 

1.0466 

3-388i 

1.0525 

3.2074 

1.0587 

3-0458 

50 

11 

.0413 

•5879 

.0467 

•3849 

.0526 

.2045 

.0588 

•0433 

49 

12 

.0413 

■5843 

.0468 

•3817 

•0527 

.2017 

.0589 

.0407 

48 

13 

.0414 

.5807 

.0469 

•3785 

.0528 

.1989 

•059 

•0382 

47 

14 

.0415 

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•047 

•3754 

.0529 

.196 

.0591 

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46 

15 

1.0416 

3-5736 

1. 0471 

3-3722 

1-053 

3-I932 

1.0592 

3-0331 

45 

16 

.0417 

•57 

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•369 

•0531 

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•0593 

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44 

17 

.0418 

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•0473 

•3659 

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•0594 

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43 

18 

.0419 

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•0474 

.3627 

•0533 

.1848 

•0595 

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42 

*9 

.042 

•5594 

•0475 

•3596 

•0534 

.182 

.0596 

•0231 

41 

20 

1.042 

3-5559 

1.0476 

3-3565 

3-I792 

1.0598 

3.0206 

40 

21 

.0421 

•5523 

•0477 

•3534 

•0536 

.1764 

•0599 

.0181 

39 

22 

.0422 

.5488 

.0478 

•3502 

•0537 

•I736 

.06  . 

•0156 

38 

23 

.0423 

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.0478 

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•0538 

. 1708 

.0601 

.0131 

37 

24 

.0424 

.5418 

.0479 

•344 

•0539 

.1681 

.0602 

.0106 

36 

25 

1.0425 

3-5383 

1.048 

3-3409 

1.054 

3-1653 

1.0603 

3.0081 

35 

26 

.0426 

•5348 

.0481 

•3378 

.0541 

.1625 

.0604 

.0056 

34 

27 

.0427 

•5313 

.0482 

•3347 

• 0542 

.1598 

.0605 

0031 

33 

28 

.0428 

•5279 

.0483 

•33i6 

•0543 

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.0007 

32 

29 

.0428 

•5244 

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2.9982 

31 

30 

1.0429 

3-5209 

1.0485 

3-3255 

I-°545 

3-I5I5 

1.0608 

2-9957 

30 

3i 

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29 

32 

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28 

33 

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27 

34 

•0433 

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.0489 

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•0549 

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26 

35 

36 

I-°434 

3-5037 

1.049 

3.3102 

1-055 

3-1379 

1.0614 

2-9835 

25 

•0435 

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.0491 

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•0551 

•1352 

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.981 

24 

37 

38 

.0436 

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.0492 

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•0552 

•1325 

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23 

•°437 

•4935 

•0493 

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•0553 

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.9762 

22 

39 

.0438 

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.2981 

•0554 

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.0618 

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21 

40 

1.0438 

3.4867 

1.0495 

3-2951 

i-o555 

3.1244 

1.0619 

2.9713 

20 

4i 

•0439 

•4833 

.0496 

.2921 

•0556 

. 1217 

.062 

.9689 

19 

42 

.044 

•4799 

.0497 

.2891 

•0557 

.119 

.0622 

.9665 

18 

43 

.0441 

.4766 

.0498 

.2861 

•0558 

.1163 

.0623 

.9641 

17 

44 

.0442 

•4732 

•0499 

.2831 

•0559 

•II37 

.0624 

.9617 

16 

45 

1.0443 

3.4698 

1.05 

3.2801 

1.056 

3- 111 

1.0625 

2-9593 

15 

46 

.0444 

.4665 

.0501 

.2772 

• 0561 

.1083 

.0626 

•9569 

14 

47 

48 

•0445 

.4632 

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•2742 

.0562 

•1057 

.0627 

•9545 

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.0446 

•4598 

•0503 

.2712 

•0363 

.103 

.0628 

•9521 

12 

49 

.0447 

•4565 

.0504 

.2683 

•0565 

. 1004 

.0629 

•9497 

11 

50 

1.0448 

3-4532 

1-0505 

3-2653 

1.0566 

3-0977 

1.063 

2.9474 

10 

5i 

.0448 

•4498 

.0506 

.2624 

•0567 

.0951 

.0632 

•945 

9 

52 

.0449 

•4465 

.0507 

•2594 

.0568 

•0925 

•0633 

.9426 

8 

53 

•°45 

•4432 

.0508 

.2565 

.0569 

.0898 

.0634 

.9402 

7 

54 

.0451 

•4399 

.0509 

•2535 

•057 

.0872 

•0635 

•9379 

6 

55 

56 

1.0452 

3*4366 

1-051 

3.2506 

1. 0571 

3.0846 

1.0636 

2-9355 

5 

•°453 

•4334 

.0511 

•2477 

•0572 

.082 

.0637 

•9332 

4 

57 

58 

•°454 

.4301 

.0512 

.2448 

•0573 

•0793 

.0638 

•9308 

3 

•°455 

.0456 

.4268 

•0513 

.2419 

•°5 74- 

.0767 

.0639 

.9285 

2 

59 

30 

.4236 

.0514 

•239 

•0575 

.0741 

.0641 

.9261 

j 

I-°457 

3.4203 

I-°5I5 

3.2361 

1.0576 

30715 

1.0642 

2.9238 

0 

Co-sec’t.  i 

73 

Secant. 

0 

Co-sec’t. 

72 

Secant. 

0 1 

Co-sec’t. 

71< 

Secant,  j 
3 1 

Co-sec’t. 

70< 

Secant. 

3 

' 

NATURAL  SECANTS  AND  CO-SECANTS. 


t 

20< 

Secant. 

, II 

Co-sec’t  1 

21c 

Secant. 

Co-sec’t. 

22' 

Secant. 

3 

Co-sec’t. 

23° 

Secant.  | Co-sec’t. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

3° 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

1.0642 

.0643 

.0644 

.0645 

.0646 

1.0647 

.0648 

.065 

.0651 

.0652 

1.0653 

.0654 

.0655 

.0656 

.0658 

1.0659 

.066 

.0661 

.0662 

.0663 

1.0664 

.0666 

.0667 

.0668 

.0669 

1.067 

.0671 

.0673 

.0674 

•0675 

1.0676 
.0677 
.0678 
.0679 
.0681 
1.0682 
.0683 
.0684 
.0685 
.0686 
1.0688 
.0689 
.069 
.0691 
.0692 
1.0694 
.0695 
.0696 
.0697 
.0698 
1.0699 
.0701 
.0702 
.0703 
.0704 
1.0705 
.0707 
.0708 
.0709 
.071 
1. 0711 

2.9238 
•92I5 
.gigi 
.9168 
•9*45 
2.9122 
.9098 
•9°75 
.9052 
.9029 
2. 9006 
.8983 
.896 
•8937 
.8915 
2.8892 
.8869  1 
.8846 
.8824 
.8801 
2.8778 
.8756 

•8733 

.8711 

.8688 

2.8666 

.8644 

.8621 

.8599 

•8577 

2.8554 

.8532 

.851 

.8488 

.8466 

2.8444 

.8422 

.84 

.8378 

.8356 

2.8334 

.8312 
.829 
.8269 
.8247 
2.8225 
.8204 
.8182 
.816 
.8139 
2.8117 
.8096 
.8074 
.8053 
. 8032 
2.801 
.7989 
.7968 
•7947 
.7925 
2.7904 

1. 0711 

.0713 

.0714 

.0715 

.0716 
1. 0717 
.0719 
.072 
.0721 
.0722 
1.0723 
.0725 
.0726 
.0727 
.0728 
1.0729 

.0731 

.0732 

.0733 

.0734 

1.0736 

.0737 

.0738 

•0739 

.074 

1.0742 

.0743 

•0744 

•0745 

.0747 

1.0748 

.0749 

•075 

•0751 

•0753 

1-0754 

•0755 

.0756 

.0758 

•0759 

1.076 

.0761 

.0763 

.0764 

.'0765 

1.0766 

.0768 

.0769 

.077 

.0771 

1.0773 

.0774 

•0775 

.0776 

.0778 

1.0779 

.078 

.0781 

.0783 

.0784 

1.0785 

2.7904 

.7883 

.7862 

.7841 

.782 

2.7799 

.7778 

•7757 

•7736 

•77i5 

2.7694 

17674 

.7653 

.7632 

.7611 

2-7591 

•757 

•755 

•7529 

•7509 

2.7488 

.7468 

•7447 

•7427 

.7406 

2.7386 

.7366 

•7346 

7325 

•7305 

2.7285 

.7265 

•7245 

.7225 

.7205 

2.7185 

•7i65 

•7145 

•7125 

•7io5 

2.7085 

.7065 

•7°45 

.7026 

.7006 

2.6986 

.6967 

.6947 

.6927 

.6908 

2.6888 

.6869 

.6849 

.683 

.681 

2.6791 

.6772 

.6752 

•6733 

.6714 

2.6695 

1.0785 
.0787 
.0788 
.0789 
.079 
1,0792 
•0793 
.0794 
.0795 
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1.0798 
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.0801 
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1.0804 
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.0807 
.0808 
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1. 0811 
.0812 
.0813 
.0815 
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1.0817 
.0819 
.082 
.0821 
.0823 
1.0824 
.0825 
.0826 
.0828 
.0829 
1.083 
.0832 
•0833 
.0834 
.0836 
1.0837 
0838 
.084 
.0841 
.0842 
1.0844 
.0845 
,0846 
.0847 
.0849 
1.085 
.0851 
.0853 
.0854 
.0855 
1.0857 
.0858 
.0859 
.0861 
.0862 
1.0864 

2.6695 

.6675 

.6656 

.6637 

.6618 

2.6599 

.658 

.6561 

.6542 

•6523 

2.6504 
6485 
.6466 
.6447 
.6428 
2.641 
.6391 
6372 
•6353 
•6335 
2.6316 
.6297 
.6279 
.626 
.6242 
2.6223 
.6205 
.6186 
.6168 
.615 
2.6131 
.6113 
.6095 
.6076 
.6058 
2.604 
.6022 
.6003 
•5985 
•5967 
2-  5949 
•593i 
•59I3 
•5B95 
•5877 

25859 

.5841 

•5823 

.5805 

•5787 

2-577 

•5752 

•5734 

•57l6 

•5699 

2.5681 

•5663 

.5646 

.5628 

1 -561 
2.5593 

1.0864 
.0865 
.0866 
*0868 
*0869 
1.087 
.0872 
.0873 
.0874 
.0876 
1.0877 
.0878 
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.0881 
.0882 
1.0884 
.0885 
.0886 
.0888 
.0889 
1.0891 
.0892 
.0893 
.0895 
.0896 
1.0897 
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1.7693 

1.227 

1.7256 

35 

26 

.1848 

.8646 

.1983 

• 815 

.2124 

.7685 

.2273 

.7249 

34 

27 

.185 

.8637 

.1985 

.8142 

.2127 

.7678 

.2276 

.7242 

33 

28 

. 1052 

.8629 

.1987 

.8134 

.2129 

.767 

.2278 

•7234 

32 

29 

•1855 

.862 

•I99 

.8126 

.2132 

.7663 

.2281 

.7227 

3i 

30 

1.1857 

1. 8611 

1. 1992 

1. 8118 

1.2134 

1-7655 

1.2283 

1.722 

30 

3i 

.1859 

.8603 

.1994 

.811 

.2136 

.7648 

.2286 

•7213 

29 

32 

.1861 

•8595 

• *997 

.8102 

•2139 

.764 

.2288 

.7206 

28 

33 

.1863 

.8586 

.1999 

.8094 

.2141 

•7633 

.2291 

.7199 

27 

34 

.1866 

.8578 

.2001 

.8086 

.2144 

.7625 

.2293 

.7192 

26 

35 

1. 1868 

1.8569 

1.2004 

1.8078 

I.  2146 

1.7618 

1.2296 

1.7185 

25 

36 

.187 

.8561 

.2006 

.807 

.2149 

.761 

.2298 

•7i78 

24 

37 

.1872 

•8552 

.2008 

.8062 

.2151 

.7603 

.2301 

.7171 

23 

38 

.1874 

•8544 

.201 

.8054 

•2153 

•7596 

.2304 

.7164 

22 

39 

.1877 

•8535 

.2013 

.8047 

.2156 

.7588 

.2306 

•7I57 

21 

40 

1.1879 

1.8527 

1. 2015 

1.8039 

I.2158 

1.7581 

1.2309 

i-7i5i 

20 

4i 

.1881 

.8519 

.2017 

.8031 

.2l6l 

•7573 

.2311 

.7144 

19 

42 

. 1883 

.851 

.202 

.8023 

.2163 

•7566 

•2314 

•7J37 

18 

43 

.1886 

.8502 

.2022 

.8015 

.2166 

•7559 

.2316 

-7I3 

17 

44 

.1888 

•8493 

.2024 

.8007 

.2l68 

•755i 

.2319 

.7123 

16 

45 

1.8485 

1.2027 

X7999 

I. 2171 

1-7544 

1.2322 

1.7116 

i5 

46 

.1892 

•8477 

.2029 

.7992 

•2173 

•7537 

.2324 

.7109 

14 

47 

.1894 

.8468 

.2031 

.7984 

•2175 

•7529 

•2327 

.7102 

13 

48 

.1897 

.846 

.2034 

.7976 

.2178 

.7522 

.2329 

•7095 

12 

49 

.1899 

.8452 

.2036 

.7968 

.2l8 

•75i4 

.2332 

.7088 

11 

50 

I.  I9OI 

1.8443 

1. 2039 

1.796 

I.2183 

1-7507 

1-2335 

1.7081 

10 

5i 

.I9O3 

•8435 

.2041 

•7953 

.2185 

•75 

•2337 

•7°75 

9 

52 

.I906 

.8427 

.2043 

•7945 

.2188 

•7493 

•234 

.7068 

8 

53 

. I908 

.8418 

.2046 

•7937 

.219 

7485 

.2342 

.7061 

7 

54 

.191 

.841 

.2048 

•7929 

.2193 

•7478 

•2345 

•7054 

6 

55 

I.  1912 

1.8402 

1.205 

1. 7921 

1.2195 

1.7471 

1.2348 

1.7047 

5 

56 

•*9*5 

•8394 

•2053 

.7914 

.2198 

•7463 

•235 

.704 

4 

57 

.1917 

•8385 

•2055 

.7906 

.22 

•7456 

•2353 

•7033 

3 

58 

.1919 

•8377 

.2057 

.7898 

.2203 

•7449 

•2355 

.7027 

2 

59 

.1921 

•8369 

.296 

.7891 

.2205 

•7442 

•2358 

.702 

1 

60 

1. 1922 

1.8361 

1.2062 

1.7883 

1.2208 

1-7434 

1.2361 

1. 7013 

0 

H 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

/ 

57° 

560 

550 

54° 

412 


NATURAL  SECANTS  AND  CO-SECANTS. 


36° 

370 

38° 

390 

' 1 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

' 

o 

1.2361 

i-7oi3 

1. 2521 

1. 6616 

1.269 

1.6243 

1.2867 

1.589 

60 

I 

•2363 

.7006 

.2524 

.661 

.2693 

.6237 

.2871 

•5884 

59 

2 

.2366 

.6999 

.2527 

.6603 

.2696 

.6231 

.2874 

•5879 

58 

3 

•2368 

•6993 

•253 

•6597 

.2699 

.6224 

.2877 

•5873 

57 

4 

•2371 

.6986 

•2532 

.6591 

.2702 

.6218 

.288 

•5867 

56 

5 

1-2374 

1.6979 

x-2535 

1.6584 

1.2705 

1.6212 

1.2883 

1.5862 

55 

6 

.2376 

.6972 

.2538 

.6578 

.2707 

.6206 

.2886 

•5856 

54 

7 

•2379 

.6965 

.2541 

.6572 

.271 

.62 

.2889 

•585 

53 

8 

.2382 

•6959 

•2543 

•6565 

•2713 

.6194 

. 2892  j 

•5845 

52 

9 

.2384 

.6952 

.2546 

•6559 

.2716 

.6188 

.2895 

•5839 

5i 

IO 

1.2387 

1.6945 

1-2549 

1-6552 

1. 2719 

1.6182 

1.2898 

1-5833 

50 

ii 

•2389 

.6938 

*'•2552 

.6546 

.2722 

.6176 

.2901  ; 

. 5828 

49 

12 

.2392 

.6ch2 

•2554 

•654 

•2725 

.617 

.29O4 

.5822 

48 

i3 

•2395 

.6925 

•2557 

•6533 

.2728 

.6164 

. 2907 

.5816 

47 

14 

•2397 

.6918 

.256 

.6527 

. -273I 

•6159 

.291  | 

.5811 

46 

i5 

1.24 

1.6912 

1-2563 

1.6521 

1-2734 

1.6153 

I-29I3 

1.5805 

45 

16 

.2403 

.6905 

•2565 

.6514 

•2737 

.6147 

.2916 

•5799 

44 

17 

.2405 

.6898 

.2568 

.6508 

•2739 

.6141 

.2919 

•5794 

43 

18 

.2408 

.6891 

.2571 

.6502 

.2742 

•6i35 

.2922 

•5788 

42 

19 

.2411 

.6885 

•2574 

.6496 

•2745 

.6129 

.2926 

•5783 

41 

20 

1. 2413 

1.6878 

1-2577 

1.6489 

1.2748 

1.6123 

I.2929 

1-5777 

40 

21 

.2416 

.6871 

•2579 

.6483 

•2751 

.6117 

.2932 

•5771 

39 

22 

.2419 

.6865 

.2582 

.6477 

•2754 

.6111 

•2935 

•5766 

38 

23 

.2421 

.6858 

•2585 

.647 

•2757 

.6105 

.2938 

•576 

37 

24 

.2424 

.6851 

.2588 

.6464 

.276 

.6099 

.2941 

•5755 

36 

25 

1.2427 

1.6845 

1.2591 

1.6458 

1.2763 

1.6093 

I.2944 

1-5749 

35 

26 

.2429 

.6838 

•2593 

.6452 

.2766 

.6087 

.2947 

■5743 

34 

27 

.2432 

.6831 

.2596 

•6445 

.2769 

.6081 

•295 

•5738 

33 

28 

•2435 

.6825 

•2599 

•6439 

.2772 

•6077 

•2953 

•5732 

32 

29 

•2437 

.6818 

.2602 

•6433 

•2775 

.607 

•2956 

•5727 

3i 

30 

1.244 

1.6812 

1.2605 

1.6427 

1.2778 

1.6064 

I.296 

1. 5721 

30 

31 

■2443 

.6805 

.2607 

.642 

.2781 

.6058 

.2963 

•57i6 

29 

32 

•2445 

.6798 

.261 

.6414 

.2784 

.6052 

.2966 

•57i 

28 

33 

.2448 

.6792 

• 2613 

.6408 

.2787 

.6046 

.2969 

•5705 

27 

34 

.2451 

.6785 

.2616 

.6402 

.279 

.604 

.2972 

•5699 

26 

35 

1-2453 

1.6779 

1.2619 

1.6396 

1-2793 

1.6034 

X-2975 

1.5694 

25 

36 

.2456 

.6772 

.2622 

.6389 

•2795 

.6029 

.2978 

.5688 

24 

37 

•2459 

.6766 

.2624 

.6383 

.2798 

.6023 

.2981 

•5683 

23 

38 

.2461 

•6759 

.2627 

•6377 

.2801 

.6017 

•2985 

•5677 

22 

39 

.2464 

.6752 

• 263 

•6371 

.2804 

.6011 

.2988 

•5672 

21 

40 

1.2467 

1.6746 

1.2633 

1-6365 

1.2807 

1.6005 

I.  299I 

1.5666 

20 

4i 

.247 

•6739 

.2636 

•6359 

.281 

.6 

.2994 

• 5661 

19 

42 

.2472 

•6733 

.2639 

•6352 

.2813 

•5994 

•2997 

•5655 

18 

43 

•2475 

.6726 

.2641 

.6346 

.2816 

-5988 

•3 

•565 

17 

44 

.2478 

.672 

.2644 

•634 

.2819 

-5982 

-30°3 

•5644 

16 

45 

1.248 

1.6713 

1.2647 

I-6334 

1.2822 

i-5976 

1.3006 

1-5639 

i5 

46 

•2483 

.6707 

.265 

.6328 

.2825 

•5971 

.301 

•5633 

14 

47 

.2486 

.67 

•2653 

.6322 

.2828 

•5965 

.3013 

.5628 

13 

48 

.2488 

.6694 

.2656 

• 6316 

.2831 

•5959 

.3016 

.5622 

12 

49 

.249 

.6687 

.2659 

.6309 

.2834 

•5953 

.3019 

•5617 

11 

50 

1.2494 

1. 6681 

1.2661 

1.6303 

1.2837 

1-5947 

1.3022 

1.5611 

10 

51 

.2497 

.6674 

.2664 

.6297 

.284 

•5942 

.3025 

.5606 

9 

52 

.2499 

.6668 

.2667 

. 6291 

.2843 

•5936 

• 3029 

•56 

8 

53 

.2502 

.6661 

.267 

.6285 

.2846 

•593 

.3032 

•5595 

7 

54 

•2505 

' -6655 

.2673 

.6279 

.2849 

.5924 

•3035 

•559 

6 

55 

1.2508 

1.6648 

1.2676 

1.6273 

1.2852 

I-59I9 

1-3038 

1-5584 

5 

56 

• 251 

.6642 

.2679 

.6267 

•2855 

•59I3 

.3041 

•5579 

4 

57 

•2513 

.6636 

.2681 

.6261 

.2858 

•59°7 

•3044 

•5573 

3 

58 

.2516 

.6629 

.2684 

•6255 

.2861 

•59QI 

.3048 

•5568 

2 

59 

.2519 

.6623 

.2687 

.6249 

.2864 

.5896 

•3051 

•5563 

1 

60 

1. 2521 

1. 6616 

1.269 

1.6243 

1.2867 

1.589 

1-3054 

1-5557 

0 

Co-sec’t. 

Secant. 

Co-skc’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

r 

530 

520 

1 51° 

1 500 

413 


NATURAL  SECANTS  AND  CO-SECANTS. 


40°  I 

41° 

42° 

439, 

' 

Secant. 

Co-sec't.  ; 

Secant. 

Co-se»c’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

' 

o 

i-3°54 

1-55.57 

i-325 

1.5242 

i-3456 

1-4945 

I-3673 

1.4663 

60 

I 

■ 3°57 

•5552 

•3253 

•5237 

•346 

•494 

•3677 

.4658 

59 

2 

.306 

•5546 

•3257 

• 5232 

.3463 

•4935 

.3681 

•4654 

58 

3 

• 3o64 

•5541 

.326 

•5227 

•3467 

•493 

.3684 

•4649 

57 

4 

• 3o67 

• 5536 

•3263 

.5222 

•347 

•4925 

.3688 

•4644 

56 

5 

i-3°7 

1-553 

1.3267 

1-5217 

1-3474 

1. 4921 

1.3692 

1.464 

55 

6 

•3073 

•5525 

•327 

.5212 

•3477 

.4916 

•3695 

•4635 

54 

7 

• 3°76 

•552 

•3274 

.5207 

.3481 

.4911 

•3699 

.4631 

53 

8 

.308 

•5514 

•3277 

.5202 

•3485 

•4906 

•3703 

.4626 

52 

9 

•3o83 

•5509 

.328 

• 5197 

.3488 

.4901 

•3707 

.4622 

5i 

IO 

1.3086 

1-5503 

1.3284 

1.5192 

1.3492 

1.4897 

I-37I 

1.4617 

50 

ii 

.3089 

•5498 

.3287 

.5187 

•3495 

.4892 

•37i4 

.4613 

49 

12 

.3092 

•5493 

•329 

.5182 

•3499 

.4887 

•37i8 

.4608 

48 

13 

.3096 

•5487 

•3294 

•5177 

• 3502 

.4882 

.3722 

.4604 

47 

14 

•3099 

.5482 

•3297 

• 5171 

•3506 

.4877 

•3725 

•4599 

46 

15 

1. 3102 

1-5477 

i-3301 

1.5166 

1-3509 

1-4873 

1.3729 

1-4595 

45 

l6 

• 3io5 

•5471 

•3304 

.5161 

•35i3 

. 4868 

•3733 

•459 

44 

17 

.3109 

.5466 

•3307 

• 5156 

•35i7 

.4863 

•3737 

.4586 

43 

l8 

.3112 

.5461 

•3311 

•5151 

•352 

.4858 

•374 

.4581 

42 

1 9 

•3ii5 

•5456 

•33I4 

.5146 

•3524 

•4854 

•3744 

•4577 

4i 

20 

1.3118 

i-545 

i-33i8 

1. 5141 

*1-3527 

1.4849 

i-3748 

1-4572 

40 

21 

.3121 

•5445 

•3321 

• 5136 

•353i 

•4844 

•3752 

.4568 

39 

22 

•3I25 

•544 

•3324 

• 5131 

•3534 

•4839 

•3756 

•4563 

38 

23 

• 3I28 

•5434 

• 3328 

.5126 

•3538 

•4835 

•3759 

•4559 

37 

24 

•3131 

•5429 

•3331 

.5121 

•3542 

•483 

•3763 

•4554 

36 

2 5 

i-3i34 

I-5424 

1-3335 

1.5116 

1-3545 

1-4825 

1-3767 

i-455 

35 

26 

•3138 

•5419 

•3338 

.5111 

•3549 

.4821 

•377i 

•4545 

34 

27 

•3M1 

•54i3 

•3342 

.5106 

•3552 

.4816 

•3774 

•454i 

33 

28 

•3i44 

.5408 

•3345 

.5101 

•3556 

.4811 

•3778 

•4536 

32 

29 

.3148 

•5403 

•3348 

.5096 

•356 

.4806 

•3782 

4532 

3i 

30 

i-3i5i 

i- 5398 

1-3352 

1. 5092 

1-3563 

1.4802 

1.3786 

1 4527 

30 

3i 

•3i54 

•5392 

•3355 

.5087 

•3567 

•4797 

•379 

•4523 

29 

32 

•3i57 

•5387 

•3359 

. 5082 

•3571 

•4792 

•3794 

.4518 

28 

33 

.3161 

•5382 

• 3362 

•5077 

•3574 

.4788 

•3797 

•4514 

27 

34 

; -3164 

•5377 

•3366 

.5072 

•3578 

•4783 

.3801 

•45i 

26 

35 

1.3167 

i- 537i 

1-3369 

1.5067 

i-358i 

1.4778 

1.3805 

i-45o5 

25 

36 

, -3!7 

•5366 

•3372 

.5062 

•3585 

•4774 

.3809 

.4501 

24 

37 

; -3I74 

536i 

•3376 

• 5057 

•3589 

•4769 

•3813 

•4496 

23 

38 

•3i77 

•5356 

•3379 

•5052 

•3592 

•4764 

.3816 

•4492 

22 

39 

.318 

•5351 

•3383 

•5047 

•3596 

•476 

.382 

•4487 

21 

40 

1.3184 

x-5345 

1.3386 

1.5042 

1.36 

1-4755 

1.3824 

1.4483 

20 

4i 

•3187 

•534 

•339 

•5037 

•3603 

•475 

.3828 

•4479 

19 

42 

•319 

•5335 

•3393 

• 5032 

•3607 

•474*6 

•3832 

•4474 

18 

43 

•3i93 

•533 

•3397 

.5027 

.3611 

.4741 

•3836 

•447 

17 

44 

•3i97 

•5325 

•34 

.5022 

.3614 

•4736 

•3839 

•4465 

16 

45 

1.32 

I-53I9 

1.3404 

1.5018 

1.3618 

1-4732 

1-3843 

1.4461 

i5 

46 

• 3203 

•53*4 

•3407 

• 5013 

•3622 

•4727 

•3847 

•4457 

14 

47 

.3207 

•5309 

•34ii 

.5008 

•3025 

•4723 

•3851 

•4452 

13 

48 

.321 

•5304 

•34H 

• 5003 

•3629 

.4718 

•3855 

•4448 

12 

49 

•3213 

•5299 

.3418 

.4998 

•3633 

•47i3 

•3859 

•4443 

11 

50 

1-3217 

I-5294 

1.3421 

1-4993 

1.3636 

1.4709 

1-3863 

1-4439 

10 

5i 

.322 

.5289 

•3425 

.4988 

•364 

•4704 

.3867 

•4435 

9 

52 

.3223 

•5283 

•3428 

•4983 

•3644 

•4699 

•387 

•443 

8 

53 

•3227 

•5278 

•3432 

•4979 

•3647 

•4695 

•3874 

.4426 

7 

54 

•323 

•5273 

•3435 

•4974 

•3651 

•469 

.3878 

.4422 

6 

55 

1-3233 

1.5268 

1-3439 

1.4969 

i-3655 

1.4686 

1.3882 

I-44I7 

5 

56 

•3237 

•5263 

•3442 

•4964 

•3658 

.4681 

.3886 

•44I3 

4 

57 

•324 

•5258 

•3446 

•4959 

.3662 

.4676 

•389 

.4408 

3 

58 

•3243 

•5253 

•3449 

•4954 

.3666 

.4672 

•3894 

•4404 

2 

59 

•3247 

.5248 

•3453 

•4949 

.3669 

•3667 

.3898 

•44 

1 

60 

1-325 

1.5242 

I-3456 

1-4945 

1-3673 

1.4663 

1.3902 

1-4395 

0 

' 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

Co-sec’t. 

Secant. 

49° 

48° 

47° 

460 

M M* 


414 


NATURAL  SECANTS  AND  CO-SECANTS. 


440 

440 

440  , 

t 

Secant. 

Co-sec’t. 

/ 

t 

Secant. 

Co-sec’t. 

/ 

/ 

Secant. 

Co-sec’t. 

/ 

0 

1.3902 

1-4395 

60 

21 

1.3984 

1-4305 

39 

4i 

1.4065 

1. 4221 

19 

I 

•3905 

•4391 

59 

22 

.3988 

.4301 

38 

42 

.4069 

.4217 

18 

2 

•39°9 

•4387 

58 

23 

•3992 

•4297 

37 

43 

•4073 

.4212 

17 

3 

•39*3 

.4382 

57 

24 

•3996 

.4292 

36 

44 

.4077 

.4208 

16 

4 

•39*7 

•4378 

56 

25 

1.4 

1.4288 

35 

45 

1.4081 

1.4204 

15 

5 

1.3921 

1-4374 

55 

26 

.4004 

.4284 

34 

46 

.4085 

.42 

14 

6 

•3925 

•437 

54 

27 

.4008 

.428 

33 

47 

.4089 

.4196 

13 

7 

3929 

•4365 

53 

28 

.4012 

.4276 

32 

48 

•4093 

.4192 

12 

8 

•3933 

.4361 

52 

29 

.4016 

.4271 

3i 

49 

.4097 

.4188 

11 

9 

•3937 

•4357 

5i 

30 

1.402 

1.4267 

30 

5o 

1. 4101 

1.4183 

10 

10 

i- 394i 

1-4352 

50 

3i 

.4024 

.4263 

29 

5i 

•4105 

.4179 

9 

11 

•3945 

•4348 

49 

32 

.4028 

•4259 

28 

52 

.4109 

•4W5 

8 

12 

•3949 

•4344 

48 

33 

.4032 

•4254 

27 

53 

•4ii3 

.4171 

7 

13 

•3953 

•4339 

47 

34 

.4036 

<425 

26 

54 

.4117 

.4167 

6 

14 

•3957 

•4335 

46 

35 

1.404 

1.4246 

25 

55 

1. 4122 

1.4163 

5 

i5 

1.396 

I-433I 

45 

36 

.4044 

.4242 

24 

56 

.4126 

•4159 

4 

16 

• 3964 

•4327 

44 

37 

.4048 

.4238 

23 

57 

•4i3 

•4154 

3 

17 

.3968 

.4322 

43 

38 

.4052 

•4333 

22 

58 

•4134 

•4*5 

2 

18 

•3972 

.4318 

42 

39 

.4056 

.4229 

21  , 

39 

.4138 

.4146 

1 

*9 

•3976 

•43i4 

41 

40 

1.406  # 

1.4225 

20 

60 

1.4142 

1. 4142 

0 

20 

1.398 

I-43I 

40 

/ 

Co-sec’t. 

Secant. 

t 

/ 

Co-sec’t. 

Secant. 

/ 

/ 

Co-sec’t. 

Secant. 

t 

45° 

450 

45° 

Preceding  Table  contains  Natural  Secants  and  Co-secants  for  every 
minute  of  the  Quadrant  to  Radius  i. 

If  Degrees  are  taken  at  head  of  column,  Minutes,  Secant,  and  Co-secant 
must  be  taken  from  head  also;  and  if  they  are  taken  at  foot  of  column, 
Minutes,  etc.,  must  be  taken  from  foot  also. 

Illustration. — 1.05  is  secant  of  17 0 45'  and  co  secant  of  72 0 15'. 

To  Compute  Secant  or  Co-secant  of  any  Angle. 
Rule. — Divide  1 by  Cosine  of  angle  for  Secant,  and  by  Sine  for  Co-secant. 
Example  i. — What  is  secant  of  250  25'? 

Cosine  of  angle  = .903  21.  Then  1 -f-  .903  21  = 1.1072,  Secant. 

2. — What  is  co  secant  of  64°  35'? 

Sine  of  angle  = .903  21.  Then  1 -4-  .903  21  = 1.1072,  Co-secant. 

To  Compute  Degrees,  IVTinntes,  and.  Seconds  of  a Secant 
or  Co-secant. 

When  Secant  is  given, 

Proceed  as  by  Rule,  page  402,  for  Sines,  substituting  Secants  for  Sines. 
Example.  —What  is  secant  for  1. 1607  ? 

The  next  less  secant  is  1. 1606,  arc  for  which  = 300  30'. 

Next  greater  secant  is  1.1608,  difference  between  which  and  next  less  is  1. 1608-— 
1. 1 606  = .0002. 

Difference  between  less  tab.  secant  and  one  given  is  1. 1607  — 1. 1606  = .0001. 

Then  .0002  : .0001  60  : 30,  which,  added  to  300  30' = 30°  30'  30". 

When  Co -secant  is  given, 

Proceed  as  by  Rule,  page  402,  substituting  Co-secants  for  Cosines. 


NATURAL  TANGENTS  AND  CO-TANGENTS. 
INTatnral  Tangents  and.  Co-tangents. 


0° 

1° 

20 

30 

' 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang* 

CO-TANG. 

Tang. 

CO-TANG. 

' 

o 

.00000 

Infinite. 

.017  46 

5l29  C 

•°34  92 

28.6363 

.05241 

I9.081I 

60 

I 

.000  29 

3437-75 

•017  75 

6.3506 

•03521 

8.3994 

•0527 

8-9755 

59 

2 

.000  58 

1718.87 

.018  04 

5-44I5 

•035  5 

8. 1664 

.05299 

8.8711 

58 

3 

.000  87 

145.92 

.01833 

4-56l3 

•035  79 

7-9372 

.053  28 

8.7678 

57 

4 

.001  16 

859-436 

.018  62 

3.7086 

.036  09 

7.7117 

•053  57 

8.6656 

56 

5 

.001  45 

687.549 

.018  91 

52.8821 

•036  38 

27.4899 

•053  87 

18.5645 

55 

6 

.001  75 

572-957 

.019  2 

2.0807 

.03667 

7-2715 

.054  16 

8.4645 

54 

7 

.00204 

491. 106 

.01949 

1.3032 

.036  96 

7.0566 

•054  45 

8-3655 

53 

8 

.002  33 

29.718 

.019  78 

0.5485 

03725 

6.845 

•054  74 

8.2677 

52 

9 

.002  62 

381.971 

.020  07 

49.8157 

•037  54 

6.6367 

•05503 

8. 1708 

51 

IO 

.002  91 

343-774 

. 020  36 

49.  IO39 

•037  83 

26.4316 

•055  33 

18.075 

5° 

n 

.003  2 

12.521 

.020  66 

8.4121 

.038  12 

6.2296 

.05562 

7.9802 

49 

12 

.003  49 

286.478 

.020  95 

7-7395 

.038  42 

6. 0307 

•055  91 

7.8863 

48 

13 

.003  78 

64.441 

.021  24 

7-0853 

.03871 

5-8348 

.0562 

7-7934 

47 

14 

.00407 

45-552 

.021  53 

6.4489 

.039 

5-6418 

.056  49 

7-7OI5 

46 

i5 

.004  36 

229. 182 

.021  82 

45.8294 

.039  29 

25-45I7 

.05678 

17.6106 

45 

16 

.004  65 

14.858 

.022  11 

5.2261 

•039  58 

5-2644 

.05708 

7-5205 

44 

i7 

.004  95 

02.219 

.022  4 

4.6386 

.039  87 

5.0798 

•057  37 

7-43I4 

43 

18 

.005  24 

190.984 

.022  69 

4.0661 

.040  16 

4.8978 

.05766 

7-3432 

42 

i9 

•005  53 

80.932 

.022  98 

3-5o8i 

. 040  46 

4.7185 

•057  95 

7-2558 

41 

20 

.005  82 

i7!.885 

.023  28 

42.9641 

.04075 

24.5418 

. 058  24 

I7-  i693 

40 

21 

.006  11 

63  -7 

•02357 

2-4335 

.041  04 

4-3675 

•058  54 

7.0837 

39 

22 

.006  4 

56.259 

• 023  86 

i-9i58 

•041  33 

4-1957 

-05883 

6.999 

38 

23 

006  69 

49.465 

•02415 

1.4106 

.041  62 

4.0263 

• 059  12 

6.915 

37 

24 

.006  98 

43-237 

.02444 

0.9174 

041  91 

3-8593 

-05941 

6.8319 

36 

25 

.007  27 

137.507 

.024  73 

40.4358 

042  2 

23.6945 

•059  7 

16.749  6 

35 

26 

.007  56 

32.219 

.025  02 

399655 

0425 

3-5321 

-059  99 

6-6681  : 

34 

27 

.007  85 

27.321 

•02531 

9-  5059 

.042  79 

3-37i8 

.060  29 

6-  5874 

33 

28 

.008  14 

22.774 

.0256 

9. 0568 

.043  08 

32137 

.060  58 

6-5075 

32 

29 

.008  44 

18.54 

.025  89 

8.6177 

•043  37 

3-0577 

.060  87 

6.4283 

31 

3° 

.008  73 

114-589 

.026 19 

38.1885 

.043  66 

22.9038 

.061  16 

16.3499 

3° 

31 

.009  02 

10. 892 

.02648 

7.7686 

•043  95 

2-7519 

.061  45 

6.2722 

29 

32 

.00931 

07. 426 

.026  77 

7-3579 

.044  24 

2.602 

•061  75 

6. 1952 

28 

33 

.009  6 

04.171 

.027  06 

6.956 

.044  54 

2.4541 

.062  04 

6. 1 19 

27 

34 

.009  89 

01. 107 

•02735 

6.5627 

.044  83 

2. 3081 

.06233 

6-0435 

26 

35 

.010 18 

98.2179 

.027  64 

36.1776 

.045  12 

22. 164 

.06262 

15.9687 

25 

36 

.01047 

5-4895 

•02793 

5. 8006 

.04541 

2.0217 

.06291 

5-8945 

24 

37 

.01076 

2. 9085 

.028  22 

5-4313 

0457 

1.8813 

. 063  2 1 

5.8211 

23 

38 

.011  05 

0-46.33 

.028  51 

5.0695 

•045  99 

1.7426 

•0635 

5-7483 

22 

39 

•OI1  35 

88. 1436 

.028  81 

4-7i5i 

. 046  28 

1.6056 

•063  79 

5.6762 

21 

40 

.011  64 

85.9398 

.029  1 

34.3678 

.046  58 

21.4704 

.06408 

15.6048 

20 

4i 

.01193 

3-8435 

•029  39 

4.0273 

.046  87 

1-3369 

•06437 

5-534 

19 

42 

.012  22 

1.847 

.029  68 

3-6935 

.047  16 

1.2049 

.06467 

5.4638 

18 

43 

.012  51 

79-9434 

.02997 

3.3662 

•047  45 

1.0747 

.064  96 

5-3943 

17 

44 

.012  8 

8.1263 

.030  26 

3-0452 

•047  74 

0. 946 

.065  25 

5-  3254 

16 

45 

.01309 

76-39 

•03055 

32.7303 

.048  03 

20.8188 

•065  54 

15-2571 

15 

46 

•01338 

4.7292 

. 030  84 

2.4213 

.04832 

0. 6932 

.06584 

5-1893 

14 

47 

• 01367 

3-139 

•031  14 

2. 1181 

.048  62 

0.5691 

.066 13 

5. 1222 

I3 

48 

•01396 

1.6151 

•03143 

1.8205 

.048  91 

0.4465 

.066  42 

5-0557 

12 

49 

.01425 

0-1533 

.031  72 

1.5284 

.0492 

0.3253 

. 066  7 1 

4.9898 

11 

5o 

01455 

68.7501 

.632  01 

31.2416 

•049  49 

20. 2056 

.067 

14.9244 

10 

5i 

.014  84 

7.4019 

•0323 

0-9599 

.049  78 

0.0872 

.0673 

4.8596 

9 

52 

.01513 

6.1055 

•03259 

0.6833 

.05007 

19.9702 

•067  59 

4-7954 

8 

53 

.01542 

4.858 

.032  88 

0.4116 

•05037 

9.8546 

.067  88 

4-73I7 

7 

54 

.01571 

3-6567 

•03317 

0. 1446 

.05066 

9-  7403 

.068  17 

4.6685 

6 

55 

.016 

62.4992 

•03346 

29.8823 

•05095 

19.6273 

.068  47 

14.6059 

5 

56 

.016  29 

1.3829 

•033  76 

9.6245 

.051  24 

9-5I56 

.068  76 

4-5438 

4 

57 

.016  58 

0. 3058 

•03405 

9-3711 

•051  53 

9.4051 

.069  05 

4. 4823 

3 

58 

.016  87 

59.2659 

•034  34 

9. 122 

.051  82 

9.2959 

.069  34 

4.4212 

2 

59 

.017  16 

8.2612 

•°34  63 

8-8771 

.052  12 

9.1879 

.069  63 

4. 3607 

60 

.01746 

57-29 

•034  92 

28.6363 

.05241 

19.0811 

. 069  93 

14.3007 

0 

' 

CO-TANG.  | 

Tang. 

Co-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

✓ 

890 

880 

870 

86° 

NATURAL  TANGENTS  AND  CO-TANGENTS. 


, 

40 

Tang.  | Co-tang. 

6 

Tang. 

D 

Co-TANG.  1 

6 

Tang. 

? .'  ■ 1 
CO-TANG. 

1 

Tang. 

> 

CO-TANG. 

' 

o 

.06993 

14.3007 

.087  49 

i«-43°  1 

.1051 

9-5I4  36 

.122  78 

8.144  35 

60 

X 

.070  22 

4. 241 1 

.087  78 

i-39i9 

■ 105  4 

•487  8l 

. 12308 

.12481 

59 

2 

.070  51 

4.1821 

.088  07 

1-354 

• 105  69 

.461  4I 

.123  38 

.105  36 

58 

3 

.070  8 

4- 1235 

.088  17 

1-316  3 

• 105  99 

•435  15 

.123  67 

.086 

57 

4 

.071 1 

4-0655 

.08866 

1.278  9 

. 106  28 

. 409  04 

•123  97 

. 066  74 

56 

5 

•071  39 

14.007Q 

.088  95 

11. 241  7 

. 106  57 

9-  383  07 

. 124  26 

8.O47  56 

55 

6 

.071  68 

3-9507 

.089  25 

1.204  8 

. 106  87 

•357  24 

.124  56 

.028  48 

54 

7 

.07197 

3-894 

•089  54 

1. 168  1 

. 107  16 

•33i  54 

.12485 

8.OO9  48 

53 

8 

.072  27 

3-8378 

.089  83 

1. 131  6 

. 107  46 

•305  99 

•12515 

7.99058 

52 

9 

.072  56 

3.7821 

.09013 

1.0954 

•107  75 

.28058 

•125  44 

.97176 

5i 

IO 

.072  85 

13.7267 

•ogo  42 

n.0594 

. 108  05 

9-255  3 

•125  74 

7.95302 

50 

ii 

•073  J4 

3.6719 

.09071 

1.0237 

.108  34 

.230 16 

. 126  03 

•934  38 

49 

12 

•07344 

3-6i74 

.091  01 

0.988  2 

.10863 

.205 16 

•126  33 

.915  82 

48 

13 

•073  73 

3-5634 

.091  3 

0.9529 

.10893 

.18028 

. 12662 

•897  34 

47 

*4 

.074  02 

3-5098 

•091  59 

0.917  8 

. 109  22 

•155  54 

. 12692 

•87895 

46 

*5 

•074  31 

13.4566 

.091  89 

10.882  9 

.10952 

9-13093 

.127  22 

7.860  64 

45 

16 

.07461 

3- 4039 

.092  18 

0.8483 

. 109  81 

. 106  46 

.12751 

.842  42 

44 

17 

.0749 

3-3515 

•09247 

0.813  9 

. no  II 

.082  11 

.127  81 

.824  28  ! 

O ZT  1 

43 

18 

•07519 

3.2996 

.092  77 

0-779  7 

.110  4 

•057  89 

.128  1 

.806  22  1 

42 

J9 

•07548 

3.248 

.093  06 

0-745  7 

.110  7 

•033  79 

.1284 

.78825 

4i 

20 

•075  78 

13.1969 

•09335 

10. 71 1 9 

.11099 

9.009  83 

. 128  69 

7- 77°  35 

40 

21 

.076  07 

3-x46i 

.09365 

0.6783 

. in  28 

8.98598 

.12899 

•752  54 

39 

22 

.07636 

3-0958 

.09394 

0.645 

.111  58 

.962  27 

. 129  29 

•734  8 

38 

23 

.076  65 

3-0458 

.09423 

0,611  8 

.11187 

.93867 

.12958 

•7i7i5 

37 

24 

•07695 

2.9962 

•094  53 

0.5789 

.11217 

.9152 

.12988 

•699  57 

36 

25 

.07724 

12.9469 

.09482 

10.546  2 

. 112  46 

8.891  85 

.13017 

7.682  08 

35 

26 

•07753 

2.8981 

.09511 

CX5136 

.112  76 

. 868  62 

•13947 

. 664  66 

34 

27 

.077  82 

2. 8496 

.09541 

0.481  3 

•11305 

•84551 

.13076 

•647  32 

33 

28 

.078 12 

2.8014 

•°95  7 

0.4491 

•H335 

.822  52 

.13106 

•63005 

32 

29 

.078  41 

2.7536 

.096 

0.417  2 

.11364 

•799  64 

.13136 

.612  87 

3i 

30 

•0787 

12.7062 

.096  29 

10. 385  4 

•II394 

8.776  89 

•13165 

7-595  75 

30 

31 

.078  99 

2,6591 

.096  58 

0.3538 

.11423 

•754  25 

13195 

.57872 

29 

32 

.079  29 

2.6124 

.096  88 

0.3224 

.11452 

•73i  72 

.13224 

• 561  76 

1 28 

33 

•079  58 

2.566 

.09717 

0.291  3 

.11482 

•70931 

•132  54 

•544  87 

i 27 

34 

.07987 

2.5199 

.097  46 

0. 260  2 

.11511 

.68701 

.132  84 

.52806 

26 

35 

.080  17 

12.4742 

.097  76 

10.2294 

•11.541 

8. 664  82 

•13313 

7-5ii  32 

1 25 

36 

.080  46 

2.4288 

.09805 

0. 198  8 

•ii57 

.64275 

•133  43 

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24 

37 

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2.3838 

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0. 168  3 

.116 

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•133  72 

.478  06 

1 23 

38 

.081 04 

2-339 

.098  64 

0. 138  1 

.11629 

•59893 

.13402 

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; 22 

39 

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2.2946 

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0. 108 

.11659 

•577i8 

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21 

40 

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12.2505 

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10.078 

.11688 

8-555  55 

.13461 

7.428  71 

j 20 

4i 

.081  92 

2.2067 

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0.048  3 

'.117*18 

•534  02 

•i349i 

.4124 

!:§ 

42 

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2. 1632 

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0.018  7 

•ii747 

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.39616 

43 

.082  51 

2. 1201 

.10011 

9.9893 

.11777 

.491  28 

•1355 

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! !7 

44 

.082  8 

2.0772 

.1004 

. 960  07 

. 1 1 8 06 

.47007 

•1358 

• 363  89 

16 

45 

.08309 

12.0346 

. 10069 

9.931  01 

.11836 

8.448  96 

.13609 

7-347  86 

15 

46 

•083  39 

1.9923 

. 100  99 

. 902  1 1 

. 118  65 

•427  95 

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14 

47 

083  68 

1.9504 

. 101  28 

•87338 

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.40705 

.136  69 

•316 

.30018 

13 

48 

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1.9087 

. 101  58 

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.11924 

• 38625 

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12 

49 

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1 8673 

. 101  87 

.816  41 

.119  54 

•365  55 

.137  28 

.284  42 

II 

50 

.084  56 

11.8262 

. 102 16 

9.788  x7 

•11983 

8-344  96 

•137  58 

7.268  73 

1° 

5i 

.084  85 

1-7853 

. 102  46 

. 760  69 

.120  13 

.32446 

•13787 

•253  1 

9 

52 

.085  14 

1.7448 

.102  75 

•732  17 

.12042 

.304  06 

.13817 

•237  54 

8 

53 

.08544 

1-7045 

• 103  05 

.70441 

.120  72 

.28376 

.13846 

.222  04 

7 

54 

•08573 

1. 6645 

• 103  34 

.6768 

.121  01 

•26355 

.13876 

.206  61 

6 

55 

. 086  02 

11.6248 

• 103  63 

9-649  35 

- 121  31 

8.243  45 

.13906 

7-i9i  25 

5 

56 

.086  32 

i i-5853 

•10393 

.62205 

. 121  6 

.223  44 

•139  35 

•175  94 

4 

57 

. 086  61 

1.5461 

. 104  22 

•594  9 

. 121 9 

.203  52 

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3 

58 

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1.5072 

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•1837 

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•145  53 

2 

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.32492  ; 

3.27085 
•267  45 
. 264  06 
260  67 

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3-25392 

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.247  19 

•24383 

.240  49 
3-237  *4 
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.23048 
•227  15 
.223  84, 
3.22053 
.217  22 
.213  92 
.210  63 
.20734 
3. 204  06 
.200  79 
•*97  52 
.194  26 
.191 

3- 187  75 
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.181  27 
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3- *7*  59 
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.165  17 
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3- *55  58 
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. 102  23 
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3.09298 

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. 086  85 
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3.07768 

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•325  24 
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•32685 

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.328  14 

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• 332  66 
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: 3.07768 

.07464 
.071  6 
.06857 
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3.06252 
•059  5 
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3-047  49 
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3.0326 
.029  63 
.02667 
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3.01783 
.014  89 
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3.003  19 
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2.99738 
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2.98868 
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2.974  3. 

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2.96004 
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2.9459 
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.93468 
2.931  89 
.9291 
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.92076 
2.91799 
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.91246 
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.906  96 
2.904  21 

•344  33 
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•345  63 
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.34661 

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.34824 
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.348  89 
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•349  54 
.34987 
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•356  08 

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•359  37 
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• 360  35 
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.362  32 
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2.90421 
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2.89055 
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2.877 

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.868  92 
. 866  24 
2.863  56 
. 860  89 
• 85822 
•855  55 
•85289 
2.85023 
•84758 
•844  94 
.842  29 

•83965 

2.83702 
•834  39 

.831 76 
.829  14 
.826  53 
2.823  9* 
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.816  1 

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2.810  91 
.808  33 
.80574 
.803 16 
. 800  59 
2.798  02 

•795  45 
.792  89 
•790  33 
•78778 

2.785  23 
.782  69 
.780 14 
•777  6i 
•775  07 

2-772  54 
.77002 

•7675 

.764  98 
.76247 
2.75996 
•757  46 
•754  96 
.75246 

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2.747  48 

60 

59 

58 

57 

56 

55 

54 

53 

52 

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50 

49 

48 

47 

46 

45 

44 

43 

42 

4* 

40 

39 

38 

37 

36 

35 

34 

33 

32 

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30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

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18 

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16 

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12 

11 

10 

9 

8 

7 

6 

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4 

3 

2 

1 

0 

1 Co-TANG.  1 

1 73' 

Tang.  ( 

0 

2o-tang. 

72( 

Tang.  ( 

3 

/O-TANG. 

71< 

Tang.  I ( 
3 1 

?0-TANG. 

70 

Tang. 

6 

' 

NATURAL  TANGENTS  AND  CO-TANGENTS. 


20° 

21° 

22° 

230  I 

Tang. 

CO-TANG. 

Tang.  I 

CO-TANG. 

Tang.  | 

Co-tang. 

Tang.  | 

CO-TANG.  | 

0 

.36397 

2.74748 

.38386 

2.605  09 

• 404  °3 

2.47509 

•424  47 

2.355  85 

I 

•364  3 

•744  99 

.3842 

. 602  83 

.404  36 

.47302 

.424  82 

•353  95 

2 

•36463 

•74251 

•384  53 

.600  57 

•404  7 

•470  95 

.425  16 

•352  05 

3 

.364  96 

.740  04 

.38487 

•598  31 

.405  °4 

.46888 

•425  5i 

•350  15 

4 

-365  29 

•737  56 

•3852 

. 596  06 

•405  38 

. 466  82 

•42585 

•348  25 

5 

•36562 

2-73509 

•385  53 

2.593  81 

•405  72 

2. 464  76 

.426 19 

2.346  36 

6 

•365  95 

•73263 

•38587 

•591  56 

. 406  06 

.4627 

.426  54 

• 344  47 

7 

.366  28 

•73017 

.3862 

.58932 

.4064 

.46065 

.426  88 

•342  58 

8 

.366  61 

.72771 

•386  54 

.587  08 

.40674 

.4586 

.427  22 

.34069 

9 

• 366  94 

.725  26 

.386  87 

.58484 

.40707 

•456  55 

•427  57 

• 33881 

10 

.36727 

2.722  81 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2-33693 

11 

-3676 

.720  36 

•387  54 

.580  38 

•407  75 

.45246 

.42826 

•335  05 

12 

•367  93 

.71792 

.38787 

•57815 

. 408  09 

•450  43 

.4286 

•33317 

13 

.368  26 

•7J5  48 

.38821 

•57S93 

•408  43 

.448  39 

.42894 

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14 

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•71305 

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.429  29 

•329  43 

15 

.36892 

2.71062 

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2-57I5 

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2-444  33 

•42963 

2.327  56 

16 

.36925 

.708 19 

.389  21 

.56928 

•40945 

•442  3 

.429  98 

•325  7 

17 

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.389  55 

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43032 

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18 

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1 9 

.37024 

.70094 

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.320 12 

20 

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2.698  53 

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2.560  46 

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2.43422 

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2.318  26 

21 

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22 

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23 

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• 432  39 

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24 

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.688  92 

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•55i  7 

.41217 

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.31086 

2 5 

.37223 

2.686  53 

.39223 

2.54952 

.41251 

2.424  18 

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2.30902 

26 

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.684 14 

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.41285 

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•433  43 

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27 

.37289 

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.420 19 

•433  78 

•305  34 

28 

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29 

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. 540  82 

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3° 

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2.674  62 

.393  91 

2.53865 

.414  21 

2.414  21 

.43481 

2.299  84 

3i 

.37422 

•672  25 

•394  25 

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•414  55 

.41223 

• 435  16 

.29801 

32 

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. 669  89 

•394  58 

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•435  5 

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33 

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•435  85 

•294  37 

34 

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•415  58 

.406  29 

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35 

•375  54 

2.662  81 

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2.527  86 

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2.404  32 

•430  54 

2.29073 

36 

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.660  46 

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37 

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. 396  26 

•523  57 

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.400  38 

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.287  1 

38 

•376  54 

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.52142 

.41694 

.39841 

•437  58 

.285  28 

39 

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.417  28 

•39645 

•437  93 

.28348 

40 

•377  2 

2.651  09 

•397  27 

2.517  i5 

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2-394  49 

.438  28 

2.281  67 

4i 

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.64875 

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.51502 

•4*7  97 

•392  53 

.43862 

.27987 

42 

•377  87 

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•397  95 

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.41831 

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43 

.3782 

.644  1 

.398  29 

.510  76 

.418  65 

.388  62 

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44 

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. 508  64 

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.439  66 

•274  47 

45 

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2-63945 

.398  96 

2.506  52 

•41933 

2.38473 

.44001 

2.272  67 

46 

•379  2 

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.382  79 

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.270  88 

47 

•379  53 

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. 502  29 

. 420  02 

. 380  84 

.44071 

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48 

.37986 

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. 420  36 

.37891 

.44105 

.2673 

49 

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.4414 

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50 

• 380  53 

2.62791 

.400  65 

2-495  97 

.42105 

2-375  04 

•44i  75 

2.263  74 

5i 

. 380  86 

.62561 

.400  98 

•493  86 

.421  39 

•373  11 

.4421 

.261  96 

52 

.381  2 

.623  32 

.401  32 

•491  77 

•421  73 

.371 18 

.44244 

.260  18 

53 

•381  53 

.621  03 

.401 66 

.48967 

.42207 

.36925 

.44279 

.2584 

54 

.38186 

.618  74 

.402 

.487  58 

.42242 

•367  33 

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.256  63 

55 

.3822 

2.61646 

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2.48549 

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2.36541 

•443  49 

2.25486 

56 

•382  53 

.614 18 

. 402  67 

.4834 

.4231 

•363  49 

•443  84 

.25309 

57 

.382  86 

.611  9 

.40301 

.481  32 

.42345 

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.44418 

.251  32 

58 

•3832 

.60963 

•403  35 

.47924 

•423  79 

•35967 

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.24956 

59 

•383  53 

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.403  69 

.47716 

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•357  76 

.44488 

.2478 

60 

.38386 

2. 605  09 

.40403 

| 2.47509 

.42447 

2.35585 

•445  23 

2.24604 

CO-TANG. 

| Tang. 

Co-tang. 

! Tang. 

C’O-TANG. 

| Tang. 

Co-TANG. 

Tang. 

690 

68° 

67° 

1 66° 

60 

59 

58 

57 

56 

55 

54 

53 

52 

5i 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

*9 

18 

17 

16 

i5 

14 

13 

12 

11 

10 

l 

7 

6 

5 

4 

3 

2 

1 

o 


NATURAL  TANGENTS  AND  CO-TANGENTS.  421 


24°  1 

25° 

26° 

27° 

* 

Tang. 

CO-TANG.  1 

Tang. 

CO-TANG. 

Tang.  | 

CO-TANG. 

Tang.  | 

CO-TANG. 

' 

•445  23 

2.24604 

.466  31 

2.144  51 

•487  73 

2.O5O3 

•509  53 

I.962  6l 

60 

I 

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.24428  : 

.466  66 

. 142  88 

. 488  09 

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59 

2 

•445  93 

.24252 

. 467  02 

.141  25 

.48845 

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•959  79 

58 

3 

.44627 

.24077  1 

•467  37 

.139  63 

.48881 

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•95838 

57 

4 

.446  62 

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•489  17 

. O44  26 

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.95698 

56 

5 

.44697 

2.237  27  1 

.468  08 

2.136  39 

•489  53 

2.042  76 

•51136 

1-955  57 

55 

6 

•447  32 

•235  53  i 

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.134  77 

.489  89 

.04125 

•5H73 

•95417 

54 

7 

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.468  79 

.J33 16 

.49026 

•039  75 

.51209 

•952  77 

53 

8 

.448  02 

.23204  | 

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. 490  62 

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.51246 

•95137 

52 

9 

•448  37 

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.4695 

.12993 

. 490  98 

•036  75 

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5i 

10 

.44872 

2.228  57 

.469  85 

2.12832 

•491  34 

2.035  26 

•513  19 

1.948  58 

50 

11 

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.126  71 

•491  7 

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•513  56 

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49 

12 

•449  42 

.225 1 

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.125  11 

.492  06 

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48 

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47 

14 

.45012 

.221 64 

.47128 

.121  9 

.49278 

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46 

15 

•450  47 

2.21992 

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2. 120  3 

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2.027  8 

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1. 941 62 

45 

16 

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44 

17 

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18 

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42 

19 

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20 

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2.211 32 

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2.11233 

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2.020  39 

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1-9347 

40 

21 

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39 

22 

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23 

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37 

24 

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36 

25 

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2.20278  - 

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2. 104  42 

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2.01302 

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1.927  82 

35 

26 

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55 

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34 

27 

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33 

28 

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32 

29 

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2. 194  3 

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2.09654 

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2.005  69 

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30 

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29 

32 

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27 

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1-999  86 

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26 

35 

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2.185  87 

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2.088  72 

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1.99841 

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1. 914  18 

25 

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24 

37 

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38 

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22 

39 

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21 

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2.17749 

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20 

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17 

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16 

45 

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2.169  17 

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1-98396 

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2.065  53 

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1.9768 

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10 

51 

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6 

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1 

60 

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2-14451 

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2.0503 

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1.962  61 

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1.880  73 

0 

Co-TANG 

Tang. 

C’O-TANG. 

. Tang. 

CO-TANG 

. 1 Tang. 

| Co-tang 

. 1 Tang. 

' 

65° 

64° 

630 

ll 

62° 

N N 


NATURAL  TANGENTS  AND  CO-TANGENTS. 


1 

28° 

29° 

300 

31° 

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Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

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5 

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. 602  84 

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||  .62487 

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0 

"7" 

CO-TANG. 

Tang. 

Co-tang  . 

Tang. 

Co-TANG. 

, 1 Tang. 

I CO-TANG . 

Tang. 

' 

610 

60° 

59° 

u 58° 

NATURAL  TANGENTS  AND  CO-TANGENTS.  423 


32° 

330 

340 

350 

-* 

Tang. 

Co-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

f 

0 

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60 

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0 

' 

Co-TANG. 

Tang. 

CO-TANG. 

1 Tang. 

Co-TANG. 

Tang. 

CO-TANG. 

Tang. 

570 

56° 

55° 

540 

424  NATURAL  TANGENTS  AND  CO-TANGENTS. 


56° 

37°  I 

38° 

39° 

' 

Tang.  >| 

Co-TANG. 

Tang. 

Co-TANG. 

Tang. 

Co-TANG. 

Tang. 

CO-TANG. 

' 

o 

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I-37638 

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1.327  04 

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I.279  94 

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58 

3 

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4 

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1.276  11 

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55 

6 

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53 

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9 

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48 

13 

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47 

14 

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1.364  66 

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46 

15 

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1-31507 

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1.268  49 

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I.22394 

45 

l6 

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44 

17 

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43 

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19 

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41 

20 

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1.359  68 

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I.-311 1 

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40 

21 

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• S^1 

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23 

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37 

24 

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.821 41 

.217  42 

36 

25 

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i-355  54, 

.76502 

1.307  16 

.79306 

1.26093 

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1.216  7 
.21598 
.215  26 

35 

26 

.738  16 

•35472' 

.76548 

•30637 

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.822  38 

34 

27 

.73861 

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28 

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29 

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3i 

30 

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30 

31 

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34 

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26 

35 

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1.299  31 

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1.209  51 
.208  79 
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.207  36 
. 206  65 

25 

36 

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24 

37 

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22 

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43 

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17 

16 

44 

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45 

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1. 291  52 

. 802  58 

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1.20237 
. 201 66 

15 

46 

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14 

47 

■ 747  64 

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13 

48 

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12 

49 

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1.198  82 

11 

5° 

•749 

i-335  11 

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1.287  64 

. 804  98 

1.242  27 

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10 

51 

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9 

52 

53 

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0 

7 

54 

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6 

55 

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1. 331  07 

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1-383  79 

• 807  38 

1.238  58 

.83662 

1. 195  28 

5 

56 

•75i 73 

• 33°  26 

•77941 

.28302 

. 807  86 

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• 193  87 
.193  16 
.192  46 

4 

57 

•752 19 

.32946 

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3 

58 

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2 

59 

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l .8386 

1 

60 

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1.32704 

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| 1-27994 

j .80978 

1-234  9 

1 -839  1 

1191  75 

0 

' 

j Co-TANG. 

1 i 

Tang. 

53° 

Co-TANG 

: | Tang. 

52° 

j j CO-TANG 

. | Tang. 
510 

I I CO-TANG. 

II  i 

Tang. 

50° 

NATURAL  TANGENTS  AND  CO-TANGENTS.  425 


40° 

41° 

420 

43° 

' 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

Tang. 

CO-TANG. 

' 

0 

•839  1 

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X-I50  37 

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I.II061 

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I.O7237 

60 

I 

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59 

2 

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58 

3 

.840  59 

. 189  64 

.87082 

.14834 

.901  99 

. 108  67 

•93415 

.070  49 

57 

4 

.841  08 

. 188  94 

•87133 

• I47  67 

.902  51 

. 108  02 

•934  69 

.069  87 

56 

5 

•841  58 

1. 188  24 

.871  84 

1.146  99 

•9°3  04 

I.  IO7  37 

•935  24 

I.069  25 

55 

6 

.84208 

•18754 

.87236 

.14632 

•9°3  57 

.10672 

•935  78 

.068  62 

54 

7 

.842  58 

. 186  84 

.872  87 

•14565 

.9041 

. 106  07 

•93633 

.068 

53 

8 

.84307 

. 186  14 

.873  38 

.14498 

.90463 

.105  43 

.936  88 

•067  38 

52 

9 

•843  57 

.18544 

.87389 

•144  3 

.905  16 

. IO4  78 

•937  42 

.066  76 

5i 

10 

.844  07 

1.18474 

.87441 

*•  x4363 

• 9°5  69 

I.  IO4  14 

•937  97 

I.066  13 

50 

11 

•844  57 

.18404 

.87492 

. 142  96 

.906  21 

• 103  49 

•938  52 

.06551 

49 

12 

.84507 

•18334 

•875  43 

.142  29 

.90674 

.102  85 

• 939  °6 

. 064  89 

48 

13 

•845  56 

. 182  64 

•87595 

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.907  27 

. 102  2 

.93961 

.064  27 

47 

14 

. 846  06 

.18194 

.876  46 

• 140  95 

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. 063  65 

46 

15 

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1. 181  25 

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I.  IOO  91 

94071 

I.063  03 

45 

16 

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44 

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41 

20 

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I-i77  77 

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1.136  94 

.91099 

I-°97  7 

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I.05994 

40 

21 

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39 

22 

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38 

23 

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37 

24 

.85107 

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36 

25 

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1.13361 

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1 • °94  5 

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35 

26 

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27 

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1. 130  29 

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3i 

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34 

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26 

35 

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1. 1 67  41 

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1. 126  99 

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25 

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23 

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40 

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20 

4i 

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18 

43 

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44 

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. 121  06 

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16 

45 

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1. 160  56 

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1. 081  79 

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46 

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• x59  87 

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• IX9  75 

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47 

. 862  67 

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48 

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• 95897 

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12 

49 

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11 

50 

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i- 157  *5 

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1.117  *3 

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1.07864 

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1. 041  58 

10 

5i 

.8647 

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. 1 16  48 

.927  63 

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9 

52 

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•155  79 

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.92817 

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.961  2 

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8 

53 

.865  72 

.x55n 

. 896  72 

• IX5 17 

.928  72 

.076  76 

.961  76 

.03976 

7 

54 

. 866  23 

•154  43 

.897  25 

.11452 

.929  26 

• 076  13 

.962  32 

.039  x5 

6 

55 

.866  74 

i- 153  75 

.89777 

1.11387 

.9298 

x-075  5 

.962  88 

1-03855 

5 

56 

.867  25 

.15308 

.8983 

.11321 

•930  34 

.074  87 

.96344 

•037  94 

4 

57 

. 867  76 

.1524 

.898  83 

.112  56 

.93088 

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.964 

•037  34 

3 

58 

.868  27 

.15172 

•899  35 

. hi  91 

•93x43 

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2 

I9 

. 868  78 

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. hi  26 

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1 

60 

. 869  29 

x-x5o  37 

.9004 

1.  no  61 

.932  52 

1.07237 

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*•035  53 

0 

' | 

Co-tang. 

Tang. 

CO-TANG. 

Tang. 

Co-tang. 

Tang. 

CO-TANG. 

Tang. 

/ 

..  J 

490 

48° 

470 

460 

N N* 


426 


NATURAL  TANGENTS  AND  CO-TANGENTS. 


440  | 

440 

r 

44° 

' 

Tang. 

CO-TANG. 

' 

' 

Tang.  I 

Co-TANG. 

' 

' 1 

Tang.  I 

CO-TANG. 

t 

0 

.96569 

1-035  53 

60 

21 

•977  56 

1.022  95 

39  ! 

41 

.98901  1 

i. on  12 

19 

1 

.96625 

1.03493 

59 

22 

•978  13 

1.022  36 

38 

42 

.98958  | 

1. 01053 

18 

2 

.966  81 

1-034  33 

58 

23 

•978  7 

1. 021  76 

37 

43 

.99016 

1.009  94 

17 

3 

.967  38 

1.033  72 

57 

24 

.97927 

1. 021  17 

36 

44 

•990  73 

1.00935 

ID 

4 

.967  94 

1.033  12 

5b 

25 

•979  84 

1.020  57 

35 

45 

•991  3i 

1.008  76 

*5 

5 

.968  5 

1.042  52 

■ 55 

26 

.98041 

I.OI998 

34 

46 

.991  89 

1.008  18 

14 

6 

.q6q  07 

I on  Q2 

54 

27 

. 980  98 

I.OI939 

33 

47 

.992  47 

1.007  59 

*3 

7 

. 969  63 

1.03132 

53 

28 

•98i55 

I.O1879 

32 

48 

•993  04 

1.007  01 

12 

8 

.9702 

1.03072 

52 

29 

.982  13 

I. Ol8  2 

31 

49 

.99362 

1.006  42 

11 

9 

.97076 

I.030  12 

5i 

3° 

.982  7 

I.OI761 

30 

50 

•994  2 

1.005  83 

10 

10 

.971  33 

1.029  52 

50 

31 

.98327 

1.017  02 

29 

51 

•994  78 

1.005  25 

9 

11 

.971  89 

1.028  92 

49 

32 

.983  84 

I.O1642 

28 

52 

•995  36 

1.00467 

8 

12 

.97246 

1.028  32 

48 

33 

.98441 

I.OI583 

27 

53 

•995  94 

1.004  08 

l 

13 

.97302 

1.027  72 

47 

34 

.984  99 

1. 015  24 

26 

54 

.996  52 

1.0035 

0 

14 

•973  59 

1.027 13 

46 

35 

• 985  56 

I.OI465 

25 

55 

.997  1 

1.002  91 

5 

15 

•974  16 

1.02653 

45 

36 

.986  13 

I.OI406 

24 

56 

.99768 

1.002  33 

4 

16 

.97472 

1.02593 

44 

37 

.986  71 

1.013  47 

23 

57 

.998  26 

1. 001  75 

3 

17 

•975  29 

1.02533 

43 

38 

.987  28 

1.012  88 

22 

58 

. 998  84 

1. 001  16 

2 

18 

.97586 

1.02474 

42 

39 

.987  86 

1. 012  29 

21 

59 

.99942 

1. 000  58 

z 

19 

•97643 

1.024  14 

4i 

40 

.98843 

1. on  7 

20 

60 

1 

1 

0 

20 

•977 

1-02355 

40 

CO-TANG. 

Tang. 

t 

/ 

CO-TANG. 

Tang. 

T7 

/ 

CO-TANG. 

Tang. 

' 

45° 

1 450 

45° 

1 

Preceding  Table  contains  Natural  Tangents  and  Co-tangents  for  every 
minute  of  the  quadrant,  to  the  radius  of  1. 

If  Degrees  are  taken  at  head  of  columns,  Minutes,  Tangents,  and  Co-tan- 
gents must  be  taken  from  head  also ; and  if  they  are  taken  at  foot  of  col- 
umns, Minutes,  etc.,  must  be  taken  from  foot  also. 

Illustration.— -.1974  is  tangent  for  n°  io',  and  co  tangent  for  78°  50'. 

To  Compute  Tangents  and.  Co-tangents  for  Seconds. 
Ascertain  tangent  or  co-tangent  of  angle  for  degrees  and  minutes  from 
Table ; take  difference  between  it  and  tangent  or  co-tangent  next  below  it. 

Then  as  60  seconds  is  to  difference,  so  are  seconds  given  to  result  required,  , 
which  is  to  be  added  to  tangent  and  subtracted  from  co-tangent. 

Illustration.— What  is  the  tangent  and  co-tangent  of  540  40'  40"? 

Tangent  of  50°  40',  per  Table  = 1.41061 ) ooo8  difference. 

Tangent  of  540  41',  k =1.41148) 

Then  60  : .00087  ::  40  : 000 58,  which,  added  to  1.41061  = 1.4x119  tangent. 

Co-tangent  of  540  40',  per  Table  = .708  91 ) ooo  difference. 

Co-tangent  of  540  41  , “ =.70848)  4:5 

Then  6o°  : .00043  *.*.  40  : 29,  which,  subtracted  from  .70891  = .70862  co-tangent. 

To  Compute  Tangent,  or  Co-tangent  of*  any  ^Angle  in 
Degrees,  Minutes,  and  Seconds. 

Divide  Sine  by  Cosine  for  Tangent,  and  Cosine  by  Sine  for  Co-tangent. 
Example.— What  is  tangent  of  250  18'? 

Sine  = .427  36  5 cosine  ==  .904  08.  Then  q§  ~ ‘4727  tangent. 

To  Compute  Number  of  Degrees,  Minutes,  and  Seconds 
of  a given  Tangent  or  Co-tangent. 

When  Tangent  is  given.— Proceed  as  by  Rule,  page  402,  for  Sines,  substi- 
tuting Tangents  for  Sines. 

Example.— What  is  tangent  for  1.411 19 ? 

Next  less  tangent  is  1.41061,  arc  for  which  is  540  40'.  Next  greatest  tangent  is 
1. 41 1 48,  difference  between  which  and  next  less  is  .000  87. 

Difference  between  less  tabular  tangent  and  one  given  is  1.41061  — 1.411 19  = .00058. 
Then  .00087  : .00058  ::  60  : 40,  which,  added  to  54°  40' = 54°  40  4°  • 

When  Co-tangent,  is  given.— Proceed  as  by  Rule,  page  402,  for  Cosines, 
substituting  Co-tangents  for  Cosines. 


AEROSTATICS. 


427 


AEROSTATICS. 

Atmospheric  Air  consists,  by  volume,  of  Oxygen  21,  and  Nitrogen  79 
parts;  and  in  10000  parts  there  are  4.9  parts  of  Carbonic  acid  gas. 
By  weight,  it  consists  of  77  parts  of  Oxygen,  and  23  of  Nitrogen. 

One  cube  foot  of  Atmospheric  Air  at  surface  of  Earth,  when  barome- 
ter is  at  30  ins.,  and  at  a temperature  of  320,  weighs  565.0964  grains  = 
.080728  lbs.  avoirdupois,  being  773.19  times  lighter  than  water. 

Specific  gravity  compared  with  water , at  62.418  = .001  293  345. 

Mean  weight  of  a column  of  air  a foot  square,  and  of  an  altitude 
equal  to  height  of  atmosphere  (barometer  30  ins.),  is  2124.6875  lbs.  = 
14.7548  lbs.  per  sq.  inch  = support  of  34.0393  feet  of  water. 

Standard  pound  is  computed  with  a mercurial  barometer  at  30  ins. ; hence, 
as  a cube  inch  of  mercury  at  6o°  weighs  .4907769  lbs.,  pressure  of  atmos- 
phere at  6o°  = 14.723  307  lbs.  per  square  inch. 

12.3873  cube  feet  of  air  weigh  a pound,  and  its  weight  varies  about 
1 gr.  for  each  degree  of  heat. 

Extreme  height  of  barometer  in  latitude  30°  to  350  N.=  30.21  ins. 

Rate  of  expansion  of  Air,  and  all  other  Elastic  Fluids  for  all  temperatures, 
is  essentially  uniform.  From  320  to  212°  they  expand  from  1000  to  1376 
volumes  = .002 088  or  ^gth  part  of  their  bulk  for  every  degree  of  heat. 
From  2120  to  68o°  they  expand  from  1376  to  2322  = .002021  for  each  de- 
gree of  heat. 

Thus,  if  volume  of  air  at  1320  is  required.  1320  — 320  = 100,  and  1000 
-f-  100  X .002088=  1209  volumes. 

Height,  at  Equator  is  estimated  at  300  feet  greater  than  at  Poles,  its 
mean  height  at  45 0 latitude. 

In  like  latitudes,  air  loses  about  i°  for  every  foot  in  height  above  level 
of  sea. 

Below'  surface  of  Earth,  temperature  increases. 

Elasticity  of  air  is  inversely  as  space  it  occupies,  and  directly  as  its  density. 

When  altitude  of  air  is  taken  in  arithmetical  proportion,  its  Rarity  will  be 
in  geometric  proportion.  Thus,  at  7 miles  above  Surface  of  Earth,  air  is  4 
times  rarer  or  lighter  than  at  Earth’s  surface;  at  14  miles,  16  times;  at  21 
miles,  64  times,  and  so  on. 

Density  of  an  aeriform  fluid  mass  at  320  and  at  t°  will  be  to  each  other 
as  1 .002  088  ( t°  — 320)  is  to  1. 

For  Volume,  Pressure,  and  Density  of  Air,  see  Heat,  page  521. 

Altitude  of  Atmosphere  at  ordinary  density  is  = a column  of  mercury  30 
ins.  in  height,  divided  by  specific  gravity  of  air  compared  with  mercury. 

Hence  30  ins.  = 2.5  feet,  which,  divided  by  .000094987,  specific  gravity 
of  air  compared  with  mercury,  = 26  319  #/ee£  = 4.985  miles. 

Gay  Lussac,  Humboldt,  and  Boussingault  estimated  it  at  a minimum  of 
30  miles,  Sir  John  Herschell  83,  Bravais  66  to  100,  Dalton  102,  and  Liais  at 
180  or  204  miles. 

The  aqueous  vapor  ahvays  existing  in  air,  in  a greater  or  less  quantity, 
being  lighter  than  air,  diminishes  its  weight  in  mixing  with  it ; and  as,  other 
things  equal,  its  quantity  is  greater  the  higher  the  temperature  of  the  air,  its 
effect  is  to  be  considered  by  increasing  the  multiplier  of  t by  raising  it  to 
.002  22. 

Glaisher  and  Coxwell,  in  1862,  ascended  in  a balloon  to  a height  of  37000 
feet. 


AEROSTATICS. 


428 

At  temperature  of  320,  mean  velocity  of  sound  is  1089  feet  per  second.  It 
is  increased  or  diminished  about  one  foot  for  each  degree  of  temperature 
above  or  below  320. 

Velocity  of  sound  in  water  is  estimated  at  4750  feet  per  second. 

Velocity  of  Sound  at  Various  Temperatures. 


0 

Per  Second. 

0 

Per  Second. 

O 

Feet. 

Feet. 

68 

5 

1056 

32 

ON 

■-O 

0 

14 

1070 

50 

1102 

77 

86 

23 

1079 

59 

1 1 12 

Per  Second. 


Feet. 

1122 

1132 

1142 


95 

104 

i?3 


Per  Second. 


Feet. 

1152 

1161 

1171 


those  of  fluids. 

Sensation  of  hearing,  or  sound,  cannot  exist  in  an  absolute  vacuum.  The 
human  voice  can  be  heard  a distance  of  3300  feet. 

Echo. At  a less  distance  than  100  feet  there  is  not  a sufficient  interval 

between  the  delivery  of  a sound  and  its  reflection  to  render  one  perceptible. 

To  Compute  Velocity  of  Sound  through  .Air. 
io89  x \3y/1 4-  [.002  088  (*  — 32)1  = v in  feet  per  second , t representing  temperature 
of  air. 

Illustration.— Flash  of  a cannon  from  a vessel  was  observed  13  seconds  before 
report  was  heard;  temperature  of  air  6o°;  what  was  distance  to  vessel? 

io89  xTffx  + [.002 088  (6o° -^2)]  = 1089 X 13 X 1.029  = 14 567- 55  flet  = 2. 76  miles. 

Theoretical  velocity  with  which  air  will  flow  into  a vacuum,  if  wholly  un- 
obstructed, is  VfJTi  = 1347-4  feet  per  second.  In  operation,  however,  it  is 
1347.4  X .707  = 952.61  feet. 

To  Compute  Velocity  of  .Air  Flowing  into  a Vacuum. 
\/VgltX  c = v in  feet  per  second , c represent  ing  coefficient  of  efflux. 

Coefficients  for  openings  are  as  follows : 

Circular  aperture  in  a thin  plate. ^ 65  to  .7 

Cylindrical  adjutage 92  1 Conical  adjutage 93 

Velocity  of  Sound  in  Several  Solids. 

Velocity  in  Air=- 1. 

3 t « a 1 Pino is.?  I Glass  ....  n. q I Steel 


Lead 

Gold 


5-6 


I Zinc  . . . 

...  9.8  I 

| Pine 

. 12.5  I Glass  . . 

..  11.9I 

| Oak 

| Copper . . 

. 11. 2 1 Pine... 

Iron . 


4- 3 
i5- 1 


To  Compute  Elevations  "by  a Barometer. 
Approximately * 60000  (log.  B-log.  b)C=keight  in  feet ; 
heights  of  barometer  at  lower  and  upper  stations,  and  C correction  due  to  T -j- 
temperatures  of  lower  and  upper  stations. 

Values  of  C or  T-f-£. 


.996 

•998 

1 

1.002 
1.004 
1.007 
1.009 
1. on 
1-013 
1. 016 


1. 018 

1.02 

1.022 

1.024 

1.027 

1.029 

1. 031 

1-033 

1.036 

1.038 


114 

116 

118 


1.04 

1.042 

1.044 

1.047 

1.049 

1. 051 

1-053 

1.056 

1.058 

1.06 


1.062 
1.064 
1.067 
1.069 
1. 07 1 

1-073 

1.076 

1.078 

1.08 

1.082 


1.084 
1.087 
1.089 
1. 091 
1.093 
1.096 
1.098 
1. 1 
1. 102 
1. 104 


1. 106 
1. 108 
1. in 

i-i*3 

1-115 

1.117 
1. 12 
1. 122 
1. 124 
1.126 


* For  more  exact  formulas,  see  Tables  and  Formulas,  by  Capt.  T.  S.  Lee,  U.  S.  Top.  Eng.,  1853. 


AEROSTATICS, 


429 


Their  values  vary  approximately  .0011  per  degree. 

Upper  Station.  Lower  Station. 

Illustration. — Thermometer  70.4  77.6 

Barometer  23.66  30.05 

0 = 77.6-1-70.4=:  1.093,  log.  B = i. 4778,  log.  6=1.374. 

Then  60000  X (1.4778  — 1.374)  X 1.093  = 6807.2  feet. 

To  Compute  Elevations  Toy-  a Thermometer. 

520  B + B2  X C=  height  in  feet.  B representing  temperature  of  water  boiling  at 
elevated  station  deducted  from  2120. 

Correction  for  temperatures  of  air  at  lower  and  upper  stations,  or  T -f- t,  to  be  taken 
from  table,  page  428,  as  before. 

Illustration.— Temperature  of  water  boiling  at  upper  station  1920;  temperature 
of  air  500  and  320.  G = 1.02. 

Then  520  X 212  — 192 + 213  — 192  X 1.02  = 11010  feet. 

To  Compute  Capacity  of  a Balloon , etc.,  see  page  218. 

Barometer. 

Elevations  by  Barometer  Readings*  (Astronomer  Royal.) 

Mean  Temperature  of  Air  50°. 

For  correction  for  temperature,  see  note  at  foot. 


Height. 

Barom. 

Height. 

Barotn. 

Height. 

Barom. 

Height. 

Barom. 

Height. 

Barom. 

. Feet. 

las. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

0 

31 

600 

30*325 

1500 

29.34 

4000 

26.769 

7 000 

23.979 

50 

30-943 

650 

30. 269 

1600 

29.233 

4250 

26.524 

7500 

23.543 

loo 

30.886 

700 

30.214 

1750 

29. 072 

4500 

26.282 

8000 

23.115 

150 

30.83 

750 

30.159 

1800 

29.019 

4750 

26  042 

8 500 

22.695 

200 

30.773 

800 

30. 103 

2000 

28.807 

5000 

25.804 

9000 

22.282 

250 

30.717 

850 

30.048 

2250 

28.544 

5250 

25.569 

9 5oo 

21.877 

300 

30.661 

900 

29.993 

2500 

28.283 

5500 

25.335 

10060 

21.479 

350 

30.604 

1000 

29.883 

2750 

28.025 

5750 

25.104 

10  500 

21.089 

400 

30.548 

1100 

29.774 

3000 

27.769 

6000 

24.875 

11 000 

20. 706 

450 

30.492 

1200 

29.665 

3250 

37.515 

6250 

24. 648 

11  500 

20.329 

500 

30.436 

1300 

29-556 

3500 

27.264 

6500 

24.423 

12  000 

19.959 

550 

30.381 

1400 

29.448 

3750 

27.015 

6750 

24. 2 

12  500 

19.952 

Barometer. 

Correction  for  Capillary  Attraction  to  be  added  in  Inches. 


Diameter  of  tube I .6  I .55  I .5  I .45  i .4  ' I .55  I .3  I .25  I .2  I .1 

Correction,  unboiled I .004  005  .007  .01  .014  .02  .025  04  059  ,087 

Correction,  boiled | .002  | .003  | 004  | 065  | .007  | 01  | .014  | .02  | 029  | .044 


To  Compute  Heiglit. 

Rule.— Subtract  reading  at  lower  station  from  reading  at  upper  station,  difference 
is  height  in  feet. 

Table  assumes  mean  temperature  of  atmosphere  to  be  500  F.  or  io°  C.  For  other 
temperatures  following  correction  must  be  applied. 

Add  together  temperatures  at  upper  and  lower  station.  If  this  sum,  in  degrees 
in  F.,  is  greater  than  ioo°,  increase  height  by  part  for  every  degree  of  excess 
above  ioo°;  if  sum  is  less  than  ioo°,  diminish  height  by  part  for  every  degree 
of  defect  from  ioo°.  Or  if  sum  in  C°  is  greater  than  200,  increase  height  by  ^-A^. 
part  for  every  degree  of  excess  above  200;  if  sum  is  less  than  200,  diminish  height 
by  g part  for  every  degree  of  defect  from  20° 

Barometer  Indications. 

Increasing  storm.— If  mercury  falls  during  a high  wind  from  S.  W.,  S.  S.  W.,  W., 
or  S. 

Violent  but  short— If  fall  be  rapid. 

Less  violent  but  of  longer  continuance.— If  fall  be  slow. 

Snow. — If  mercury  falls  when  thermometer  is  low. 

Improved  weather. — When  a gradual  continuous  rise  of  mercury  occurs  with  a 
falling  thermometer. 


AEROSTATICS. 


430 

Heavy  gales  from  N. — Soon  after  first  rise  of  mercury  from  a very  low  point. 

Unsettled  weather.— With  a rapid  rise  of  mercury. 

Settled  weather.— With  a slow  rise  of  mercury. 

Very  fine  weather.— With  a continued  steadiness  of  mercury  with  dry  air. 

Stormy  weather  with  rain  (or  snow).— With  a rapid  and  considerable  fall  of  mer- 

^Threatening,  unsettled  weather.— With  an  alternate  rising  and  falling  of  mercury. 

Lightning  only.— When  mercury  is  low,  storm  being  beyond  horizon. 

Fine  weather.— With  a rosy  sky  at  sunset. 

Wind  and  rain.— When  sky  has  a sickly  greenish  hue. 

Rain.— When  clouds  are  of  a dark  Indian  red. 

Foul  weather  or  much  wind.— When  sky  is  red  in  morning. 

"Weather  Grlasses. 

Explanatory  Card.  Vice-Admiral  Fitzroy , F.  R.  S. 

Barometer  Rises  for  Northerly  wind  (including  from  N.  W.  by  N.  to  E.),  for  dry, 
or  less  wet  weather,  for  less  wind,  or  for  more  than  one  of  these  changes— 

Except  on  a few  occasions  when  rain,  hail,  or  snow  comes  from  N.  with  strong  wind. 

Barometer  Falls  for  Southerly  wind  (including  from  S.  E.  by  S.  to  W.),  for  wet 
weather,  for  stronger  wind,  or  for  more  than  one  of  these  changes — 

Except  on  a few  occasions  when  moderate  wind  with  rain  (or  snow)  comes  from  N. 

For  change  of  wind  toward  Northerly  directions,  a Thermometer  falls. 

For  change  of  wind  toward  Southerly  directions,  a Thermometer  rises . 

Moisture  or  dampness  in  air  (shown  by  a Hygrometer)  increases  before  rain,  fog, 
or  dew. 

Add  one  tenth  of  an  inch  to  observed  height  for  each  hundred  feet  Barometer  is 
above  half-tide  level. 

Average  height  of  Barometer,  in  England,  at  sea-level,  is  about  29.94  inches;  and 
average  temperature  of  air  is  nearly  50  degrees  (latitude  of  London). 

Thermometer  falls  about  one  degree  for  each  300  feet  of  elevation  from  ground, 
but  varies  with  wind. 

“ When  the  wind  shifts  against  the  sun, 

Trust  it  not,  for  back  it  will  run.” 

First  rise  after  very  low  I Long  foretold— long  last, 

Indicates  a stronger  blow.  | Short  notice — soon  past. 

Rarefaction  of  Air . 

In  consequence  of  rarefaction  of  air,  gas  loses  of  its  illuminating  power  1 cube 
inch  for  each  2.69  feet  of  elevation  above  the  sea.  (M-  Bremond.) 

Clouds. 

Classification.— 1.  Cirrus— Like  to  a feather,  commonly  termed  Mare's 
tails.  2.  Cirro-cumulus  — Small  round  clouds,  termed  mackerel  sky. 
3.  Cirro-stratus — Concave  or  undulated  stratus.  4.  Cumulus— Conical, 
round  clusters,  termed  wool-packs  and  cotton  balls.  5^ Cumulostratus  - 

Two  latter  mixed.  6.  Nimbus — A cumulus  spreading  out  in  arms,  and 
precipitating  rain  beneath  it.  7.  Stratus — A level  sheet. 

Note. — Cirrus  is  most  elevated. 

Height. — Clouds  have  been  seen  at  a greater  height  than  37000  feet. 

Velocity. — At  an  apparent  moderate  speed,  they  attain  a velocity  of  80 
miles  per  hour. 

lAglvtiiing. 

Classification.— i.  Striped  or  Zigzag— Developed  with  great  rapidity. 
2.  Sheet — Covering  a large  surface.  3.  Globular — When  the  electric 
fluid  appears  condensed,  and  it  is  developed  at  a comparatively  lower 
velocity.  4.  Phosphoric  — When  the  flash  appears  to  rest  upon  the 
edges  of  the  clouds. 


AEROSTATICS. — ATMOSPHERIC  AIR, 


431 


WEATHER  INDICATIONS. 


Weather. 

Clouds. 

Sky. 

Fine  and 

Soft  or  delicate-looking  and  in- 

Gray in  morning  and  light, 

Fair. 

definite  outlines. 

delicate  tints  and  low  dawn. 

Wind. 
Wind  only. 

Hard-edged,  oily  - looking,  and 
tawny  or  copper-colored,  and  the 
more  hard,  “greasy,”  and  ragged, 
the  more  wind. 

Light  scud  alone. 

High  dawn,  and  sunset  of  a 
bright  yellow. 

Rain. 

Small  and  inky. 

Sunset  of  a pale  yellow. 

Wind  and 
Rain. 

Light  scud  driving  across  heavy 
masses. 

Orange  or  copper  color. 

Rain  and 
Wind. 
Change  of 
Wind. 

Hard  defined  outlines. 

High  upper,  cross  lower  in  a di- 
rection different  to  their  course  or 
that  of  wind. 

GeneraL 

Gaudy  unusual  hues. 

Fair.— When  sea-birds  fly  early  and  far  out,  when  dew  is  deposited,  and  when'a 
leech,  confined  in  a bottle  of  water,  will  curl  up  at  the  bottom. 

Rain.— Clear  atmosphere  near  to  horizon  and  light  atmospheric  pressure,  or  a 
good  “hearing  day,”  as  it  is  termed. 

Storm.— When  sea-birds  remain  near  to  shore  or  fly  inland. 

Rain,  Snow,  or  Wind. — When  a leech,  confined  in  a bottle  of  water,  will  rise  ex- 
citedly to  the  surface. 


Thunder.— When  a leech,  confined  as  above,  will  be  much  excited  and  leave  the 
water. 


Value  of*  Indications  of*  Hair  Weather,  in  Days,  Com- 
pared to  one  of  Plain. 

From  an  extended  series  of  observations.  (Lowe.y 


Profuse  Dew. 

White  Stratus  in  a valley. . . 
Colored  Clouds  at  sunset . . , 

Solar  Halo 

Sun  red  and  rayless 

Sun  pale  and  sparkling 

White  Frost 

Lunar  Halo 

Lunar  burr,  or  rough-edged, 

Moon  dim 

Moon  rising  red 


4. 5 Mock  Sun  or  Moon. . , . . 

7.2  Stars  falling  abundant. . 

2.9  Stars  bright 

1.9  Stars  dim 

10.3  Stars  scintillated 

1 Aurora  borealis 

4.2  Toads  in  evening 

1 Landrails  noisy.  

2.8  Ducks  and  Geese  noisy. 

2 Fish  rising 

7 Smoke  rising  vertically. 


3-3 

3-2 

3-4 

1- 5 
6 

1.8 

2.4 

13 

2- 3 
i-5 
5 


For  weather-foretelling  plants,  see  page  185. 


ATMOSPHERIC  AIR. 

Very  pure  air  contains  Oxygen  20.96,  Nitrogen  79,  and  Carbonic  Acid  .04. 

Air  respired  by  a human  being  in  one  hour  is  about  15  cube  feet,  produc- 
ing 500  grains  of  carbonic  acid,  corresponding  to  137  grains  carbon,  and 
during  this  time  about  200  grains  of  water  will  be  exhaled  by  the  lungs. 

During  this  period  there  would  be  consumed  about  415  grains  of  oxygen. 

In  one  hour,  then,  there  would  be  vitiated  73  cube  feet  pure  air. 

A man,  weighing  150  lbs.,  requires  930  cube  feet  of  air  per  hour,  in  order 
that  the  air  he  breathes  may  not  contain  more  than  1 per  1000  of  carbonic 
acid  (at  which  proportion  its  impurity  becomes  sensible  to  the  nose);  he 
ought,  therefore,  to  have  800  cube  feet  of  well  ventilated  space. 


ATMOSPHERIC  AIR. — ANIMAL  POWER. 


432 

An  adult  human  being  consumes  in  food  from  145  to  165  grains  of  carbon 
per  hour,  and  gives  off  from  12  to  16  cube  feet  of  carbonic  acid  gas. 

An  assemblage  of  1000  persons  will  give  off  in  two  hours,  in  vapor,  8.5 
gallons  water,  and  nearly  as  much  carbon  as  there  is  in  56  lbs.  of  bitumi- 
nous coal. 

proportion  of  Oxygen,  and.  Carbonic  Acid  at  following 
Locations. 

Pure  Air  represented  by  Oxygen  20. 96. 


Street  in  Glasgow 20.895 

Regent  Street,  London 20.865 

Centre  Hyde  Park 21.005 


Metropolitan  Railway  (underground) . . 20.6 

Pit  of  a Theatre 20.74 

Gallery  of  a Theatre. 20.63 


Carbonic  Acid  .04  Per  cent. 


Open  field,  Manchester 0383 

Churchyard 0323 

Market,  Smithfield 0446 

Factory  mills 283 

School- rooms 097 

Pitt  of  theatre,  n P.  M 32 

Boxes  “ i,2  “ 218 

Gallery  “ 10  “ ..101 

* Roscoe. 


Top  of  Monument,  London 0398 

Hyde  Park ' 0334 

Metropolitan  Railway  (underground)..  .338 

Lake  of  Geneva 046 

Boys1  school 31* 

Girls’  “ 723t 

Horse  stable 7 

Convict  prison °45 

\ Pelteuhoffer. 


Consumption  of  Atmospheric  Air.  ( Coathupe .) 

One  wax  candle  (three  in  a lb.)  destroys,  during  its  combustion,  as  much 
oxygen  per  hour  as  respiration  of  one  adult. 

A lighted  taper,  when  confined  within  a given  volume  of  atmospheric  air, 
will  become  extinguished  as  soon  as  it  has  converted  3 per  cent,  of  grven 
volume  of  air  into  carbonic  acid. 

Carbonic  Acid  Exhaled  per  Minute  by  a Man.  [Or.  Smith.) 

During  sleep  4.99  per  cent.,  lying  down  5.91,  walking  at  rate  of  2 miles 
per  hour  18. 1,  at  3 miles  25.83,  hard  labor  44-97* 


animal  power. 

Work. 

Work  is  measured  by  product  of  the  resistance  and  distance  through 
which  its  point  of  application  is  moved.  In  performance  of  work  by 
means  of  mechanism,  work  done  upon  weight  is  equal  to  work  done  by 

^ Unit  of  Work  is  the  moment  or  effect  of  1 pound  through  a distance 
of  1 foot,  and  it  is  termed  a foot-pound. 

In  France  a kilogrammetre  is  the  expression,  or  the  pressure  ot  a 
kilogramme  through  a distance  of  1 meter  = 7.233  foot-pounds. 

Result  of  observation  upon  animal  power  furnishes  the  following  as  maximum 
daily  effect: 

1.  When  effect  produced  varied  from  .2  to  .33  of  that  which  could  be  produced 
without  velocity  during  a brief  interval. 

2.  When  the  velocity  varied  from  .16  to  .25  for  a man,  and  from  .08 _to  .060  for  a 
horse,  of  the  velocity  which  they  were  capable  for  a brief  intenal,  and  not  involv- 
ing any  effort. 

3.  When  duration  of  the  daily  work  varied  from  . 33  to  . 5 for - a brief  interval, 
during  which  the  work  could  be  constantly  sustained  without  Prejudice s tc » bea1^ 
of  man  or  animal;  the  time  not  extending  beyond  18  hours  per  day  however  lim- 
ited may  be  the  daily  task,  so  long  as  it  involved  a constant  attendance. 


ANIMAL  POWER. 


433 


Mien. 

Mean  effect  of  power  of  men  working  to  best  practicable  advantage,  is 
raising  of  70  lbs.  1 foot  high  in  a second,  for  10  hours  per  day  4200  foot- 

pounds per  minute. 

Windlass. — Two  men,  working  at  a windlass  at  right  angles  to  each  Other, 
can  raise  70  lbs.  more  easily  than  one  man  can  30  lbs. 

Labor.— A man  of  ordinary  strength  can  exert  a force  of  30  lbs.  for  10 
hours  in  a day,  with  a velocity  of  2.5  feet  in  a second  = 4500  lbs.  raised  one 
foot  in  a minute  = .2  of  work  of  a horse. 

A man  can  travel,  without  a load,  on  level  ground,  during  8.5  hours  a day, 
at  rate  of  3.7  miles  an  hour,  or  31.25  miles  a day.  He  can  carry  in  lbs. 
11  miles  in  a day.  Daily  allowance  of  water,  1 gallon  for  all  purposes;  and 
he  requires  from  220  to  240  cube  feet  of  fresh  air  per  hour. 

A porter  going  short  distances,  and  returning  unloaded,  can  carry  135  lbs. 
7 miles  a day,  or  he  can  transport,  in  a wheelbarrow,  150  lbs.  10  miles  in  a 
day. 

Cram. — The  maximum  power  of  a man  at  a crane,  as  determined  by  Mr. 
Field,  for  constant  operation,  is  15  lbs.,  exclusive  of  frictional  resistance, 
which,  at  a velocity  of  220  feet  per  minute  = 3300  foot-pounds,  and  when 
exerted  for  a period  of  2.5  minutes  was  17.329  foot-pounds  per  minute. 

Pile-driving. — G.  B.  Bruce  states  that,  in  average  work  at  a pile-driver,  a 
laborer,  for  10  hours,  exerts  a force  of  16  lbs.,  plus  resistance  of  gearing,  and 
at  a velocity  of  270  feet  per  minute;  making  one  blow  every  four  minutes. 

Rowing. — A man  rowing  a boat  1 mile  in  7 minutes,  performs  the  labor 
of  6 fully-worked  laborers  at  ordinary  occupations  of  10  hours  per  day. 

Drawing  or  Pushing. — A man  drawing  a boat  in  a canal  can  transport 
110000  U)s;  for  a distance  of  7 miles,  and  produce  156  times  the  effect  of  a 
man  weighing  154  lbs.,  and  walking  31.25  miles  in  a day ; and  he  can  push 
on  a horizontal  plane  20  lbs.  with  a velocity  of  2 feet  per  second  for  10  hours 
per  day. 

Tread-mill. — A man  either  inside  or  outside  of  a tread-mill  can  raise  30 
lbs.  at  a velocity  of  1.3  feet  per  second  for  10  hours,  = 1 404  000  foot-pounds. 

Pulley—  A man  can  raise  by  a single  pulley  36  lbs.,  with  a velocity  of  .8 
of  a foot  per  second,  for  10  hours. 

Walking.— A man  can  pass  over  12.5  times  the  space  horizontally  that  he 
can  vertically,  and,  according  to  J.  Robison,  by  walking  in  alternate  directions 
upon  a platform  supported  on  a fulcrum  in  its  centre,  he  can,  weighing  165 
lbs.,  produce  an  effect  of  3 984  000  foot-pounds,  for  10  hours  per  day. 

Pump , Crank , Bell , and  Rowing. — Mr.  Buchanan  ascertained  that,  in  work- 
ing a pump,  turning  a crank,  ringing  a bell,  and  rowing  a boat,  the  effective 
' power  of  a man  is  as  the  numbers  100,  167,  227,  and  248. 

Pumping.— A practised  laborer  can  raise,  during.  10  hours,  1000000  lbs. 
water  1 foot  in  height,  with  a properly  designed  and  constructed  pump. 

Crank.  A man  can  exert  on  the  handle  of  a screw-jack  of  11  inches  ra- 
dius for  a short  period  a force  of  25  lbs.,  and  continuously  15  lbs.,  a net 
power  of  20  lbs.  Mr.  J.  Field’s  tests  gave  11.5  lbs.  as  easily  attained,  17.3  as 
difficult,  and  27.6  with  great  difficulty. 

Mowing. — A man  can  mow  an  acre  of  grass  in  1 day. 

Reaping. — A man  can  reap  an  acre  of  wheat  in  2 days. 

\ Ploughing. — A man  and  horse  .8  of  an  acre  per  day. 

O o 


434 


ANIMAL  POWER. 


Day’s  "Worltk  {D.  K.  Clark.) 

Laborer.  — Carrying  bricks  or  tiles,  net  load  106  lbs.  = 6oo  lbs.  i mile. 

Carrying  coal  in  a mine,  liet  load  95  to  115  lbs.  = 342  lbs.  1 mile. 

Loading  coke  into  a wagon,  net  load  100  lbs.  = 270  lbs.  1 mile.  • 

Loading  a boat  with  coal,  net  load  190  lbs.  = 1230  lbs.  1 mile,  or  20  cube  yards  of 

eapiggino-  stublde  land  .055  of  an  acre  per  day,  or  2000  cube  feet  of  superficial  earth. 
Breaking  1.5  cube  yards  hard  stone  into  2 inch  cubes. 

Quarrying.— A man  can  quarry  from  5 to  8 tons  of  rock  per  day. 

A foot-soldier  travels  in  1 minute,  in  common  time,  90  steps  ==70  yards. 

He  occupies  in  ranks  a front  of  20  inches,  and  a depth  of  13,  without  a kuapsack; 
interval  between  the  ranks  is  13  inches.  , . „ 

Average  weight  of  men,  150  lbs.  each,  and  five  men  can  stand  m a space  of  1 
square  yard. 

Effective  Power  of  NEeix  for*  a,  Sliort  IPeriod. 


Manner  of  Application. 


Force. 


Bench-vice  or  Chisel 

Drawing-knife  or  Auger. . 

Hand-plane 

Hand  saw. 


Lbs. 

72 

100 

50 

36 


Manner  of  Application. 


Force. 


Screw-driver,  one-liand  . 

Small  screw  driver 

Thumb  and  fingers, 

Windlass  or  Pincers 


Lbs. 

84 

14 

H 

60 


The  muscles  of  the  human  jaw  exert  a force  of  534  lbs. 

Mr.  Smeaton  estimated  power  of  an  ordinary  laborer  at  ordinary  work  was  equiv- 
alent to  3762  foot-pounds  per  minute.  But,  according  to  a particular  case  made  by 
him  in  the  pumping  of  water  4 feet  high,  by  good  English  laborers,  their  power  was 
equivalent  to  3904  foot-pounds  per  minute;  and  this  he  assigned  as  twice  that  of 
ordinary  persons  promiscuously  operated  with. 

Mr  J Walker  deduced  from  experiments  that  the  power  of  an  ordinary  laborer,  in 
turning  a crank,  was  13  lbs.,  at  a velocity  of  320  feet  per  minute  for  8 hours  per  day. 

Amount  of  Labor  produced  by  a IVIan.  {Morin.) 


MANNER  OF  APPLICATION. 

- I 

Power,  i 

•! 

Velocity 

per 

Second. 

Weight  I 
raised.  | 
Feet  per  j 
Minute,  j 

I-P 

for 

Period 

given. 

Throwing  earth  with  a shovel,  a height  of  5 feet. . 
Wheeling  a loaded  barrow  up  an  inclined  plane, 

Lbs. 

. . 6, 

132 

Feet. 

i-33 

.625 

Lbs. 

480 

495° 

No. 

8.7 

90 

Raising  and  pitching  earth  in  a shpvel  13  feet 

6 

2.25 

810 

14-7 

Pushing  and  drawing  alternately  in  a vertical 

13 

2-5 

1950 

35-5 

Transporting  weight  upon  a barrow,  and  return- 

132 

1 

7 920 

144 

For  8 Hours  per  Day. 

62 

• j : ^ ~ 0i;,At  nl  \ ad  nnlondod 

i43 

26 

.5 

4290 

Ascending  a siigni  eiev  (iiiuii,  ...... . 

Walking,  and  pushing  or  drawing  in  a horizontal 

2 

3120 

45-2 

18 

2.5 

2790 

39 

140 

• 5 

4200 

61. 1 

26 

5 

7 800 

1 13 

For  7 Hours  per  Day. 

88 

2.5 

13  200 

160.5 

Walking  with  a load  upon  his  back 

For  6 Hours  per  Day. 

Transporting  a weight  upon  his  back,  and  return- 

140 

1-75 

14700 

160.5 

Transporting  a weight  upon  his  back  up  a slignt 

140 

.2 

1 680 

19 

elevation,  and  returning  unloaded.  ••••• 

44 

1 -5 

1 320 

14.4 

individual  case,  at  140  lbs;,  at  a ve- 


* Morin  gives  amount  of  labor,  of  a man  upon  tread-mill,  in  an  L— -----  feet  JIS 

looity  of  .5  feet  per  second  for  8 hours  per  day  = 70  lbs.  at  1 foot  per  second,  hence  70  . 3 


ANIMAL  POWER, 


To  Compute  Number  of  IVLeix  to  Perform  Work  vipon 
a Tread-mill  or  P^e-driver, 

Rule.— To  product  of  weight  to  be  raised  and  radius  of  crank,  add  fric- 
tion of  wheel,  and  divide  sum  by  product  of  power  and  radius  of  wheel. 

Fyaaiplf  —How  many  men  are  required  upon  a tread-mill,  20  feet  in  diameter, 
to  raise  a weight  of  9233.33  U*.  crank  9 inches  in  length,  weight  of  wheel  and  its 

load  estimated  at  5000  lbs.,  and  friction  at  .015. 

Weight  of  a man  assumed  at  25  lbs.  Radius  of  crank  .7 5 feet. 

Effect  of  a man  on  a treadmill,  page  433,  30  lbs.  at  a velocity  of  1.3  feet  per  second, 
— 1. 3 X 60  = 78  feet  per  minute. 

9233- 33  X .75  -j-  5000  X .015  = 7000  lbs.  resistance  of  load  and  wheel,  and  7000-4- 
78  -X  10  X 3°  

18.8  men. 


— — 7000  = load  and  weight  -4- product  of  power  increased  by  its 

20  X 3- 1416  ^ J ...  * 7 

velocity  over  load,  radius  of  wheel  and  power  z 


- 7000-f- 1. 241  X 10  X 3° 1 


Horse. 

Amount  of  Labor  produced  by  a Horse  under  different 
Circumstances.  {Morin.) 


For  10  hours  per  day. 


MANNER  OF  APPLICATION. 

Power. 

Velocity 

per 

Second. 

Weight 
drawn. 
Feet  per 
Minute. 

l P 
for 

Period 

given. 

Drawing  a 4- wheeled  carriage  at  a walk 

Lbs. 

Feet. 

Lbs.  I 

No. 

154 

’ T 

27  720 

504 

With  load  upon  his  back  at  a walk 

264 

3-75 

59  4oo 

1080 

Transporting  a loaded  wagon,  and  return  ng  un- 

184  800 

336o 

1540 

2 

Drawing  a loaded  wagon  at  a walk 

1540 

3-75 

346  500 

6300 

For  8 Hours  per  Day. 

Upon  a revolving  platform  at  a walk 

100 

3 

18000 

260.8 

For  4.5  Hours  per  Day. 

218.7 

Upon  a revolving  platform  at  a trot 

66 

6.75 

26730 

Drawing  an  unloaded  4-wheeled  carriage  at  a trot. 

97 

7-25 

43  195 

353-5 

Drawing  a loaded  4- wheeled  carriage  at  a trot 

770 

7-25 

334  950 

2741 

If  traction  power  of  a horse,  when  continuously  at  a walk,  is  equal  to  120  lbs., 
and  grade  of  road  1 in  30,  resistance  on  a level  being  one  thirtieth  of  load,  he  can 
draw  a load  of  120  x 30  -4-  2 = 1500  lbs. 


Street  Rails  or  Tramways. 
Cars,  26  lbs.  per  ton,  or  1 to  86  as  a mean. 

Performance  of  Horses  in  France. 


[Henry  Hughes .) 

[M.  Charie-Marsaines.) 


SEASON. 

Road. 

Weight 

per 

Horse. 

Spaed 

per 

Hour. 

Work  per 
Hour,  drawn 
One  Mile. 

Ratio  of 
Pavement  to 
Macadam. 

Winter 

) Pavement 
( Macadam 
| Pavement 
j Macadam 

Tons. 

1.306 

.851 

1-395 

Miles. 

2.05 

Ton-miles. 

2.677) 
1.62$  ) 
3.027) 
2.464) 

1.644  to  1 

Summer 

1. 91 
2.17 

r.229  to  1 

1.141 

2.  10 

Average  daily  work  of  a Flemish  horse  in  North  of  France,  where  country  is  flat 
and  loads  heavy,  is,  on  same  authority,  as  follows: 

Winter,  21.82  ton-miles  per  day. ; 

Summer,  27.82  k‘  “ 


i Mean  for  the  year,  25. 


given  in  example  = 53.8  lbs.,  from  which  a deduction  is  to  be  made  for  excess  of  amount  of  labor  that 
can  be  performed  in  8 hours  over  10.  Or,  as  10  : 8 : : 53-8  : 43-04  lbs.,  which  does  not  essentially  differ 
from  effect  of  30  lbs.  for  that  of  an  average  performance. 


ANIMAL  POWER. 


436 


Greatest  mechanical  effect  of  an  ordinary  horse  is  produced  in  operating  a 
gin  or  drawing  a load  on  a railroad,  when  travelling  at  rate  of  2.5  miles  per 
hour,  where  he  can  exert  a tractive  force  of  150  lbs.  for  8 hours  per  day. 


At  a speed  of  10  miles  per  hour,  a horse  will  perform  13  miles  per  day  for 
3 years.  In  ordinary  staging,  a horse  will  perform  15  miles  per  day. 

To  Compute  Tractive  Power  of  a Horse  Team , see  Traction , page  848. 
Assuming  maximum  load  that  a horse  can  draw  on  a gravel  road  as  a 
standard,  he  can  draw, 

On  best-broken  stone  road 2 to  3 times. 

On  a well-made  stone  pavement 3 to  5 “ 

On  a stone  trackway 7 to  8 “ 

On  plank  road 4 to  12  “ 

On  a railway 18  to  20  “ 

Note.— Track  of  an  iron  railway  compared  with  a plank-road  is  as  27  to  10. 

To  Compute  Power  of  Draught  of  a Horse  at  Different 


Hence,  r represents  force  which  horse  must  over- 
come to  move  his  own  weight. 

h 0 

Then,  by  similar  triangles,  A C or  l : B C or  h : : 0 : r.  Or,  -j-  = r. 

If  t represents  tractive  power  of  horse,  upon  a level,  of  100  lbs.,  t'  tractive 
power  upon  a plane  of  inclination,  and  r that  part  of  force  exerted  by  horse 

which  is  expended  upon  his  own  body,  then  t ' = £ — or  t ^ = t 'in  lbs. 

Illustration. — If  inclination  is  1 in  50. 

Assume  t = 100,  weight  of  horse  900  lbs.,  and  l = 50.01. 


Assuming  load  that  a horse  can  draw  on  a level  at  100,  he  can  draw  upon 
inclinations  as  follows : 


On  his  bade  a horse  can  carry  from  220  to  390  lbs.,  or  about  27.5  per  cent, 
of  his  weight. 

Labor . — The  work  of  a horse  as  assigned  by  Boulton  & Watt,  Tredgold, 
Rennie,  Beardmore,  and  others,  ranges  from  20  000  to  39  320  foot-pounds  per 
minute  for  8 hours,  a mean  of  27  750  lbs. 

A horse  can  travel,  at  a walk,  400  yards  in  4.5  minutes ; at  a trot,  in  2 
minutes ; and  at  a gallop,  in  1 minute.  He  occupies  in  ranks,  a front  of  40  j 
ins.,  and  a depth  of  10  feet;  in  a stall,  from  3.5  to  4.5  feet  fron^;  and  at  a 
picket,  3 feet  by  9 ; and  his  average  weight  = 1000  lbs. 

Carrying  a soldier  and  his  equipments  (225  lbs.)  he  can  travel  25  miles 
in  a day  of  8 hours. 

A draught-horse  can  draw  1600  lbs.  23  miles  a day,  weight  of  carriage  in- 
cluded. 


Horse  upon  Turnpike  Road. 


Elevations. 


B Let  ABC  represent  an  inclined  plane,  0 ^weight 
of  a horse  which,  being  resolved  into  two  com- 
ponent forces,  one  of  which,  n , is  perpendicular  to 


r plane  of  inclination,  and  other,  r,  is  parallel  to  it. 

^ U vnrvvncnnto  -pAVOD  wllioh  Vl ArCA  mncf  AVOl1 


1 in  100 91  1 in  75 88  1 in  50. 

1 “ 90 90  1 “ 70 87  1 “ 45, 

1 “ 80 89  1 “ 60 85  1 “ 40. 


82  1 in  35 74  1 in  20 55 

80  1 “ 30 70  1 “ 15 40 

77  i “ 25 64  1 “ 10 10 


AXIMAL  POWER.  437 

Ordinary  work  of  a horse  may  be  stated  at  22  500  lbs.,  raised  1 foot  in  a 
minute,  for  8 hours  per  day. 

In  a mill,  he  moves  at  rate  of  3 feet  in  a second.  Diameter  of  track  should  not 
be  less  than  25  feet. 

Rennie  ascertained  that  a horse  weighing  1232  lbs.  could  draw  a canal-boat 
at  a speed  of  2.5  miles  per  hour,  with  a power  of  108  lbs.,  20  miles  per  day. 
This  is  equivalent  to  a work  of  23  760  foot-lbs.  per  minute,  He  estimated 
that  the  average  work  of  horses,  strong  and  weak,  is  at  the  rate  of  22  000 
foot-lbs.  per  minute. 

From  results  of  trials  upon  strength  and  endurance  of  horses  at  Bedford,  Eng.,  it 
was  determined  that  average  work  of  a horses  20000  foot-lbs.  per  minute.  A good 
horse  can  draw  1 ton  at  rate  of  2.5  miles  per  hour,  from  10  to  12  hours  per  day. 

Expense  of  conveying  goods  at  3 miles  per  hour,  per  horse  teams  being  1,  expense 
at  4.33  miles  will  be  1.33,  and  so  on,  expense  being  doubled  when  speed  is  5.125  miles 
per  hour. 

Strength  of  a horse  is  equivalent  to  that  of  5 men,  and  his  daily  allowance  of 
water  should  be  4 gallons. 

Amount  of  Labor  a Horse  of  average  Sfrengtli  is  capa- 
ble of  performing,  at  different  Velocities,  011  Canal, 
Railroad,  and  Turnpike. 


Traction  estimated  at  83.3  lbs. 


Veloci- 

Dura- 

Useful Effect,  drawn  i Mile. 

Veloci- 

Dura- 

Useful Effect,  drawn  1 Mile. 

ty  per 

tion  of 

On  a 

On  a Rail- 

On a Turn- 

ty per 

tion  of 

On  a 

On  a Rail- 

On a lurn- 

Hour. 

Work. 

Canal. 

road. 

pike. 

Hour. 

Work. 

Canal. 

road. 

pike. 

Miles. 

Hours. 

Tons. 

Tons. 

Tons. 

Miles. 

Hours. 

Tons. 

Tons. 

Tons. 

2-5 

n-5 

520 

115 

14 

6 

2 

30 

48 

6 

3 

8 

243 

92 

12 

7 

i-5 

19  0 

4i 

5-1 

4 

4-5 

102 

72 

9 

8 

1125 

12.8 

36 

4-5 

5 

2-9 

52 

57 

7.2 

10 

•75 

6.6 

28.8 

3-6 

Actual  labor  performed  by  horses  is  greater,  but  they  are  injured  by  it. 


Tractive  Power  of  a horse  decreases  as  his  speed  is  increased,  and  within  limits 
of  low  speed,  or  up  to  4 miles  per  hour,  it  decreases  nearly  in  an  inverse  ratio. 


For  10  Hours  per  Day. 


Miles. 

Traction. 

Miles. 

Traction. 

Miles. 

Traction. 

Miles. 

Traction. 

Per  Hour. 

Lbs. 

Per  Hour. 

Lbs. 

Per  Hour. 

Lbs. 

Per  Hour. 

Lbs. 

75 

33° 

i-5 

165 

2.25 

IIO 

3 

82 

250 

1-75 

140 

2-5 

100 

3-5 

70 

1.25 

200 

2 

125 

2-75 

90 

4 

62 

For  Ordinary  or  Short  Periods.  ( Molesworth .) 

Miles  per  hour 2 3 3-5  4 4-5  5 

Power  in  lbs 166  125  104  83  62  41 


Miles  per  hour 2 3 3-5  4 4-5  5 

Power  in  lbs 166  125  104  83  62  41 


Mule.  (D.K.  Clark.) 

Load  on  back , 170  to  220  lbs.  day’s  work  = 6400  lbs.  1 mile ; 400  lbs.  at  2.9 
miles  per  hour  =.  5300  lbs.  1 mile,  and  330  lbs.  at  2 miles  per  hour  = 5000  lbs. 
1 mile. 

Upon  a revolving  platform,  at  a velocity  of  3 feet  per  second,  = 11  880  lbs.  raised 
one  foot  per  minute,  or  172.2  EP  for  8 hours  per  day 

Ass. 

Load  on  back , 176  lbs.  carried  19  miles  day’s  work  = 3300  lbs.  1 mile. 

In  Syria  an  ass  carries  450  to  550  lbs.  grain. 

Upon  a revolving  platform,  at  a velocity  of  2.75  feet  per  second,  = 5280  lbs.  raised 
one  foot  per  minute,  or  76.5  EP  for  8 hours  per  day. 

0 0* 


438 


ANIMAL  POWER. 


Ox. 

An  Ox,  walking  at  a velocity  of  2 feet  in  a second  (1.36  miles  per  hour), 
exerts  a power  of  154  lbs.,  = 18480  lbs.  raised  one  foot  per  minute,  or 
268.8  IP  for  8 hours  per  day. 

A pair  of  well-conditioned  bullocks  in  India  have  performed  work  = 8000  foot-lbs. 

perminUte'  Camel. 

Load  on  back.  550  lbs.  carried  30  miles  per  day  for  4 days,  4 days’  work 
16  500  lbs.  1 mile,  for  5 days  13000  lbs.  1 mile  = 44  IP  for  10  hours  per  day. 

Load  of  a Dromedary , 770  lbs. 

Llama. 

Load  on  bach , no  lbs.,  day’s  work  2000  to  3000  lbs.  1 mile  = .5  to  .75  IP 
for  10  hours  per  day. 

Birds  and  Insects. 

Area  of  their  wing  surface  is  in  an  inverse  ratio  to  their  weight. 

Assuming  weight  of  each  of  the  following  Birds  to  be  one  pound,  and  each  Insect 
one  ounce,  the  relative  area  of  their  wing  surface  proportionate  to  that  of  their  act- 
ual weight  would  be  as  follows  ( M . De  Lucy): 


Sq.  ft. 

Swallow 4.85 

Sparrow 2.7 

Turtle-dove..  2.x 3 


Sq.  ft. 

Pigeon 1-27 

Vulture 82 

Crane,  Australia,  .41 


Sq.  ft. 

Gnat 3.05 

Dragon-fly,  sm’ll,  1.83 
Ladybird 1.66 


Sq.  ft. 

Cockchafer ...  32 

Bee 33 

Meat-fly 35 


Crocodile  and  Bog. 

The  direct  power  of  their  jaws  is  estimated  at  120  lbs.  for  the  former  and 
44  for  the  latter,  which,  with  the  leverage,  will  give  respectively  6000  and 
1500  lbs. 

PERFORMANCES  OF  MEN,  HORSES,  ETC. 

Following  are  designed  to  furnish  an  authentic  summary  of  the  fastest  or 
most  successful  recorded  performances  in  each  of  the  feats,  etc.,  given. 

MAN.  Walking. 

T874  Wm.  Perkins , London,  Eng.,  .5  mile,  in  2 min.  56  sec.;  1,  in  6 min.  23  sec.; 
2 in  13  min.  30  see.;  1876,  8,  in  59  min.  5 sec.;  1877,  20,  in  2 hours  39  min.  57  sec. 

1 i83o  T.  Smith , London,  Eng.,  12  miles,  in  1 hour  31  min.  42.4  sec. 

1881’  C.  A.  Harriman , Chicago,  111.,  530  miles,  in  5 days  20  hours  47  min. 

1851’,  J.  Smith , London,  Eng.,  25  miles,  in  3 hours  42  min.  16  sec 
1878  W.  Howes , London,  Eng.,  50  miles,  in  7 hours  57  mm.  44  sec.,  1880,  75  miles, 
in  13  hours  7 min.  27  sec.,  and  100,  in  18  hours  8 min.  15  sec. 

x88o  John  Dobler , Buftalo,  N.  Y.,  150  miles  850  yards,  in  24  hours. 

1801’  Cant.  R.  Barclay , Eng.,  country  road,  90  miles,  in  20  hours  22  min.  4 sec.,  in- 
cluding rests;  1803,  .2/ mile;  in  56  sec.,  and  Charing  Cross  to  Newmarket,  64,  in  10 
hours ^including  rests;  1806, 100,  in  19  hours , including  1 hour  30 min.  in  rests;  1809, 
1000  in  1000  consecutive  hours , walking  a mile  only  at  commencement  of  each  hour. 
1877  D O'1  Leary,  London,  Eng.,  200  miles,  in  45  hours  21  min.  33  sec. 
i8i8|  Jos.  Eaton]  Stowmarket,  Eng.,  4032  quarter  miles,  in  4032  consecutne  quar- 

t0\siy™Wm.  Gale,  London,  Eng.,  1500  miles,  in  1000  consecutive < >«»”■?•  j-S  nllles 
each  hour;  and  4000  quarter  miles,  in  4000  consecutive  periods  of  10  minutes. 

1882  C/ias.  Rowell,  New  York,  N.  Y.,  89  miles  1640  yards,  in  12  hours. 

1882’,  Geo.  Hazael , New  York,  N.  Y.,  600  miles  220  yards,  in  6 days. 

Ptviani  ng. 

1710  Levi  Whitehead,  Branham  Moor,  Eng.,  4 miles,  in  19  min. 

1844’,  Geo.  Seward,  of  U.  S.,  Manchester,  Eng.,  100  yards,  in  9.25  sec. 
i860,  Geo.  Forbes , Providence,  R.  I..  150  yards,  in  15  sec. 

i8si  Chas.  Westhall , Manchester,  Eng.,  150  yards,  in  15  sec.,  and  200,  in  19.5  sec. 
1864,  Jas.  Nuttall , Manchester,  Eng.,  600  yards,  in  1 min.  13  sec. 
no,  r r \fvrr<t  New  York.  N.  Y.,  1000  yards,  in  2 min.  13  sec. 

1863’,  Wm.  Lang,  Newmarket,  Eng.,  1 mile,  in  4 min.  2 sec.,  descending  grouhd; 
Manchester,  2,  in  9 min.  11.5  sec.;  1865,  n miles  1660 yards,  in  1 hour  2 mm.  2.3 sec. 


ANIMAL  POWER. 


439 


i8«;2  Wm.  Howitt,  “American  Deer,”  London,  Eng.,  10  miles,  in  51  mm.  34  sec., 
walking  last  200  yards,  time,  if  run,  51  min.  20  sec.;  and  15,  in  1 hour  22  mm. 

1863  L.  Bennett.  “Deerfoot,” Hackney  Wick,  Eng.,  12  m.,  in  1 hour  2 mm.  2.5  sec. 
1870  Patrick  Byrnes , Halifax,  N.  S.,  20  miles,  in  1 hour  54  sec. 

1880  D Donovan , Providence,  R.  I.,  40  miles,  in  4 hours  48  mm.  22  sec. 

1879I  G.  Hazael , London,  Eng.,  50  miles,  in  6 hours  15  min.  57  sec. 
i7_’  a Courier , East  Indies,  102  miles,  in  24  hours. 

Jumping,  Leaping,  etc. 

i8a8  P Ml  Neely,  Petersburg,  Ky.,  10  jumps,  standing,  no  feet  4 ins. 

1854!  J.  Howard , Chester,  Eng.,  1 jump,  board  raised  4 ms.  in  front,  running  start, 
with  dumb-bells,  5 lbs.,  29  feet  7 ins.  , , , 

1868,  Geo.  M.  Kelley , Corinth,  Miss.,  running,  and  from  a spring  board,  leaped  over 

17  horses  standing  side  by  side.  .,  . . 

1874  J.  Lane  Dublin,  Ireland,  running  start,  1 jump,  without  aid,  23  feet  1.5  ms. 
1878  E.  W.  Johnson , Baltimore,  Md.,  standing  leap,  5 feel  3 ms 
1870’  G W.  Hamilton , Romeo,  Mich.,  dumb-bells,  22  lbs.,  standing  jump,  14  feet 
S S ins  ; and'  1880,  dumb-bells,  12  lbs.,  3 standing  jumps,  39  feet  1 inch. 

1880,  P.  Davin , Dublin,  Ireland,  running  leap,  6 feet  2.75  ms. 


Lifting. 

1825,  Thomas  Gardner , of  New  Brunswick,  N.  S.,  a barrel  of  pork,  320  lbs.,  under 
each  arm:  also  transported  across  a pier  an  anchor,  1200  lbs. 

1868.  Wm.  B.  Curtis , New  York,  N.  Y.,  3239  lbs.,  in  harness. 

1881,  D.  L.  Dowd , Springfield,  Mass.,  by  hands,  1317  lbs. 


Throwing  Weiglits. 

1870  D.  Dinnie , New  York,  N.  Y.,  light  stone,  18  lbs.,  heavy  stone,  24  lbs 

34  feet  6 tns.;  heavy  hammer,  24  lbs.,  83  feet  8 ins. ; 1872,  Aberdeen,  Scotland,  light 
hammer,  138  feet ; run,  16  lbs.,  162  feet.  . 

1877,  M.  Davin,  Dublin,  Ireland,  run,  56  lb.  weight,  30  feet  2 ms. 


Swimming. 

1835,  S.  Bruck,  15  miles,  in  a rough  sea,  in  7 hours  30  min. 

1846,  A Native,  off  Sandwich  Islands,  7 miles  at  sea,  with  a live  pig  under  one  arm. 

1878,  E.  T.  Jones , London,  Eng.,  100  yards,  in  1 min.  8.5  sec 

1870,  Pauline  Bohn,  Milwaukee,  Wis.,  650  feet,  still  water,  in  2 mm.  43  sec. 

1881,  Wm.  Beckwith,  London,  Eng.,  1000  yards,  in  15  min.  8.5  sec. 

1872,  J.  B.  Johnson , Hendon,  Eng.,  open  water,  1 mile,  in  28  mm.  24.6  sec.  ; Agri- 
cultural Hall,  London,  Eng.,  remained  under  water,  3 min.  35  sec. 

1875,  Capt.  M.  Webb,  Dover,  Eng.,  to  Calais,  France,  23  miles,  crossing  two  full 
and  two  half  tides  = 35  miles,  in  21  hours  45  min. 

1880,  J.  Strickland , Melbourne,  Australia,  plunged  r$feet  1 inch. 


Skating. 

1854,  Wm.  Clark,  Madison,  Wis.,  1 mile,  in  1 min.  56  sec. 

1868,  John  Conyers , Lake  Simcoe,  Can.,  8 miles,  in  18  win.  40.5  sec. 

1876,  E.  St.  Clair  Milliard,  Chicago,  111.,  50  miles,  in  4 hours  57  min.  3 sec. 

1877,  John  Ennis,  Chicago,  111.,  100  yards,  calm,  in  n.75  sec. ; 9 laps  to  a mile, 
100  miles,  in  n hours  37  min.  45  sec.  ; and  145  inside  of  19  hours. 

Note.— The  Sporting  Magazine , London,  vol.  ix.,  page  135,  reports  a man  in  1767  to  have  skated  a 
mile  upon  the  Serpentine,  Hyde  Park,  London,  in  57  seconds. 


HORSE.  Trotting. 

1878,  “Controller,”  San  Francisco,  Cal.,  io  miles,  harness,  in  27  win,  27.25  sec., 
and  20  miles,  wagon,  in  58  min.  57  sec. 

1875,  “Steel  Grey,”  Yorkshire,  Eng.,  10  miles,  saddle,  in  27  min.  56.5  sec. 

1867,  “John  Stewart,”  Boston,  Mass.,  half-mile  track,  20  miles,  harness,  in  58 
min.  5.75  sec.,  and  20.5  miles  in  59  min.  31  sec. 

1830,  “Top  Gallant,”  Philadelphia,  Penn.,  12  miles,  harness,  in  38  min. 

1829,  “Tom  Thumb,”  Sunbury  Common,  Eng.,  16.5  miles,  harness,  248  lbs.,  in  56 
min.  45  sec.;  and  100  miles,  in  10  hours  7 min.,  including  37  min.  in  rests. 

1869,  “Morning  Star,”  Doncaster,  Eng.,  18  miles,  harness  (sulky  100  lbs.),  in  57 
min.  27  sec. 

1835,  “ Black  Joke,”  Providence,  R.  L,  50  miles,  saddle,  175  lbs.,  in  3 hours  57  mm. 


ANIMAL  POWER. 


440 


1855,  “Spangle,”  Long  Island,  N.  Y.,  50  miles,  wagon  and  driver  400  lbs.,  in  3 
hours  c.q  min.  4 sec.  . • ■ ' ” 

1837,  “Mischief,”  Jersey  City,N.  J.,  to  Philadelphia,  Penn.,  84.25  miles,  harness, 
very  hot  day  and  sandy  road,  in  8 hours  30  min. 

1853,  “Conqueror,”  Long  Island,  N.  Y.,  100  miles,  harness,  in  8 hours  55  mm.  53 
sec.,  including  15  short  rests.  , , 

1873,  M.  Delaney's  mare,  St.  Paul’s,  Minn.,  200  miles,  race  track,  harness,  m 44 
hours  20  min.,  including  15  hours  49  min.  in  rests. 

1834,  “Master  Burke  ” and  “ Robin,”  Long  Island,  N.  Y.,  100  miles,  wagon,  in  10 
hours,  17  min.  22  sec.,  including  28  min.  34  sec.  in  rests. 

Stage-coacliing. 

1750  By  the  Duke  of  Queensberry,  Newmarket,  Eng.,  19  miles,  in  53  min.  24  sec. 

1830’  London  to  Birmingham , Eng.,  “Tally-ho,”  109  miles,  in  7 hours  50  mm., 
including  stop  for  breakfast  of  passengers. 

Leaping.* 

1821,  A horse  of  Mr.  Mane,  at  Loughborough,  Leicestershire,  Eng.,  173  lbs.,  over  a 
hedge  6 feet  in  height,  35  feet. 

1821  A horse  of  Lieut.  Green,  Third  Dragoon  Guards,  at  Incliinnan,  Eng.,  ridden 
by  a heavy  dragoon,  over  a.  wall  6 feet  in  height  and  1 foot  in  width  at  top. 

1839,  “Lottery,”  Liverpool,  Eng.,  over  a wall,  33  feet. 

1847,  “Chandler,”  Warwick,  Eng.,  over  water,  37  feet. 

Note.— The  maximum  stride  of  a horse  is  estimated  to  be  28  feet  9 ins.;  “Eclipse”  has  covered  25 
feet.  The  maximum  stride  of  an  elk  is  34  feet,  and  of  an  elephant  14  feet. 


Running. 

1701,  Mr.  Sinclair,  on  the  Swift  at  Carlisle,  a gelding,  1000  miles,  in  1000  consecu- 

tiVi7  3r UGeo.  Osbaldeston,  Newmarket,  156  lbs.,  100  miles,  by 16  horses,  in  4 hours  19 
mm  40  see.,  and  200.  by  28  horses,  in  8 hours  39  min.,  including  1 hour  2 mm.  56  see. 
in  rests:  1 horse,  “Tranby,”  16  miles,  in  33  mm.  15  sec. 

1 7 c2  Speddina's  mare,  100  miles,  in  12  hours  30  min.,  for  2 consecutive  days. 

1754!  A Galloway  mare  of  Daniel  Corker’s,  Newmarket,  300  miles,  by  one  rider, 

67  lbs.,  in  64  hours  20  min.  , 

1761,  John  Woodcock,  Newmarket,  100  miles  per  day,  by  14  horses,  one  each  day, 

for  20  consecutive  days.  . 

1814  An  Officer  of  14 th  Dragoons , Blackwater,  12  miles,  1 horse,  m 25  mtn.11  sec. 
1868  N.  H.  Mowry,  San  Francisco,  Cal.,  race  track,  160  lbs.,  300  miles  by  30 horses 
(Mexican),  in  14  hours  9 min.,  including  40  minutes  for  rests;  the  first  200,  in  8 
hours  2 min.  48  sec.,  and  the  fastest  mile  in  2 min.  8 sec.  , .. 

1869,  Nell  Coher , San  Pedro,  Texas,  61  miles,  in  2 hours  55  min.  15  sec.,  including 

1870  John  Faylor,  Carson  City,  Nevada,  50  miles,  by  18  horses,  in  1 hour  58  min. 
23  sec. : and  Omaha,  Neb.,  56  miles,  in  2 hours  26  mm.,  including  rests. 

1876,  John  Murphy , New  York,  N.  Y.,  155  miles,  20  horses,  in  6 hours  45  mm. 

7 sec.  . , 

1878,  Capt.  Salvi,  Bergamo  to  Naples,  Italy,  580  miles,  in  10  days. 

1880  “ Mr  Brown,”  Rancocas,  N.  J.,  aged , 160  lbs.,  10  miles,  in  26  min.  18  sec. 
1828!  “Chapeau  de  Paille”  (Arabian),  India,  1.5  miles,  115  lbs.,  in  2 min.  53  sec. 
183-!  Capt.  Horne  (Arabians),  Madras  to  Bungalore,  India,  200  miles,  in  less  than 
10  hours. 

DOGS.  Conrsing  and.  Chasing. 

A Greyhound  and  Hare  ran  12  miles  in  30  min. 

1794,  A Fox,  at  Brende.  Eng.,  ran  50  miles  in  6.5  hours. 

A Greyhound,  at  Bushy  Park,  Eng.,  leaped  over  a brook  30  feet  6 ins. 

BIRDS.  Flying. 

Vulture,  150  miles;  Wild  Goose  and  Swallow,  90  miles;  Crow,  25  miles  per  hour. 

1870,  Carrier  Pigeons.  Pesth  to  Cologne,  Germany,  600  miles,  in  8 hours. 

1875,  Carrier  Pigeon,  Dundee  Lake  to  Paterson,  N.  J.,  3 miles,  in  3 mm.  24  sec. 


* A Salmon  can  leap  a dam  14  feet  in  height.— Sporting  Magazine,  London,  vol.  xin,  page  79. 


HORSE-POWER. BELTS  AND  BELTING. 


441 


HORSE -POWER. 


Horse-poicer. — IP  is  the  principal  measure  of  rate  at  which  work  is  per- 
formed. One  horse-power  is  computed  to  be  equivalent  to  raising  of  33  000 
lbs.  one  foot  high  per  minute,  or  550  lbs.  per  second.  Or,  33000  foot-lbs.  of 
work,  and  it  is  designated  as  being  Nominal,  Indicated,  or  Actual. 

A BP  in  work  is  estimated  at  33000  lbs.,  raised  1 foot  in  a minute;  but  as  a horse 
can  exert  that  force  for  only  6 hours  per  day,  one  work  IP  is  equivalent  to  that  of 
4. 5 horses. 

Cheval-vapeur  of  France  is  computed  to  be  equivalent  to  75  kilogram- 
meters  of  work  per  second,  or  7.233  foot-lbs.,  or  75  x 7.233  = 542.5  foot-lbs., 
which  is  1.37  per  cent,  less  than  American  or  English  value. 


BELTS  AND  BELTING. 

Capacity  of  belts  to  transmit  power  is  determined  by  extent  of  their 
adhesion  to  surface  of  pulley,  and  it  is  very  limited  in  comparison  with 
tensile  strength  of  belt. 

Resistance  of  a belt  to  slipping  depends  essentially  upon  character 
of  surface  of  pulley,  its  degree  of  tension,  and  width,  and  as  adhesion 
is  in  proportion  to  pressure  on  surface  of  pulley,  long  belts,  by  having 
greater  weight,  give  greater  adhesion. 

Tensile  strength  of  Belting  per  square  inch  of  section  ranges  as  follows : 

Tanned  Leather , .186  inch  thick,  from  2846  to  5000  lbs.,  or  from  530  to 
930  lbs.  per  inch  of  width ; when  spliced  385  lbs.,  and  when  laced  210  lbs. 

Taking  .3  as  a factor  of  safety,  70  and  128  lbs.  represent  resistance  per 
sq.  inch  that  belts  in  operation  may  be  subjected  to,  and  they  have  been  run 
successfully  at  these  tensions. 

Raw  hide  has  a tensile  strength  of  1.5  times  that  of  tanned. 

By  Experiments  of  H.  It.  Towne  and  Mr.  Kirkaldy.  ( England.) 

Tensile  strength  of  Single  leather  belting  per  square  inch  of  section. 

Laced,  960  lbs.  Riveted,  1740  lbs.  Solid,  3080  lbs. 

Norris  Co.— Double,  2 ins.,  2942  lbs. ; 6 ins.,  5603  lbs. ; 12  ins.,  14861  lbs. 
Single,  3.5  ins.,  3007  lbs. ; 5 ins.,  4060  lbs. ; 10  ins.,  8846  lbs. 

Spill's  belting,  from  flax,  saturated  with  an  endurable  substance,  gave  ten- 
sile strength  per  inch  of  width  as  follows : 

No.  1,  5 ins.  wide,  1254  Ms.  No.  2,  5 ins.  wide,  1489  lbs.  No.  3,  10  ins. 
wide,  1663  lbs. 

At  a velocity  of  1000  feet  per  minute,  a width  of  leather  belt  Of  r inch  will  trans- 
mit power  of  1 horse,  and  at  a velocity  of  1800  feet,  .56  of  an  inch  will  transmit  a 
like  power,  pulley  being  fully  three  feet  in  diameter,  equal  to  a stress  of  lbs.  per 
inch  of  width  of  belt  of  ordinary  thickness. 


To  Compute  Width,  of  a X^eatlier  Belt. 

Assuming  a well-defined  case  (where  limit  of  adhesion  was  ascertained), 
a belt  of  ordinary  construction  (laced),  and  9 inches  in  width,  transmitted 
the  power  of  15  horses  over  a pulley  4 feet  in  diameter,  at  a velocity  of  1800 
feet  per  minute,  with  an  arc  ot  adhesion  of  2100,  or  of  .6  or  7.54  feet  of  cir- 
cumference, and  with  an  area  of  95  square  feet  of  belt  per  H?. 

U 4400  to  5000  IP 

nence’  gfy =zw;  W representing  width  of  belt  in  inches , d di- 


ameter of  pulley  in  feet,  and  v velocity  of  belt  in  feet  per  minute. 

Note. — Thickness  of  belt  should  be  added  to  diameter  of  pulley.  Applying  thes< 
elements  to  the  formulas  of  13  different  authors,  the  result  varies  from  7.8s  to  i-i.t 
ins.,  mean  of  which  is  10.675.  For  double  belting  width  = .6  w. 


BELTS  AND  BELTING. 


442 


Illustration.— If  IP  25,  diameter  of  pulley  4 feet,  and  velocity  2250  feet;  what 
should  be  width  of  belt? 

4500  X 25  _ I2  - ins.  for  ordinary  thickness  of . 1875  in. 

4 X 2250 

To  Compute  Elements  of*  Belting. 

J,w_  IP  33«»_t>.  33  000  H* „r  W_  A 

1000  ’ v W ’ V ' t ' It 

P representing  power  or  stress  transmitted,  W weight  or  stress  on  belt, 
t thickness  of  belt , S stress  on  belt  per  inch  of  width , A and  a areas  of  coil 
and  eye , and  l length  in  feet. 

Note.— 70  square  feet  of  good  belting  are  capable  of  transmitting  an  indicated  IP. 


India  Rubber  Belting.  ( Vulcanized .) 

Results  of  Experiments  upon  Adhesion  of  India  Rubber  and  Leather  Belting.— 
(J.  H.  Cheever). 


Rubber. 


Leather. 


Leather  belt  slipped  on  iron  pulley  at  48 
“ “•  leather  “ 64 

“ “ rubber  ‘‘  128 


Rubber  belt  slipped  on  iron  pulley  at  90 
“ “ leather  “ 128 

“ “ rubber  “ 183 

Hence  it  appears  that  a Rubber  Belt  for  equal  resistances  with  a Leather  Belt 
may  be  reduced  respectively  46,  50,  and  30  per  cent. 

Iron  Wire.— A wire  rope  .375  inch  in  diameter,  over  a pulley  4 feet  in 
diameter,  and  running  at  a velocity  of  1250  feet  per  minute,  will  transmit 

4-5  . .. 

Diameter  of  pulley  should  not  be  less  than  140  times  diameter  of  rope,  in  order 
to  avoid  undue  bending  of  wires. 

A sheet-iron  belt  7 inches  in  width  proved  more  effective  than  one  of  leather  of 
like  width. 

General  Notes. 

Leather  Belts— Are  best  when  oak  tanned,  should  be  frequently  oiled,*  and  when 
run  with  hair  side  over  pulley  will  give  greatest  adhesion. 

Ordinary  thickness  .1875  inch,  and  weight  60  lbs.  per  cube  foot 


Relative  effect  of  different  pulleys  and  belts: 

Pulleys.—  Leather  surface 1.  I Turned  iron. 64 

Rough  iron 41  I Turned  wood 7 

Tensile  strength  of  calf  and  sheep  skins  is  about  one  half  that  of  beeve  and  horse. 
Morin  assigns  50  lbs.  as  a proper  stress  per  inch  of  width  of  good  belting. 
Presence  of  small  holes  in  a belt  will  prevent  its  slipping  or  squealing. 

Rubber  Belts.— Best  vulcanized  rubber  is  stronger  than  leather,  and  its  resistance 


is  from  50  to  85  per  cent,  greater. 

To  increase  adhesion,  coat  driving  surface  with  boiled  oil  or  cold  tallow,  and  then 
apply  powdered  chalk.  ; 

When  new,  cut  them  .1875  inch  short  for  each  foot  in  length  required,  to  admit 
of  the  stretch  that  occurs  in  their  early  operation. 

They  should  be  kept  free  from  contact  with  an  animal  oil. 

Three  ply,  .1875  inch  thick,  has  a tensile  resistance  of  600  lbs.  per  inch  of  width,  j 

Relative  slipping  of  a vulcanized  belt,  over  smooth  or  turned  leather  or  rubber- 
faced iron  pulleys  is  as  .5,-7,  and  1. 

Rubber,  Gutta  percha,  and  Canvas  belts  will  stretch  continuously. 

Memoranda. 

Belts  should  be  set  as  near  horizontal  as  practicable,  in  order  that  the  sag  may 
increase  adhesion  on  pulley,  and  hence  power  should  be  communicated  through 
under  side. 

The  “creeping”  or  lost  speed  by  belts  is  about  2 per  cent.,  hence,  to  maintain  a 
uniform  or  required  speed,  driver  must  be  increased  in  diameter  pro  rata  with  slip. 


* See  Cements , etc.,  page  871,  for  compositions,  etc. 


BELTS  AND  BELTING. — BLASTING. 


443 


A belt,  ii  ins.  in  width,  over  a driver  4 feet  in  diameter,  running  from  1200  to  2250 
feet  per  minute,  will  transmit  the  power  from  two  steam  cylinders,  6 ins.  in  diam- 
eter and  11  ins.  stroke,  averaging  125  revolutions  per  minute,  with  a pressure  of 
60  lbs.  per  sq.  inch. 

A double  belt,  75  ins.  in  width  and  153.5  feet  in  length,  transmitted  650  IIP. 

Pulleys  should  have  a slight  convexity  of  surface.  Authorities  differ,  from  .5  inch 
per  foot  of  breadth  to  .1  of  breadth.  Belts  run  at  a high  speed  are  less  liable  to  slip 
than  at  low  speed. 

The  best  speeds  for  economy  are  from  1200  to  1500  feet  per  minute,  and  the  best 
for  result  not  to  exceed  1800. 

Belts.— Leather,  hair-side 1 I Leather,  flesh- side. . . .74  | Rubber 51 

Gutta  percha .44  | Canvas 35 

Coefficient  of  Friction  of  a Belt  in  operation  is  assumed  to  be  .423. 

Smooth  surface  belts  are  most  endurable  and  soft  most  adherent. 

Round  belts  .25  and  .5  inch  in  diameter  are  fully  equal  in  operation  to  flat  of  1 
and  3 ins.,  and  grooves  in  their  pulleys  should  be  angular  or  V shaped. 

The  neutral  point  of  a rope  belt  is  at  .33  of  diameter  from  inside  surface. 

Friction  of  driving  and  pulley  bearings  is  about  .025. 

A fan-blower  No.  6*,  driven  by  a belt  3.875  ins.  in  width  and  .18  in  thickness,  at 
a velocity  of  2820  revolutions  per  minute,  requires  power  of  9.7  horses. 

Area  of  belts  per  IP  varies  essentially,  ranging  from  25  to  100  square  feet;  the 
mean  is  75. 


BLASTING. 


y 


In  Blasting,  rock  requires  from  .25  to  1.5  lbs.  gunpowder  per  cube 
yard,  according  to  its  degree  of  hardness  and  position.  In  small  blasts 
2 cube  yards  have  been  rent  and  loosened,  and  in  very  large  blasts  2 to 
4 cube  yards  have  been  rent  and  loosened,  by  1 lb.  of  powder. 

Tunnels  and  shafts  require  1.5  to  2 lbs.  per  cube  yard  of  rock. 

Gunpowder  has  an  explosive  force  varying  from  40000  to  90000 
lbs.  per  sq.  inch.  That  used  for  blasting  is  much  inferior  to  that  used  for 
projectiles,  the  proportion  being  fully  one  third  less. 

NTitro-glycerin-e  is  an  unctuous  liquid,  which  explodes  by  concussion, 
an  extreme  pressure  (2000  lbs.  per  sq.  inch),  or  a temperature  exceeding  6oo° 
if  quickly  applied  to  it ; it  will  inflame,  however,  and  burn  gradually. 

At  a temperature  below  40°  it  solidifies  in  crystals. 

Its  explosion  is  so  instantaneous  that  in  rock-blasting  tamping  is  not  nec- 
essary ; its  explosive  power  by  weight  is  from  4 to  5 times  that  of  gun- 
powder. 

Dynamite  is  nitro-glycerine  75  parts,  absorbed  in  25  parts  of  a sili- 
ceous earth  termed  kieselguhr;  it  also  explodes  so  instantaneously  as  to 
render  tamping  in  blasting  quite  unnecessary. 

It  is  insoluble  in  water,  and  may  be  used  in  wet  holes ; it  congeals  at  40°, 
is  rendered  ineffective  at  2120,  and  has  an  explosive  force  by  weight  of  3 
times  that  of  gunpowder,  and  by  bulk  4.25  times. 

Gun-cotton  is  insoluble  in  water,  and  has  an  explosive  force  by 
weight  of  from  2.75  to  3 times  that  of  gunpowder,  and  by  bulk  2.5  times. 
It  may  be  detonated  in  a wet  state  with  a small  quantity  of  dry  material. 

Tonite  is  nitrated  gun-cotton,  and  is  known  also  as  cotton  powder.  It 
is  produced  in  a granulated  form. 

T-jitLo-fracteur  is  a nitr o-glycerine  compound  in  which  a portion  of 
the  base  or  absorbent  material  is  made  explosive  by  the  admixture  therein 
of  nitrate  of  baryta  and  charcoal. 

* For  a table  of  Belts  for  Fan-blowers,  etc.,  see  J.  H.  Cooper,  in  “ Jour.  Franklin  Inst.,”  vol.  66,  p.  409. 


BLASTING. 


444 


Cellulose  Dynamite  is  when  gun-cotton  is  used  as  the  absorbent 
for  nitre-glycerine;  it  will  explode  frozen  dynamite,  and  is  more  sensitive  to 
percussion  than  it. 

To  Compute  Charge  of*  Gunpowder  fbr  Hoche  Blasting. 

Rule. — Divide  cube  of  line  of  least  resistance  by  25,  as  for  limestone,  to 
32  for  granite,  and  remainder  will  give  charge  of  powder  in  lbs. 

Or,  L3  -4-  32  = lbs. 

Example.— When  line  of  least  resistance  is  6 feet,  what  is  charge  required? 

63  -4-  32  = 6. 75  lbs. 

Line  of  least  resistance  should  not  exceed  .5  depth  of  hole. 

Tamping. — Dried  clay  is  the  most  effective  of  all  materials  for  tamping;  Broken 
Brick  the  next,  and  Loose  Sand  the  least. 

Relative  Costs  of  a Tunnel  and  Shaft  in  England.  (Sir  John  Burgoyne.) 


Iron  and  steel 8.98 

Smiths  and  coal 6 

Fuses 7- 


Powder 29.04 

Labor 


Weight  of  Explosive  Materials  in  Holes  of  Different  Diameters . 


Diam. 

Powder 
or  Gun- 
cotton. 

Dynamite. 

Diam. 

Powder 
or  Gun- 
cotton. 

Dynamite. 

Diam. 

Powder 
or  Gun- 
cotton. 

Dynamite. 

Ins. 

1 

1.25 

x-5 

Oz. 

.419 

•654 

.942 

Oz. 

.67 

1.046 

I-507 

Ins- 

i-75 

2 

2.25 

Oz. 

1.283 

I-675 

2.12 

Oz. 

2.053 

2.68 

3-392 

Ins. 

2-5 

2-75 

3 

Oz. 
2.618 
3. 166 
3-769 

Oz. 

4.189 

5.066 

6.03 

Boring  Holes  in  Granite. 

Depth 
of 

Hole. 


Diam. 

of 

Jumper. 

Depth 

of 

Hole. 

Men. 

Depth  bored 
per  Day. 

Ham- 

mer. 

Diam. 

of 

Jumper. 

Ins. 

Ins. 

No. 

Feet. 

Lbs. 

Ins. 

1 

1 to  2 

1 

8 

6 

2.25 

*•75 

2. 5 to  6 

3 

12 

*4 

2-5 

2 

4 t0  7 

3 

8 

14 

3 

Ins. 

5 to  10 
9 tO  12 
9 to  15 


Men. 


Depth  bored 
per  Day. 


Feet. 

6 

5 

4 


Lbs. 

16 

16 

18 


Drill.  - 


-Width  of  bit  compared  to  stock  .625. 

ORarges  of*  Powder. 

Usual  practice  of  charging  to  one  third  depth  of  hole  is  erroneous,  inasmuch  as 
volume  of  charge  increases  as  square  of  diameter  of  hole.  Hence  holes  ofl*5*n 
...  .1 r.f  nA„ni  dontho  uyuiIh  rami  ire  chareres  in  proportion  of  2. 25  and  4. 


Line  of 
least  re- 
sistance. 

Powder. 

Line  of  | 
least  re- 
sistance. 

Powder. 

Line  of 
least  re- 
sistance. 

Powder. 

Line  of 
least  re- 
sistance. 

Powder. 

Feet. 

1 

2 

Oz. 

•75 

4 

Feet. 

3 

4 

Lbs.  Oz. 
13-5 
2 

Feet. 

5 

6 

Lbs.  Oz. 
3 14-5 

6 12 

Feet. 

7 

8 

Lbs.  Oz. 
xo  11. 5 
16 

Effects. 

Gunpowder.  — From  its  gradual  combustion,  rends  and  projects  rather  than 

ShA  Mta  5. 5 ins.  in  diameter  and  „ feet  7 ins.  in  depth,  filled  to  8 feet  ro  ins.  with 
75  lbs.  powder,  has  removed  and  rent  1200  cube  yards,  equal  to  2400  to  s. 
labor  expended  was  that  of  3 men  for  14  days. 

Temperature  of  gases  of  explosion  40000. 

Gun-cotton.—  From  the  rapidity  of  its  combustion,  shatters. 

Dynamite.  — From  the  greater  rapidity  of  its  combustion  over  gun-cotton,  is  more 
shattering  in  its  explosion. 


BLASTING. BLOWING  ENGINES. 


445 


C 


Drilling. 

Churn- drilling.— A.  churn-driller  will  drill,  in  ordinary  hard  rock,  from  8 to  12 
feet,  2 inch  holes  of  2.5  feet  depth,  per  day,  and  at  a cost  of  from  12  to  18  cents  per 
foot,  on  a basis  of  ordinary  labor  at  $1  per  day.  Drillers  receiving  $2.50. 

One  man  can  bore,  with  a bit  1 inch  in  diameter,  from  50  to  100  inches  per  day 
of  10  hours  in  granite,  or  300  to  400  inches  per  day  in  limestone. 

Tamping.— Two  strikers  and  a holder  can  bore,  with  a bit  2 inches  in  diameter, 
10  feet  in  a day  in  rock  of  medium  hardness. 

Composition  for  waterproof  charger  or  fuse  consists  by  weight  of  Pitch,  8 parts; 
Beeswax  and  Tallow  each  1 part. 

Alining.  (Lefroy’s  Handbook.) 

In  demolition  of  walls  line  of  least  resistance  L = half  thickness,  and  C is  a co- 
efficient depending  on  structure. 

Charge  in  lbs.  = C X L3. 

In  a wall  without  counterforts,  where  interval  between  the  charge  is  2 L,  C — .15. 

In  a wall  with  counterforts  the  charge  to  be  placed  in  centre  of  each  counterfort 
at  junction  with  wall,  C = .2. 

Where  the  charge  is  placed  under  a foundation,  having  equal  support  on  both 
sides,  C = .4. 

A leather  bag,  containing  50  to  60  lbs.  powder,  hung  or  supported  against  a gate 
or  like  barrier,  will  demolish  it. 

For  ordinary  mines  in  average  rock  charge  in  ounces  = L3-r- 160. 


BLOWING  ENGINES. 

For  Smelting. 

Volume  of  oxygen  in  air  is  different  at  different  temperatures.  Thus, 
dry  air  at  85°  contains  10  per  cent,  less  oxygen  than  when  it  is  at  tem- 
perature of  320;  and  when  it  is  saturated  with  vapor,  it  contains  12 
per  cent.  less.  If  an  average  supply  of  1500  cube  feet  per  minute  is 
required  in  winter,  1650  feet  will  be  required  in  summer. 

Smelting  of  Iron  Ore. 

Colce  or  Anthracite  Coal. — 18  to  20  tons  of  air  are  required  for  each  ton 
of  Pig  Iron,  and  with  Charcoal  17  to  18  tons  are  required. 

(1  ton  of  air  at  340  = 29  751,  and  at  6o°  = 31 366  cube  feet.) 

Pressure . — Pressure  ordinarily  required  for  smelting  purposes  is  equal  to 
a column  of  mercury  from  3 to  10  inches,  or  a pressure  of  1.5  to  5 lbs.  per 
square  inch. 

Reservoir. — Capacity  of  it,  if  dry,  should  be  15  to  20  times  that  of  cylin- 
der if  single  acting,  and  10  times  if  double  acting. 

Pipes. — Their  area,  leading  to  reservoir,  should  be  .2  that  of  blast  cylinder, 
and  velocity  of  the  air  should  not  exceed  35  feet  per  second. 

A smith’s  forge  requires  150  cube  feet  of  air  per  minute.  Pressure  of 
blast  .25  to  2 lbs.  per  square  inch.  A ton  of  iron  melted  per  hour  in  a cu- 
pola requires  3500  cube  feet  of  air  per  minute.  A finery  forge  requires 
100000  cube  feet  of  air  for  each  ton  of  iron  refined.  A blast  furnace  re- 
quires 20  cube  feet  per  minute  for  each  cube  yard  capacity  of  furnace. 

A Ton  of  Pig  Iron  requires  for  its  reduction  from  the  ore  310000  cube 
feet  of  air,  or  5.3  cube  feet  of  air  for  each  pound  of  carbon  consumed 
Pressure,  .7  lb.  per  square  inch. 

Pp 


446  blowing  engines. 

To  Compute  Power  Required,  to  Drive  a Blowing 
Engine. 


.000  050  9 


»=./Z_ 

V .93  x . 


.93  X .7854  x V 


'r’(£+fL.b*”o“=B>' 

v representing  velocity  of  air  in  feet  per  sec - 


ond,  d and  d'  diameters  of  pipe  and  of  nozzle  in  feet,  —\f- 


35 


93  X .7854  X 500 


= -309- 

Illustration  —What  should  he  power  of  a steam-engine  to  drive  35  cube  feet  of 
air  at  a velocity  of  500  feet  per  second,  through  a pipe  1 foot  in  diameter  and  300 
feet  in  length?  . 77  . , , 

c = ratio  -between  power  employed  and  effect  produced  by  it  = in  a well-constructed 
engine  . 5,  and  C = . 93.  d = . 2974,  assumed  at . 3. 

.0000509  , 

^5  ‘ 


d C = .93.  a = .2974,  afcbuuiuu  at,  .j. 

^ 253  ^^  + 60-7-33000  = 22631.625  X 60  — 33000  = 41.15  EP. 


To  Compute  Required  Dower  of  a Blowing  Engine. 


F + fX  a v__  jp  p representing  pressure  of  blast  in  lbs.  per  sq.  inch; 

a arm  of  cylinder  in  sq.  ins. ; v velocity  of  piston  in  feet  per  minute;  f fac- 
tion of  piston  and  from  curvatures , etc.,  estimated  at  1.25  per  sq.  inch  oj 
piston. 

Note.— If  cylinder  is  single  acting,  divide  result  by  2. 


Illustration.— Assume  area  of  blast  cylinder  5600  sq.  ins.,  pressure  of  blast  2.25 
lbs.  per  sq.  inch,  and  velocity  of  piston  96  feet  per  second. 

2.25  + 1.25  X 5600.X  96  _ 1 881 600  _ ^ Worses,  the  exact  power  developed  in 
33000  33000 

this  case. 


To  Compute  Dimensions  of  a Driving  Engine. 

Rule  i. — Divide  power  in  lbs.  by  product  of  mean  effective  pressure  upon  - 
piston  of  steam  cylinder  in  lbs.  per  sq.  inch,  and  velocity  of  piston  in  feet 
per  minute,  and  quotient  will  give  area  of  cylinder  in  sq.  ins. 

2. — Divide  velocity  of  piston  by  twice  number  of  revolutions,  and  quotient 
will  give  stroke  of  piston  in  feet. 

Volume  of  air  at  atmospheric  density  delivered  into  reservoir,  in  consequence  of 
escape  through  valves,  and  partial  vacuum  necessary  to  produce  a current,  will  be 
about . 2 less  than  capacity  of  cylinder. 

Example. — Assume  elements  of  preceding  case,  with  a pressure  of  50  lbs.  steam, 
cut  off  at  .375,  and  with  12  revolutions  of  engine  per  minute,  what  should  be  area 
of  cylinder  of  a non-condensing  engine  ? 

Mean  effective  pressure  of  steam  with  5 per  cent,  clearance  = 50  lbs.,  and  50 
/*_j_I4.7  — 50  — 2.5  + 3.33  + 14.7  = 29.47  lbs. , and  velocity  of  piston  *=  192  feet 

5600  X 2.25  + 1.25  X 96  _ 1881600  ins  and  = 8 feet  strode. 

29.47  X 192  5658  12  X 2 

Area  of  cylinder  in  this  case  was  324  sq.  ins. 

For  Volume,  Pressure,  and  Density  of  Air,  see  Heat,  page  521. 


* See  formula  and  note  for  power  of  non-condensing  engine,  page  733. 


BLOWING  ENGINES. 


447 


To  Compute  Elements  of  a Blowing  Engine. 
Single  Stroke. 


V P+f  „r  AsnP+/_ 

— Or  — rj:  , 


A ni  F + / „ . V V + io  L 


230  33 000 

D2  s n TJp 

= a 


3 


D2  s n Tr  , -p,  . , 

- — — = Y ; and  34  P -f  32  = t. 
92 


Y representing  volume  of  air  in  cube  feet  per  minute,  P pressure  of  air  and 
f f rictional  resistance  in  lbs.  per  sq;  inch,  A area  of  cylinder  and  a area 
of  its  valves  in  sq.  ins.,  s stroke  of  piston  in  feet,  n number  of  single  strokes 
of  piston  per  minute,  L length  of  air-pipe  from  reservoir  to  discharge  in  feet, 
d diameter  of  air  or  blast  pipe  and  I)  diameter  of  cylinder  in  ins.,  v velocity 
of  blast  in  feet  per  second,  and  t temperature  of  blast  consequent  upon  cow - 
2>ression  in  degrees. 

Illustrations.  — Assume  blowing  cylinder  50  ins.  in  diam.,  stroke  of  piston  10 
feet,  number  of  single  strokes  10  per  minute,  pressure  by  mercurial  manometer 
6.12  ins.,  frictional  resistance  .4  lb.,  length  of  pipe  25.25  feet,  and  area  of  valves 
95  sq.  ins. 

Y = 1363. 54  cube  feet , P = 3 lbs. , A 1963. 5 sq.  ms. 


H and  h representing  height  of  barometer  and  pressure  of  blast  in  ins.  of 
mercury ; t temperature  of  blast ; and  v velocity  in  feet  per  second. 

Illustration. — A furnace  having  2 tuyeres  of  5 ins.  diameter,  pressure  and  tem- 
perature of  blast  3 ins.  and  3500,  and  barometer  30  ins. ; what  is  volume  of  air  trans- 
mitted per  minute? 

C for  a conical  opening  ==?  .94. 


feet  velocity  per  second. 

Then,  area  5 ins.  = 19. 635,  which  X 2 = 39. 27  ins. , and  39. 27  X 1 5- 14  X 60  -4- 144  = 
247. 73  cube  feet. 

To  Compute  Pressure  of  Blast  from  Water  or  jMCercarial 
Grange. 

Rule. — Divide  Water  and  Mercurial  Gauge  in  ins.  by  27.67  and  2.04  re- 
spectively, and  quotient  will  give  pressure  in  lbs.  per  sq.  inch. 


Proportions  of  Parts.  Blades. — Their  width  and  length  should  be  at  least 
equal  to  .4  or  .5  radius  of  fan. 

Openings. — Inlet  should  be  equal  to  radius  of  fan ; and  outlet,  or  dis- 
charge, should  be  in  depth  not  less  than  .125  diameter,  its  width  being  equal 
to  width  of  fan. 

Eccentricity. — .1  of  diameter  of  fan.  Journals,  4 diameters  of  shaft. 


and  1963,5  X 10  X 10  X 3 + -4  _ 


= 20.23  IP* 


33000 


—95  sq.  ins. 


To  Compute  Volume  of  Air  transmitted  by  air  Engine. 
When  Pressure,  Temperature , etc.,  are  given. 


Then  av  x 60= Y in  cube  feet  per  minute. 


•94  = 34*5 


Fan-blowers. 


BLOWING  ENGINES. 


448 


By  the  experiments  of  Mr.  Buckle,  he  deduced 

1.  That  velocity  of  periphery  of  blades  should  be  .9  that  of  their  theoretical 
velocity ; that  is,  velocity  a body  would  acquire  in  falling  height  of  a homo- 
geneous column  of  air  equivalent  to  required  density. 

2.  That  a diminution  of  inlet  from  proportions  here  given  involved  a 
greater  expenditure  of  power  to  produce  same  density. 

3.  That  greater  the  depth  of  blade,  greater  the  density  of  air  produced 
with  same  number  of  revolutions. 

To  Compute  Elements  of  a Fan-blower. 


v representing  velocity  of  periphery  of  fan  in  feet  per  second , d inches  of 
■ mercury , V volume  of  air  in  cube  feet , and  a area  of  discharge  in  sq.  ins. 

Illustration. — Assume  velocity  of  periphery  of  fan  123  feet  per  second,  density 
of  blast  .25  inch,  volume  of  air  1845  cube  feet,  and  area  of  discharge  40  sq.  ins. 


.242  X 4°  X 123  _ 2g7  independent  of  friction  of  blast  in  pipes  and  tuyeres. 


To  Compute  Power  of  a Centrifugal  Fan. 

Y2  -4-  97  300  = P.  V representing  velocity  of  tips  of  fan  in  feet  per  second . 


Operation  of  a blower  requires  about  2.5  per  cent,  of  power  of  attached 
boiler. 

A11  increase  in  number  of  blades  renders  operation  of  fan  smoother,  but 
does  not  increase  its  capacity. 

Pressure  or  density  of  a blast  is  usually  measured  in  ins.  of  mercury,  a 
pressure  of  1 lb.  per  sq.  inch  at  6o°  = 2.0376  ins. 

When  water  is  used,  a pressure  of  1 lb.  = 27.671  ins. 

Cupola  blast  .8  lbs.,  and  Smith's  forge  .25  to  .3  lbs.  per  sq.  inch. 

An  ordinary  Eccentric  Fan,  4 feet  in  diameter,  with  5 blades  10  ins.  wide 
and  14  in  length,  set  1.5  ins.  eccentric,  with  an  inlet  opening  of  17.5  ins.  in 
diameter,  and  an  outlet  of  12  ins.  square,  making  870  revolutions  per  min- 
ute, will  supply  air  to  40  tuyeres,  each  of  1.625  ins.  in  diameter,  and  at  a 
pressure  per  sq.  inch  of  .5  inch  of  mercury. 

An  ordinary  eccentric  fan-blower,  50  ins.  in  diameter,  running  at  1000  revolutions 
per  minute,  will  give  a pressure  of  15  ins.  of  water,  and  require  for  its  operation  a 
power  of  12  horses.  Area  of  tuyere  discharge  500  sq.  ins. 

A non-condensing  engine,  diameter  of  cylinder  8 ins.,  stroke  of  piston  1 foot,  press- 
ure of  steam  18  lbs.  (mercurial  gauge),  and  making  100  revolutions  per  minute,  will 
drive  a fan,  4 feet  by  2,  opening  2 feet  by  2,  500  revolutions  per  minute. 

Such  a blower  was  applied  as  an  exhausting  draught  to  smoke-pipe  of  steamer 
Keystone  State , cylinder  80  ins.  by  8 feet,  and  evaporation  was  doubled  over  that 
of  when  wind  was  calm. 

In  French  blowing  engines,  volume  of  air  discharged  75  per  cent,  that  of 
volume  of  piston  space  in  cylinder,  stroke  equal  diameter  of  cylindci,  and 
velocity  of  piston  from  100  to  200  feet  per  minute. 

Area  of  admission  valves  from  .066  to  .083  of  that  of  cylinder  for  speeds 
of  100  to  150  feet  per  minute,  and  from  .1  to  .111  for  higher  speeds.  Area 
of  exit  valves  from  .066  to  .05  of  cylinder.  (M.  Claudel.) 


40  x 123  X 60 


= 1845  cub.  ft. 


Memoranda, 


BLOWING  ENGINES. — CENTRAL  FORCES.  449 


By  some  experiments  lately  concluded  in  England  with  boilers  of  two 
steamers,  to  determine  relative  effects  of  natural  and  forced  draught  furnaces, 
the  results  were  as  follows  (R.  J.  Butler ) : 

Per  Sq.  Foot  of  Grate  Surface,— Natural  Draught , 10  to  10.87  IBP;  Steam 
Blast , 12.5  to  13;  Forced  or  Blast  Draught , 15  to  16. 

Heating  Surface  per  IIP. — Natural  Draught , 2.44  to  2.61 ; Steam  Blast , 
1. 71  to  2.86;  Forced  or  Blast  Draught , 1.56  to  2.5. 

Tube  Surface  per  IIP  in  Sq.  Feet— Natural  Draught , 2.03  to  2.18 ; Steam 
Blast , 2.02  to  2.08 5 Forced  or  Blast  Draught , 1.3  to  2.8. 

IIP  per  Sq.  Foot  of  Grate  in  these  Trials.  — Natural  Draught , 10.15  to 
10.87 ; Steam  Blast,  12.76  to  13. 1 ; Forced  or  Blast  Draught,  10.6  to  16.9. 

Root's  Rotary  Blower—  Is  constructed  from  .125  to  14  nominal  IP,  supplying 
from  150  to  10800  cube  feet  of  air  per  minute.  Delivery  pipe  2.5  to  19  ms. 
in  diameter.  Efficiency  65  to  80  per  cent,  of  power. 

For  Ventilation  of  Mines — From  40  to  280  revolutions  per  minute,  equal 
to  discharge  of  12  500  to  200  000  cube  feet  of  air  per  minute.  15.5  to  189  IP. 

Steam  cylinder  from  14  X 18  ins.  to  28  X 48  ins. 

For  other  details  of  Blowing  Engines  see  page  898. 


CENTRAL  FORCES. 

All  bodies  moving  around  a centre  or  fixed  point  have  a tendency  to 
fly  off  in  a straight  line:  this  is  termed  Centrifugal  Force ; it  is  op- 
posed to  & Centripetal  Force,  or  that  power  which  maintains  a body  in 
its  curvilineal  path. 

Centrifugal  Force  of  a body,  moving  with  different  velocities  in  same 
circle,  is  proportional  to  square  of  velocity.  Thus,  centrifugal  force  of 
a body  making  10  revolutions  in  a minute  is  4 times  as  great  as  centrif- 
ugal force  of  same  body  making  5 revolutions  in  a minute.  Hence,  in 
equal  circles,  the  forces  are  inversely  as  squares  of  times  of  revolution. 

If  times  are  equal,  velocities  and  force§  are  as  radii  of  circle  of  revolution. 

The  squares  of  times  are  as  cubes  of  distances  of  centrifugal  force  from 
axis  of  revolution. 

Centrifugal  forces  of  two  unequal  bodies,  having. same  velocity,  and  at  same  dis- 
tance from  central  body,  are  to  one  another  as  the  respective  quantities  of  matter 
in  the  two  bodies. 

Centrifugal  forces  of  two  bodies,  which  perform  their  revolutions  in  same  time, 
the  quantities  of  matter  of  which  are  inversely  as  their  distances  from  centre,  are 
equal  to  one  another. 

Centrifugal  forces  of  two  equal  bodies,  moving  with  equal  velocities  at  different 
distances  from  centre,  are  inversely  as  their  distances  from  centre. 

Centrifugal  forces  of  two  unequal  bodies,  moving  with  equal  velocities  at  different 
distances  from  centre,  are  to  one  another  as  their  quantities  of  matter,  multiplied  by 
their  respective  distances  from  centre. 

Centrifugal  forces  of  two  unequal  bodies,  having  unequal  velocities,  and  at  differ- 
ent distances  from  their  axes  are  in  compound  ratio  of  their  quantities  of  matter, 
squares  of  their  velocities,  and  their  distances  from  centre. 

Centrifugal  force  is  to  weight  of  body,  as  double  height  due  to  velocity  is  to  radius 
of  rotation. 

A Radius  Vector  is  a line  drawn  from  centre  of  force  to  moving  body. 

P P* 


450 


CENTRAL  FORCES. 


To  Compute  Centrifugal  Force  of  any-  Body. 

Rule  i.— Divide  its  velocity  in  feet  per  second  by  4.01,  also  square  of 
quotient  by  diameter  of  circle ; this  quotient  is  centrifugal  force,  assuming 
the  weight  of  body  as  1.  Then  this  quotient,  multiplied  by  weight  of  body, 
will  give  centrifugal  force  required. 

Example.— What  is  the  centrifugal  force  of  the  rim  of  a fly-wheel  having  a diam- 
eter of  10  feet,  and  running  with  a velocity  of  30  feet  per  second? 

30-^4.01  = 7.48,  and  7.482  = 10  = 5.59,  or  times  weight  of  rim, 

Or  W ft2  VR2-j~^2  __  0 r representing  radius  of  inner  diameter  of  ring. 

’ 4100 

Note.— Diameter  of  a fly-wheel  should  be  measured  from  centres  of  gravity  of  rim. 

When  great  accuracy  is  required,  ascertain  centre  of  gyration  of  body,  and 
take  twice  distance  of  it  from  axis  for  diameter. 

Rule  2. — Multiply  square  of  number  of  revolutions  in  a minute  by  diam- 
eter of  circle  of  centre  of  gyration  in  feet,  and  divide  product  by  constant 
number  5217 ; quotient  is  centrifugal  force  when  weight  of  body  is  1.  Then, 
as  in  previous  Rule,  this  quotient,  multiplied  by  weight  of  body,  is  centrif- 
ugal force  required. 

Or  n ^ — W.  n representing  number  of  revolutions  per  minute , d diameter  of 

’ 5217 

circle  of  gyration  in  feet,  and  W weight  of  revolving  body  in  lbs. 

Example.— What  is  centrifugal  force  of  a grindstone  weighing  1200  lbs.,  42  inches 
in  diameter,  and  turning  with  a velocity  of  400  revolutions  in  a minute? 

Centre  of  gyration  = rad.  (42 = 2)  X 7071  = 14.85  ins.,  which  -7-12  and  X2  = 
2. 475  feet  — diameter  of  circle  of  gyration.  Then  - ^ 47  X 1200  = 91  080  lbs. 


Formulas  to  Determine  "Various  Elements. 


C*  = 


W v 2 


W Rn2 


R = 


2930  C _ 


32.166  R ’ 
Wv 


: W R v'  1. 225 ; W = 


C 32.166  R 


Wn2 


2930 

/2930  C _ _ /C  R 32.166 

32.166 c ’ * v w 


= 6.28  vf  R. 


vv  n 32.  iuu  v v ” **  * 

C representing  centrifugal  force,  W mass  or  weight  of  revolving  body , both  in  lbs., 
R radius  of  circle  of  revolving  body  in  feet,  n number  of  revolutions  per  minute,  and 
v and  v linear  or  circumferential  and  angular  velocities  of  body  in  feet  per  second. 

. . /*  1 /» ..  lho  vnrrvhrir 


1 ana  v w ^ ^ ~ 

Illustration. -What  is  centrifugal  force  of  a sphere  weighing  30  lbs.,  revolving 
around  a centre  at  a distance  of  5 feet,  at  30  revolutions  per  second . 


5X2X3-  1416X30  _ feet  Then  C 3°  = 46.04  Ms. 

6o  — ** 1 J 32.166X5 


Centrifugal  forces  of  two  bodies  are  as  radii  of  circles  of  revolution  directly,  and 
as  squares  of  times  inversely. 

Ir  lustration If  a fly-wheel,  12  feet  in  diameter  and  3 tons  in  weight,  revolves 

in  8 seconds, 'and  anoUnfr  of  like  weight  revolves  in  6,  what  should  he  the  dmmetcr 
of  the  second  when  their  centrifugal  forces  are  equal  ? 

Then  3:3::“  : % ; or  » = =6.7sfeet,x  = unknown  element. 


5 ••  g2 


Centrifugal  forces  of  two  bodies,  when  weights  are  unequal , are  directly  as  squares 
of  times. 


j umts.  . 

Illustration.— What  should  be  the  ratio  of  the  weights  of  the  wheels  m the  pre 
ceding  case,  their  forces  being  equal? 


Then  3 : 62  : 82,  or  a;  = = 5- 333  tons- 


Molesworth  gives  .000  34  W R n2  = C. 


CENTRAL  FORCES. — FLY-WHEEL. 


451 


FLY-WHEEL. 

A Fly-wheel  by  its  inertia  becomes  a reservoir  as  well  as  a regulator 
of  force,  and  to  be  effective  should  have  high  velocity,  and  its  diameter 
should  be  from  3 to  4 times  that  of  stroke  of  driving  engine. 

Co-efficient  of  fluctuation  of  energy  in  a machine  ranges  from  .015 
to  .035. 

Weight  of  a fly-wheel  in  engines  that  are  subjected  to  irregular  mo- 
tion, as  in  a cotton-press,  rolling-mill,  etc.,  must  be  greater  than  in  others 
where  so  sudden  a check  is  not  experienced,  and  its  diameter  should 
range  from  3.5  to  5 times  length  of  the  crank. 

A single  acting  engine  requires  a weight  of  wheel  about  2.5  times  greater 
than  that  for  a double  acting,  and  5 times  for  double  engines  of  double  action. 

To  Compute  YV'eigh.t  of  11  i m.  of  a,  ITly- wheel. 

Rule. — Multiply  mean  effective  pressure  upon  piston  in  lbs.  by  its  stroke 
in  feet,  and  divide  product  by  product  of  square  of  number  of  revolutions, 
diameter  of  wheel,  and  .000  23. 

Note. — If  a light  wheel  is  required,  multiply  by  .0003;  and  if  a heavy  one,  by 


Example  i. — A non-condensing  engine  (double  acting),  having  a diameter  of  cyl- 
inder of  14  ins.,  and  a stroke  of  piston  of  4 feet,  working  full  stroke,  at  a pressure 
of  65  lbs.  mercurial  gauge,  and  making  40  revolutions  per  minute,  develops  about 
65  IP;  what  should  be  the  weight  of  its  fly-wheel,  when  adapted  to  ordinary  work? 

Area  of  cylinder  154  sq.  ins.  Mean  pressure  assumed  50  lbs.  per  sq.  inch.  Diam- 
eter of  wheel  4 feet  stroke  x 3 5 = 14  feet. 

50  X 154  X 4 = 30  800,  which  -4-  402  X 14  X .00023  = 5978  lbs. 

2. — If  a fly-wheel,  16  feet  in  diameter  and  4 tons  in  weight,  is  sufficient  to  regulate 
an  engine  (double  acting)  when  it  revolves  in  4 seconds,  what  should  be  the  weight 
of  a wheel,  12  feet  in  diameter,  revolving  in  2 seconds,  so  that  it  may  have  like  cen- 
trifugal force? 

Note.— The  centrifugal  forces  of  two  bodies  are  as  the  radii  of  the  circles  of  revo- 
lution directly,  and  as  squares  of  times  inversely. 


Then  = nr  _ 4 X .6  X 2*^4  X 16  X 4 

4 2 22  12X42  12X16 

Assume  elements  of  example  i. 


= 1. 333  tons. 


5978  X »i  -r- 13.25  = 45.12  square  ins. 


To  Compute  Dimensions  of  Rim. 


Rule.  Multiply  weight  of  wheel  in  lbs.  by  .1,  and  divide  product  by 
mean  diameter  of  rim  in  feet ; quotient  will  give  sectional  area  of  rim  in 
square  inches  of  cast  iron. 

PS  vy 

— W,  and  — ^ = A.  P representing  pressure  on  piston  and  TV  weight  of 

xoheel  in  lbs.,  S stroke  of  piston  and  D mean  diameter  of  wheel,  both  infect,  and  A 
area  of  section  of  rim  in  sq.  ins. 


Or, 


1 i6nPSC 
60  D 


DO  u C representing  coefficient  varying  from  3 to.  4 ordinarily, 

and  increasing  to  6 when  great  regularity  of  speed  is  required , and  n number  of  revo- 
lutions per  minute. 


Note.— Maximum  safe  velocity  for  cast  iron  is  assumed  at  80  feet  per  second. 

For  engines  at  high  expansion  of  steam,  or  with  irregular, loads,  as  with  a rolling- 
mill,  multiply  W by  1.5,  or  put  W 100  lbs.  for  each  IIP.  (Molesworth.) 

In  corn  or  like  mills,  the  velocity  of  periphery  of  fly-wheel  should  exceed  that  of 
the  stones. 


45  2 CENTRAL  FORCES. — GOVERNORS.— PENDULUMS. 


GOVERNORS. 

A Governor  or  Conical  Pendulum  in  its  operation  depends  upon  the 
principles  of  Central  Forces. 

When  in  a Ball  Governor  the  balls  diverge,  the  ring  on  vertical  shaft 
raises  and  in  proportion  to  the  increase  of  velocity  of  the  balls  squared, 
or  the  square  roots  of  distances  of  ring  from  fixed  point  of  arms,  cor- 
responding to  two  velocities,  will  be  as  these  velocities. 

Thus,  if  a governor  makes  6 revolutions  in  a second  when  ring  is  16 
ins.  from  fixed  point  or  top,  the  distance  of  ring  will  be  5.76  ins.  when 
speed  is  increased  to  10  revolutions  in  same  time. 

For  10  : 6 : : V 16  : 2.4,  which,  squared  = 5.76  ins.,  distance  of  ring 
from  top . Or,  62  : 102  5-7^  • *6  ins. 

A governor  performs  in  one  minute  half  as  many  revolutions  as  a 
pendulum  vibrates,  the  length  of  which  is  perpendicular  distance  be- 
tween plane  in  which  the  balls  ipove  and  the  fixed  point  or  centre  of 
suspension. 

To  Compute  Number  of  Revolutions  of  a Ball  Governor 

per  Minute  to  maintain  Balls  at  any  given  Height. 

j88  .4-  y/YL  ==  revolutions.  H representing  vertical  height  between  plane  of  balls 
and  points  of  their  suspension  in  ins. 

Illustration. -If  the  rise  of  the  halls  of  a centrifugal  governor  is  22  ins.,  what 
are  the  number  of  revolutions  per  minute  ? 

j88-4-  V22  ==.  4o-°9  revolutions. 

To  Compute  Vertical  Height  between  Plane  of  Balls 
and  their  Points  of  Suspension. 

(l8S  + r)*  = vertical  height  in  ins.  r representing  number  of  revolutions  per  minute. 

Illustration.— If  number  of  revolutions  of  a centrifugal  governor  is  100,  what 
will  be  rise  of  balls  ? 


188  -r- 100=  1.882  — 3-53  ins • 

To  Compute  Angle  of  Arms  or  Blane  of  Balls  with 
Centre  SLaft. 

r -i-l  = sin.  / . r representing  distance  of  balls  from  plane  of  centre  shaft,  and  l 
distance  betweU  balls  and  point  of  suspension  measured  m plane  of  shaft. 

Illustration. -Distance  of  balls  from  plane  of  centre  shaft  is  .0  inches,  and 
their  distance  from  point  of  suspension  is  25;  what  is  the  angle . 

10  -r-  25  = .4,  and  sin.  .4=5  23°  35'. 

(54.l6-7-7l)2  y 

When  Number  of  Revolutions  are  given.  — ; — = c05- 


l 


Illustration. — Revolutions  of  a governor  per  minute  are  50,  and  length  of  its 
arms  2 feet;  what  is  their  angle  with  plane  of  shalt . 

(54. 16  -r-  5°)2  _ ^173  = 5865  __  cos.  54o  6'. 


PENDULUMS. 

Pendulums  are  Simple  or  Compound , the  former  being  a material 
point,  or  single  weight  suspended  from  a fixed  point,  about  which  it 
oscillates,  or  vibrates,  by  a connection  void  of  weight ; and  the  latter, 
a like  body  or  number  of  bodies  suspended  by  a rod  or  connection. 
Any  such  body  will  have  as  many  centres  of  oscillation  as  theie  are 
given  points  of  suspension  to  it.,  and  when  any  one  of  these  centres  are 
determined  the  others  are  readily  ascertained. 


CENTRAL  FORCES. — PENDULUMS. 


453 


Thus,  sox  s g = a constant  product , and  sr=VsoXsg,  s g o and  r 
representing  points  of  suspension,  gravity,  oscillation,  and  gyration. 

Or,  any  body,  as  a cone,  a cylinder,  or  of  any  form,  regular  or  irregular 
so  suspended  as  to  be  capable  of  vibrating,  is  a compound  pendulum,  and 
distance  of  its  centre  of  oscillation  from  any  assumed  point  of  suspension  is 
considered  as  the  length  of  an  equivalent  simple  pendulum. 

The  Amplitude  of  a simple  pendulum  is  the  distance  through  which  it 
passes  from  its  lowest  position  to  its  farthest  on  either  side. 

Complete  Period  of  a pendulum  in  motion  is  the  time  it  occupies  in  making 
two  vibrations.  b 

All  vibrations  of  same  pendulum,  whether  great  or  small,  are  performed 
very  nearly  in  same  time. 

Number  of  Oscillations  of  two  different  pendulums  in  same  time  and  at 
same  place  are  in  inverse  ratio  of  square  roots  of  their  lengths. 

Length  of  a Pendulum  vibrating  seconds  is  in  a constant  ratio  to  force  of 
gravity. 

Time  of  Vibration  is  half  of  a complete  period,  and  it  is  proportional  to 
square  root  of  length  of  pendulum.  Consequently,  lengths  of  pendulums  for 
different  vibrations  are— 


Latitude  of  Washington , 
39.0958  ins.  for  one  second. 

9.774  ins.  for  half  a second. 


4.344  for  third  of  a second. 
•.4435  for  quarter  of  a second. 


lengths  of  Pendulums  vibrating  Seconds  at  Level  of 
tlie  Sea  in  several  Places. 


Equator.. 


Ins. 


ins. 

. 39.0152  New  York 39-1017  Paris 

on  nntR  T.nt  . rO  r . 


London. 


Ins. 

39.1284 
39- 1 393 


xwnv 39.101 

Washington 39-0958  I Lat,  45° 39.127 

To  Compute  Length  of  a Simple  Pendulum  for  a giver 
Latitude. 

39. 127  — .099  82  cos.  2 L — l.  L representing  latitude. 

Illustration.— Required  the  length  of  a simple  pendulum  vibrating  seconds  ir 
toe  latitude  of  500  31  . 

L = 50°  3i/ cos.  2 L =3 2 x 50°  3 1 ' — cos.  180°  — 500  31' x 2 = cos.  78°  58'r=.i9i  3S 
39- 127  + • 191  38  X .099  82  ( two  — or  negative  ==  an  affirmative  or  -f-)  — 39. 1461  ins 


To  Compute  Length  of  a Simple  Pendulum  for  a giver 
Nnmber  of  Vibrations. 

dulum  ^inins^  rePresenl™9  length  for  latitude , t time  in  seconds , and  l length  of  pen 

Illustration.— Required  vibrations  of  a pendulum  in  a minute  at  New  York  an 
60;  what  should  be  its  length?  ’ 

39. 1017  x i2=39.  1017,  Or,  --  = 1.  n representing  number  of  vibrations  per  second 


To  Compute  Number  of  Vibrations  of  a Simple  Pendu- 
lum in  a given  Time. 


y/L't  _ t 

— n)  ~ representing  time  of  one  vibration  in  seconds. 


To  Compute  Centre  of  Gravity  of  a Compound  Pendu- 
lnm  of  Two  Weights  connected  in  a Right  Line. 

When  Weights  are  both  on  one  Side  of  Point  of  Suspension. 

zw+rw_ 

— 0 — distance  oj  centre  of  gravity  from  point  of  suspension. 


W + w 


454 


CENTRAL  FORCES. — PENDULUMS. 


When  Weights  are  on  Opposite  Sides  of  Point  of  Suspension. 


I w — l'w 


- q — distance  of  centre  of  gravity  of  greater  weight  from  point  of  sus- 

W + w 

pension. 

^ote.— To  obtain  strictly  isochronous  vibrations,  the  circular  arc  must  be  sub- 
stituted for  the  cycloid  curve,  which  possesses  the  property  of  having  an  inclina- 
tion, the  sine  of  which  is  simply  proportional  to  distance  measured  on  the  curve 
from  its  lowest  point. 

For  construction  of  a Cycloidal  pendulum,  see  Deschaniel’s  Physics,  Fart  I.,  pp. 
71-2. 


To  Compute  Length  of  a Simple  Pendulum,  -Vibrations 
of  which  will  be  same  in  Number  as  Inches  m its 
Length. 

-^(60  -v/L)2  = i in  inches. 

Illustration.— What  will  be  length  of  a pendulum  in  New  York,  vibrations  of 
which  will  be  same  number  as  the  ins.  in  its  length  ? 

V W 39.1017  X 60)2  = 7.21 12  = 52  ins. 


To  Compute  Time  of  Vibration  of  a Simple  Pendulum, 
Length  being  given. 


y/l^-L  = tin  seconds. 


Illustration.—  Length  of  a pendulum  is  156.4  ins. ; what  is  the  time  of  its  vibra- 
tion in  New  York? 

I5^’4_  _ 2 seconds. 


h 


J 39.IOI7 

Or  /—  x 3. 1416  = t.  I representing  length  of  a pendulum  vibrating  seconds  in 

^ \ Q • • 

ins.,  g measure  of  force  of  gravity,  and  t time  of  one  oscillation. 


Illustration.— Length  of  a simple  pendulum  vibrating  seconds,  and  measure  of 
force  of  gravity  at  Washington,  are  39.0958  ins.,  and  32.155  feet. 


3.1416 


39.0958 


i55  X 12 


= 3.1416  x y/ 1.013  = 3- *4*6  X .3i83  = i second. 


To  Compute  Number  of  Vibrations  of  a Simple  Pen- 
dulum in  a given  Time. 

Xt  = n.  n representing  number  of  vibrations. 


ft 


Illustration.— The  length  of  a pendulum  in  New  York  is  156.4  ins.,  and  time  of 
its  vibration  is  2 seconds;  what  are  number  of  its  vibrations? 


/39-IOI7  w2_  / 53.  X2  = . 5X2  = 1 vibration.  Hence,  1 X — = 3°  vl~ 

v 156.4  V 12.506  2 

brations  per  minute. 


To  Compute  Measure  of  Grravity,  Length  of  Pendulum 
and.  Number  of  its  Vibrations  being  given. 

.82246  ln_  __  g g representing  measure  of  gravity  in  feet. 


To  Compute  Number  of  Revolutions  of  a Conical  Pen- 
dulum per  NIinute. 


/2933:5  _ n representing  distance  between  point  of  suspension  and  plane  oj 
V h 

revolutions  in  ins. 


Note.— Number  of  revolutions  per  minute  are  constant  for  any  given  height,  and 
the  time  of  a revolution  is  directly  as  square  root  of  height. 


CRANES. 


455 


CRANES. 

Usual  form  of  a Crane  is  that  of  a right-angled  triangle,  the  sides 
being  post  or  jib,  and  stay  or  strut,  which  is  hypothenuse  of  triangle. 

When  jib  and  post  are  equal  in  length,  and  stay  is  diagonal  of  a square, 
this  form  is  theoretically  strongest,  as  the  whole  stress  or  weight  is  borne  by 
stay,  tending  to  compress  it  in  direction  of  its  length ; stress  upon  it,  com- 
pared to  weight  supported,  being  as  diagonal  to  side  of  square,  or  as  1.4142 
to  1.  Consequently,  if  weight  borne  by  crane  is  1000  lbs.,  thrust  or  com- 
pression upon  stay  will  be  1414.2  lbs.,  or  as  a e to  e W,  Fig.  1. 

When  Post  is  Supported  at  "both.  Head  and.  Foot,  as 
Fig.  1. 

Weight  W is  sustained  by  a rope  or  chain, 
and  tension  is  equal  upon  both  parts  of  it ; that 
is,  on  two  sides  of  square,  i a and  e W.  Conse- 
quently jib,  i a,  has  no  stress  upon  it,  and  serves 
merely  to  retain  stay,  a e. 

If  foot  of  stay  is  set  at  w,  thrust  upon  it,  as 
compared  with  weight,  will  be  as  an  to  aw ; 
and  if  chain  or  rope  from  i to  a is  removed,  and 
weight  is  suspended  from  a,  tension  on  jib  will 
be  as  i a to  a W. 

# W foot  of  stay  is  raised  to  0 , thrust,  as  compared  with  weight,  will  be  as 
line  a 0 is  to  a W,  and  tension  on  jib  will  be  as  line  ar. 

By  dividing  line  representing  weight,  as  a W or  a w)  into  equal  parts,  to 
represent  tons  or  pounds,  and  using  it  as  a scale,  stress  upon  any  other  part 
may  be  measured  upon  described  parallelogram. 

Thus,  as  length  of  a W,  compared  to  a e,  is  as  1 to  1.4142 : if  a W is  di- 
vided into  10  parts  representing  tons,  a e would  measure  14.142  parts  or  tons. 

'W'h.erL  Post  is  Supported:  at  Foot  only. 

If  post  is  wholly  unsupported  at  head,  and  its  foot  is  secured  up  to  line. 
0 \\ , then  W,  acting  with  leverage,  e W,  will  tend  to  rupture  post  at  e,  with 
same  effect  as  if  twice  that  weight  was  laid  upon  middle  of  a beam  equal  to 
twice  length  of  e W,  e being  at  middle  of  beam,  which  is  assumed  to  be  sup-* 
ported  at  both  ends,  and  of  like  dimensions  to  those  of  post. 

Or,  force  exerted  to  rupture  post  will  be  represented  by  stress,  W,  multi- 
plied by  4 times  length  of  lever,  e W,  divided  by  depth  of  post  in  line  of 
stress,  squared,  and  multiplied  by  breadth  of  it  and  Value  * of  its  material. 

Post  of  such  a crane  is  in  condition  of  half  a beam  supported  at  one  end, 
weight  suspended  from  other ; consequents,  it  must  be  estimated  as  a beam 
ot  twice  the  length  supported  at  both  ends, “stress  applied  in  middle. 

To  Compute  Stress  on  Jib,  and  on  Stay  or  Stmt. -Fig.  2. 

On  diagram  of  crane,  Fig.  2,  mark  off  on  line  of 
chain,  a W,  a distance,  a b , representing  weight  on 
chain ; from  point  b draw  a line,  b c,  parallel  to  jib, 
a e , and  where  this  intersects  stay  or  strut,  draw  a 
vertical  line,,  c 0,  extending  to  jib,  and  distances 
from  a to  points  & c and  0 c,  measured  upon  a scale 
of  equal  parts,  will  represent  proportional  strain. 

Illustration. — In  figure,  weight  being  10  tons,  stress 
on  stay  or  strut  compressing,  a c,  will  be  31  tons,  and 
on  jib  or  tension-rods,  a 0,  26  tons. 


* For  Value  of  Materials,  see  page  779. 


456 


CRANES. 


To  Compute  Dimensions  of  E*ost  of  a Crane. 

When  Post  is  Supported  at  Feet  only.  Rule— Multiply  weight  or  stress 
to  be  borne  in  lbs.  by  length  of  jib  in  feet  measured  upon  a horizontal 
plane ; divide  product  by  Value  of  material  to  be  used,  and  product,  divided 
by  breadth  in  ins.,  will  give  square  of  depth,  also  in  ins. 

Example. — Stress  upon  a crane  is  to  be  22  400  lbs.,  and  distance  of  it  from  centre 
of  post  20  feet;  what  should  be  dimension  of  post  if  of  American  white  oak? 

Value  of  American  white  oak  50.  Assumed  breadth  12  ins. 

2?  409  -X  20  — 3q60j  and  — 746.67.  Then  ^746.67  = 27.32  ins. 

50  12 

When  Post  is  Supported  at  both  Ends.  Rule.— Multiply  weight  or  stress 
to  be  borne  in  lbs.  by  twice  length  of  jib  in  feet  measured  upon  a horizontal 
plane  • divide  product  by  Value  of  material  to  be  used,  and  product,  divided 
bv  four  times  breadth  in  ins.,  will  give  square  of  depth,  also  in  ins. 

Example.— Take  same  elements  as  in  preceding  case.  Assumed  breadth  10  ins. 

iy  ?2—  — 448,  and  ^448  — 21.  *66  ins. 

4 X 10 

In  Fig.  3,  angle  a b e and  ebc  being  equal,  chain  or  rope  is  represented 
by  a b c , and  weight  by  W ; stress  upon  stay  b d,  as 
compared  with  weight,  is  as  b d to  a b or  b c. 

In  practice,  however,  it  is  not  prudent  to  consider 
chain  as  supporting  stay ; but  it  is  proper  to  disregard 
chain  or  rope  as  forming  part  of  system,  and  crane 
should  be  designed  to  support  load  independent  of  it. 
It  is  also  proper  that  angles  on  each  side  of  diagonal 
stay,  in  this  case,  should  not  be  equal.  If  side  a b is 
formed  of  tension-rods  of  wrought  iron,  point  a should 
be  depressed,  so  as  to  lengthen  that  side,  and  decrease 
angle  a be;  but  if  it  is  of  timber,  point  a should  be 
raised,  and  angle  a b e increased. 


22  400  x 20  x 2 

Then  — 1 = 17  92°> 

50 


Fig-  3* 


Fig.  4. 


Fig.  4 shows  a form  of  crane  very  generally  used; 
angles  are  same  as  in  Fig.  3,  and  weight  suspended  from 
it,  being  attached  to  point  d,  is  represented  by  line  b d. 
The  tension,  which  is  equal  to  weight,  is  shown  by  length 
of  line  b c,  and  thrust  by  length  of  line  b ^measured  by 
a scale  of  equal  parts,  into  which  line  b d,  representing 
weight,  is  supposed  to  be  divided. 

But  if  b e be  direction  of  jib,  then  b g will  show  ten- 
sion and  bf  the  thrust  (df  being  taken  parallel  to  b e), 
both  of  them  being  now  greater  than  before;  line  b d 
representing  weight,  and  being  same  in  both  cases. 

To  Ascertain  Stress  on  Jit),  on  Strnt 
of  a Crane.— Fig.  5. 

Through  a draw  a s , parallel  to  jib  or  tension-rod 
0 ?%  and  also  s u parallel  to  strut  a r ; then  1 s is  a 
diagonal  of  parallelogram,  sides  of  which  are  equal  to  r a and  r u. 

If  then  r s represents  a stress  of  20  lbs., 
the  two  forces  into  which  it  is  decom- 
posed are  shown  by  r u and  r a ; 0 r is 
equal  to  r «,  as  each  of  them  is  equal  to 
a s,  and  r s is  equal  to  0 a.  Hence,  20 
represented  by  a 0,  stress  on  jib  will  be 
represented  by  0 r,  and  that  on  strut  by 
r a. 

Assuming  then  or  3 feet,  a r 3.5,  and 
0 a 1,  stress  on  jib  will  be  60  lbs.,  and  on  strut  70. 


CRANES. 


457 


Thus,  in  all  cases,  stress  on  jib  or  tension-rod  and  on  strut  can  be  deter- 
mined by  relative  proportions  of  sides  of  triangle  formed. 

To  Compute  Stress  vipon  Strxxt  of  a Crane. 

Rule  —Multiply  length  of  strut  in  feet  by  weight  to  be  borne  in  lbs. ; di- 
vide product  by  height  of  jib  from  point  of  bearing  of  strut  m feet,  and 
quotient  will  give  stress  or  thrust  in  lbs. 

Fxample.—  Length  of  strut  of  a crane  is  28.284  feet,  height  of  post  is  26.457  feet, 
and  weight  to  be  borne  is  22  400  lbs. ; what  is  stress? 

28.284X22400  = 633  56l6  = 7 Z&5. 

26.457  26.457 

Chains  and.  Tropes. 

Chains  for  Cranes  should  be  made  of  short  oval  links,  and  should  not  ex- 
ceed 1 inch  in  diameter. 

^hort-linked  Crane  Cliains  and  Ropes  showing  Di- 
melons  and  Weight  of  each,  and  Proof  of  Chain 
in  Tons 


Diam. 

of 

Chain9. 

Weight 

per 

Fathom. 

Proof 

Strain. 

Circumf. 

of 

Rope. 

Weight 
of  Rope 
per  Fath. 

Diam. 

of 

Chains. 

Weight 

per 

Fathom. 

Proof 

Strain. 

Circumf. 

of 

Rope. 

Weight 
of  Rope 
per  Fath. 

Ins. 

.3l25 

•375 

•4375 

.5625 

.625 

Lbs. 

6 

8.5 

II 

14 

18 

24 

Tons. 

•75 

1-5 

2.5 

3- 5 

4- 5 

5- 25 

Ins. 

2- 5 

3- 25 
4 

4- 75 

5- 5 
6.25 

Lbs. 

1- 5 

2- 5 

3- 75 
5 

7 

8.7 

Ins. 

.6875 

•75 

.8125 

• 875 

•9375 

1 

Lbs. 

28 

32 

36 

44 

50 

56 

Tons. 

6- 5 

7- 75 
9- 25 

io-75 

12.5 

14 

Ins. 

7 

7-5 

8.25 

9 

9-5 

10 

Lbs. 

10.5 

12 

15 

17-5 

I9-5 

22 

Ropes  of  circumferences  given  are  cousiutueu  iu  uc  ut  " 

the  chains,  which,  being  short-linked,  are  made  without  studs. 

A crane  chain  will  stretch,  under  a proof  of  15  tons,  half  an  inch  per  fathom. 


Machinery-  of  Cranes. 

To  attain  greater  effect  of  application  of  power  to  a crane,  the  wheel- work 
must  be  properly  designed  and  executed. 

If  manual  labor  is  employed,  it  should  be  exerted  at  a speed  of  220  feet 
per  minute. 

Proportions. — Capacity  of  Crane , 5 tons. 

Radius  of  winch  or  handle  15  to  18  ins.  Height  of  axle  from  floor  36  to  39. 

1st  pinion,  n teeth,  1.25  ins.  pitch.  I 2d  pinion,  12  teeth,  1.5  ins.  pitch. 

1st  wheel,  89  “ 1.25  “ “ | 2d  wheel,  96  “ 

Barrel  8 ins.  X n teeth  X 12  teeth  X n 200  lbs.  — 30  800  _ ^ ^ ^ _ gtatical  re_ 

Winch  17  ins.  X 89  teeth  x 96  teeth  x 4 men  ==  1513 
s'.stance  to  each  of  the  4 men  at  winches. 

An  experiment  upon  capacitv  of  a crane,  geared  1 to  105,  developed  that 
a strong  man  for  a period  of  2.5  minutes  exerted  a power  of  27562  foot- 
pounds per  minute,  which,  when  friction  of  crane  is  considered,  is  fully  equal 
to  the  power  of  a horse  for  one  minute. 

In  practice  an  ordinary  man  can  develop  a power  of  15  lbs.  upon  a crane, 
handle  moved  at  a velocity  of  220  feet  per  minute,  which  is  equivalent  to 
3300  foot-pounds. 

For  Treatise  on  Cranes,  see  Woales’  Series,  No.  33. 

Q Q 


458 


COMBUSTION. 


COMBUSTION. 

Combustion  is  one  of  the  many  sources  of  heat,  and  denotes  combi- 
nation of  a body  with  any  of  the  substances  termed  Supporters  of  Com- 
bustion ; with  reference  to  generation  of  steam,  we  are  restricted  to  but 
one  of  these  combinations,  and  that  is  Oxygen. 

All  bodies,  when  intensely  heated,  become  luminous.  When  this  heat 
is  produced  by  combination  with  oxygen,  they  are  said  to  be  ignited ; 
and  when  the  body  heated  is  in  a gaseous  state,  it  forms  what  is  termed 
Flame. 

Carbon  exists  in  nearly  a pure  state  in  charcoal  and  in  soot.  It  com- 
bines with  no  more  than  2.66  of  its  weight  of  oxygen.  In  its  combus- 
tion, 1 lb.  of  it  produces  sufficient  heat  to  increase  temperature  of  14  500 
lbs.  of  water  i°. 

Hydrogen  exists  in  a gaseous  state,  and  combines  with  8 times  its 
weight  of  oxygen,  and  1 lb.  of  it,  in  burning,  raises  heat  of  50000  lbs. 
of  water  10.* 

An  increase  in  the  rapidity  of  combustion  is  accompanied  by  a dimi- 
nution in  the  evaporative  efficiency  of  the  combustible. 

Mr.  D.  K.  Clark  furnishes  the  following:  When  coal  is  exposed  to  heat  in  a fur- 
nace, the  carbon  and  hydrogen,  associated  in  various  chemical  unions,  as  hydrocar- 
bons, are  volatilized  and  pass  off.  At  lowest  temperature,  naphthaline,  resins,  and 
fluids  with  high  boiling-points  are  disengaged;  at  a higher  temperature,  volatile 
fluids  are  disengaged;  and  still  higher,  olefiant  gas,  followed  by  light  carburetted 
hydrogen,  which  continues  to  be  given  off  after  the  coal  has  reached  a low  red  heat. 
As  temperature  rises,  pure  hydrogen  is  also  given  off,  until  finally,  in  the  fifth  or 
highest  stage  of  temperature  for  distillation,  hydrogen  alone  is  discharged.  What 
remains  after  distillatory  process  is  over,  is  coke,  which  is  the  fixed  or  solid  carbon 
of  coal,  with  earthy  matter  or  ash  of  the  coal. 

The  hydrocarbons,  especially  those  which  are  given  off  at  lowest  temperatures, 
being  richest  in  carbon,  constitute  the  flame-making  and  smoke-making  part  of  the 
coal.  When  subjected  to  heat  much  above  the  temperatures  required  to  vaporize 
them,  they  become  decomposed,  and  pass  successively  into  more  and  more  perma- 
nent forms  by  precipitating  portions  of  their  carbon.  At  temperature  of  low  red- 
ness none  of  them  are  to  be  found,  and  the  olefiant  gas  is  the  densest  type  that 
remains,  mixed  with  carburetted  and  free  hydrogen.  It  is  during  these  trans- 
formations that  the  great  volume  of  smoke  is  made,  consisting  of  precipitated  car- 
bon passing  off  uncombined.  Even  olefiant  gas,  at  a bright  red  heat,  deposits  half 
its  carbon,  changing  into  carburetted  hydrogen;  and  this  gas,  in  its  turn,  may 
deposit  the  last  remaining  equivalent  of  carbon  at  highest  furnace  heats,  and  be 
converted  into  pure  hydrogen. 

Throughout  all  this  distillation  and  transformation,  the  element  of  hydrogen 
maintains  a prior  claim  to  the  oxygen  present  above  the  fuel;  and  until  it  is  satis- 
• fled,  the  precipitated  carbon  remains  unburned. 

Summary  of  Products  of  Decomposition  in  tlie  Furnace. 

Reverting  to  statement  of  average  composition  of  coal,  page  485,  it  ap- 
pears that  the  fixed  carbon  or  coke  remaining  in  a furnace  after  volatile 
portions  of  coal  are  driven  off,  averages  61  per  cent,  of  gross  weight  of  the 
coal.  Taking  it  at  60  per  cent.,  proportion  of  carbon  volatilized  in  com- 
bination with  hydrogen  will  be  20  per  cent.,  making  total  of  80  per  cent,  of 
constituent  carbon  in  average  coal. 

Of  the  5 per  cent,  of  constituent  hydrogen,  1 part  is  united  to  the  8 per 
cent,  of  oxygen,  in  the  combining  proportions  to  form  water,  and  remaining 
4 parts  of  hydrogen  are  found  partly  united  to  the  volatilized  carbon,  and 
partly  free. 


Mean  effect. 


COMBUSTION. 


4 


These  particulars  are  embodied  in  following  summary  of  condition  of 
elements  of  ioo  lbs.  of  average  coal,  after  having  been  decomposed,  and  prior 
to  entering  into  combustion — 

ioo  Lbs.  of  Average  Coal  in  a Furnace. 


Composition  Lbs. 


Carbon  { Voiatiiized. . . 

. 20 

Hydrogen 

• 5 

Sulphur 

. 1.25 

Oxygen  

. 8 

Nitrogen 

Ash,  etc 

• 4-55 

► forming 


Lbs.  Decomposition. 

60  fixed  carbon. 

24  hydrocarbons  and  free  hydrogen. 
1.25  sulphur. 

. 85.25 

9 water  or  steam. 

1.2  nitrogen. 

4.55  ash,  etc. 


IOO 


IOO 


showing  a total  useful  combustible  of  85.25  per  cent.,  of  which  25.25  per 
cent,  is  volatilized.  While  the  decomposition  proceeds,  combustion  proceeds, 
and  the  25.25  per  cent,  of  volatilized  portions,  and  the  60  per  cent,  of  fixed 
carbon,  successively,  are  burned. 

It  may  be  added  that  the  sulphur  and  a portion  of  the  nitrogen  are  dis- 
engaged in  combination  with  hydrogen,  as  sulphuretted  hydrogen  and  am- 
monia. But  these  compounds  are  small  in  quantity,  and,  for  the  sake  of 
simplicity,  they  have  not  been  indicated  in  the  synopsis. 

Volume  of  Air  chemically  consumed  in  complete  Combustion  of  Coal. 

Assume  100  lbs.  of  average  coal.  Then,  by  following 

80  + 3 ^5  — ^ + -4  X 1.25  X 152  = 14  060  cube  feet  of  air  at  62°  for  100  lbs.  coal. 

For  volatilized  portion,  Hydrogen  (H),  4 lbs.  x 457  = 1 828  cube  feet. 

Carbon  (C),  20  “ X 152=  3040  “ “ 

Sulphur  (S),  1.25  “ X 57  = 71  “ “ 

4939  u u 

For  fixed  portion,  Carbon,  60  lbs.  x 152=  9120  “ “ 

Total  useful  combustible,  85.25  “ 14059  “ “ for  com- 

plete  combustion  of  100  lbs.  coal  of  average  composition  at  62°. 

To  Compute  Volume  of  Air  at  62°,  under  One  At- 
mosphere, chemically  consumed  in  Complete  Com- 
bustion of  1 Lb.  of  a given  ITnel. 

Rule. — Express  constituent  carbon,  hydrogen,  oxygen,  and  sulphur,  as 
percentages  of  whole  weight  of  fuel ; divide  oxygen  by  8,  deduct  quotient 
from  hydrogen,  and  multiply  remainder  by  3 ; multiply  sulphur  by  .4 ; add 
products  to  the  carbon,  and  multiply  sum  by  1.52.  Final  product  is  volume 
of  air  in  cube  feet. 

To  compute  iveight  of  air  chemically  consumed. — Divide  volume  thus  found 
by  13.14;  quotient  is  weight  of  air  in  lbs. 


Or,  1. 52  (C  -f-  3 (H  — ~ ) -f . 4 S)  = Air.  0 Oxygen. 

Note.— In  ordinary  or  approximate  computations,  sulphur  may  be  neglected. 
Example.— Assume  1 lb.  Newcastle  coal.  0 = 82.24,  11  = 5.42,  0 = 6.44,  and 
S = i-35- 


— — =4. 805,  5-42  — .805  = 4.615  X 3 = i3- 84S,  1-35  X .4  = -54»  i3- 845 + -54 +82. 24 
= 96.625,  and  96.625  X 1.52  = 146.87  cube  feet. 

Then  146.87-7-13.14  = 11.18  lbs. 


o 


COMBUSTION. 


To  Compute  Total  Weight  of  Gaseous  Products  of  Com- 
plete Combustion  of  1 I/b.  of  a given  Enel. 

Rule.— Express  the  elements  as  per-centages  of  fuel;  multiply  carbon  by 
.126,  hydrogen  by  .358,  sulphur  by  .053,  and  nitrogen  by  .01,  and  add  prod- 
ucts together.  Sum  is  total  weight  of  gases  in  lbs. 

Or,  .126  C + .358  H + .053  S + .oi  N = Weight. 

Example.  —Assume  as  preceding  case.  N = 1. 61. 

82.24  X 1.264-5.42  X .358  + i-35  X 053  + 1.61  X .01  = 12.39  lbs. 

To  Compute  Total  -Volume,  at  62°,  of  Gaseous  Products 
of  Complete  Combustion  of  1 Eh.  of  given  Fuel. 
Rule.— Express  elements  as  per-centages;  multiply  carbon  by  1.52,  hy- 
drogen by  5.52,  sulphur  by  .567,  and  nitrogen  by  .135,  and  add  products 
together.  Sum  is  total  volume,  at  62°  F.,  of  gases,  in  cube  feet. 

Or,  1.52  C + 5.52  H + .567  S -J- . 135  N = Volume. 

To  Compute  Volume  of  the  several  Gases  separately 
from  their  Respective  Quantities. 

Rule.— Multiply  weight  of  each  gaseous  product  by  volume  of  1 lb.  in 
cube  feet  at  62°,  as  below. 


Volume  of  1 Lb.  of  Gases  at  62°  under  a Pressure  of  14.7  Lbs. 

Cube  feet.  Cube  feet.  Cube  feet. 


Aqueous  Vapor  or| 
Gaseous  Steam . j 


21.125  | 
Air. 


Nitrogen 13- 5GI 

Carbonic  Acid 8.594 


I Oxygen 11.887  I 

I Hydrogen 190  | 

13. 1 41  cube  feet. 

For  a lb.  of  oxygen  in  combustion,  4.35  lbs.  air  are  consumed;  or,  by  volume,  for 
a cube  foot  of  oxygen  4.76  cube  feet  of  air  are  consumed. 

1 lb.  Hydrogen  consumes 34- 8 lbs.,  or  457  cube  feet,  at  62°. 

1 “ Carbon,  completely  burned,  consumes 11.6  “ ‘ 152  * u 

x « “ partially  “ “ ....  5-8  “ 76  ,t 


Sulphur  consumes. . 


4-35  ' 


Composition 


and.  Equivalents  of  Gases, 
Combustion  of  Euel. 


57 

combined  in 


Oxygen  . . . 
Hydrogen. 

Carbon 

Sulphur. . . 
Nitrogen . . 


COMPOUNDS. 

^Atmospheric  Air 
(mech.  mixture) . . 
Aqueous  Vapor  or 
Water 


Equiv- 
alents. 
O.  I 

H.  1 
C.  1 
S.  I 
N.  1 


0.  23 

N.  77 

O.  1 
H.  1 


By 

Weight. 


6 

16 

14 


26.8! 


GASES. 

Elements. 

By 

Weight. 

COMPOUNDS. 

Equiv- 

alents. 

Light  Carburetted  ) 

C.  2 

I21 

Hydrogen J 

1 

H.  4 

A) 

l\ 

Carbonic  Oxide j 

1 

0.  1 
C.  1 

Carbonic  Acid ] 

1 

0.  2 
C.  I 

1] 

Olefiant  Gas  (Bi-car-  ] 

l 

C.  4 

24 1 

buretted  Hyd. . . . ; 

1 

H.  4 

4l 

Sulphurous  Acid. . . J 

\ 

0.  2 
S.  I 

16) 
16  } 

Weights  of  products  in  combustion  of  1 lb.  of  given  fuel,  are— 

C ==  .0366.  H = .09.  S = .o2.  N = . 0893  C + . 268  H + . 0335  S + . 01  N. 


Cube  Feet. 

.02  X 5- 85  = .117  volume  sulph.  acid. 
.0893  + -268  + .0335  + .01  X i3-5°i  =3 
5. 409  volume  nitrogen. 


Cube  Feet. 

.0366  X 8.59=  .315  volume  carbonic 

acid. 

.09  X 190  =17.1  “ steam. 

Volume  of  Air  or  Gases  at  higher  temperatures  than  here  given  (62°)  is  ascer- 

tained  by  V V 461  = V'.  V representing  volume  of  air  or  gas  at  temperature  t, 
u t +461 
and  V'  at  temperature  t'. 


By  Volume  1 Oxygen,  3.762  Nitrogen. 


COMBUSTION. 


46l 


Chemical  Composition  of  some  Compound  Com- 
bustibles. 


Carbonic  oxide 

Light  carburetted  hydrogen. . . 
Olefiant  gas,  Bicarburetted  hyd. 

Sulphuric  ether 

Alcohol.. . 

Turpentine . 

Wax 

Olive  oil 

Tallow 


Combining  equivalents. 


Car. 

Hyd. 

Oxy. 

Car. 

Hyd. 

Oxy. 

Per  Cent. 

Per  Cent. 

Per  Cenl 

1 

— 

1 

42.9 

— 

57-i 

2 

4 

— 

75 

25 

— 

4 

4 

— 

85-7 

J4-3 

— 

4 

5 

1 

64.8 

13-5 

21.7 

4 

6 

2 

52.2 

13 

34-8 

20 

16 

■ — 

88.2 

11. 8 

— 

— 

— 

81.6 

i3-9 

4-5 

— 

— . 

— 

77.2 

13-4 

9.4 

— 

— 

— 

79 

11  -7 

9-3 

Heating  powers  of  compound  bodies  are  approximately  equal  to  sum  of 
heating  powers  of  their  elements. 

Thus,  carburetted  hydrogen,  which  consists  of  two  equivalents  of  carbon  and  four 
of  hydrogen,  weighing  respectively  2X6  = 12  and  1 x 4 = 4,  in  proportion  of  3 to  1, 
or  .75  lb.  of  carbon  and  .25  lb.  of  hydrogen  in  one  lb.  of  gas.  Elements  of  heat  of 
combustion  of  one  lb.  are,  then- 

units  of  heat. 

For  carbon 14  544  X .75  = 10908 

For  hydrogen 62  032  X .25  = 15  508 

Total  heat  of  combustion,  as  computed 26416 

Total  heat,  by  direct  trial 23513 


Heating  Powers  of  Coinbus  1 i L>  1 e s . 
(MM.  Favre  and  Silbermann , D.  K.  Clark  and  others.) 


Hydrogen 

Carbon,  making ) 
carbonic  oxide,  j 
Carbon,  making  1 
carbonic  acid. . ) 
Carbonic  oxide. . . . , 
Light  carburetted  1 

hydrogen j 

Olefiant  gas 

Sulphuric  ether 

Alcohol 

Turpentine 

Sulphur 

Tallow 

Petroleum 

Coal  (average) 

Coke,  desiccated. . . 
Wood,  desiccated  . . 
Wood  - charcoal,  1 

desiccated j 

Peat,  desiccated 

Peat-charcoal,  de- ) 

siccated j 

Lignite 

Asphalt 


Oxygen 
consumed 
per  lb.  of 
Com- 
bustible. 


Lbs. 


i-33 

2.66 

•57 

4 

3-43 


j.6 

2.78 
3- 29 


2-95 

4.12 


2-5 

1.4 


2.25 

I*75 

2.28 


2.03 

2-73 


Weight  and  Volume 
of  Air  consumed  per 
lb.  of  Combustible. 


Lbs. 
34-8 
5-8 
11. 6 
2.48 
17.4 
i5 


i4-3 

4-35 

12.83 

17-93 

10.7 

10.9 


9.8 
7.6 

9.9 

8.85 

11.87 


Cube  Feet 
at  62°. 

457 

76 


152 

33 

229 

196 

149 

159 


57 


235 

141 

143 

80 


129 

100 


129 

116 

156 


Total  Heat 
of  Combus- 
tion of  1 lb. 
of  Combus- 
tible. 


Units. 
62  032 
4 452 


14  500 
4 325 
23  513 


21343 
16  249 
12929 
19  534 
4032 
18  028 
27  531 
M 133 
13  550 
7792 


13  309 
9 951 
12325 


11  678 
16655 


Equivalent  evaporative 
Power  of  1 lb.  of  Com- 
bustible, under  one  At- 
mosphere. 


Lbs.  of  wa- 
ter at  62°. 
55-6 


13 

3-88 

21.07 


19. 12 
14.56 
11.76 

W-5 
3.61 
16. 1 q 


12.67 
12. 14 
6.98 


11.92 

8.91 

11.04 


Lbs.  of  wa- 
ter at  2120. 
64.2 


4.61 


15 

4.48 

24-34 

22.09 

16.82 

I3-38 

20.22 

4.17 

18.66 

28.5 

14.62 

14.02 
8.07 

*3-i 3 

10.3 
12.76 

12. 1 
17.24 


When  carbon  is  not  completely  burned,  and  becomes  carbonic  oxide,  it  produc 
less  than  a third  of  heat  yielded  when  it  is  completely  burned.  For  heating  pow 
of  carbon  an  average  of  14  500  units  is  adopted. 

Q Q* 


COMBUSTION”. 


463 


To  Compute  Heating  Power  of  1 Lt>.  of  a given  Com- 
bustible. 


When  proportions  of  Carbon,  Hydrogen , Oxygen,  and  Sulphur  are  given . 
Rule. — Ascertain  difference  between  hydrogen  and  .125  of  oxygen;  multi- 
ply remainder  by  4.28 ; multiply  sulphur  by  .28,  add  products  to  the  carbon, 
multiply  sum  by  14  500,  divide  by  100,  and  product  is  total  heating  power 

in  units  of  heat.  

Or,  145  (C  + 4.28  H — Ox  125  + .28  S ) = heat. 

Illustration. — Assume  as  preceding  case. 

5.42  ^82.28  X -125  X 4.28  + 1.35  x .28  + 82.28  X i4  5°o-^-ioo— 15 005. 

To  Compute  Evaporative  Power  of  1 Lb.  of  a Griven. 
Combustible. 

When  Proportions  of  Carbon , Hydrogen , Oxygen,  and  Sulphur  are  given . 
Rule. — Ascertain  difference  between  hydrogen  and  .125  of  oxygen,  multiply 
remainder  by  4.28 ; multiply  sulphur  by  .28,  add  products  to  the  carbon,  and 
multiply  sum  by  .13,  when  water  is  supplied  at  62°,  and  .15  when  at  2120; 
product  is  evaporative  power  in  lbs.  of  water  at  2120. 

Or,  When  total  heating  power  is  known,  divide  it  by  1116  when  water  is 
at  62°,  or  996  when  at  2120. 

Illustration. — By  table,  heating  power  of  Tallow  is  18028  units. 

Hence,  18028  -P  1 116  = 16.15  lbs.  water  evaporated  at  62°. 


Temperature  of’  CoiTADvistiorL. 

Temperature  of  combustion  is  determined  by  product  of  volumes  and 
specific  heats  of  products  of  combustion. 

Illustration.— 1 lb.  carbon,  when  completely  burned,  yields  3.66  lbs.  carbonic 
acid  and  8.94  of  nitrogen.  Specific  heats  .2164  and  .244. 

3.66  X .2164  =•.  .792  units  of  heat  for  i°. 

8.94  X -244  ==  2.x8i  “ “ “ i°. 

12.6  2.973  “ “ “ i°. 

Consequently,  products  of  combustion  of  1 lb.  carbon  absorbs  2.973  units  of  heat 
in  producing  i°  temperature. 

Weiglit  and  Specific  Heat  of  Products  of  Cornbnistion, 
and  Temperature  of  Combustion.  (2>.  K.  Clark.) 


1 Lb.  of  Combustible. 


Gaseous  Products  for  1 Lb.  of  Combustible. 


Weight. 

Mean 

specific 

Heat. 

Heat  to  raise 
the  Tempera- 
ture i°. 

Temperature  of 
Combustion. 

Lbs. 

Water  = 1. 

Units. 

0 

Ratio. 

bo 

.302 

10.814 

5744 

100 

11.97 

. 256 

3- 063 

5305 

92 

15-9 

•257 

4.089 

5219 

9i 

13.84 

.256 

3-54 

5093 

88.7 

11.94 

.246 

2-935 

4879 

85 

12.6 

.236 

2-973 

4877 

85 

15.21 

•257 

3-9*4 

4826 

84 

10.09 

.27 

2.68 

4825 

84 

18.4 

.268 

4-933 

4766 

83 

5-35 

.211 

1.128 

3575 

62 

12. 18 

•257 

3.127 

3470 

60 

22.64 

.242 

5-478 

2614 

45 

Hydrogen 

Sulphuric  ether 

Olefiant  gas  (Bi-carburetted  hyd.) 

Tallow 

Coal  (average) 

Carbon,  or  pure  coke 

Wax 

Alcohol 

Light  carburetted  hydrogen 

Sulphur 

Turpentine  

Coal,  with  double  supply  of  air. . 

Whence  it  appears,  that  mean  specific  heat  of  products  of  combustion,  omitting 
hydrogen  .302  and  sulphur  .211,  is  about  .25. 


Hence,  To  Ascertain  Temperature  of  Combustion. — Divide  total  heat  of 
combustion  in  units  by  units  of  heat  for  x°,  and  quotient  will  give  tem- 
perature. 


COMBUSTION. 


463 

Illustration.— What  is  temperature  of  combustion  of  coal  of  average  composi- 
tion? 

Gaseous  products  as  per  preceding  table  11.94,  which  X .246  specific  heat  = 2.935 
units  of  heat  at  i°. 

Hence,  14  133  units  of  combustion  (from  table,  page  461)  -4-  2.935  = 48129  temper- 
ature of  combustion  of  average  coal. 

If  surplus  air  is  mixed  with  products  of  combustion  equal  to  volume  of  air  chem- 
ically combined,  total  weight  of  gases  for  one  lb.  of  this  coal  is  increased  to  22.64. 
See  following  table,  having  a mean  specific  heat  of  .242. 

Then  22.64  X .242  = 5.478  units  for  i°. 

Hence,  14  133  total  heat  of  combustion  -4-  5.478  = 2614°  temperature  of  combus- 
tion, or  a little  more  than  half  that  of  undiluted  products. 

Taking  averages,  it  is  seen  that  the  evaporative  efficiency  of  coal  varies 
directly  with  volume  of  constituent  carbon,  and  inversely  with  volume  of 
constituent  oxygen  ; and  that  it  varies,  not  so  much  because  there  is  more  or 
less  carbon,  as,  chiefly,  because  there  is  less  or  more  oxygen.  The  per-cent- 
ages  of  constituent  hydrogen,  nitrogen,  sulphur,  and  ash,  taking  averages, 
are  nearly  constant,  though  there  are  individual  exceptions,  and  their  united 
effect,  as  a whole,  appears  to  be  nearly  constant  also. 

Ileat  of  ComlDvistion. 

Or,  number  of  times  in  combustion  of  a substance , its  equivalent  weight  of  water 
would  be  raised  i°,  by  heat  evolved  in  combustion  of  substance. 

Alcohol 12930  I Ether 16  246  I Olefiant  gas 21  340 

Charcoal 14  545  I Olive  oil 17750  | Hydrogen 62030 

Combustion  of  IT.tiel, 

Constituents  of  coal  are  Carbon , Hydrogen , Azote , and  Oxygen. 

Volatile  products  of  combustion  of  coal  are  hydrogen  and  carbon,  the 
unions  of  which  (relating  to  combustion  in  a furnace)  are  Carburetted 
hydrogen  and  Bi-carburetted  hydrogen  or  Olefiant  gas , which,  upon  com- 
bining with  atmospheric  air,  becomes  Carbonic  acid  or  Carbonic  oxide , 
Steam,  and  uncombined  Nitrogen. 

Carbonic  oxide  is  result  of  imperfect  combustion,  and  Carbonic  acid 
that  of  perfect  combustion. 

Perfect  combustion  of  carbon  evolves  heat  as  15  to  4.55  compared 
with  imperfect  combustion  of  it,  as  when  carbonic  oxide  is  produced. 

1 lb.  carbon  combines  with  2.66  lbs.  of  oxygen,  and  produces  3.66  lbs. 
of  carbonic  acid. 

Smoke  is  the  combustible  and  incombustible  products  evolved  in  combustion  of 
fuel,  which  pass  off  by  flues  of  a furnace,  and  it  is  composed  of  such  portions  of 
hydrogen  and  carbon  of  the  fuel  gas  as  have  not  been  supplied  or  combined  with 
oxj7gen,  and  consequently  have  not  been  converted  either  into  steam  or  carbonic 
acid;  the  hydrogen  so  passing  away  is  invisible,  but  the  carbon,. upon  being  sepa- 
rated from  the  hydrogen,  loses  its  gaseous  character,  and  returns  to  its  elementary 
state  of  a black  pulverulent  body,  and  as  such  it  becomes  visible. 

Bituminous  portion  of  coal  is  converted  into  gaseous  state  alone,  carbonaceous 
portion  only  into  solid  state.  It  is  partly  combustible  and  partly  incombustible. 

To  effect  combustion  of  1 cube  foot  of  coal  gas,  2 cube  feet  of  oxj7gen  are  required* 
and,  as  10  cube  feet  of  atmospheric  air  are  necessary  to  supply  this  volume  of  oxy- 
gen, 1 cube  foot  of  gas  requires  oxygen  of  10  cube  feet  of  air. 

In  furnaces  with  a natural  draught,  volume  of  air  required  exceeds  that 
when  the  draught  is  produced  artificially. 

An  insufficient  supply  of  air  causes  imperfect  combustion : an  excessive 
supply,  a waste  of  heat. 


COMBUSTION. 


464 

Volume  of  atmospheric  air  that  is  chemically  required  for  combustion  of 
1 lb.  of  bituminous  coal  is  150.35  cube  feet.  Of  this,  44.64*  cube  feet  com- 
bine with  the  gases  evolved  from  the  coal,  and  remaining  105.71  cube  teet 
combine  with  the  carbon  of  the  coal. 

Combination  of  gases  evolved  by  combustion  gives  a resulting  volume 
proportionate  to  volume  of  atmospheric  air  required  to  furnish  the  oxygen, 
as  11  to  10.  Hence  the  44.64  cube  feet  must  be  increased  m this  proportion, 
and  it  becomes  44.64  + 4.46  = 49.1. 

Gases  resulting  from  combustion  of  the  carbon  of  coal  and  oxygen  of  the 
atmosphere,  are  of  same  bulk  as  that  of  atmospheric  air  required  to  furnish 
the  oxygen,  viz.,  105.71  cube  feet.  Total  volume,  then,  of  the  atmospheric 
air  and  gases  at  bridge  wall,  flues,  or  tubes,  becomes  105.71  + 49;1  — *54-8i 
cube  feet,  assuming  temperature  to  be  that  of  the  external  air.  Gonse- 
quentlv,  augmentation  of  volume  due  to  increase  of  temperature  of  a fur- 
nace is  to  be  considered  and  added  to  this  volume,  hi  the  consideration  of  the 
capacity  of  flue  or  calorimeter  of  a furnace. 

There  is  required,  then,  to  be  admitted  through  the  grates  of  a furnace  for 
combustion  of  1 lb.  of  bituminous  coal  as  follows  : 

Coal  containing  Soper  cent,  of  cartoon,  or  .7047  per  cent  of  coke. 

1 lb.  coal  X 44.64  cube  feet  of  gas = 44- 64 

.7047  lb.  carbon  x 150  cube  feet  of  air  . . . = 105- 71 

150.35  cube  feet 

For  anthracite,  by  observations  of  W.  R.  Johnston,  an  increase  of  30  per 
cent,  over  that  for  bituminous  coal  is  required  = 195.45  cube  feet. 

Coke  does  not  require  as  much  air  as  coal,  usually  not  to  exceed  108  cube 
feet,  depending  upon  its  purity. 

Heat  of  an  ordinary  furnace  may  be  safely  considered  at  iooo°  ; hence  air 
entering  ash-pit  and  gases  evolved  in  furnace  under  general  law  of  expan- 
sion of  permanently  elastic  fluids  of  ^ygths  of  its  volume  (or  .002087)  for 
each  degree  of  heat  imparted  to  it,  the  154.81  is  increased  m volume  from 
ioo°  (assumed  ordinary  temperature  of  air  at  ash-pit)  to  1000  — 9°°  » tnen 
900X  .002  087  = 1.8783  times,  or  154.81  + 154-81  X 1.8783  = 445.59  cube  feet. 

If  the  combustion  of  the  gases  evolved  from  coal  and  air  was  complete, 
there  would  be  required  to  give  passage  to  volume  of  but  445-59  cube  teet 
over  bridge  wall  or  through  flues  of  a furnace;  but  by  experiments  it  ap- 
pears that  about  one  half  of  the  oxygen  admitted  beneath  grates  of  a furnace 
passes  off  uncombined : the  area  of  the  bridge  wall,  or  flues  or  tubes,  must  con- 
sequently be  increased  in  this  proportion,  hence  the  445.59  becomes  691.10. 

Velocity  of  the  gases  passing  from  furnace  of  a proper-proportioned  boiler 
* rTI1  891.18 

may  be  estimated  at  from  30  to  36  feet  per  second.  Then  ^ — 

.00687  sq.  feet,  or  .99  sq.  ins.,  of  area  at  bridge  wall  for  each  lb.  of  coal  con- 
sumed per  hour. 

A limit,  then,  is  here  obtained  for  area  at  the  bridge  wall,  or  of  flues  or 
tubes  immediately  behind  it,  below  which  it  must  not  be  decreased,  or  com- 
bustion will  be  imperfect.  In  ordinary  practice  it  will  be  found  advan- 
tageous to  make  this  area  .014  sq.  feet,  or  2 sq.  ins.  for  every  lb.  ot  bitu- 
minous coal  consumed  per  sq.  foot  of  grate  per  hour,  and  so  on  111  proportion 
for  any  other  quantity. 

Volumes  of  heat  evolved  are  very  nearly  same  for  same  substance,  what- 
ever temperature  of  combustible. 


COMBUSTION. 


465 


Relative  Volumes 


Lbs.  ! 

Warlich’s  patent.. . . 13.1  I 

Charcoal n.16 

Coke 11.28  I 


of  Air  required  for  Combustion  of  Fuels. 

Lbs. 


Lbs. 


Anthracite  Coal 12.13 

Bituminous  “ 10.98 

Bitum.  Coal,  average  ic.7 


Bitum.  Coal,  lowest. . 5.92 


Peat,  dry 7.08 

Wood,  dry 6 


Perfect  combustion  of  1 lb.  of  carbon  requires  11.18  lbs.  air  at  62  , and 
total  weight  = 12.39  lbs.  Total  heat  of  combustion  of  1 lb.  carbon  or  char- 
coal is  14500  thermal  units;  mean  specific  heat  of  products  of  combustion 
is  .25,  which,  multiplied  by  12.39  as  above  = 3-0975,  and  14  500*  --  3.0975  = 
4681°  temperature  of  a furnace,  assuming  every  atom  ot  oxygen  that  was 
ignited  in  it  entered  into  combination.  ^ 

If  however,  as  in  ordinary  furnaces,  twice  volume  of  air  enters,  then 
products  of  combustion  of  1 lb.  of  coal  will  be  12.39  + I1:?8.— 2^57’  whlch’ 
multiplied  by  its  specific  heat  of  .25  as  before,  and  if  divided  into  14500, 
quotient  will  be  2641°,  which  is  temperature  of  an  ordinary  furnace. 

Ratio  of  Combustion— Quantity  of  fuel  burned  per  hour  per  sq.  foot  of 
grate  varies  very  much  in  different  classes  of  boilers.  In  Cornish  boileis  it 
is  3.5  lbs.  per  sq.  foot ; in  ordinary  Land  boilers,  10  to  20  lbs. ; (English)  13 
to  14  lbs. ; in  Marine  boilers  (natural  draught),  10  to  24  lbs. ; (blast)  30  to 
60  lbs. ; and  in  Locomotive  boilers,  80  to  120  lbs. 

Volumes  of  air  and  smoke  for  each  cube  foot  of  water  converted  into 
steam,  is  for  coal  and  coke  2000  cube  feet,  for  wood  4000  cube  feet ; and  for 
each  lb.  of  fuel  as  follows : 

Coal 207  | Canuel  coal...  315  | Coke 216  | Wood 173 

Calorific  power  of  1 lb.  good  coal  = 14  000  X 772  = 10  808  000  lbs- 


Relative  Evaporation  of*  Several  Conabnstibles  in.  Lbs. 
of  Water,  Heated  1°  by  1 Lb.  of  Material. 


Combustible. 

Composition. 

Water. 

Lbs. 

Alcohol 812 

(Hyd.  .12) 
i Carb.  .45j 

8 120 

Bituminous  coal. . . 

(Hyd.  .04) 
I Carb.  .75  J 

9830 

Carbon 

14  220 
9028 
5o854 

Cokfi 

Carb.  .84 

Hydrogen  (mean). . 

Oak  wood,  dry 

(Hyd.  .06) 
(Carb.  .53} 

6 018 

“ “ green... 

(Hyd.  .o8> 
( Carb,  .37/ 

5662 

Combustible.  I Composition. 


Olive  oil 

Peat,  moist 

“ dry 

Pine  wood,  dry. . . . 
Sulphuric  ether.  .7 
Tallow 


Hvd.  .13 
Carb.  .77 
( Hyd.  .04 
(Carb.  .43 
f Hyd.  .06 
(Carb.  .58 
Hyd.  .06 
Carb.  .7 
Hyd.  .13 
Carb.  .6 


Water. 

Lbs. 

14560 

3481 

3900 

3618 

8680 

14  560 


1 lb.  Hydrogen  will  evaporate  62.6  lbs.  water  from  2120  = 60.509  lbs.  heated  i°. 

1 lb.  Carbon  “ 14.6  lbs.  2120,  or  raise  12  lbs.  water  at 

6o°  to  steam  at  120  lbs.  pressure. 

1 lb.  of  Oxygen  will  generate  same  quantity  of  heat  whether  in  combustion  with 
hydrogen,  carbon,  alcohol,  or  other  combustible. 


Relative  Volumes  of  Gases  or  Products  of  Combustion  per  Lb.  of  Fuel. 


Supply  of  Air  per  lb. 

of  Fuel. 

Supply  of  Air  per  lb. 

of  Fuel. 

Temp. 

12  lbs. 

18  lbs. 

24  lbs. 

Temp. 

12  lbs. 

18  lbs. 

24  lbs. 

Air. 

Volume 

Volume 

Volume 

Air. 

Volume 

Volume 

Volume 

per  lb. 

per  lb. 

per  lb. 

per  lb. 

per  lb. 

per  lb. 

O 

Cube  Feet. 

Cube  Feet. 

Cube  Feet. 

O 

Cube  Feet. 

Cube  Feet. 

Cube  Feet. 

32 

150 

225 

300 

572 

3H 

471 

628 

68 

161 

241 

322 

752 

369 

553 

738 

104 

172 

258 

344 

1 1 12 

479 

718 

957 

212 

205 

3°7 

409 

1472 

588 

882 

1176 

392 

259 

389 

5i9 

2500 

906 

1359 

1812 

* Mean  of  all  experiments  13964. 


466  COMBUSTION. EXCAVATION  AND  EMBANKMENT. 


To  Compute  Consumption  of  Fuel  to  Heat  ^Air. 

Rule.— Divide  volume  of  air  to  be  heated  by  volume  of  1 lb.  of  it,  at  its 
temperature  of  supply ; multiply  result  by  number  of  heat-units  necessary 
to  raise  1 lb.  air  through  the  range  of  temperature  to  which  it  is  to  be  heated, 
and  product,  divided  by  number  of  heat-units  of  fuel  used,  will  give  result 
in  lbs.  per  hour. 


Example.— What  is  required  consumption  per  hour  of  coal  of  an  average  compo- 
sition to  heat  776400  cube  feet  of  air  at  540  to  1140? 

Coal  of  an  average  composition  (Table,  page  461)  = 14 133  heat-units.  Volume  of 

461  + 54. 


1 lb.  air  at  540  (see  formula,  page  522)  = - 


39- * 


- = 12.94  cube  feet.  1 X 1 14  — 54 


X *2377  (specific  heat  of  air)  = 14. 262  heat-units. 

776  4°°  X 14- 262  -i- 14 133  — 60. 55  Ms. 


12.94 


Loss  of  heat  by  conduction  of  it  to  walls  of  apartment  is  to  be  added  to  this. 


excavation  and  embankment. 

LaPor  aii cl  Work  upon  Excavation  and  Embankment. 

Elements  of  Estimate  of  Worlc  and  Cost. 

Per  Day  of  10  Hours. 


Cart.— One  horse.  Distance  or  lead  assumed  at  100  feet,  or  200, feet  for 
a trip , at  a speed  of  200  feet  per  minute . 


, V,  V ^ 

Earths  — Of  gravelly,  loam,  and  sandy,  a laborer  will  load  per  day  into  a 
cart  respectively  io,  12,  and  14  cube  yards  as  measured  in  embankment,  and 
if  measured  in  excavation,  .11  more  is  to  be  added,  in  consequence  of  the 
greater  density  of  earth  when  placed  in  embankment  than  in  excavation. 

Note. Earth  when  first  loosened,  increases  in  volume  about  .2,  but  when  settled 

in  embankment’it  has  less  volume  than  when  in  bank  or  excavation 


Carting. — Descending,  load  .33  cube  yard,  Level,  .28,  and  Ascending  .25, 
measured  in  embankment ; and  number  of  cart-loads  in  a cube  yard  of  em- 
bankment  are,  Gravelly  earth  3,  Loam  3.5,  and  Sandy  earth  4. 


Loosening. — Loam,  a three-horsed  plough  will  loosen  from  250  to  800  cube 
yards  per  day. 


Trimming— Cost  of  trimming  and  superintendence  1 to  2 cents  per  cube 
yard, 


Scooping.  — A scoop  load  measures  about  .1  cube  yard  in  excavation; 
time  lost  in  loading,  unloading,  and  turning,  1. 125  mmutes  per  load ; in 
double  scooping  it  is  1 minute.  Time  occupied  for  every  100  feet  of  dis 
tance  from  excavation  to  embankment,  1.43  minutes. 


Time  —Time  occupied  in  loading,  unloading,  awaiting,  etc.,  4 minutes  per 
load. 


To  Compute  Number  of  Loads  or  Trips  in  Cube  Yards  > 
per  Cart  per  Lay . 

/ 60  \ h+-v  = n.  E representing  average  distance  of  carting  from  em - 

\E  -T- 100  + 4/  _ - - 


hankmenufstalions  of  roofed  each,  y number  of  cart-loads  to  cube  yard  of  excava- 
tion, and  n number  of  cube  yards  in  embankment,  hauled  by  a cart  per  day  to  dis- 
tance  E. 


EXCAVATION  AND  EMBANKMENT. 


467 


Illustration.  — What  is  number  of  cube  yards  of  loam  that  can  be  removed  by 
one  cart  from  an  embankment  on  level  ground  for  an  average  distance  of  250  feet  ? 


Substituting  for  3,  3.5,  and  4 number  of  cart-loads  in  a cube  yard  of  embank* 
ment,  20, 17.14,  and  15,=  60  minutes,  divided  respectively  by  these  numbers. 


ascending,  h representing  number  of  hours  actually  at  work. 

To  Compute  Cost  of  Excavating  and.  Embanking  per 
Cube  Yard. 


in  different  earths , as  10,  12,  and  14,  c of  one  cart  and  driver  per  day , l cost  of  loosen- 
ing material  per  cube  yard , and  s cost  of  trimming  and  superintendence , both  per 
cube  yard , and  all  in  cents. 

Illustration.— Volume  of  excavation  in  loam  30000  cube  yards.  Level  carting 
650  feet  = 6.  s trips  or  courses.  Loosening  by  plough  1.7  cents  per  cube  yard, 
laborers  106  cents  per  day,  carts  160,  and  trimming  and  superintendence  1.5  cents 
per  cube  yard. 


Then  x*7  + i-5  = 8-833  + 9-797  + 1*7  + 1.5  = 21.83  cents  per  cube 


By  Carts. — A laborer  can  load  a cart  with  one  third  of  a cube  yard  of  sandy 
earth  in  5 minutes,  of  loam  in  6,  and  of  heavy  soil  in  7.  This  will  give  a result,  for 
a day  of  10  hours,  of  24,  20,  and  17.2  cube  yards  of  the  respective  earths,  after  de- 
ducting the  necessary  and  indispensable  losses  of  time,  which  is  estimated  at  .4. 

It  is  not  customary  to  alter  the  volume  of  a cart-load  in  consequence  of  any  dif- 
ference in  density  of  the  earths,  or  to  modify  it  in  consequence  of  a slight  inclina- 
tion in  the  grade  of  the  lead. 

In  a lead  of  ordinary  length  one  driver  can  operate  4 carts.  With  labor  at  $1 
per  day,  the  expense  of  a horse  and  cart,  including  harness,  repairs,  etc.,  is  $1.25 
per  day. 

A laborer  will  spread  from  50  to  100  cube  yards  of  earth  per  day. 

The  removal  of  stones  requires  more  time  than  earth. 

The  cost  of  maintaining  the  lead  in  good  order,  the  wear  of  tools,  superintend- 
ence, trimming,  etc.,  is  fully  2.5  cents  per  cube  yard. 

By  Wheel-barrows. — A laborer  in  wheeling  travels  at  the  rate  of  200  feet  per  min- 
ute, and  the  time  occupied  in  loading,  emptying,  etc.,  is  about  1.25  minutes,  with- 
out including  lead.  The  actual  time  of  a man  in  wheeling  in  a day  of  10  hours  is  .9 
or  2.25  minutes  per  lead  of  100  feet.  Hence, 

To  Compute  Number  of*  Barrow-Loads  removed  by  a 


By  Carts.—  Quarried  rock  will  weigh  upon  an  average  4250  lbs.  per  cube  yard, 
and  a load  may  be  estimated  at  .2  cube  yard,  and  weighing  a very  little  more  than 
a load  of  average  earth. 

Hence,  the  comparative  cost  of  carting  earth  and  rock  is  to  be  computed  on  the 
basis  of  a cube  yard  of  earth  averaging  3.5  loads  and  one  of  rock  5 loads,  with  the 
addition  of  an  increase  in  time  of  loading,  and  wear  of  cart. 


E = 250  -f- 100  = 2. 5,  and  y = 3. 5. 

X 10  -f-  3.5  = 26.37  cube  yards. 
6-5 


+ + 5 — L representing  pay  of  laborers , v value  or  result  of  loading 


yard. 


E ar tbxvor  Is , 


Laborer  per  Day. 


10  X 60  X -Q 

— ■ ■ = n.  n representing  number  of  leads  of  100  feet. 

1.25  w 

A barrow- load  is  about  .04  of  a cube  yard. 

Rock. 


468 


excavation  and  embankment. 


Labor. 

For  labor  of  a man,  see  Animal  Power,  pp.  433-34-  . mb.  feet  of 

By  Wheel-barrow.  - A barrow-load  may  be  assumed  at  .75  «*•  - a cube 

^Blasting  - When  labor  is  * . per  day,  hard  rock  in  ordinary  position  may  be 
blld"  hweve^in^onsequence'of  condition,  position,  etc., may  vary  from  so 
cents  to  $ 1. 

i^cubTyards^of  hard  rock  may  be  carted  per  day  over  a lead  of  roo  feet,  at  a cost 
Stone. 

Hauling  Stone.- A cart  drawn  by  horses  over  an  ordinary  road  will  travel  ,.15 

and  ^unloading1  by  hand^vhen  'labcmte*  $ 1 as  per  day  and  a horse  75  cents,  ,s  a5 
cents' per  pe4=L4-75  cube  whicb  he  moves  is  ,a5 

Work  done  by  an  ammans  not  impeded,  and  force  then  exerted  .45 

ff  S^ffoTc^he  animaS  exert  at  a dead  pull. 

Earthwork.  ( Molesworth .) 


proportion  vj 
culated  at  50  yards  run. 


Gett’s.|  Fill’s,  j Wheel’s. 


In  loose  earth,  sand,  etc. 

“ Compact 

“ Marl 


'■I 

Gett’s. 

Fill’s. 

Wheel’s. 

J 

In  Hard  clay 

I 

1.25 

1.25 

“ Compact  gravel 

I 

1 

1 

1 

“ Rock,  from 

1 3 

1 

z 

Sand  . 


Average  Weight  of  Earths,  Rocks,  etc. 
Per  cube  yard. 

Lbs. 


Marl 

Lbs.  1 

Sandstone  . . 
1 minlp  . . . . 

Lbs. 

. 4368 
. 4480 

Granite . 
Trap 

Clay 

J Chalk  . . . . 

1 Quartz 

Slate 

Lbs. 


4700 


4700 
4710  ; 


liworii,  

sumed  at  1. 
When  in  Embanlcment. 


Rock,  large. 

Medium 

Metal 


I Sand  and  gravel. 

. J of 


T c ^ana  auu  ; \ A 

,'L  to  1.3  Clay  and  earth  after  subsidence. . 

3 3 \ u u before 

1.2  1 


1.07 

1.08 
, 1.2 


\ 


FRICTION. 


469 


FRICTION. 

Friction  is  the  force  that  resists  the  bearing  or  movement  of  one  sur- 
face over  another,  and  it  is  termed  Sliding  when  one  surface  moves 
over  another,  as  on  a slide  or  over  a pin ; and  Rolling  when  a body  ro- 
tates upon  the  surface  of  some  other,  as  a wheel  upon  a plane,  so  that 
new  parts  of  both  surfaces  are  continually  being  brought  in  contact  with 
each  other. 

The  force  necessary  to  abrade  the  fibres  or  particles  of  a body  is 
termed  Measure  of  friction  ; this  is  determined  by  ascertaining  what 
portion  of  the  weight  of  a moving  body  must  be  exerted  to  overcome 
the  resistance  arising  from  this  cause. 

Coefficient  of  Friction  expresses  ratio  between  pressure  and  resistance  of 
one  surface  over  or  upon  another,  or  of  surfaces  upon  each  other. 

Angle  of  Repose  is  the  greatest  angle  of  obliquity  of  pressure  between 
two  planes,  consistent  with  stability,  the  tangent  of  which  is  the  coefficient 
of  friction. 

Experiments  and  Investigations  have  adduced  the  following  observations 
and  results : 

1.  Amount  of  friction  in  surfaces  of  like  material  is  very  nearly  propor- 
tioned to  pressure  perpendicularly  exerted  on  such  surfaces. 

2.  With  equal  pressure  and  similar  surfaces,  friction  increases  as  dimen- 
sions of  surfaces  are  increased. 

3.  A regular  velocity  has  no  considerable  influence  on  friction ; if  velocity 
is  increased  friction  may  be  greater,  but  this  depends  on  secondary  or  inci- 
dental causes,  as  generation  of  heat  and  resistance  of  the  air. 

M.  Morin’s  experiments  afford  the  principal  available  data  for  use.  Though  con- 
stancy of  friction  holds  good  for  velocities  not  exceeding  15  or  16  feet  per  second 
yet,  for  greater  velocities,  resistance  of  friction  appears,  from  experiments  of  M.’ 
Poiree,  in  1851,  to  be  diminished  in  same  proportion  as  velocity  is  increased. 

4.  Similar  substances  excite  a greater  degree  of  friction  than  dissimilar. 
If  pressures  are  light,  the  hardest  bodies  excite  least  friction. 

5.  In  the  choice  of  unguents,  those  of  a viscous  nature  are  best  adapted  for 
• rough  or  porous  surfaces,  as  tar  and  tallow  are  suitable  for  surfaces  of  woods, 

i and  oils  best  adapted  for  surfaces  of  metals. 

6.  A rolling  motion  produces  much  less  friction  than  a sliding  one. 

7.  Hard  metals  and  woods  have  less  friction  than  soft. 

8.  Without  unguents  or  .lubrication,  and  within  the  limits  of  33  lbs.  press- 
ure per  sq.  inch,  the  friction  of  hard  metals  upon  each  other  may  be  esti- 
mated generally  at  about  one  sixth  the  pressure. 

9.  Within  limits  of  abrasion  friction  of  metals  is  nearly  alike. 

10.  With  greatly  increased  pressures  friction  increases  in  a very  sensible 
ratio,  being  greatest  with  steel  or  cast  iron,  and  least  with  brass  or  wrought 
iron. 

11.  With  woods  and  metals,  without  lubrication,  velocity  has  very  little 
influence  in  augmenting  friction,  except  under  peculiar  circumstances. 

1 2.  When  no  unguent  is  interposed,  the  amount  of  the  friction  is,  in  every 
case,  independent  of  extent  of  surfaces  of  contact ; so  that,  the  force  with 
which  two  surfaces  are  pressed  together  being  the  same,  their  friction  is  the 
same,  whatever  may  be  the  extent  of  their  surfaces  of  contact. 

13.  Friction  of  a body  sliding  upon  another  will  be  the  same,  whether  the 
body  moves  upon  its  face  or  upon  its  edge. 

Hr 


FRICTION. 


470 


14.  When  fibres  of  materials  cross  each  other,  friction  is  less  than  when 
they  run  in  the  same  direction. 

15.  Friction  is  greater  between  surfaces  of  the  same  character  than  be- 
tween those  of  different  characters.  # ... 

36  'With  hard  substances,  and  within  limits  of  abrasion,  friction  is  as 
pressure,  without  regard  to  surfaces,  time,  or  velocity. 

17  The  influence  of  duration  of  contact  (friction  of  rest)  varies  with  the 
nature  of  substances;  thus,  with  hard  bodies  resting  upon  each  other,  the 
£ a maximum  very  quickly ; with  soft  bodies,  very  slowly ; with 

wood  upon  wood,  the  limit  is  attained  in  a few  minutes  5 and  with  metal  on 
wood,  the  greatest  effect  is  not  attained  for  some  days. 

Coefficient  of  Friction  of  Journals. 

Diameters  from  2 to  4 ins.  Speeds  varied  as  1 to  4.  Pressure  up  to  2 tons. 
(From  data  of  M.  Morin.) 

Coefficient  * 


Surfaces  of  Contact. 


Journals. 

Cast  iron  on  cast  iron. 


Bearings. 


Cast  iron  on  gun  metal 

Cast  iron  on  lignum-vitse. . . . . 
Wrought  iron  on  cast  iron  . . 
Wrought  iron  on  gun  metal. 


Wrought  iron  on  lignum-vitse. 

Gun  metal  on  gun  metal 

Lignum-vitse  on  cast  iron 


Lubrication. 


( Olive  oil,  or  tallow. . 
(Unctuous  and  wet. . 
( Olive  oil,  or  tallow. . 
(Unctuous  and  wet.. 


pressure  = 
Ordinary 
Lubrication. 


(Slightly  unctuous . 
{Oil,  ( 


or  lard. 

(Lard  and  plumbago. 
Olive  oil,  or  tallow . . . 

! Olive  oil,  or  tallow . . 
Unctuous  and  wet.. 
Slightly  unctuous. . . 

( Oil. . . .* 

(Unctuous 

Oil 

Unctuous.. 


.07  to  .08 
.14 

.07  to  .08 
.16 
.18 


.14 

.07  to  .08 
.07  to  .08 
.19 

•25 


* Continuous  lubrication  reduces  the  coefficients  fully  one  half. 
Surfaces  of  Contact. 


Oak  on  oak 

Wrought  iron  on  oak  . 


Cast  iron  on  oak. . 


Leather  on  oak 

Leather  belt  on  oak  (flat). . . 
u “on  oak  pulley. 


Disposition  of  Fibres  and 
Lubrication. 


Parallel 


and  soaped 
“ wet 

it  “ soaped 

11  “ wet 

u “ soaped 

“ “ wet 

u “ dry 

Perpendicular  u “ 

.47  of  pressure,  and  over  turned  cast-iron  pulleys 


Coefficient 
pressure  = i. 


.16 

.26 


.22 

.19 

.29 

.27 

•47 


Leather  belts  over  wood  drums 
.28  of  pressure. 

Coefficients  of  Friction  of  Motion. 

Condition  of  Surfaces  and  Unguents. 


Hemp  cords,  etc.  . 
Metal  upon  wood. 


Wood  upon  wood . 


P. 

id 

>>-g 

r% 

| 

0 

02 

>» 

■2 

9 

•g 

s 

S-  -3 

0 g 

Q 

O 

H 

Q 

On  wood 

•45 

•33 

— 

— 

— 

— 

— 

On  iron. 

— 

•15 

— 

.19 

Mean . . . 

.18 

•3i 

.07 

.09 

.09 

.2 

•13 

; Raw 

•54 

• 3 6 

. 16 

— 

.2 

l Dry 

Mean . . . 

•34 

.42 

•31 

.24 

.14 

.06 

-07 

.14 

.08 

.2 

.14 

1 .36 

•25 

— 

1 -°7 

.07 

•15 

.12 

FRICTION. 


471 


Relative  Value  of  TJnguents  to  Reduce  Friction. 


Unguents. 

Wood 

upon 

Wood. 

Wood 

upon 

Metals. 

Metals 

upon 

Metals. 

Unguents. 

Wood 
upon 
W ood. 

Wood 

upon 

Metals. 

Metals 

upon 

Metals. 

Dry  soap 

•4 

.82 

•32 

.85 

.67 

.27 

•7 

1 -96 

Olive  oil 

Tallow 

1 

1 

•93 

.24 

1 

.8 

Lard  and  plumbago. 

Water 

.22 

.18 

To  Determine  Coefficient  of  Friction  of  Bodies. 

Place  them  upon  a horizontal  plane,  attach  a cord  to  them,  and  lead  it  in 
a direction  parallel  to  the  plane  over  a pulley,  and  suspend  from  it  a scale  in 
which  weights  are  to  be  placed  until  body  moves. 

Then  weight  that  moves  the  body  is  numerator,  and  weight  of  body  moved 
is  denominator  of  a fraction,  which  represents  coefficient  required. 

Illustration.— If,  by  a pressure  of  320  lbs.  friction  amounts  to  80  lbs.,  its  coeffi- 
cient of  friction  in  this  case  would  be  80  -4-  320  = . 25. 

Hence,  if  coefficient  of  friction  of  a wagon  over  a gravel  road  was  . 25,  and  the  load 
8400  lbs.,  the  power  required  to  draw  it  would  be  8400  X 25  = 2100  lbs. 

Coefficients  of  Axle  Friction.  ( M . Morin.) 


Condition  of  Surfaces  and  Unguents. 


Substances. 

Dry  and 
a iittle 
Greasy. 

Greasy 
and  wet 
with 
Water. 

Oil,  Tallov 

In  usual 
way. 

>r,  or  Lard. 

Continu- 

ously. 

Very  soft 
and  puri- 
fied Car- 
riage 
Grease. 

Bell  met  ,1  upon  bell  metal 

.097 

Cast  iron  upon  bell  metal 

.194 

.161 

•075 

•054 

.065 

Cast  iron  upon  cast  iron 

.079 

•075 

•054 

Cast  iron  upon  lignum -vitse 

.185 

. 1 

.092 

.109 

Wrought  iron  upon  bell  metal 

.251 

.189 

•075 

•054 

.09 

Wrought  iron  upon  cast  iron 

•075 

•054 

Wrought  iron  upon  lignum-vitae 

1 188 

.125 

Friction  of  a journal  of  an  axle  which  presses  on  one  side  only,  as  in  a 
worn  bearing,  is  less  than  when  it  presses  at  all  points,  the  difference  being 
about  .005. 

Friction  of  Axles. — With  axles,  friction  of  motion  has  alone  been  experi- 
mented upon.  When  weight  upon  axle  and  radius  of  its  journal  is  given, 
mechanical  effect  of  friction  may  be  readily  determined. 

The  mechanical  effect  absorbed  by,  or  of  friction,  increases  with  pressure 
or  weight  upon  journal  of  axle  and  number  of  revolutions. 

Friction  of  an  axle  is  greater  the  deeper  it  lies  in  its  bearing. 

If  journal  of  an  axle  lies  in  a prismatic  bearing,  as  in  a triangle,  etc., 
friction  is  greater,  as  there  is  more  pressure  on,  and  consequently  greater 
friction  in  contact:  in  a triangular  bearing  it  is  about  double  that  of  a cyl- 
indrical bearing. 

To  Compute  Mechanical  Effect  of  Friction,  on  Journal 
of  an  Axle. 

— f.  n representing  number  of  revolutions , and  r radius  of  journal 

. , 30 

in  feet. 

Illustration.— Weight  of  a wheel,  with  its  axle  or  shaft  resting  on  its  journals, 
is  360  lbs. ; diameter  of  Journals  2 ins. ; and  number  of  revolutions  30  ; what  is  me- 
chanical effect  of  the  friction,  the  coefficient  of  it  being  .16? 

3,1416  X 30  X X 360  X * _ 452.4  _ o8  lbc 

'30 


30 


FRICTION. 


472 

Bv  amplication  of  friction-wheels  (rollers)  friction  is  much  reduced  and 
mechamcal  effect  then  becomes,  when  weights  of  friction-wheels  are  disre- 
garded, 

p nf  W r r'  __  p rf  representing  radii  of  axles  of  friction-wheels, 

a’  radii  of  frMioTwltls,  and  a angle  of  lines  of  direction  between  axis  of  roller 
and  axis  of  friction-wheels.  !J)r„  _ F = F,  r 

When  a single  friction-wheel  is  used,  ^ X/W  — F,  an  r> 

and  its  shaft,  making  5 revolutions  per 

2.  wSls  ^chS^^^  ** 

use  of  the  friction- wheel? 

x;  32-^2Xi2_^  ^ circum.  of  wheel  = z 8.4  times  of  axle. 

Coefficient  of  friction  assumed  at  .075.  Hence  — 3g*  7-  = 58-59  lbs.= power 


2 10  x 3-Uij  „ 2 618  feet  = distance  passed 


at  circum.  to  overcome  friction  at  axle 
by  friction. 

Consequently, 2 618  X_1  - .2181  feet  = distance  passed  by  friction  in  one  second. 

Hence,  .2181  X 2250  (30000  X .075)  = 49°J25-  3-  * faction refer  red  to  circum. 

axle+by  radius  of  friction-wheel,  and  38. 4 X • 2 = 7- 68  -F xctlon  reJei  rea 

of  wheel,  and  « 98.145  = mechanical  effect  by  application  of  friction-wheel 

= a reduction  of  four  fifths. 

Friction  of  Pivots. 

Friction  on  Pivots  is  independent  of  their  velocity,  inches  m a 
degree  than  their  pressures,  and  approximates  very  nea  S 

and  axle  friction. 

Friction  on  Conical  Bearings  is  greater  than  with  like  elements  on  plan 

^Figure  of  point  of  a pivot,  as  to  its  acuteness,  affects  friction : with  great 
pressure  the  most  advantageous  angle  for  the  figure  ranges  from  30  to  45  , 
with  less  pressure  it  may  be  reduced  to  10  ana  12  . 

Relative  Value  of  Angles  of  Rivots. 

6o x | 1 50 66  | 45° 39 

Relative  Values  of  different  Materials  for  use  as  Pivots. 

83  I Granite i I Tempered  steel 44 

55  | Rock  crystal 76  I 

Friction  and.  Rigidity  of  Cordage. 
Experiments  by  Amonton  and  Coulomb,  with  an  apparatus  of  Amontons, 
furnish  the  following  deductions : ...  . 

i.  That  resistance  caused  by  stiffness  of  cords  about  the  same  or  like  pul- 
leys varies  directly  as  the  suspended  weight.  . 

2 That  resistance  caused  by  stiffness  of  cords  increases  not  only  in  direct 
propor^n  of  suspended  weights,  but  also  in  direct  proportion  of  dtameter 
of  the  cords. 


Agate. 
Glass . 


FRICTION. 


473 


Consequently,  that  resistance  to  motion  over  the  same  or  like  pulleys, 
arising  from  stiffness  of  cords,  is  in  direct  compound  proportion  of  suspend- 
ed weight  and  diameter  of  cords. 

3.  That  resistance  to  bending  varied  inversely  as  diameter  of  sheave  or 
drum. 

g I C t 

4.  That  complete  resistance  is  represented  by  expression  — - — . S rep- 
resenting constant  for  each  rope  and  sheave , expressing  stiffness  of  rope ; T 
tension  of  rope  which  is  being  bent , expressed  by  C T ; C constant  for  each 
rope  and  sheave ; and  d diameter  of  sheave , including  diameter  of  rope. 

5.  That  stiffness  of  tarred  ropes  is  sensibly  greater  than  that  of  white  ropes. 

Extending  results  obtained  by  Coulomb,  Morin  furnishes  following  for- 
mulas : 

For  White  Ropes:  12  w-f-d  (.002  15 -f-.ooi  77  n-\-. 0012  W)  = R.  For  Tarred 
Ropes : 12  ra-r-  d (.010  54  -j-  .0025  n -J-  .0014  W)  — R.  R representing  rigidity  in  lbs., 
n number  of  yarns,  d diameter  of  sheave  in  ins.  and  rope  combined , and  W weight 
in  lbs. 

Illustration. — Wliat  is  value  of  stiffness  or  resistance  of  a dry  white  rope  hav- 
ing a diameter  of  60  yarns,  which  runs  over  a sheave  6 ins.  in  diameter  in  the 
groove,  with  an  attached  weight  of  1000  lbs.  ? 

Assume  diameter  for  60  yarns  to  be  1.2  ins.  Then  12  x (.002 15  -f-.ooi  77  x 

7*  ^ 

60  -f-  .0012  X 1000)  = 100  X 1*308  35  = 130.835  lbs. 

Value  of  natural  stiffness  of  ropes  increases  as  the  square  of  number  of 
threads  nearly,  and  value  of  stiffness  proportional  to  tension  is  directly  as 
number  of  threads,  being  a constant  number.  Hence,  having  the  rigidity  for 
any  number  of  threads,  the  rigidity  for  a greater  or  lesser  number  is  readily 
ascertained. 

Wire  Ropes. 

W eisbach  deduced  from  his  experiments  on  wire  ropes  that  their  rigidity 
for  diameters  capable  of  supporting  equal  strains  with  hemp  ropes  is  con- 
siderably less. 

Wire  ropes,  newly  tarred  or  greased,  have  about  40  per  cent,  less  rigidity 
than  untarred  ropes. 

Rolling  Friction. 

Rolling  Friction  increases  with  pressure,  and  is  inversely  as  diameter  of 
rolling  body. 

For  rolling  upon  compressed  wood,./  = .019  to  .031. 

When  a Body  is  moved  upon  Rollers  and  Power  applied  at  the  Base  of  the  Body , 
(/+/')  — = F.  / andf'  representing  coefficients  of  friction  of  two  surfaces  upon 
which  rollers  act. 

When  Power  is  applied  at  Circumference  of  Roller,  f W — r =.  F. 

When  Power  is  applied  at  Axis  of  Roller,  / W -4-  r -r-  2 :=  F. 


Bearings  for  Bropeller  Shaft.  (Mr.  John  Penn.) 


Bearings. 

Pressure 

per 

Sq. Inch. 

Time 
of  Op- 
eration. 

Bearings. 

Pressure 

per 

Sq.  Inch. 

Time 
of  Op- 
eration. 

Babbit’s  metal  on  iron*. . . 

Lbs. 

1600 

Min. 

8 

Brass  on  ironf 

Lbs. 

675 

4480 

4000 

4000 

1250 

Min. 

60 

Box  on  brass 

4480 

448 

448 

448 

5 

Brass  on  iron  J 

Box  on  iron 

QO 

Lignum-vitse  on  brass  . . 
Snake-wood  on  brass  . . . 
Lignum-vitse  on  iron  . . . 

Brass  on  brass 

QO 

5 

Brass  on  iron 

3° 

5 

2160 

* Rolled  out.  f Abraded.  i Set  fast. 

R R* 


FRICTION. 


474 


Result  Of  Experiments  upon  Friction  of  Several  Instru- 
rn.en.ts.  (iv-  o.  Hail. ) 


Instrument. 

Friction. 

Velocity  ratio. 

Mechanical  1 
efficiency.  | 

Useful 

effect. 

F L 

2.21  +.5453 

2 

1.8 

Per  Cent. 

V 

64 

36 

2.36  +.238 
3.87  -f“  • I51 
.0  +.014 

6 

4 

a 3 sheaves. .•••••••••••• 

44  differential • . 

16 

6.1 

193 

70 

36 

Inclined  plane,  angle  170  2'. . . . 

.09  +.55 
.66  -f.007 

3-4 

414 

1.72 

116 

5i 

28 

WViof»l  and  Axlfi • • 

.204 + .043 
•5  + -i69 

2.46  +.  21 
.0 

31 

22 

70 

u u Barrel 

5-95 

5-55 

93 

u u Pinion 

8 

4.1 

23 

18 

1 87 

^r?t  

. 185 -j-. 008 

137 

03 

F representing  friction,  and  L load. 

Illustration  i.-If  it  is  required  to  ascertain  powCT  necessary  to raise  aoojba 

2 feet,  by  a single  movable  pulley,  200  X .5453  + *•"—  ' Hence,  for  appli- 

t&btl&bs.  mHencet?oPrPapplicaUon  of  I56.96  lbs.,  roo  or  63.7.  per  cent, 
are  effectively  employed. 


re  ejjtciiutby 

, _The  velocity  ratio  of  a crane  being  137,  and  its  mechanical  efficiency  8?’  a 
man  applying  26  lbs.  to  it  can  raise  87  X 26  = 2262  lbs. 


Application  of  preceding  Results. 
illustration.  - If  a vessel,  including  beTe 

iron  rollers,  running  on  cast-iron  rails? 

1000  X 10  _ iqo  fons  =power  required  to  draw  vessel  independent  of  faction. 


Ratio  of  friction  to  pressure  of  wrought  iron  on  cast,  in  an  axle  and  bearing, 

o„  Ratio  of  ditto  of  cast  iron  upon  cast,  say  .005. 

' Hence  .075  + .005  = .08  of  1000  tons  = 80  tons,  which,  added  to  100  tons  before  de- 
ducted,  gives  180  tons,  or  resistance  to  be  overcome. 


Power  or  effect  lost  by  friction  in  axles  and  their  bearing  may  be  ex- 
pressed  by  formula 

W/dr  = p / representi„g  coefficient  of  friction,  d diameter  of  axle  in  ins. , and 

23°  . . 

r number  of  revolutions  per  minute. 


8 ins. ; what  is  the  effect  ot  friction  ? 

20  000  X .07  X 8X_20‘_  973  qi  ibS' 


23° 

Hence  P u 4-  33000  = H>.  v representing  circumference  of  shaft  in  feet  x by  revo ■ , 
lutions  per  minute. 

The  power  or  effect  lost  by  friction  in  guides  or  slides  may  be  expressed 
by  following  formula : 

W/s  r ...  _.p  s representing  stroke  of  cross-head,  and  l length  of  con- 


6oXV(5  i2-*2)' 
necting  rod  in  feet. 


FRICTION. 


475 


Frictional  Resistances. 


Friction  of*  Steam-engines. 

Friction  of  Condensing  Engines  in  Lbs.  per  Sq.  Inch, 
of  Piston. 


Diameter 

of 

Cylinder. 

Oscillating 

and 

Trunk. 

Beam 

and 

Geared. 

Direct- 
acting  and 
Vertical. 

Diameter 

of 

Cylinder. 

Oscillating 

and 

Trunk. 

Beam 

and 

Geared. 

Direct- 
acting  and 
Vertical. 

10 

5 

6 

7 

50 

2-5 

2.7 

3-3 

15 

4 

5 

6 

60 

2.4 

2.6 

3 

20 

3-5 

4 

5 

70 

2-3 

2.5 

2.7 

25 

3 

3-6 

4-5 

80 

2 

2.3 

2.6 

30 

•3 

3-5 

4 

100 

1.6 

2.2 

2-5 

35 

2.6 

3 

3-5 

IIO 

i-5 

2 

2.1 

Experiments  upon  different  steam-engines  have  determined  that  friction, 
when  pressure  on  piston  is  about  12  lbs.  per  sq.  inch,  does  not  exceed  1.5  lbs., 
or  about  one  tenth  of  power  exerted. 

Friction  of  double  cylinder  (50-inch  diam.)  direct-acting  condensing  pro- 
peller engine  is  1.25  lbs.  per  sq.  inch  of  piston  = 10.3  per  cent,  of  total  power 
developed ; friction  of  load  is  .9  lbs.  per  sq.  inch  of  piston  = 7.5  per  cent,  of 
total  pressure;  and  friction  of  propeller  is  1.3  lbs.  per  sq.  inch  of  piston  = 
10.8  per  cent,  of  total  power  = 28.6  per  cent. 

Friction  of  double  cylinder  (70-inch  diam.)  inclined  condensing  water- 
wheel engine  with  its  load  is  15  per  cent,  of  total  power  developed. 

In  general,  when  engines  are  in  good  order,  their  efficiency  ranges  from  80 
per  cent,  for  small  engines  to  93  per  cent,  for  large. 

Power  required  to  work  air-pumps  is  5 per  cent.,  and  to  work  feed-pumps 
1 per  cent. 

Results  of  Experiments  upon  Friction,  of  Machinery. 

(Davison. ) 

Steam-engine,  vertical  beam,  one  tenth  its  power ; 190  feet  horizontal,  and 
180  feet  vertical  shafting,  with  34  bearings,  having  an  area  of  3300  sq.  ins., 
with  11  pair  of  spur  and  bevel  wheels ; 7.65  IP. 

Set  of  three-throw  Pumps,  6 ins.  in  diam.,  delivering  5000  gallons  per  hour 
at  an  elevation  of  165  feet;  4.7  EP,  or  about  13  per  cent. 

Two  pair  iron  Rollers  and  an  elevator,  grinding  and  raising  320  bushels 
malt  per  hour ; 8.5  IP. 

Ale-mashing  Machine,  800  bushels  malt  at  a time ; 5.68  IP. 

Archimedes  Screw  (ninety-five  feet),  15  ins.  in  diameter,  and  an  elevator 
conveying  320  bushels  malt  per  hour  to  a height  of  65  feet ; 3.13  IP. 

Friction  Clutch. — Driven  by  a leather  belt  14  ins.  in  width ; face  of  clutch 
5 ins.  deep ; broke  a cast-iron  shaft  6.5  ins.  in  diameter. 

Flax  Mill  (M.  Cornut,  1872). — Two  condensing  engines,  cylinders,  12.9 
ins.  x 44.3  ins.  stroke,  and  22  ins.  X 59.8  ins.  stroke.  Pressure  of  steam, 
50  lbs.  per  sq.  inch ; revolutions,  25  per  minute.  Friction  of  entire  machin- 
ery, 20  per  cent. 

With  vegetable  oil  and  hand  oiling  a steam  pressure  of  62  lbs.  per  sq. 
inch  was  required,  and  with  mineral  oil  and  continuous  oiling  a pressure  of 
50  lbs.  only  was  required. 

By  continuous  oiling,  a saving  of  44  per  cent,  was  effected  over  hand 

oiling. 


476 


FRICTION. 


ZETax  Mill. 

Power  required  to  Drive  Engine,  Shafting,  and.  entire 
Machinery.  {M.  Cornut.) 

Indicated  Horse-power. 

One  Machine 


Parts. 


Engines,  shafting,  and  belts 

4 cards •••••••, ; 

14  drawing  frames  (29  heads  or  156 

slivers) 

4 combing  machines. 

6 roving  frames  (330  spindles) 

20  spinning  frames. 

Dry  (1480  spindles) 

Wet  (2080  “ ) 

Total  1 50. 1 1 IP. 


Total. 

at  work. 

empty. 

Machines. 

30.41 

— 

— 

— 

8.42 

2.105 

1.423 

32 

7.19 

•093  4 

.0794 

15 

2.22 

•555 

.151 

78 

7.78 

.026  27* 

2-434  . 

7-3 

47-5 

.032 1* 

2.515 

21.6 

46-59 

.022  4* 

1.613 

19 

Effect  of 


* Per  100  spindles. 


Estimate  of  Horse's  Power—  2080  spindles,  wet,  34.4  per  IP,  long  fibre. 

640  “ dry,  20.1 

J40  “ “ 2±5  “ “ iow- 

3560  “ average,  23.7  “ “ 

The  IP  per  100  spindles  varies  inversely  as  sq.  root  of  their  number. 

Winding  Engine  ( G . H.  Daglish). 

Shafts  7^8  to  1740  feet  in  depth  ; cylinder  65  X 84  ins.  stroke  ; pressure^  of  steam, 
Ibsper^sq.  inch;  revolutions  12.5  per  minute;  mean  diamete ) of  di  um , 26  feet. 
E>  313.4;  effect  235  = 75  Per  cent. 

Tools.  {Dr.  Hartig). 


Single  shearing,  . . + ^ = IP  *>  drive  tool,  n representing  number  of 
cuts  per  minute,  t thickness  of  plate,  and  H>  to  shear,  a representing 


v/m-oo  - - 1980000 

area  of  surface  cut  or  punched  per  hour  in  sq.  and  F(ii66  + 1691 1)  a factor  ex- 

2>ressing  work  required  to  cut  or  shear  a surface  of  1 inch  squat  e. 

Illustration. — A shearing  machine  cutting  4648  sq.  ins.  of  surface  per  hour,  in 
plates  4 inch  thick,  required  .68  H>  to  run  and  4.3  to  operate  it,  equal  to  5 horses. 


Iron  Plate-bending 


85  000  b t2  l 


. 11  300  b t2  l 
— P for  cold  plates , and  


_ p for  red-hot  plates.  b,  t,  and  l representing  breadth , thickness , and  length  of  plate, 
r radius  of  curvature,  all  in  ins.,  and  P net  power  of  bending. 

Power  for  large  rolls  when  running  only  .5  to  6 IP. 

Ordinary  Cutting  Tools,  in  jVIeta.1. 

Materials  of  a brittle  nature,  as  cast  iron,  are  reduced  most  economically  in  power 
consumed  by  heavy  cuts;  while  materials  which  yield  tough  curling  sha1 ® 
more  economically  reduced  by  thinner  cuttings.  Following  formulas  apply  to  g 
cutting  work: 

Power  required  to  plane  cast  iron  is—  t 

I>lallillg  Cast  iron,  W lmss  + £L^=km  W representing  weight  of  cast 
iron  removed  per  hour,  in  lbs  , and  s average  sectional  area  of  shavings,  in  sq.  ins. 

Steel,  Wrought  iron,  and  Gun-metal,  with  cuts  of  an  average  character- 

stoel 112  W = EP  | Wrought  iron,  .o52W=IP  | Gun-metal,  .0127  IP 

N 

Planing  and  Molding. -Run  without  cutting.  — = I?-  N reP' 

resenting  sum  of  revolutions  of  all  the  shafts  per  minute. 


FRICTION. 


477 


Molding.  — Pine,  .0566  -f  — , and  Red  Beech,  088  95  -j-  ■ 00 J 31  = IP.  h rep- 

resenting depth  of  wood  cut  down  to  form  molding. 


Turning. — Steel,  .047  W = EP;  Wrought  iron,  .0327  W = 2*;  Cast  iron, 
.0314  W = BP. 

For  turning  off  metals,  power  required  is  less  than  for  planing,  and  it  is  ascer- 
tained that  greater  power  is  required  for  small  diameters  than  large. 


Light  Lathes , .05  -f- .0005  EP;  1 or  2 shafts,  .05  -f-  .0012  n = IP;  3 or  4 shafts, 

.05 -f- .05  n = EP.  Heavy  Lathes,  ,025 -J- .0031  n;  .025 -|- .053  w;  .025-]-.  18  n. 
n representing  number  of  revolutions  of  spindle  per  minute. 


Drilling.— Power  required  to  remove  a given  weight  of  metal  is  greater  than 
in  planing.  Volume  being  taken  in  place  of  weight. 

Holes  from  .4  to  2 ins.  in  diameter. 

Cast  iron,  dry.  V ^.0168 -f  :°°°67^=IP.  Wrought  iron, oil.  V ^.0168  + '-^^= IP. 

V representing  volume  removed  in  cube  ins.  per  hour , and  d diameter  of  hole. 

Without  gearing,  .0006  n-j-.ooos  n'\  with  gearing,  .0006  w-f-.ooi  n';  radial 
drills  without  gearing,  .0006  n + .004  n';  radial  drills  with  gearing,  .04  -j-  .0006  n -f- 
.004  n'.  n representing  number  of  revolutions  per  minute  of  gearing  shaft , and  n' 
of  drill. 

Slotting. — Stroke  8 ins.  .045  -j — IP.  n representing  number  of  strokes 

i per  minute , and  s stroke  in  ins. 

"Wood-sawing,  Circular — A cube  foot  of  soft  wood  and  half  a cube 
[ foot  of  hard,  reduced  to  sawdust,  requires  1 IP. 


Hard  wood,  = BP'.  Soft  wood,  — — = BP'.  A representing  area  in  sq.  feet 
and  EP'  horse-power  per  sq  foot,  both  cut  per  hour , and  c width  of  cut  in  ins. 

From  .4  to  4 ins.  in  diameter.—  Pine.  V ^ 000 125  -f-  - °Q^  = EP. 

Sc  ' 

Dry  pine  timber.  .004  28  -{-.0065  — = IP'.  S representing  stroke  of  saw  in  feet, 
and  f feed  per  cut  in  ins. 

= H?  for  horse  power  to  run  only  without  cutting,  d representing  diameter 
of  saw  in  ins. , and  n number  of  revolutions  per  minute. 


Net  power  required  to  cut  with  a circular  saw  is  proportional  to  volume  of  ma- 
terial removed.  For  a saw  cutting  hot  iron,  at  a circumferential  speed  of  7875  feet 

per  minute,  and  making  a cut  .14  inch  wide,  power  is  expressed  by  formulas 

.702  A = EP,  for  red-hot  iron.  1.013  A:=EP,  for  red-hot  steel. 

* A representing  sectional  area  of  surface  cut  through , in  sq.  feet. 

Vertical  Saw.  .004  28 + .0065  ~=IP  in  dry  pine  timber  per  sq.  foot 

per  hour.  S representing  stroke  of  saw  in  feet,  c width  of  cut  in  ins.,  and  f feed  of 
cut  in  ins. 

Band  Saw.  0034  + — EP'm Pine.  .00483+ '-^-^4=  IP 'inOak. 

10000/  T J 10000 / 

• 005  76  + 'ioooo  f ~ IP' in  Beech,  v representing  velocity  of  saw,  and  f rate  of  feed, 
in  feet  per  minute. 

Screw  Catting.  Screws,  = IP-  Taps, — = IP.  d representing 
. . . , , 04  '29  0 

diameter  m ins. , and  l length  cut  in  feet  per  hour. 

1 Machine  of  medium  dimensions,  .2  IP. 


478 


FRICTION. 


arindstones.  pC-  = IP.  p representing  pressure  upon  stone , v circum- 
ferential velocity  of  stone  iTfeet  per  minute , and  C coefficient  of  friction. 

Coefficients  of  Friction  between  Grindstones  and  Metals . 

Cast  iron,  .22  at  high  speed,  .72  at  low  speed;  Wrought  iron,  .44  at  high  speed, 
j at  low;  Steel,  .29  at  high  speed,  .94  at  low. 

Power  required  to  run  them  alone. 

Small. . 


Large 000  040  gdv  = W 

or 000 128  d 2 n t=  IP 


.16 + .000  089  5 d v — IP 
.16-]-. 000 28  d2  n = IP 


Grrain  Conveyers. 

, Mg  tssa?  «? 

55?5FBJ&,Srjp.*  A™. 

!fper Tent  0 ?Sof  s4crewP  At  speeds  above  60  turns  per  minute,  the  gram  will 
not  advance,  but  will  revolve  with  screw. 


Sectional  area  of  body  of  grain  moved 
Dove  60  turns  per  minute,  the  g " 

Screw  Steamer.  {Vice-admiral  C.  R.  Moorsom,  E.  N.) 


Moving  friction  of  hull 07 

Moving  friction  of  load 003 

Moving  friction  of  rotation  of)  . . .09 
blades  of  screw ) 


Slip  of  screw *7* 

Resistance  of  hull boo 


Side  Lever  Steam-engine.  (J.  V.  Merrick.) 
In  Pressure  of  Steam. 

Friction  to  work  air-pump 

Friction  of  weight  of  parts 

Friction  of  cylinder  packing 

£1^0^  


.585  to 

.7  lb. 

•5  “ 

’5  a 

•I5, 

•3 

.046 

'°9l 

, .169  u 

.258  “ 

i-45 

1.85  lbs. 

Hencc  = ,.65  lbs.  per  sq.  inch.  If  journals  are  kept  constantly  lubri- 


1.45  + 1.85 . 


cated,  as  with  automatic  lubrications,  friction  of  weight ^1  .^reducedto  .*h«jd 

engln^without.6 load^^Frfction "of  ^ioadf  according  as  journals  are  lubricated,  ends  ; 
keyed  up,  etc.,  will  range  from  2 to  5 per  cent. 

Locomotives  and  Railway  Trains.  See  Railways,  page  682. 

Friction  developed  in  Launching  of  Vessels. 

°“ne  Railway.- To  draw  3000  tons  upon  greased  slides  a power  of  250  tons  was 
no"  toStovS  it,  but  when  started  i5o  tons  would  draw  ,t. 

Woollen  Machinery.  (Dr.  Hartig.)  When  runn.ng  empty  8.15  IIP,  and  at  work 

^The  efficiency  of  the  various  machines  averaging  60.5  per  cent. 

Friction  of  a Non-condensing  Steam-engine. 

Friction  of  an  Engine.  Diameter  of  cylinder  20  ins.  by  40  ins.  stroke  of  p.ston 

Revolutions,  15  to  70  per  minute.  — T 86  to  8.60  IP. 

Fngine,  unloaded,  2 lbs.  per  sq.  inch — . ••  _ ‘ 6 t0  IQ()1  “ 

Shafting,  unloaded,  2.5  to  45  lbs.  per  sq.  inch — ^ 9-  « 

Total  4. 5 to  6.5  lbs.  per  sq.  inch - 4 22  3 


FUEL. 


479 


FUEL. 


With  equal  weights,  where  each  kind  is  exposed  under  like  advan- 
tageous circumstances,  that  which  contains  most  hydrogen  ought,  in  its 
combustion,  to  produce  greatest  volume  of  flame.  Thus,  pine  wood  is 
preferable  to  hard,  and  bituminous  to  anthracite  coal. 

When  wood  is  used  as  a fuel,  it  should  be  as  dry  as  practicable. 
To  produce  greatest  quantity  of  heat,  it  should  be  dried  by  direct  ap- 
plication of  heat ; usually  it  has  about  25  per  cent,  of  water  combined 
with  it,  heat  necessary  for  evaporation  of  which  is  lost. 

Different  fuels  require  different  volumes  of  oxygen ; for  different 
kinds  of  coal  it  varies  from  1.87  to  3 lbs.  for  each  lb.  of  coal.  60  cube 
feet  of  air  is  necessary  to  furnish  1 lb.  of  oxygen ; and,  making  a due 
allowance  for  loss,  nearly  go  cube  feet  of  air  are  required  in  furnace  of 
a boiler  for  each  lb.  of  oxygen  applied  to  combustion. 


Lignite.  Brown  Coal  or  Bituminous  Wood. — Presents  a distinct  woody 
structure ; is  brittle,  and  burns  readily,  leaving  a white  ash,  and  contains 
and  absorbs  moisture  in  some  cases  fully  40  per  cent. 

Caking. — Fractures  uneven,  and  when  heated  breaks  into  small  pieces, 
which  afterwards  agglomerate  and  form  a compact  body.  When  the  pro- 
portion of  bitumen  is  great,  it  fuses  into  a pasty  mass.  This  coal  is  unsuit- 
ed where  great  heat  is  required,  as  the  draught  of  a furnace  is  impeded  by 
its  caking.  It  is  applicable  for  production  of  gas  and  coke. 

Splint  or  Hard. — Color  black  or  brown-black,  lustre  resinous  and  glisten- 
ing. It  kindles  less  readily  than  caking  coal,  but  when  ignited  produces  a 
clear  and  hot  fire. 

Cherry  or  Soft. — Alike  to  splint  coal  in  fracture,  but  its  lustre  is  more 
splendent.  Does  not  fuse  when  heated,  is  very  brittle,  ignites  readily,  and 
produces  a bright  fire  with  a yellow  flame,  but  consumes  rapidly. 

Cannel. — Color  jet,  or  gray  or  brown-black,  compact  and  even  texture,  a 
shining,  resinous  lustre.  Fractures  smooth  or  flat,  conclioidal  in. every  di- 
rection, and  polishes  readily. 

Experiments  upon  practical  burning  of  this  description  of  coal  in  furnace  of  a 
steam-boiler  give  an  evaporation  of  from  6 to  10  lbs.  of  fresh  water,  under  a pressure 
of  30  lbs.  per  sq.  inch  per  lb.  of  coal;  Cumberland  (Md.,  U.  S.)  coal  being  most  ef- 
fective, and  Scotch  least. 

Limit  of  evaporation  from  2120  for  1 lb.  of  best  coal,  assuming  all  of  heat 
evolved  from  it  to  be  absorbed,  would  be  14.9  lbs. 

Coals  that  contain  sulphur,  and  are  in  progress  of  decay,  are  liable  to  spontaneous 
combustion. 

There  are  very  great  variations  in  the  chemical  composition  and  proper- 
ties of  coals. 


Cannel. 

Shaly. 

Asphalt. 

Hard. 


Semi  or  gaseous. 


13  it  it  m i o s Coal. 


American. 

Carbon,  from  75  to  80  per  cent. 


Hydrogen,  from  5 to  6. 
Oxygen,  from  4 to  10. 
Nitrogen,  from  1 to  2. 
Sulphur,  from  .4  to  3. 
Ash,  from  3 to  10. 


Carbon,  from  70  to  91  per  cent. 
Hydrogen,  from  3. 5 to  nearly  7. 
Oxygen,  from  about  .5  to  20. 
Nitrogen,  from  a mere  trace  to  2.2. 
Sulphur,  from  o to  5. 

Ash,  from  .2  to  15. 

Coke,  from  49  to  93. 


British. 


Coke,  from  48.5  to  79.5. 


For  Volume  of  Air,  etc.,  see  Combustion,  page  465. 


480 


FUEL. 


Coal. 

Anthracite. 

Anthracite  or  Glance  Coal , or  Culm— Is  hard,  compact,  lustrous,  and  some- 
times iridescent,  most  perfect  being  entirely  free  from  bitumen;  it  ignites 
with  difficulty,  and  breaks  into  fragments  when  heated. 

Evaporative  power,  in  furnace  of  a steam-boiler  and  under  pressure,  is 
from  7.5  to  9.5  lbs.  of  fresh  water  per  lb.  of  coal. 

Coal  from  one  pit  will  sometimes  vary  6 per  cent,  in  evaporative  value. 


Elements  of  Various  American  Coals. 


Illinois,  Warren  Co 

Bureau  “ 

Mercer  “ 

Indiana,  Clay  “ 

Coopriders 

Pennsyl*  \ Connellsville. . . 
vania  j Youghiogheny . 

Fayette  Co 

Kentucky,  Sardric 

Mud  River 

Ohio,  Nelsonville 

Colorado,  Carbon  City 

Washington  Territory 


Specific 

Gravity. 


1.23 

1.32 

1.26 
1.28 
1.28 

1.28 
i-3 

1.29 

1.32 

1.28 

1.27 
1.21 
1.32 


Fixed  I Volatile  j 
Carbon.  I Matter.  | 


Moist- 

ure. 


Earthy 

Matter. 


Per 

Cent. 

5i-7 

57-6 

54-8 

56.5 

50.5 
65 
58.4 

58 

5i 

57 

58.4 

56.8 

58.25 


Per 

Cent. 

43-1 

28.8 

31.2 

32- 5 
42.5 
24 
35 
34 
42-5 
37 

33- 05 

34- 2 
31-75 


8-5 

3 

4-5 


3- 5 
6.65 

4- 5 
7 


Per 

Cent. 


II. 2 

8.4 


Per 

Cent. 

2.4 
5-6 
2-5 

4 

6.5 
5-6 

5 

4-5 

2-5 

1.9 

4-5 

3 


Per 

Cent. 

5-2 


Coke. 

Coke. — Coking  in  a close  oven  will  give  an  increase  of  yield  of  40  per  cent, 
over  coking  in  heaps,  gain  in  bulk  being  22  per  cent.  Coals  when  coked  in 
heaps  will  lose  in  bulk. 

Cannel  and  Welsh  (Cardiff)  coals  when  coked  in  retorts  will  gain  from  10 
to  30  per  cent,  in  bulk  and  lose  36.5  per  cent,  in  weight. 

Relative  costs  of  coal  and  coke  for  like  results,  as  developed  by  an  ex- 
periment in  a locomotive  boiler,  are  as  1 to  2.4. 

Evaporative  power  in  furnace  of  a steam-boiler  and  under  pressure,  is 
from  7.5  to  8.5  lbs.  of  fresh  water  per  lb. 

Bituminous  coal  will  vield  from  60  to  80  per  cent,  of  coke.  Averaging . 
66  per  cent.  It  is  capable  of  absorbing  15  to  20  per  cent,  of  moisture. 

Heat  of  combustion  lost  in  coking  of  bituminous  coal  40  per  cent. 

Charcoal. 

Charcoal  properly  termed,  is  not  made  below  a temperature  of  536°.  The 
best  quality  is  made  from  Oak,  Maple,  Beech,  and  Chestnut. 

Wood  will  furnish,  when  properly  burned,  about  23  per  cent,  of  coal. 

Charcoal  absorbs,  upon  an  average  of  the  various  kinds,  from  .8  Per  c^t* » 
of  water  for  Beech,  to  16.3  for  Black  Poplar,  Oak  absorbing  about  4.28,  and 
Pine  8.9.  . „ 

Evaporative  power,  in  furnace  of  a boiler  and  under  pressure,  is  5.5  los.^ 
of  fresh  water  per  lb.  of  coal. 

Volume  of  air  chemically  required  for  combustion  of  1 lb.  of  charcoal  is, 
when  it  consists  of  79  carbon,  129  cube  feet  at  62°. 

138  bushels  charcoal  and  432  lbs.  limestone,  with  2612  lbs.  of  ore,  will  pro- 
duce 1 ton  of  pig  iron. 


FUEL. 


481 


Produce  of  Charcoal  from  Various  Woods  dried  at  300°  and  Carbonized 
at  5720.  {M.  Violette.) 


Wood. 

Weight. 

Wood. 

Weight. 

Cork 

Per  Cent. 

62.8 

46.09 
44-25 
41.48 

40.9 

Larch 

Per  Cent. 
40.31 
36.06 
34-69 
34-59 
34-17 

Oak 

Chestnut .... 

Beech 

Apple .... 

Pine 

Elm 

Birch 

Poplar  roots 

Poplar 31.12  per  cent. 

In  a Green  or  Ordinary  State.  {Weight per  cent.) 


Wood. 

Weight. 

Maple 

Per  Cent. 

*2  7.  7 S 

Willow 

DD*  / D 

oo.  n a 

Black  elder. . . . . 
Ash 

DO*  /4 

33- 61 
33-28 
31.88 

Pear 

23-8  1 

I Birch 

1 Oak 

. 22.85 

I Red  Pine  . . , 

2 6-7 

Elm 

I “ young.. 

• 33-3 

White  Pine  . 

21. 1 | 

1 Maple 

1 Poplar 

. 20. 5 

1 Willow 

Apple . . , 

Ash 

Beech. . . 

It  appears  from  this  that  cork,  the  lightest  of  woods,  yields  largest  percentage 
of  charcoal,  about  63  per  cent. ; and  that  poplar  yields  lowest,  about  31  per  cent. 
There  does  not  appear  to  be  any  definite  relation  between  density  of  wood  and 
volume  of  yield.  J 

Produce  by  a slow  process  of  charring  is  very  nearly  50  per  cent,  greater  than  bv 
a quick  process.  3 * & y 

Lignite. 

Lignite  is  an  imperfect  mineral  coal.  It  is  distinguished  from  coal  by 
its  large  proportion  of  oxygen,  being  from  13  to  29  per  cent.  Its  specific 
gravity  ranges  from  1.12  to  1.35.  1 

Elements  of  Various  American  Lignites.  {W.  M.  Barr.) 

Location. 


Kentucky 

Blandville  . 
Washington  Terr’y . . 
Vancouver’s  Island. . 
Colorado,  Carbon  City 
Canon  City 

Arkansas 

Texas,  Robertson  Co. 


Spec. 

Grav. 


1. 17 

1.27 

1.28 

1.23 


Per  Cent. 
40 
3i 

S8.25 

62 

4I-25 

56.8 

34-5 

45 


Per  Cent. 
23 
48 

3i-75 

3i 

46 

34-2 

28.5 

39-5 


Water. 


Per  Cent. 
30 
H-5 
7 
4 

3' 5 
4-5 
32 


Per  Cent. 
7 

9-5 

3 

3 

9-25 

4'5 

5 

4-5 


Per  Cent. 
53 
59-5 
38.75 
35 

49- 5 

38.7 

60.5 

50- 5 


Per  C 
47 
40.  i 
61.5 

65 

50.= 
61.3 
39- } 
49-' 


Asplialt. 

Asphalt , alike  to  Lignite,  contains  a large  proportion  of  oxygen. 
"Wood. 

Wood , as  a combustible,  is  divided  into  two  classes,  the  hard,  as  Oak,  Ash 
Llm  Beech,  Maple,  and  Hickory,  and  soft,  as  Pine,  Cotton,  Birch,  Sycamore 
and  Chestnut.  J 

Green  wood  subjected  to  a temperature  ranging  from  340°  to  4400  wil] 
lose  30  to  45  per  cent,  of  its  weight.  44 

At  a temperature  of  300°,  Oak,  Asli,  Elm,  and  Walnut,  in  a comparatively 
seasoned  state,  lost  from  16  to  18  per  cent.  ^ 

Woods  contain  an  average  of  56  per  cent,  of  combustible  matter. 

an  ?naMs  ?f  M*  Violette  it  appears  that  composition  of  wood  is  about 
Qflmn  throu»h.out  tree^  that  of  the  bark  also ; that  wood  and  bark  have  about 
same  proportion  of  carbon  (49  per  cent.),  but  that  bark  has  more  ash  than  wood 

5 and7  per  cem  T°°^  haVG  ^ carbon  than  W00(i  (45  per  cent.),  and  more  ash| 

Leaves  when  dried  at  2120  lost  60  per  cent,  of  water,  and  branches  45  per  cent. 

S s 


FUEL. 


482 

Evaporative  power  of  1 cube  foot  of  pine  wood  is  equal  to  that  of  1 cube 
foot  of  fresh  water ; or,  in  the  furnace  of  a steam-boiler  and  under  pressure, 
it  is  4.75  lbs.  fresh  water  for  1 lb.  of  wood. 

Northern  Wood.— One  cord  of  hard  wood  and  one  cord  of  soft  wood,  such 
as  is  used  upon  Lakes  Ontario  and  Erie,  is  equal  in  evaporative  effects  to 
2000  lbs.  of  anthracite  coal. 

Western  Wood. — One  cord  of  the  description  used  by  the  river  steamboats 
is  equal  in  evaporative  qualities  to  12  bushels  (960  lbs.)  of  Pittsburgh  coal. 
9 cords  cotton,  ash,  and  cypress  wood  are  equal  to  7 cords  of  yellow  pme. 

Solid  portion  (lignin)  of  all  woods,  wherever  and  under  whatever  circum- 
stances of  growth,  are  nearly  similar,  specific  gravity  being  as  1.46  to  1.53. 
Densest  woods  give  greatest  heat,  as  charcoal  produces  greater  heat  than 

^For  every  14  parts  of  an  ordinary  pile  of  wood  there  are  n parts  of  space; 
or  a cord  of  wood  in  pile  has  71.68  feet  of  solid  wood  and  56.32  feet  of  voids. 

Trees  in  the  early  part  of  April  contain  20  per  cent,  more  water  than  they 
do  in  the  end  of  January. 

Ash. 


Woods. 

Wood. 

Leaves. 

Woods. 

Wood,  j 

Leaves. 

Per  Cent. 

Per  Cent. 

Elm 

Per  Cent. 
1.88 

Per  Cent. 

II. 8 

•35 

•34 

5-4 

5 

Oak 

.21 

4 

Birch 

Pitch  Pine 

•25 

3*15 

Feat. 

Peat  is  the  organic  matter,  or  soil,  of  bogs,  swamps,  and  marshes— decayed 
moss,  sedge,  coarse  grass,  etc.— in  beds  varying  from  i to  40  feet  in  depth. 
That  near  the  surface,  and  less  advanced  in  transformation,  is  light,  spongy , 
and  fibrous,  of  reddish-brown  color;  lower  down,  it  is  more  compact,  of  a 
darker  brown  color ; and,  in  lowest  strata,  it  is  of  a blackish  brown,  or  almost 
black,  of  a pitchy  or  unctuous  surface,  the  fibrous  texture  nearly  or  alto- 
gether transformed.  . A 0 , . . „ 

Peat  in  its  natural  condition,  contains  from  75  to  80  per  cent,  of  water. 
Occasionally  its  constituent  water  amounts  to  85  or  90  per  cent.,  in  which 
case  peat  is  of  the  consistency  of  mire.  It  shrinks  very  much  in  drying; 
and  its  specific  gravity  varies  from  .22  to  1.06,  surface  peat  being  lightest, 
and  deepest  peat  densest.  .....  .x  « . • • A„r 

When  peat  is  milled,  so  that  its  fibre  is  broken  up,  its  contraction  m dry- 
ing is  much  increased,  and  in  this  condition  it  is  termed  condensed.  # 

When  ordinarily  air  dried,  it  will  contain  20  to  30  per  cent,  of  moisture, 
and  when  effectively  dried  at  least  15  per  cent. 

Products  of  Distillation  of  Peat. 

Water  31.4.  Tar  2.8.  Gas  36.6.  Charcoal  29.2. 

The  distillation  of  the  tar  will  yield  paraffine,  oil,  gas,  water,  and  char- 
coal, and  the  water  acetic  acid,  wood  spirit,  and  chloride  of  ammonia. 

Evaporative  power,  in  furnace  of  a steam-boiler  and  under  pressure,  is 
from  3.5  to  5 lbs.  of  fresh  water  per  lb.  of  fuel. 

Taix. 

Tan,  oak  or  hemlock  bark,  after  having  been  used  in  the  process  of  tan- 
ning, is  combustible  as  a fuel.  It  consists  of  the  fibre  °f  bark,  an  , 
according  to  M.  Peclet,  5 parts  of  bark  produce  4 parts  of  dry  tan;  ana 
heating  power  of  it  when  perfectly  dry,  or  containing  but  15  per  cent,  of 
ash,  is  6100  units ; while  that  of  tan  in  an  ordinary  state  of  dryness,  con- 
taining  30  per  cent,  of  water,  is  4284.  Weight  of  water  evaporated  at  212 
by  1 lb.,  equivalent  to  these  units,  is  6.31  lbs.  for  dry,  and  4.44  for  moist. 


FUEL, 


483 


Relative  "Valxies  of*  different  Fuels. 


Description. 

Lbs.  of  Steam  from 
Water  at  2120 
by  1 lb.  of  Fuel. 

Relative  Evapora- 
tive Power  for 
equal  Weights. 

Relative  Evapora- 
tive Power  for 
equal  Volumes. 

Relative  Rapidi- 
ties of  Ignition. 

Relative  Freedom 
from  Waste. 

Relative.  Com- 
pleteness of 
Combustion. 

Relative 

Weights. 

Anthracites. 

Peach  Mountain,  Pa 

10.7 

X 

I 

•505 

•633 

•725 

•945 

Beaver  Meadow 

9.88 

•923 

.982 

.207 

.748 

.6 

1 

Bituminous. 

Newcastle 

8.66 

.809 

.776 

•595 

.887 

•346 

•9°4 

Pictou 

8.48 

.792 

•738 

.588 

.418 

1 

.876 

Liverpool 

7.84 

•733 

.663 

.581 

1 

•333 

.852 

Cannelton,  Ind 

7-34 

.686 

.616 

1 

.984 

•578 

. 848 

Scotch 

6-95 

.649 

.625 

.521 

•499 

.649 

•9°9 

Pine  wood,  dry 

4.69 

•436 

•175 

— 

16.417 

— 

— 

■Weights,  Evaporative  Rowers  per  Weight  and  Bulk, 
etc.,  of*  different  Fuels.  [W.  R.  Johnson  and  others.) 


Fuel. 

Specific 

Gravity. 

Weight 

per 

Cube  Foot. 

Steam  from 
Water  at 
2X2°  by  1 lb. 
of  Fuel. 

Clinker 
from  100  lbs. 

Cube  Feet 
in  a Ton. 

Bituminous. 

Lbs. 

Lbs. 

Lbs. 

No. 

Cumberland,  maximum 

I-3I3 

52.92 

10.7  . 

2.13 

42.3 

“ minimum 

1-337 

54-29 

9.44 

4-53 

41.2 

Duffryn 

1.326 

53-22 

10.14 

— 

42.09 

Cannel,  Wigan 

1.23 

48.3 

7.7 

— 

46.37 

Blossburgh 

1.324 

53-05 

9.72 

3-4 

42.2 

Midlothian,  screened 

1.283 

45-72 

8-94 

3-33 

49 

“ average 

1.294 

54-04 

8-39 

8.82 

41.4 

Newcastle,  Hartley 

1-257 

50.82 

8.76 

3-i4 

44 

Pictou 

i-3l8 

49-25 

8.41 

6.13 

45  Q 

Pittsburgh 

1.252 

46.81 

8.2 

•94 

47.8 

Sydney 

i-338 

47-44 

7-99 

2.25 

47.2 

Carr’s  Hartley 

1.262 

47.88 

7.84 

1.86 

46.7 

Clover  Hill,  Va 

1.285 

45-49 

7.67 

3.86 

49.2 

Cannelton,  Ind 

1-273 

47-65 

7-  34 

1.64 

47  0 

Scotch,  Dalkeith 

i-5i9 

51.09 

7.08 

5-63 

43-8 

Chili 

— 

— 

5-72 

— 

— 

Japan 

1-231 

48.3 

— 

— 

Anthracite. 

Peach  Mountain 

1.464 

53-79 

10. 1 1 

3-03 

41.6 

Forest  Improvement 

i-477 

53-66 

10.06 

.81 

41.7 

Beaver  Meadow 

i-554 

56.19 

9.88 

.6 

39-8 

Lackawanna 

1. 421 

48.89 

9-79 

1.24 

45-8 

Beaver  Meadow,  No.  3 

1. 61 

54-93 

9.21 

1. 01 

40.7 

Lehigh 

i-59 

55-32 

8-93 

1.08 

40.5 

Coke. 

Natural  Virginia 

1-323 

46.64 

8.47 

5-3i 

48.3 

Midlothian 

— 

32-7 

8.63 

10.51 

68.5 

Cumberland  

— 

31.6 

8.99 

3-55 

70.9 

Miscellaneous. 
Charcoal,  Oak 

i-5 

24 

5-5 

Ash. 

3.06 

104 

Peat 

•53 

30 

5 

— 

75 

Warlich’s  fuel 

i- 15 

6g.i 

xo.4 

2.91 

32-44 

Wylam’s  “ 

65 

8.9 

. — 

- — 

Pine  wood,  dry 

— 

21 

4-7 

•3i 

106.6 

484 


FUEL. 


Weights  and  Comparative  Values  of  different  Woods. 


Woods. 


Shell-bark  Hickory  . . . 
Red-heart  Hickory  . . . 

White  Oak 

Red  Oak 

Virginia  Pine 

Southern  Pine 

Hard  Maple 


Cord. 

Value. 

Woods. 

Cord. 

Value. 

Lbs. 

4469 

1 

New  Jersey  Pine 

Lbs. 

2137 

•54 

37°5 

.81 

Yellow  Pine 

I9°4 

•43 

3821 

.81 

White  Pine 

1868 

.42 

3254 

.69 

Beech 

— 

• 7 

2689 

.61 

Spruce  

— 

•52 

3375 

•73 

Hemlock 

— 

•44 

•33 

2878 

.6 

Cottonwood 

Liquid.  Fuels. 

Petroleum. 

Petroleum  is  a hydro-carbon  liquid  which  is  found  in  America  and  Europe. 
According  to  analysis  of  M.  Sainte-Claire  Deville,  composition  of  15  petro- 
leums from  different  sources  was  found  to  be  practically  constant.  Average 
specific  gravity  was  .87.  Extreme  and  average  elementary  composition  was 
as  follows : 


Carbon 82  to  87. 1 per  cent.  Average,  84.7  per  cent. 

Hydrogen 11.21014.8  “ “ 13.1  “ 

Oxygen 5 to  5.7  “ “ 2.2  “ 


100 

Its  heat  of  combustion  is  20240,  and  its  evaporative  power  at  2120  20.33. 

Petroleum  Oils — Are  obtained  by  distillation  from  petroleum,  and  are  com- 
pounds of  carbon  and  hydrogen,  in  average  proportion  of  72.6  and  27.4. 

Boiling-point  ranges  from  86°  to  495 °. 

Schist  Oil — Consists  of  carbon  80.3  parts,  hydrogen  11.5,  and  oxygen  8.2. 

Pine  Wood  Oil  — Consists  of  carbon  87.1  per  cent.,  hydrogen  10.4,  and 
oxygen  2.5. 

Coal-gas. 


Coal  Gas — As  furnished  by  Chartered  Gas  Co.  of  London  is  composed  as 
follows ; 


Carbon. 

Hydrogen. 

Oxygen. 

Hydrogen. 

Nitrogen. 

Olefiant  Gas,  ) 
Bi-carb.hyd. ) * * 
Marsh  gas,  ) 

Carb.  hyd.  ( * • • • 
Carbonic  oxide.... 

3.096 

•434 

Hydrogen ..... 
Oxygen 

.08 

51.8 

- 

26.445 

8.815 

Nitrogen 

.38 

3-84 

5-ii 

Total.. 

Heat  of  combustion  at  2120  52  961  units,  and  evaporative  power  47.51  lbs. 


Coal-gas.  (F  Harcourt.) 


Carb. 

Hyd. 

Oxy. 

Nit. 

Carb. 

Hyd. 

Oxy. 

Nit. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Olefiant  gas 

10.5 

*•7 

— 

— 

Hydrogen 

— ■ 

8.1 

— 

— 

Marsh  gas 

39-7 

13.2 

— 

— 

Nitrogen 

— 

— 

5-8 

Carbonic  oxide. . 

5-9 

7-9 

— 

Oxygen  

— 

— 

•3 

Carbonic  dioxide 

1.9 

— 

5 

— • 

Total 

58 

23 

13.2 

One  lb.  of  this  gas  had  a volume  of  30  cube  feet  at  62°  ; heat  of  combus- 
tion 22684  units;  and  of  one  cube  foot  756  units,  which  is  equivalent  to 
evaporation  of  .68  lb.  of  water  from  62°,  or  of  .78  lb.  from  2120  per  cube  foot. 


FUEL. 


485 


Average  C omp o s it  i on.  of  Panels . 


Specific 

Grav- 

ity. 

Carbon. 

Hydro- 

gen. 

Nitro- 

gen. 

Oxygen. 

Sul- 

phur. 

Ash. 

Bituminous  Coals. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Australian 

X-3X 

— 

— 

— 

— 

•5 

8.38 

Borneo 

1.28 

64.52 

4-74 

.8 

20.75 

i-45 

7-74 

British,  lowest 

— 

68.72 

4.76 

— 

18.63 

x-35 

— 

Boghead,  dry,  average 

1. 18 

63-94 

8.86 

.96 

4-7 

•32 

21.22 

Chili,  Conception  Bay 

1.29 

70-55 

5-76 

•95 

13.24 

1.98 

7-52 

“ Chiriqui 

— 

38.98 

4.01 

.58 

13-38 

6.14 

36.91 

Cannel,  Wigan 

1.23 

79- 23 

6.08 

1. 18 

7.24 

x-43 

4.84 

Cumberland.  Md 

1.31 

93.81 

1.82 

— 

2-77 

— 

1.6 

Coke,  Garesfield 

— 

97.6 

— 

— 

— 

.85 

i-55 

“ Durham 

— 

89-5 

— 

— 

— 

1.25 

9-25 

“ Average 

— 

93-44 

4.66 

— 

— 

1.22 

5-34 

Duffryn 

i-33 

88.26 

i-45 

.6 

1.77 

3.26 

Formosa  Island 

1.24 

78.26 

5-7 

.64 

10.95 

•49 

3-96 

88.56 

87-73 

4.88 

( 4-38) 
( 5-65  ) 

— 

2. 19 

“ caking 

1.29 

5.08 

— 

i-54 

“ long  flame 

i-3 

82.94 

5-35 

( 8.63) 

— 

3.08 

“ average* 

i-3i 

85 

4-5 

( 7 

) 

— 

3-5 

Indian,  average 

47-3 

— 

— I 

— 

— 

22.9 

“ Kotbec 

— 

90 

— 

— 

— 

— 

4 

Patagonia 

— 

62.25 

5-05 

•63 

1 17-54 

x-i3 

x3-4 

Russian,  Miouchit 

— 

9I-45 

4-5 

( 4-05) 

— ' 

— 

Sydney,  S.  W 

— 

82.39 

5-32 

1.27  | 

I 8.32 

.07 

2.04 

Splint,  Wvlam 

il  Glasgow 

— 

74.82 

6.18 

( 5.09) 

— 

I3-91 

— 

82.92 

5-49 

(10.46) 

— 

x*x3 

“ Cannel,  Lancashire 

— 

83-75 

5-66 

( 8.04) 

— 

2-55 

**  “ Edinburgh 

— 

67.6 

5-4 

( 12.43) 

■ — 

x4-57 

“ Cherry,  Newcastle. 

— 

84.85 

5-o5 

( 8.43) 

— 

1.67 

“ Caking,  Garesfield 

— 

87-95 

5-24 

( 5.42) 

— 

x-39 

“ Ebbro  Vale,  Welsh 

“ Llangenneck  u 

— 

89.78 

5-i5 

2. 16 

•39 

1.02 

x-5 

— 

84.97 

4.26 

i-45 

3-5 

.42 

5-4 

Vancouver’s  Island 

— 

66.93 

5-32 

if  02 

8-7 

2.2 

x5-83 

Anthracites. 

Anthracite 

i-5 

88.54 

— 

— 

— 

•52 

8.67 

French 

i-5 

86. 17 

2.67 

( 2.85) 

— 

8. 56 

Russian 

96.66 

x-35 

( 1 

99) 

— 

— 

Woods. 

Beech 

— 

50.17 

6.12 

1.05 

40.38 

— 

1.77 

Birch 

— 

48.12 

6-37 

1.15 

43-95 

. 

.48 

Oak 

— 

48.13 

5-25 

.82 

44-5 

. — 

x-3 

White  Pine 

— 

49-95 

6.41 

— 

43.65 

— 

.31 

Woods,  average 

-r- 

49-7 

6.06 

1.05 

4J-3 

— 

1.8 

Charcoal. 

Oak 

— 

87.68 

2.83 

— 

6-43 

— 

3.06 

Pine 

— 

71-36 

5-95 

— 

22.19$ 

— 

•4 

Maple 

— 

70.07 

4.61 

— 

24.89$ 

— 

•43 

Miscellaneous. 

Asphalt 

1.06 

79.18 

9-3 

( 8.72) 

— 

2.8 

Lignite,  perfect 

“ imperfect 

1.29 

69.02 

5-05 

( 20 

.12) 

• — 

5.82 

1.25 

60. 18 

5-29 

(29.03) 

— ■ 

5-57 

“ bituminous 

1. 18 

74. 82 

7-36 

(13.38) 

— ■ 

4-45 

“ Colorado 

1.28 

56.8 

— 

— 

— 

4-5 

“ Kentucky 

1.2 

40 

— 

— 

— 

— 

7 

“ Arkansas 

— 

34-5 

— 

— 

-r- 

— 

5 

Peat,  dense 

— 

61.02 

5-77 

.81 

32.4 

— 

“ Irish,  average 

.528 

58.18 

5-96 

1.23 

31.21 

— 

3-43 

Patent,  Warlich 

“ Wylam’s 

x-xS 

90.02 

5-56 

— 

— 

1.62 

2.91S 

1. 1 

79-91 

5-69 

1.68 

6.63 

1.25 

4.84 

* Heat  of  Combustion  of  1 Lb.  14  723. 

X Including  Nitrogen. 

Ss* 

+ Heat  of  Combustion  of  1 Lb.  15  651. 
§ Including  Oxygen. 

486 


FUEL. 


Average  Composition  of  Coals  and.  Fuels,  Heat  of  Com- 
bustion, and  Evaporative  Power. 

Deduced  from  analysis  and  experiments  of  Messrs.  De  La  Beche,  Playfair , and  Peclet. 


Coals  an©  Fuels. 

Specific 

Gravity. 

Carbon. 

Hydro- 

gen. 

Compos 

Nitro- 

gen. 

5ITI0N. 

Sul- 

phur. 

Oxy- 

gen. 

Ash. 

Heat  of  Com- 
bustion of  1 
lb. 

Evaporation 
from  water 
at  2120. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Per  ct. 

Units. 

Lbs. 

Derbyshire  and) 
Yorkshire ) 

I.29 

79.68 

4.94 

1.41 

1. 01 

0 

M 

OO 

2.65 

13  860 

14-34 

Lancashire 

I.27 

77-9 

5-32 

1-3 

1.44 

9-53 

4.88 

i39l8 

14.56 

Newcastle 

1.26 

82.12 

5-3i 

i-35 

1.24 

5-69 

3-77 

14  820 

15-32 

Scotch 

1.26 

78.53 

5-6i 

1 

1. 11 

9.69 

4-03 

14  164 

14-77 

Welsh 

1.32 

83.78 

4-79 

.98 

i-43 

4-i5 

4.91 

14858 

15-52 

Average  of  British. 

1.28 

80.4 

5-i9 

1. 21 

1.25 

7.87 

4-05 

14320 

14. 82 

Patent  fuels 

1. 17 

83-4 

4 97 

1.08 

1.26 

2.79 

5.- 93. 

15000 

15.66 

Van  Diemen’s  Land 

65.8 

3-5 

i-3 

1. 1 

5-58 

22.71 

11  320 

11.83 

Chili 

— 

63-56 

5-43 

.82 

2-5 

14.84 

13-31 

11 030 

11.68 

Lignite,  Trinidad. . 
“ French  Alps 

— 

65.2 

4-25 

i-33 

.69 

21.69 

6.84 

10438 

10.87 

1.28 

70. 02 

5-2 

— 

— 

— 

3.01 

11  790 

12. 1 

“ Bitum.,Cuba 

1.2 

75-85 

7-25 

— 

— . 

— 

3-94 

14562 

14.96 

“ Wash.  Ter.*. 

— 

67 

4-55 

— 

1 

— 

3-1 

12538 

12.91 

Asphalt 

I.06 

79.18 

9-3 

— 

— 

— 

2.8 

16655 

17.24 

Petroleum 

.87 

84.7 

I3- 1 

— 

— 

2.2 

— 

20240 

20.33 

“ oils 

•75 

— 

— 

— 

— 

27  530 

28.5 

vOak  bark  Tan,  dry. 

— 

— 

— 

— 

— 

i5 

6 100 

6.31 

“ “ moist 



— 

— 

— 

— 

— 

i5 

4284 

4.24 

"Charcoal  at  3020. . . 

i-5 

47-51 

6.12 

( 0 and  N 46.29  ) 

.8 

8130 

8.4 

“572°... 

1.4 

73-24 

4-25 

( 0 and  N 21.96) 

•57 

11 861 

12.27 

“ “ 8ioc. . . 

1. 71 

81.64 

4.96 

(0  and  N 15.24  ) 

1. 61 

14916 

15-43 

Peat,  dry,  average. 

•53 

58. 18 

5-96 

1.23 

1 FT. 

1 31-21 

3-43 

995i 

10.3 

“ moist,  t “ 

43- 1 

4-3 

( 0 and  N 21.4  ) 

3-3 

8917 

9.22 

Coal-gas 

42 

33-38 

66.16 

.38 

1 - 

| .08 

— 

52961 

47-51 

* Water  7.  Oxygen  and  Nitrogen  17.36.  t Moisture  27.8.  Sulphur  .2. 


Elements  of  Enels  not  included  in  Preceding  Tables. 
Heat  of  ' ~ 

Combustion 


Fuel. 


of  1 lb. 


Evaporative 
Power  of  1 
lb.  at  2120. 


Weight  ! 
of  1 I 
Cub.  Foot. 


Volume  of 
1 Ton. 


Bituminous  Coal. 

Welsh 

Newcastle 

Lancashire 

Scotch 

Boghead 

British,  average. 

Irish,  lowest 

Cumberland,  Md 

American,  average 

French,  average 

Australian 

Anthracite. 

American 

French 

Miscellaneous. 


Units. 
14858 
14  820 
13918 
14  164 
14478 
14  133 


14723 


14038 


Lbs. 

9-05 

8.01 

7-94 

7-7 

7.87 

8.13 

9-85 


Per  cent. 

73 

61 

58 

54 

30-94 

61 

9° 

83-7 

82.5 

64.2 

68.27 

94.82 

88.83 


Lbs. 

82 

78.3 

79-4 

78.6 

79.8 

99.6 
84-  93 
87-54 


93-  78 


Cube  Feet. 
42.7 
45-3 
45-2 
42 

44-52 

35-7 

42.4 

43-49 

40 


42-35 


Warlich’s  fuel 

Coke Mickley 

Virginia,  average 

Charcoal 

Lignite,  perfect 

“ imperfect 

u Russian 

Asphalt 

Woods,  dry,  average 


16495 

15  600 

13550 
12325 
11 678 

9834 

15837 

16555 

7792 


14.02 

12. 1 
10. 18 

17.24 

8.07 


47 

37-5 

9 


73-5 

45 


34-5 

80 


69.8 

12.76 


114 


FUEL. — GRAVITATION. 


487 


Miscellaneous. 

Experiments  undertaken  by  Baltimore  and  Ohio  R.  R.  Co.  determined 
evaporating  effect  of  1 ton  of  Cumberland  coal  equal  to  1.25  tons  of  anthra- 
cite, and  1 ton  of  anthracite  to  be  equal  to  1.75  cords  of  pine  wood;  also 
that  2000  lbs.  of  Lackawanna  coal  were  equal  to  4500  lbs.  best  pine  wood. 

One  lb.  of  anthracite  coal  in  a cupola  furnace  will  melt  from  5 to  10  lbs.  of  cast 
iron ; 8 bushels  bituminous  coal  in  an  air  furnace  will  melt  1 ton  of  cast  iron. 

Small  coal  produces  about  .75  effect  of  large  coal  of  same  description. 

Experiments  by  Messrs.  Stevens,  at  Bordentown,  N.  J.,  gave  following  results: 

Under  a pressure  of  30  lbs.,  1 lb.  pine  wood  evaporated  3.5  to  4.75  lbs.  of  water. 
1 lb.  Lehigh  coal,  7.25  to  8.75  lbs. 

Bituminous  coal  is  13  per  cent,  more  effective  than  coke  for  equal  weights;  and 
in  England  effects  are  alike  for  equal  costs. 

Radiation  from  Fuel.—  Proportion  which  heat  radiated  from  incandescent  fuel 
bears  to  total  heat  of  combustion  is, 

From  Wood 29  | From  Charcoal  and  Peat 5 

Least  consumption  of  coal  yet  attained  is  1.5  lbs.  per  IIP.  It  usually  varies  in 
different  engines  from  2 to  8 lbs. 

Volume  of  pine  wood  is  about  5.5  times  as  great  as  its  equivalent  of  bituminous 
coal. 


GRAVITATION. 


Gravity  is  an  attraction  common  to  all  material  substances,  and 
they  are  affected  by  it  directly,  in  exact  proportion  to  their  mass,  and 
inversely,  as  square  of  their  distance  apart. 

This  attraction  is  termed  terrestrial  gravity , and  force  with  which  a 
body  is  drawn  toward  centre  of  Earth  is  termed  the  weight  of  that  body. 


Force  of  gravity  differs  a little  at  different  latitudes : the  law  of  variation, 
however,  is  not  accurately  ascertained ; but  following  theorems  represent  it 
very  nearly : 

g (1  — .002  837  cos.  2 lat. ) 'j  , representing  force  of  gravity  at  lati- 

•tf  |:±:£! $ ft ‘the  equator  j =*  ,4  £ 
r,  32. 171  (lat.  450)  (1  + .005  133  sin.  L)  ^1  — = £ 


Or, 


- g.  L representing  latitude , 


H height  of  elevation  above  level  of  sea,  and  R radius  of  Earth,  both  in  feet. 


Note.— If  2 L exceeds  900,  put  cos.  180  — 2 L,  and  R at  Equator  = 20 926 062,  at, 
Poles  20853429,  and  mean  20889746. 


Illustration. — What  is  force  of  gravity  at  latitude  450,  at  an  elevation  of  209 
feet,  and  radi us  = 20  900  000  feet  ? 

32.171  (14-.005133  sin.  450)  (x  — 2o~^oq = 32.171  X 1.00363  X .99998  = 32.287. 


Gravity  at  Various  Locations  at  Level  of  Sea. 

Equator 32.088  I New  York 32.161  1 London 32.189 

Washington 32.155  | Lat.  450 32-171  J Poles 32:253 

In  bodies  descending  freely  by  their  own  weight,  their  velocities  are  as 
times  of  their  descent,  and  spaces  passed  through  as  square  of  the  times. 

Times , then,  being  1,  2,  3,  4,  etc.,  Velocities  will  be  1,  2,  3,  4,  etc. 

Spaces  passed  through  will  be  as  square  of  the  velocities  acquired  at  end 
of  those  times,  as  1,  4,  9, 16,  etc. ; and  spaces  for  each  time  as  1,  3,  5,  7,  9,  etc. 


GRAVITATION. 


488 


A body  falling  freely  will  descend  through  16.0833  feet  in  first  second  of 
time,  and  will  then  have  acquired  a velocity  which  will  carry  it  through 
32.166  feet  in  next  second. 

If  a body  descends  in  a curved  line,  it  suffers  no  loss  of  velocity,  and  the 
curve  of  a cycloid  is  that  of  quickest  descent. 

Motion  of  a falling  body  being  uniformly  accelerated  by  gravity,  motion 
of  a body  projected  vertically  upwards  is  uniformly  retarded  in  same  manner. 

A body  projected  perpendicularly  upwards  with  a velocity  equal  to  that 
which  it  would  have  acquired  by  falling  from  any  height,  will  ascend  to 
the  same  height  before  it  loses  its  velocity.  Hence,  a body  projected  up- 
wards is  ascending  for  one  half  of  time  it  is  in  motion,  and  descending  the 
other  half. 

Various  Formulas  here  given  are  for  Bodies  Projected  Upwards  or 
Falling  Freely , in  Vacuo. 

When , however,  weight  of  a body  is  great  compared  with  its  volume , and  velocity 
of  it  is  low , deductions  given  are  sufficiently  accurate  for  ordinary  purposes. 

In  considering  action  of  gravitation  on  bodies  not  far  distant  from  surface  of  the 
Earth,  it  is  assumed,  without  sensible  error,  that  the  directions  ian  which  it  acts  are 
parallel,  or  perpendicular  to  the  horizontal  plane. 

A distance  of  one  mile  only  produces  a deviation  from  parallelism  less  than  one 
minute,  or  the  60th  part  of  a degree. 


Relation,  of  Time,  Space,  and.  "Velocities. 


Time  from 
Beginning  of 
Descent. 

Velocity  acquired 
at  End  of  that 
Time. 

Squares 

of 

Time. 

Space  fallen 
through  in  that 
Time. 

Spaces 

for 

this  Time. 

Space  fallen 
through  in  last 
Second  of  Fall. 

Seconds. 

Feet. 

Seconds. 

Feet. 

No. 

Feet. 

1 

32.166 

1 

16.083 

1 

16.08 

2 

64-  333 

4 

64-333 

3 

48.25 

3 

96-5 

9 

144-75 

5 

80.41 

4 

128.665 

16 

257-33 

7 

112.58 

5 

160.832 

25 

402.08 

9 

144-75 

6 

193 

36 

579 

11 

176.91 

7 

225. 166 

49 

788.08 

13 

209.08 

8 

257-333 

64 

1029.33 

i5 

241.25 

9 

289.5 

81 

1302.75 

17 

273.42 

10 

321.666 

100 

1608.33 

19 

305-58 

and  in  same  manner  this  Table  may  be  continued  to  any  extent. 


‘Velocity  acquired  due  to  given  Height  of  Fall  and 
Height  due  to  given  "Velocity. 


v2  * 

8.0 ^y/h  — v',  32.2  t = v)  - — =zh ; and  16.083  t2  ==  h. 

04.4 

h representing  height  of  fall  in  feet , v velocity  acquired  in  feet  per  second , and  t 
time  of  fall  in  seconds. 


To  Compute  ^Action  of  G-ravity. 


Time. 

When  Space  is  given . Rule.— Divide  space  by  16.083,  and  square  root 
of  quotient  will  give  time. 

Example. —How  long  will  a body  be  in  falling  through  402.08  feet? 

V402.08  -5-  16.083  = 5 seconds. 

When  Velocity  is  given.  Rule.  — Divide  given  velocity  by  32.166,  and 
quotient  will  give  time. 

Example.— How  long  must  a body  be  in  falling  to  acquire  a velocity  of  800  feet 
per  second  ? 800  -r*  32. 166  = 24. 87  seconds. 


GRAVITATION. 


489 


Velocity. 

When  Space  is  given.  Rule.  — Multiply  space  in  feet  by  64.333,  and 
square  root  of  product  will  give  velocity. 

Example.— Required  velocity  a body  acquires  in  descending  through  579  feet. 

V 579  X 64.333  = 193  feet- 

Velocity  acquired  at  any  penod  is  equal  to  twice  the  mean  velocity  during 
that  period. 

Illustration.— If  a ball  fall  through  2316  feet  in  12  seconds,  with  what  velocity 
will  it  strike? 

2316  -4-  12  = 193,  mean  velocity , which  X 2 = 386  feet  = velocity. 

When  Time  is  given.  Rule. — Multiply  time  in  seconds  by  32.166,  and 
product  will  give  velocity. 

Example.— What  is  velocity  acquired  by  a falling  body  in  6 seconds? 

32. 166  X 6 = 192.996  feet. 

Space. 

When  Velocity  is  given.  Rule.— Divide  velocity  by  8.04,  and  square  of 
quotient  will  give  distance  fallen  through  to  acquire  that  velocity. 

Or,  Divide  square  of  velocity  by  64.33. 

Example.  — If  the  velocity  of  a cannon-ball  is  579  feet  per  second,  from  what 
height  must  a body  fall  to  acquire  the  same  velocity? 

579  8.04  = 72.014,  and  72.0142  = 5186.02  feet. 

When  Time  is  given.  Rule.  — Multiply  square  of  time  in  seconds  by 
16.083,  and  it  will  give  space  in  feet. 

Example.— Required  space  fallen  through  in  5 seconds. 

52  — 2 5,  and  25  X 16.083  = 402.08  feet. 

Distance  fallen  through  in  feet  is  very  nearly  equal  to  square  of  time  in  fourths 
of  a second. 

Illustration  i. — A bullet  dropped  from  the  spire  of  a church  was  4 seconds  in 
reaching  the  ground;  what  was  height  of  the  spire? 

4 X 4 — 16,  and  162  = 25 6 feet. 

By  Rule,  4 X 4 X 16.0833  = 257-33  feet- 

2.— A bullet  dropped  into  a well  was  2 seconds  in  reaching  bottom;  what  is  the 
depth  of  the  well  ? 

Then  2X4  = 8,  and  82  — 64  feet. 

By  Rule,  2 X 2 X 16.0833  = 64.33  feet- 

By  Inversion.—  In  what  time  will  a bullet  fall  through  256  feet? 

V 256  = 16,  and  16  -7-  4 = 4 seconds. 

Space  fallen  tlirongli  in  last  Second  of  Fall. 

When  Time  is  given.  Rule.— Subtract  half  of  a second  from  time,  and 
multiply  remainder  by  32.166. 

Example.— What  is  space  fallen  through  in  last  second  of  time,  of  a body  falling 
for  10  seconds  ? 0 

10  — . 5 x 32. 166  = 305. 58  feel. 

Promiscuous  Examples. 

1.  If  a ball  is  1 minute  in  falling,  how  far  will  it  fall  in  last  second? 

Space  fallen  through  = square  of  time,  and  1 minute  — 60  seconds. 

602  X 16.083  = 57  898  feet  for  60  seconds. 

592  X 16.083  = 55984  “ “ 59  “ 

19x4  1 

descended11^  °^£enera^D£  a velocity  of  193  feet  per  second,  and  whole  space 
I93~j“  32-166  = 6 seconds;  62  X 16.083  = 579/^. 


GRAVITATION. 


49° 


3.  If  a body  was  to  fall  579  feet,  what  time  would  it  be  in  falling,  and  how  far 
would  it  fall  in  the  last  second? 


579  X 2 _ , 6 _ 6 seconds,  and  6 — . 5 X 32- 166  = 5-  5 X 32- 166  ==  *76-91  feet- 
32. 166 

Formulas  to  determine  the  various  Elements. 

S V 2 s . 


/ S V 2 S /2 » . _ 

T_ v ’ ~ s'  ~ v ’ y 
2.  s=(jLy.  =1!;  =1*;  =£1!;  =T*.Si 

V25  g)  2 9 2 2 


h±=T-.5g. 


v = Vsx.2fli;  =Tr 


jV.5S,s;  = 


T representing  time  of  falling  in  seconds,  V udoctty  acquired  in  feet  per  second 
S space  or  vertical  height  in  feet,  h space  fallen  through  in  last  second,  g 32.166  and 
.5  g and  .25  g representing  16.083  and  8.04. 


Retarded.  Nlotion. 

A body  projected  vertically  upward  is  affected  inversely  to  its  motion 
when  falling  freely  and  directly  downward,  inasmuch  as  a like  cause  retards 
it  in  one  case  and  accelerates  it  in  the  other. 

In  air  a ball  will  not  return  with  same  velocity  with  which  it  started.  In 
vacuo  it  would.  Effect  of  the  air  is  to  lessen  its  velocity  both  ascending  and 
descending.  Difference  of  velocities  will  depend,  upon  relative  specific  grav- 
ity of  ball  and  density  of  medium  through  which  it  passes.  Ihus,  greater 
weight  of  ball,  greater  its  velocity. 

To  Compute  ^Action  of  Grravity  by  a Body  projected 

Upward  or  Downward  with,  a given.  Velocity. 

Space. 

When  projected  Upward.  Rule.— From  the  product  of  the  given  velocity 
and  the  time  in  seconds  subtract  the  product  of  32.166,  and  half  the  square 
of  the  time,  and  the  remainder  will  give  the  space  in  feet. 

Or,  Square  velocity,  divide  result  by  64.33,  and  quotient  will  give  space 
in  feet. 

Example.— If  a body  is  projected  upward  with  a velocity  of  96.5  feet  per  second, 
through  what  space  will  it  ascend  before  it  stops? 

96. 5^-32.166  = 3 seconds  = time  to  acquire  this  velocity. 

Then,  96.5  X 3 — ^32.166  X = 289.5  — 144*75  = *44-75 
Time. 

Rule. — Divide  velocity  in  feet  by  32.166,  and  quotient  will*give  time  in 
seconds. 

Example.— Velocity  as  in  preceding  example. 

96. 5 -4-  32. 166  = 3 seconds. 

Velocity. 

Rule.— Multiply  time  in  seconds  by  32.166,  and  product  will  give  velocity 
in  feet  per  second. 

Example.— Time  as  in  preceding  example. 

3 x 32. 166  = 96. 5 feet  velocity. 

Space  fallen  throxigh  in.  last  Second. 

Rule.— Subtract  .5  from  time,  multiply  remainder  by  32.166,  and  product 
will  give  space  in  feet  per  second. 

Example. — Time  as  in  preceding  example. 

3 5 X 32. 166  = 2. 5 X 32.  166  = 80.41 6 feet. 


GRAVITATION. 


491 


When  projected  Downward. 

Space. 

Rule. — Proceed  as  for  projection  upwards  and  take  sum  of  products. 

Example  l— If  a body  is  projected  downward  with  a velocity  0196.5  feet  per  sec- 
ond, through  what  space  will  it  fall  in  3 seconds? 

96-  5 X 3 + (32.  i66  X = 289. 5 -f  144. 75  = 434. 25  feet. 

Or,  t2  x 16.083  + v x t = s. 

2.  — If  a body  is  projected  downward  with  a velocity  of  96.5  feet  per  second, 
through  what  space  must  it  descend  to  acquire  a velocity  of  193  feet  per  second? 

96.5  -r-  32. 166  = 3 seconds , time  to  acquire  this  velocity. 

1 93  — i—  32.166  = 6 seconds , time  to  acquire  this  velocity. 

Hence  6 — 3 = 3 seconds , time  of  body  falling. 

Then  96.5  X 3 = 289.5  = product  of  velocity  of  projection  and  time. 

16.083  X 32  = 144-75  — product  of  32. 166,  and  half  square  of  time. 

Therefore  289.5-}-  144.75  = 434.25/^. 

Time. 

Rule.— Subtract  space  for  velocity  of  projection  from  space  given,  and 
remainder,  divided  by  velocity  of  projection,  will  give  time. 

Example. — In  what  time  will  a body  fall  through  434.25  feet  of  space,  when  pro- 
jected with  a velocity  of  96. 5 feet  ? 

Space  for  velocity  of  96. 5 = 144.75/ee^ 

Then,  434.25  — 144.75  = 96.5  = 289.5  — 96.5  = 3 seconds. 

V^elocity-. 

Rule.— Divide  twice  space  fallen  through  in  feet  by  time  in  seconds. 

Example. — Elements  as  in  preceding  example. 

Space  fallen  through  when  projected  at  velocity  of  96.5  feet—  144.75  feet,  and  434.25 
feet  — space  fallen  through  in  3 seconds. 

Then,  144.754434.25  = 579  feet  space  fallen  through,  and  V579  = 16.083  = 6 
seconds. 

Hence,  579  X 2 = 6 = 1158  = 6 = 193  feet. 

Space  Fallen.  tlirougTi  in  last  Second. 

Rule.— Subtract  .5  from  time,  multiply  remainder  by  32.166,  and  product 
will  give  space  in  feet  per  second. 

Example.— Elements  as  in  preceding  example. 

6 — .5  X 32.166  = 5.5  X 32.166  = 176.91  feet. 

Ascending  bodies,  as  before  stated,  are  retarded  in  same  ratio  that  descending 
bodies  are  accelerated.  Hence,  a body  projected  upward  is  ascending  for  one  half 
of  the  time  it  is  in  motion,  and  descending  the  other  half. 

Illustration  i. — If  a body  projected  vertically  upwards  return  to  earth  in  12 
seconds,  how  high  did  it  ascend  ? 

The  body  is  half  time  in  ascending.  12  = 2 = 6. 

Hence,  by  Rule,  p.  489,  62  X 16.083  = 579  feet = product  of  square  of  time  and 
16.083. 

2.— If  a body  is  projected  upward  with  a velocity  of  96.5  feet  per  second,  it  is 
required  to  ascertain  point  of  body  at  end  of  10  seconds. 

96-5“f  32  I66  = 3 seconds,  time  to  acquire  this  velocity , and  32x  16.083  = 144.75 
feet,  height  body  reached  with  its  initial  velocity. 

Then  10  — 3 = 7 seconds  left  for  body  to  fall  in. 

Hence,  by  Rule,  as  in  preceding  example,  7 2 X 16.083  = 788.07,  and  788.07  — 

I *44-  75  — 643. 32  feet  = distance  below  point  of  projection. 

\ 0r>  102  X 16.083  = 1608.3  feet,  space  fallen  through  under  the  effect  of  gravity,  and 

1 96. 5 X 10  = 965  feet , space  if  gravity  did  not  act.  Hence  1608. 3 — 965  = 643. 3 feet. 


492 


GRAVITATION. 


3. — A body  is  projected  vertically  with  a velocity  of  135  feet;  what  velocity  will 
it  have  at  60  feet  ? 

I352-4-  64.33  = 283.3  feet  space  projected  at  that  velocity , 135 -4- 32. 16  = 4. 197  sec- 
onds = time  of  projection,  and  283.3  — 60=  223.3  = space  to  he  passed  through  after 


attainment  of  60  feet. 
= 283.3/^. 


Hence,  V 223. 3 X 64. 33  = 119.85  feet  velocity , and  223.34-60 


119.85.  Hence,  1*9'8--  — 223.3  fat  space , and  283.3  - 

64-33 


By  Inversion. — Velocity  : 

223. 3 = 60  feet. 

Formulas  to  Determine  Elements  of  Retarded  IVtotioia. 
s gt 


1.  v=.V  — gt. 


V = - 


t — 


t 

V — 
9 


+94- 


S.  s = Xt-9—. 


3.  V = v-f  gt, 

6.  s = tv 

2 

9.  h — T — t — t'—.sg. 


n , V /V2  2S 

8.  t — — • — . f —z . 

9 \ 92  9 

v representing  velocity  at  expiration  of  time , t any  less  time  than  T,  t'  less  time  than 
t,  s space  through  which  a body  ascends  in  time  t,  V,  T,  S,  and  h as  in  previous  formulas, 
page  490. 

Illustration.— A body  projected  upwards  with  a velocity  of  193  feet  per  second, 
was  arrested  in  5 seconds. 

T = 6,  t'=i. 

1.  What  was  its  velocity  when  arrested  ? (1.) 

2.  What  was  the  time  of  its  passing  through  562.92  feet  of  space  ? (7.) 

3.  What  space  had  it  passed  through  ? (5.) 

4.  What  was  the  time  of  its  projection,  when  it  had  a velocity  of  96.5  feet?  (4.) 

5.  What  was  the  height  it  was  projected  in  the  last  second  of  time?  (8.) 


1.  193  — 32. 166  X 5 = 32. 17  feet. 


562.92  32. 166  X 5 


5 

i93  X 5 


= 193  velocity. 


32.166  X 5s 


= 562.92/ee^ 


193 

32. 166 


3.  32. 17  + 32. 166X 5 — J93  velocity. 

193  — 96- 5 _ 

32. 166 

6 193  — 32-17 . 

32. 166 


- = 3 seconds. 

- = 5 seconds. 


1932  2 x 562.9= 

32. 1662  32.166 

!.  6 — 5 — 1 — . 5 x 32. 166 


= 6 — V36 — 35  = 5 seconds. 

48.25  feet. 

Grravity-  and  ZVTotion  at  an  Inclination. 

If  a body  freely  descend  at  an  inclination,  as  upon  an  inclined  plane,  by 
force  of  gravity  alone,  the  velocity  acquired  by  it  when  it  arrives  at  ter- 
mination of  inclination  is  that  which  it  would  acquire  by  falling  freely 
through  vertical  height  thereof.  Or,  velocity  is  that  due  to  height  of  in- 
clination of  the  plane. 

Time  occupied  in  making  descent  is  greater  than  that  due  to  height,  in 
ratio  of  length  of  its  inclination,  or  distance  passed,  to  its  height. 

Consequently,  times  of  descending  different  inclinations  or  planes  of  like 
heights  are  to  one  another  as  lengths  of  the  inclinations  or  planes. 

Space  which  a body  descends  upon  an  inclination,  when  descending  by 
gravity , is  to  space  it  would  freely  fall  in  same  time  as  height  of  inclination 
is  to  its  length ; and  spaces  being  same,  times  will  be  inversely  in  this  pro- 
portion. 

If  a body  descend  in  a curve,  it  suffers  no  loss  of  velocity. 

If  two  bodies  begin  to  descend  from  rest,  from  same  point,  one  upon  an  in- 
clined plane,  and  the  other  falling  freely,  their  velocities  at  all  equal  heights 
below  point  of  starting  will  be  equal. 


GRAVITATION. 


493 


Illustration.— What  distance  will  a body  roll  down  an  inclined  plane  300  feet 
long  and  25  feet  high  in  one  second,  by  force  of  gravity  alone? 

As  300  : 25  ::  16.083  : 1.34025  feet. 

Hence  if  proportion  of  height  to  length  of  above  plane  is  reduced  from  25  to  300 
to  25  to  600,  the  time  required  for  body  to  fall  1.34025  feet  would  be  determined  as 
follows : 

As  25  • 600  ” 1.34025  : 32.166,  and  32.166  = 16.083  X 2 = twice  time  or  space  in 
which  it  would  fall  freely  required  for  one  half  proportion  of  height  to  length. 

Or  as  1.34025  : 32.166,  as  above. 

’ 25  25 

Impelling  or  accelerating  force  by  gravitation  acting  in  a direction  paral- 
lel to  an  inclination,  is  less  than  weight  of  body,  in  ratio  of  height  of  in- 
clination to  its  length.  It  is,  therefore,  inversely  in  proportion  to  length  of 
inclination,  when  height  is  the  same. 

Time  of  descent,  under  this  condition,  is  inversely  in  proportion  to  accel- 
erating force. 

If,  for  instance,  length  of  inclination  is  five  times  height,  time  of  making 
freely  descent  at  inclination  by  gravitation  is  five  times  that  in  which  a 
body  would  freely  fall  vertically  through  height ; and  impelling  force  down 
inclination  is  .2  of  weight  of  body. 

Wlien  bodies  move  down  inclined  planes,  the  accelerating  force  is  ex- 
pressed by  h -r- 1 , quotient  of  height  length  of  plane  $ or,  what  is  equivalent 
thereto,  sine  of  inclination  of  plane,  i.  e.,  sin.  a. 

Illustration. — An  inclined  plane  having  a height  of  one  half  its  length,  the  space 
fallen  through  in  any  time  would  be  one  half  of  that  which  it  would  fall  freely. 

Velocity  which  a body  rolling  down  such  a plane  would  acquire  in  5 seconds  is 
80.416  feet. 

Thus,  32.166  X 5 = 160.833  feet,  and  an  inclined  plane,  having  a height  one  half 
of  its  length,  has  an  angle  or  sine  of  300.  Hence,  sin.  300  = .5,  and  160.833  X .5  = 
80.41 6 feet. 

Formulas  to  Determine  various  Elements  of  Gravita- 
tion 011  an  Inclined  Plane. 

*•  S = .5SrT*«n.a;  =-5™  4-  V^orTrtn.  «. 


. 2 I 

2.  V = g T sin.  a ; = V(2  9 S sm.  a)  ; = — 


6.  H = 


l2 

:-50T2- 


3-  =Wh; 


5.  S = V T . 5 g T2  sin.  a. 


Or, 


2 g sin.  a 

v representing  velocity  of  projection  in  feet  per  second , S space  or  vertical  height 
of  velocity  and  projection,  a angle  of  inclination  of  plane,  l length,  and  H height  of 
plane. 

Illustration.  — Assume  elements  of  preceding  illustration.  V = 80.416,  T = 5, 
and  H = 201.04. 

1.  .5  X 32.166  X 52  X -5  — 201.04 /ee£.  2.  32.166  X 5 X .5  = 8o.4i6/ee£. 

3.  / = / V25  = 5 seconds. 

V \ 32. 166  x. 5/  V \16.083 / 


283.422 


- = 201.04  feet. 


— — — — - — — — - — ^1.^,/w.  7.  4 X 5 X V20I-°4  — 283.42 /ee^. 

.5  X 16.083  X 5 

If  projected  downward  with  an  initial  velocity  of  16.083  feet  per  second.  V-f-^. 

4.  16.083-f-32.166x  5 X .5  = 9 6. 5 feet. 

5.  80.416-}-  16.083  X 5 — -5  X 32.166  X 52  X .5  = 281.46/eefc 

t t 


GRAVITATION. 


494 


Illustration. —What  time  will  it  take  for  a ball  to  roll  38  feet  down  an  inclined 
plane,  the  angle  a=  120  20',  and  what  velocity  will  it  attain  at  38  feet  from  its  start- 
ing-point? 


X • 2136  = 22.88  feet  per  second. 

When  a body  is  projected  upward  it  is  retarded  in  the  same  ratio  that  a 
descending  body  is  accelerated. 

Illustration.— If  a body  is  projected  up  an  inclined  plane  having  a length  of 
twice  its  height,  at  a velocity  of  96.5  feet  per  second, 

Then,  T = 96. 5 -7-32.166  = 3 seconds.  S = . 5 32-j66  X 32  X .5  = 72.375  feet.  v=z 
32.166  X 3 X.  5 = 48-25/^. 


Problems  on  descent  of  bodies  on  inclined  planes  are  soluble  by  formulas 
1 to  9,  page  495,  for  relations  of  accelerating  forces.  As  a preliminary  step, 
however,  accelerating  force  is  to  be  determined  by  multiplying  weight  of 
descending  body  by  height  of  plane,  and  dividing  product  by  length  of  plane. 

Illustration. —If  a body  of  15  lbs.  weight  gravitate  freely  down  an  inclined 
plane,  length  of  which  is  five  times  height,  accelerating  force  is  15^5  = 3 lbs.  If 
length  of  plane  is  100  feet  and  height  20,  velocity  acquired  in  falling  freely  from  top 
to  bottom  of  plane  would  be  


Whereas,  for  a free  vertical  fall  through  height  of  20  feet,  time  would  be, 


which  is  .2  of  time  of  making  descent  on  inclined  plane. 

Velocities  acquired  by  bodies  in  falling  down  planes  of  like  height  will  all  be 
equal  when  arriving  at  base  of  plane. 

When  Length  of  an  Inclined  Plane  and  Time  of  Free  Descent  are  given. 

Rule. — Divide  square  of  length  by  square  of  time  in  seconds  and  by  16 ; 
the  quotient  is  height  of  inclined  plane. 

Example. — Length  of  plane  is  100  feet,  and  time  of  descent  is  5.59  seconds;  then 
vertical  height  of  descent  is 


If  an  Accelerating  or  Retarding  force  is  greater  than  gravity,  that  us, 
weight  of  the  body,  the  constant,  g,  or  32.166,  is  to  be  varied  in  proportion 
thereto,  and  to  do  this  it  is  to  be  multiplied  by  the  accelerating  force,  and 
product  divided  by  weight  of  body. 

Thus,  Let  f represent  accelerating  force,  and  w weight  of  body. 


The  same  rules  and  formulas  that  have  been  given  for  action  of  gravity  alone 
are  applicable  to  the  action  of  any  other  uniformly  accelerating  or  retarding  lorce, 
the  numerical  constants  above  given  being  adapted  to  the  force. 


j 2 S _ 
V g sin.  a~ 


2 X 38 


= 3. 33  seconds.  V = g T sin.  a = 32. 166  X 3-  33 


32. 166  X -2136 


Inclined  Plane, 


Time  occupied  in  making  descent, 


making 


32.166 


5-592  X 16.08 


==  20  feet. 


Accelerated  and  Retarded  NIotion 


Then  64'333-^  or  or  1-6  °8^  become  the  constants. 

’ w ’ W 10 


GRAVITATION. 


495 


Average  Velocity-  of  a Moving  Body  uniformly  Accel- 
erated or  Retarded. 

Average  velocity  of  a moving  body  uniformly  accelerated  or  retarded, 
during  a given  time  or  in  a given  space,  is  equal  to  half  sum  of  initial  and 
final  velocities ; and  if  body  begin  from  a state  of  rest  or  arrive  at  a state  of 
rest,  its  average  speed  is  half  the  final  or  initial  velocity,  as  the  case  may  be. 

Thus,  in  example  of  a ball  rolling,  initial  speed  or  velocity  is,  in  either 
case,  60  feet  per  second,  and  terminal  speed  is  nothing ; average  speed  is 

therefore  -6o  ° namely,  one  half  of  that,  or  30  feet  per  second. 

When  a cannon-ball  is  projected  at  an  angle  to  horizon,  there  are  two  forces  act- 
ing on  it  at  same  time— viz.,  force  of  charge,  which  propels  it  uniformly  in  a right 
line,  and  force  of  gravity,  which  causes  it  to  fall  from  a right  line  with  an  accel- 
erated motion;  these  two  motions  (uniform  and  accelerated)  cause  the  ball  to  move 
in  the  curved  line  of  a Parabola. 


w representing  weight  of  hall  and  P of  powder  in  lbs.  ; l time  of  flight  in  seconds  • 
h horizontal  range , and  h vertical  height  of  range  of  projection  of  ball  in  feet. 

Illustration.  — A cannon  loaded  to  give  a ball  a velocity  of  900  feet  per  second 
the  angle  a = 450;  what  is  horizontal  range,  the  time  t and  height  of  range  h ? ’ 


Note.— As  distance  b will  be  greatest  when  angle  a = 45°,  product  of  sine  and 
cosine  is  greatest  for  that  angle.  Sin.  450  x cos.  450  = . 5. 

24  lb.  ball  with  a velocity  of  2000  feet  per  second  at  450  range  7300  feet. 

G-eneral  Bormnlas  for  Accelerating  and  Retarding 
Forces, 


Formulas  for  Flight  of  a Cannon-ball. 


V2  sin.  a,  cos.  a ^ 

n J 


V sin.  a 


9 


; 


900 2 x sin.  45°  X COS.  45O  _ 9002  X -5 


-^^  = 12590  feet 


32.  l66 


. 900  X.  7071 

t = tv — = iQ-  70  seconds  : h ~ 

32.166  ; 


900  2 x .7071 2 
2 X 32.166 


= 6295  feet. 


GRAVITATION. 


496 


Again  same  result  may  be  arrived  at,  according  to  Note  1,  by  multiplying  con- 
stant 64!  333,  in  Rule,  page  494,  for  gravity,  by  ratio  of  force  and  weight,  which  in 
this  case  is  and  64.333  X 3^  = 6-4333-  Substituting  6.4333  for  64.333  in  that 
rule,  formula  becomes 

~ V2  602  . . 

S = = = 559-  59  feeL 

6-4333  6.4333 

The  question  may  be  answered  more  directly  by  aid  of  table  for  falling  bodies, 
page  488.  Height  due  to  a velocity  of  60  feet  per  second,  is  55.9  feet;  which  is  to 
be  multiplied  by  inverse  ratio  of  accelerating  force  and  weight  of  body,  or  or  10; 
that  is,  55.9  x 10  = 559  feet 

If  the  question  is  put  otherwise— What  space  will  a weight  move  over  before  it 
comes  to  ci  sttite  of  rest,  with,  an  initial  velocity  of  60  feet  per  second,  allowing  fric- 
tion  to  be  one  tenth  weight?  The  answer  is  that  friction,  which  is  retarding  force, 
bein"  one  tenth  of  weight,  or  of  gravity,  space  described  will  be  10  times  as  great  as 
is  necessary  for  gravity,  supposing  the  weight  to  be  projected  vertically  upwards  to 
bring  it  to  a state  of  rest.  The  height  due  to  velocity  being  55.9  feet;  then 
55.9X10  = 559  feet. 

Average  velocity  of  a moving  body,  uniformly  accelerated  or  retarded  during  a 
given  period  or  space,  is  equal  to  half  sum  of  initial  and  final  velocities. 


To  Compute  "Velocity  of  a,  [Falling  Stream  of  Water  per 
Second,  at  End  of  any-  given  Time. 

When  Perpendicular  Distance  is  given. 

Example.— What  is  the  distance  a stream  of  water  will  descend  on  an  inclined 
plane  10  feet  high,  and  100  feet  long  at  base,  in  5 seconds? 

52  X 16. 083  = 402.08  feet  = space  a body  will  f reedy  fall  in  this  time. 

Then,  as  100  : 10  ::  402.08  : 40.21  feet  = proportionate  velocity  on  a plane  of  these 
dimensions  to  velocity  when  falling  freely. 


[Miscellaneous  Illnstrations. 


j.—What  is  the  space  descended  vertically  by  a falling  body  in  7 seconds. 

S = 5 g X t2.  Then  16.083  X 72  = 788.067  feet. 

2.  —What  is  the  time  of  a falling  body  descending  400  feet,  and  velocity  acquired 
at  end  of  that  time? 

t = ~.  Then  l6°--  = 4.98  sec.  v = V2  p X S.  Then  V64.333  X 4°°  = 160.4  feet, 
g 32.166 

3. — If  a drop  of  rain  fall  through  176  feet  in  last  second  of  its  fall,  how  high  was 
the  cloud  from  which  it  fell? 

S = — . Then  -^1^482.75/^. 

2 g 64. 166 

a. If  two  weights,  one  of  5 lbs.  and  one  of  3,  hanging  freely  over  a sheave,  are 

set  free,  how  far  will  heavier  one  descend  or  lighter  one  rise  in  4 seconds. 

x 16.083  x 42  = l X 257.328  = 64.33  feet. 

5 + 3 8 


5. — if  length  of  an  inclined  plane  is  100  feet,  and  time  of  descent  of  a body  is  6 
seconds,  what  is  vertical  height  of  plane  or  space  fallen  through? 


62X  -5  l 


579 


- = 17.27/eek 


6.— If  a bullet  is  projected  vertically  with  a velocity  of  135  feet  per  second,  what 
velocity  will  it  have  at  60  feet? 


Formula  7,  page  492. 


i35  / i352 

32.166  V 32.1662 


2 * _ .j  seconds. 

32 .166 


GUNNERY. 


497 


GUNNERY. 

A heavy  body  impelled  by  a force  of  projection  describes  in  its  flight 
or  track  a parabola,  'parameter  of  which  is  four  times  height  due  to 
velocity  of  projection. 

Velocity  of  a shot  projected  from  a gun  varies  as  square  root  of 
charge  directly,  and  as  square  root  of  weight  of  shot  reciprocally. 

To  Compute  ‘Velocity  of  a Shot  or  Shell. 

Rule.— Multiply  square  root  of  triple  weight  of  powder  in  lbs.  by  1600; 
divide  product  by  square  root  of  weight  of  shot ; and  quotient  will  give  ve- 
locity in  feet  per  second. 

Example.— What  is  velocity  of  a shot  of  196  lbs.,  projected  with  a charge  of  o lbs 
of  powder?  ^ 

V 9 X 3 X 1600  -r-  y/ 196  = 8320  = 14  ==  594.3  lbs. 

To  Compute  Range  for  a Charge,  or  Charge  for  a Range. 

When  Range  for  a Charge  is  given. — Ranges  have  same  proportion  as 
charges  of  powder ; that  is,  as  one  range  is  to  its  charge,  so  is  any  other 
range  to  its  charge,  elevation  of  gun  being  same  in  both  cases.  Consequently , 

To  Compute  Range. 

Rule. — Multiply  range  determined  by  charge  in  lbs.  for  range  required 
divide  product  by  given  charge,  and  quotient  will  give  range  required.  * 

Example.  If,  with  a charge  of  9 lbs.  of  powder,  a shot  ranges  4000  feet,  how  far 
will  a charge  of  6.75  lbs.  project  same  shot  at  same  elevation? 

4000  X 6. 75  -r-  9 = 3000  feet. 

To  Compute  Charge. 

. Rule.— Multiply  given  range  by  charge  in  lbs.  for  range  determined, 
divide  product  by  range  determined,  and  quotient  will  give  charge  required. 

Example.— If  required  range  of  a shot  is  3000  feet,  and  charge  for  a range  of  4000 
feet  has  been  determined  to  be  9 lbs.  of  powder,  what  is  charge  required  to  project 
same  shot  at  same  elevation  ? 

3000  X 9 4°°°  — 6-75  lbs. 

To  Compute  Range  at  one  Elevation,  when  Range  for 
another  is  given. 

Rule. — As  sine  of  double  first  elevation  in  degrees  is  to  its  range,  so  is 
sine  of  double  another  elevation  to  its  range. 

Example.— If  a shot  range  1000  yards  when  projected  at  an  elevation  of  45°  how 
far  will  it  range  when  elevation  is  300  16',  charge  of  powder  being  same?  1 
Sine  of  450  X 2 = 100  000 ; sine  of  30°  16'  X 2 = 87  064. 

Then,  as  100000  : 1000  87064  : 870.64  feet. 

To  Compute  Elevation  at  one  Range,  when  Elevation 
for  another  is  given. 

Rule. — As  range  for  first  elevation  is  to  sine  of  double  its  elevation,  so 
is  range  for  elevation  required  to  sine  for  double  its  elevation. 

Example.— If  range  of  a shell  at  450  elevation  is  3750  feet,  at  what  elevation 
must  a gun  be  set  for  a shell  to  range  2810  feet  with  a like  charge  of  powder? 

Sine  of  450  X 2 100000. 

Then,  as  3750  : 100000  ::  2810  : 74933  = 51716  for  double  elevation  — 2 16'. 

Approximate  Rule  for  Time  of  Flight. 

Under  4000  yards,  velocity  of  projectile  900  feet  in  one  second ; under 
6000  yards,  velocity  800  feet ; and  over  6000  yards,  velocity  700  feet. 

Guns  and  Howitzers  take  their  denomination  from  weights  of  their  solid 
shot  in  round  numbers,  up  to  the  42-pounder;  larger  pieces,  rifled  guns,  and 
mortars,  from  diameter  of  their  bore. 

T T* 


GUNNERY. 


498 

Initial  'Velocity  and.  Ranges  of  Shot  and  Shells. 

The  Range  of  a shot  or  shell  is  the  distance  of  its  first  graze  upon  a horizontal 
~i ~ *r>r»nnfArl  nnnn  its  nroner  carriage. 

Range. 


Arms  and  Ordnance.  J 

Projectile. 

Description.  | Weight.  | 

Powder. 

Initial  | 
Velocity.| 

Time  of  1 
Flight.  1 

Eleva 

tion. 

Rifle  Musket 

Elongated. 

Grains. 

5io 

Grains. 

60 

Feet. 

963 

Seconds. 



Musket,  1841 

Round. 

412 

no 

1500 

— 

— 

6-Pounder 

.<< 

Lbs. 

6.15 

Lbs. 

1.25 

— 

5 

u 

12.3 

2-5 

1826 

i-75 

1 

24  “ 

32  

u 

24.25 

6 

1870 

— 

2 

u 

32-3 

8 

1640 

— 

1 

42  “ 

42.5 

10.5 

— 

— 

5 

8-inch  Columbiad. . . 

“ 

65 

10 

— 

14.19 

i5 

10  “ u • • • 

“ 

127-5 

15 

— 

14.32 

i5 

10  “ Mortar 

Shell. 

98 

10 

— 

36 

45 

13  “ “ 

“ 

200 

20 

— 

— 

45 

k “ Columbiad... 

u 

302 

40 

— ■ 

— 

7 

15  “ “ 

u 

3i5 

50 

— 

23.29 

25 

RIFLED. 

10-pounder  Parrott. . 
20  u u 

“ 

9-75 

1 

— 

21 

20 

“ 

*9 

2 

— 

17.25 

i5 

30  “ u 

“ 

29 

3-25 

— 

27 

25 

* it  u 

IOO 

Elongated. 

100 

10 

— 

29 

25 

100  “ “ 

Shell. 

IOI 

10 

1250 

28 

25 

200  “ “ 

“ 

150 

16 

— 

— 

4 

1 2- inch  Rodman 

— 

50 

1154 

5-5 

Hall’s  Rockets 

3- inch. 

16 

- 

- 

47 

Yards. 


1523 

575 

1147 

7i3 

1955 

3224 

3281 

4250 

4325 

1948 

4680 

5000 

4400 

6700 

6910 

6820 

2200 


1720 


Penetration  of  Shot  and  Shell. 


Mean  Penetration. 

Mean  Penetration . 

Ordnance. 

S, 

gS 

1 

A 

2 

e 

Ordnance. 

C/ 

tL 

3 

-2  • 

2 

‘2 

0 

s 

ffig 

5 

O 

5 

£5 

Lbs. 

Yds. 

Ins. 

Ins. 

Ins. 

Lbs. 

Yds. 

Ins. 

Ins. 

Ins. 

32  Lbs.  Shot. 

8 

880 

15-25 

3-5 

8-inch  Howitz.* 

6 

880 

— 

8-5 

1 

n 

IOO 

60 

8 “ Columb.if 

12 

200 

— 

— 

— 

32 

42  ‘ 

10.5 

IOO 

54-75 

18 

4 

10  “ “ t 

iS 

1 14 

63-5 

44 

7-75 

42  “ Shell. 

7 

100 

40-75 

— 

— 

10  “ “ * 

18 

IOO 

56.75 

— 

— 

1 24  ins.  of  Concrete. 

* Shell.  t Shot 

Solid  shot  broke  against  granite,  uut  nut  agamai*  ^ ~ 

effect  is  less  upon  brick  than  upon  granite. 

Shells  broke  into  small  fragments  against  each  of  the  three  materials. 
Penetration  in  earth  of  shell  from  a 10- inch  Columbiad  was  33  ins. 

Experiments — England.  ( Holley . ) 


Ordnance. 

Charge. 

Projectile. 

Weight. 

Velocity. 

Range. 

Target  and  Effects. 

Lbs. 

Lbs. 

Feet. 

Yards. 

Iron  plates,  14  ins. 
— loosened. 

n-inch  U.  S.  Navy. 

30 

Shot. 

169 

1400 

50 

15-inch  Rodman. . . 

60 

“ 

400 

1480 

50 

Iron  plates,  6 ins. — 
destroyed. 

RIFLED. 

7-inch  Whitworth. . 

25 

Shot. 

150 

1241 

200 

Inglis'st— destr’d. 

10.5-inch  Armstrong 
13-inch  “ 

45 

90 

* 41 

3°7 

344-5 

1228 

1760 

200 

200 

Solid  plates,  n ins. 
thick— destr'd. 

* Steel.  t 8-inch  vertical  an 
Blabs,  9X5  ins.  ribs  and  3-inch  ribs. 


GUNNERY.  499 

Elements  of  Report  of  Board  of  Engineers  for  Fortifications , U~.  S.  A. 

Professional  Papa's  RTo.  25.  ( Brev . Maj.-Gen.  Z.  B.  Tower.) 

Experimental  firings  for  penetration  during  tlie  past  twenty  years  have 
determined  that  wrought  iron  and  cast  iron,  unless  chilled,  are  unsuitable  for 
projectiles  to  be  used  against  iron  armor ; that  the  best  material  for  that 
purpose  is  hammered  steel  or  Whitworth’s  compressed  steel. 

2.  That  cast-iron  and  cast-steel  armor-plates  will  break  up  under  the  im- 
pact of  the  heaviest  projectiles  now  in  service,  unless  made  so  thick  as  to 
exclude  their  use  in  ship-protection. 

3.  That  wrouglit-iron  plates  have  been  so  perfected  that  they  do  not  break 
up,  but  are  penetrated  by  displacement  or  crowding  aside  of  the  material  in 
the  path  of  the  shot,  the  rate  of  penetration  bearing  an  approximately  deter- 
mined ratio  to  the  striking  energy  of  the  projectile,  measured  per  inch  of 
shot’s  circumference,  as  expressed  by  the  following  formula  : 


2'°V7 — — = penetration  in  ins.  V representing  velocity  in 

V 2#X2  r it  X 2240  X • 86  J 

feet  per  second , P weight  of  shot  in  lbs .,  and  r radius  of  shot  in  ins. 

That  such  plates  can  therefore  be  safely  used  in  ship  construction,  their 
thickness  being  determined  by  the  limit  of  flotation  and  the  protection 
needed, 

4.  That,  though  experiments  with  wrouglit-iron  plates,  faced  with  steel, 
have  not  been  sufficiently  extended  to  determine  the  best  combination  of 
these  two  materials,  we  may  nevertheless  assume  that  they  give  a resistance 
of  about  one  fourth  greater  than  those  of  homogenous  iron. 

5.  That  hammered  steel  in  the  late  Spezzia  trials  proved  superior  to  anv 
other  material  hitherto  tested  for  armor-plates.  The  19-inch  plate  resisted 
penetration,  and  was  only  partially  broken  up  by  4 shots,  three  of  which  had 
a striking  energy  of  between  33  000  and  34000  foot-tons  each.  Not  one  shot 
penetrated  the  plate.  Those  of  chilled  iron  were  broken  up,  and  the  steel 
projectile,  though  of  excellent  quality,  was  set  up  to  about  two  thirds  of  its 
length. 


"Velocity  and.  Ranges  of  Shot.  [Krupff s Ballistic  Tables.) 
IPenetration  in.  Wrought  Iron. 

. / V2P 

V ~~  — Z7c * = penetration  in  ins.  C = 2. 53. 

V 2 g X 2 r it  X.2240  x C x 


Gun. 

Cali- 

ber. 

Powder. 

Shot. 

V 

at 

Muzzle 
per  Sec. 

elocity 

Rat 

3000 

tge. 

6000 

at 

Muzzle 

Penet: 

600 

ration 

Range 

3900 

6000 

Tons. 

Ins. 

Lbs. 

Lbs. 

Feet. 

Yds. 

Yds. 

Ins. 

Ins. 

Ins 

Ins. 

Armstrong,  100.. 

17-75 

550 

2022 

1715 

1424 

1191 

34-76 

33-2 

27-55 

22.04 

“ “ . . 

17-75 

776 

2000 

1832 

1518 

1259 

37-52 

35-  3 1 

29.66 

23-47 

Woolwich,  81.. 

16 

445 

1760 

1657 

1393 

1181 

32.6 

31-23 

26. 24 

21-35 

Krupp,  71.. 

15-75 

485 

1715 

1703 

1434 

1211 

33-52 

32. 12 

27.04 

21.89 

18. . 

9-45 

165 

474 

1688 

I35i 

1113 

20.42 

I9-31 

15.46 

12. 14 

U.  S.*  8- inch 

8 

35 

180 

1450 

1036 

840 

10.23 

9.22 

6.72 

5-i7 

* Unchambered. 


Target.— For  100-ton  gun,  steel  plate  22  ins.  thick,  backed  with  28.8  ins.  of  wood 
2 wrought-iron  plates  1.5  ins.  thick,  and  the  frame  of  a vessel. 

Effect. — Total  destruction  of  steel  plate,  and  backing  entered  to  a depth  of  22  ins 
but  not  perforated.  * * 


500 


GUNNERY. 


Summary  of  Record  of  3Practi.ce  in  Europe  witli  Heavy 
Armstrong,  Woolwich,  and.  Krupp  Guns. 

Board  of  Engineers  for  Fortifications , U.  S.  A. , Professional  Papers  No.  25. 


Gun. 

Powder. 

Projectile. 

Charge  of 
Powder. 

Weight  of 
Projectile. 

Initial  Velocity 
per  Second. 
V. 

Initial 
P V2 

En 

N 

per  inch  01  <5 

circumference  5^ 
of  shot. 

P V 2 

1 

N 

O. 

Cl 

Lbs. 

Lbs. 

Feet. 

Ft. -tons. 

Foot-tons. 

Armstrong,  1 

1. 5- inch  cubes. . 

Shot.... 

330 

2000 

1446 

28990 

544-05 

100  Tons,  caliber  1 

Waltham  Abbey 

u 

375 

2000 

1543 

33000 

023 

17  ins.,  bore  30.5  f 

Fossano 

u 

400 

2000 

1502 

31  282 

5°5-74 

feet.  J 

“ ....... 

11 

776 

2000 

1832 

46  580 

835-32 

Woolwich,  81  "l 

.75-inch  cubes. 

“ 

170 

1258 

1393 

16922 

37i-5 

Tons,  caliber  14.5  V 

1.5  “ “ 

a 

220 

M5° 

1440 

20  842 

457-57 

ins.,  bore  24  feet.  J 

2 “ “ 

“ . . . . 

250 

1260 

1523 

20  259 

444.78 

caliber  16  ins. . . . . 

1.5  “ “ 

u 

310 

1466 

1553 

24  508 

520.4 

38  Tons,  'l 

1.5  “ • “ 

Pall,  shell 

130 

800 

1451 

ii  668 

297.64 

caliber  12.5  ins.,  'r 

1.5  “ “ 

“ 

200 

800 

1421 

11  210 

2o5-4 

bore  16.5  feet.  J 

1.5  “ “ . 

u. 

180 

800 

1504 

12545 

3I9-4 

Krupp,  71  Tons,  1 

Prism  A 

Plain  . . . 

298 

1707 

1184 

16602 

335-42 

caliber  15.75  ins.,  ^ 

“ H 

Shrapnel 

485 

i725 

1703 

34  503 

697.91 

bore  28. 58  feet.  J 

“ 2 inch. . . 

Shell. . . . 

441 

1419 

1761 

30  484 

616.14 

18  Tons,  1 

»!  1 hole... 

Plain  . . . 

132 

300 

1873 

7 298 

246.03 

caliber  9.45  ins.,  y 

“ 2 inch. .. 

Shrapnel 

145 

474 

1688 

9 367 

315.66 

bore  17.5  feet.  J 

Shell. . . . 

165 

300 

1991 

1 8 

244 

277.69 

Penetration  in  Ball  Cartridge  Paper,  No.  1. 

Musket,  with  134  grains,  at  13.3  yards 653  sheets. 

Common  rifle,  92  grains,  at  13.3  yards 5°°  sheets. 


Penetration  of*  Lead  13 alls  in  Small  Arm s . 
Experiments  at  Washington  Arsenal  in  1839,  and  at  West  Point  in  1837. 

Distance. 


Musket 

Common  Rifle 

Hall’s  rifle 

Hall’s  carbine,  musket 
caliber 

Pistol 

Rifle  musket 

Altered  musket 

Rifle,  Harper’s  Ferry 
Pistol  carbine. ...... 

Sharpe’s  carbine 

Burnside’s  “ .... 


Diameter 
of  Bali. 


Inch. 

(.64 

[.64 


■5775 


•5775 

.685 

•5775 

•5775 

•55 

•55 


Charge 

Powder. 


Grains. 

134 

144 

100 

92 

100 

70 

70 

80 

90* 

100* 

51 

60 

70 

40 

60 

55 


Yards. 

9 

5 

5 

9 

5 

9 

5 

5 

5 

5 

5 

200 

200 

200 

200 

30 

30 


Weight 
of  Ball. 


Penetration. 

White  Oah.  White  Pine. 


Grains. 

397-5 

397*5 

219 

219 

219 

219 


219 

500 

730 

500 

450 

463 

350 


* Charges  too  great  for  service. 


Ins. 

1.6 

3 

2.05 

1.8 

2 

.6 

i-7 

.8 

1. 1 

1.2 
•725 


11 

10.5 

9-33 

5-75 

7.17 

6.15 


Musket  discharged  at  9 yards  distance,  with  a charge  of  134  grains,  1 ball  and 
buckshot,  gave  for  ball  a penetration  of  1.15  ins.,  buckshot,  .41  inch. 


GUNNERY. 


501 


Loss  cf  Force  "by  Windage. 

A comparison  of  results  shows  that  4 lbs.  of  po,wder  give  to  a ball  without  wind- 
age nearly  as  great  a velocity  as  is  given  by  6 lbs.  having  . 14  inch  windage,  which 
is  true  windage  of  a 24-lb.  ball;  or,  in  other  words,  this  windage  causes  a loss  of 
nearly  one  third  of  force  of  charge. 

Vents. — Experiments  show  that  loss  of  force  by  escape  of  gas  from  vent 
of  a gun  is  altogether  inconsiderable  when  compared  with  whole  force  of 
charge. 

Diameter  of  Vent  in  U.  S.  Ordnance  is  in  all  cases  .2  inch. 


Eject  of  dijerent  Waddings  with  a Charge  of  77  Grains  of  Powder. 


Wad. 

Velocity  of  Ball 
per  Second. 

Ball  wrapped  in  cartridge  paper  and  crumpled 

Feet. 
13  77 

1 felt  wad  upon  powder  and  1 upon  ball 

2 felt  wads  upon  powder  and  r upon  ball 

*34^ 

T A Ro 

1 elastic  wad  upon  powder  and  1 upon  ball 

1132 

2 pasteboard  wads  upon  powder. 

2 elastic  wads  upon  powder 

I IOO 

Felt  wads  cut  from  body  of  a hat,  weight  3 grains. 

Pasteboard  wads  . 1 of  an  inch  thick,  weight  8 grains. 

Cartridge  paper  3 X 4.5  ins.,  weight  12.82  grains. 

Elastic  wads , “Baldwin’s  indented,”  a little  more  than  .1  of  an  inch  thick 
weight  5.127  grains.  * 

Most  advantageous  wads  are  those  made  of  thick  pasteboard,  or  of  or- 
dinary cartridge  paper. 

In  service  of  cannon , heavy  wads  over  ball  are  in  all  respects  injurious. 

For  purpose  of  retaining  the  ball  in  its  place,  light  grommets  should  be  used. 

On  the  other  hand,  it  is  of  great  importance,  and  especially  so  in  use  of  small 
arms,  that  there  should  be  a good  wad  over  powder  for  developing  full  force  of 
charge,  unless,  as  in  the  rifle,  the  ball  has  but  very  little  windage.  [Capt.  Mordecai  ) 


Weight  and.  Dimensions  of  Lead  JBalls. 
Number  of  Balls  in  a Lb.,  from  1.3125  to  .237  of  an  Inch  Diameter. 


Diam. 

No. 

Diam. 

No. 

Diam. 

No.- 

Diam. 

No. 

Diam. 

No. 

Diam. 

No. 

Ins. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

1.67 

1 

•75 

11 

•57 

25 

.388 

80 

.301 

170 

•259 

270 

1.326 

2 

•73 

12 

•537 

30 

•375 

• 88 

•295 

180 

.256 

280 

*•*57 

1-051 

3 

4 

•71 

•693 

*3 

x4 

•51 

•505 

35 

36 

•372 

•359 

90 

IOO 

•29 

.285 

190 

200 

.252 

.249 

290 

300 

•977 

5 

.677 

x5 

.488 

40 

•348 

no 

.281 

210 

• 247 

9 in 

•9X9 

6 

.662 

16 

.469 

45 

.338 

120 

.276 

220 

T / 

.244 

320 

•873 

•835 

7 

8 

.65 

•637 

17 

18 

•453 

.426 

50 

60 

•329 

.321 

130 

I40 

.272 

.268 

230 

240 

.242 

•239 

330 

340 

.802 

9 

.625 

*9 

•405 

70 

•3X4 

150 

.265 

250 

■237 

350 

•775 

10 

.615 

20 

•395 

75 

•307 

l6o 

.262 

260 

Heated  shot  do  not  return  to  their  original  dimensions  upon  coolin^ 
a permanent  enlargement  of  about  .02  per  cent,  in  volume. 


but  retain 


A A 
A 

B B, 


Number  of  Pellets  in  an  Ounce 

40  I B 75.1  No.  3.. 

50  No.  1 82  4. . 

58  I 2 1 12  I 5. . 

No.  14. . . 


of  Lead  Shot  of  the  dijerent  Sizes. 


135 

177 

218 


I No.  6 280 


No.  c 


• 34i 
. 600 


- 3i5o 


984 

1726 

2140 


502 


GUNNERY. 


Proportion  of  Powder  to  Sliot  for  following  Numbers 
of  Sliot. 


No. 

Shot.  . 

Powder. 

No.  | 

Shot. 

Powder. 

No. 

Shot. 

Powder. 

"t  ' 

Oz. 

Drains. 

I 

Oz. 

Drams. 

Oz. 

Drams. 

2 

2 

i-5 

4 

i-5 

1-875 

6 

1.25 

2-375 

3 

i-75 

1.625 

;5  J 

i-375 

2.125 

7 

1. 125 

2.625 

Note. 2 oz.  of  No.  2 shot,  with  1.5  drams  of  powder,  produced  greatest  effect. 

Increase  of  powder  for  greater  number  of  pellets  is  in  consequence  of  increased 
friction  of  their  projection. 

Numbers  of  Percussion  Caps  corresponding  with  Birmingham  Numbers. 


Eley’s 1 

5 I 

6 

7 1 

8 1 

9 

24 

10 

11 

18  I 

12  I 

13 1 

14 

Birmingham. . | 

43 

44 

46 

i 48  j 

49 

[5o 

| 51  and  52 

53  and  54  1 

[ 55  and  56 

! 57 

1 58 

1 58 

Where  there  are  two  numbers  of  Birmingham  sizes  corresponding  with  only  one 
of  Eley’s,  it  is  in  consequence  of  two  numbers  being  of  same  size , varying  only  in 
length  of  caps. 

Comparison  of  Force  of  a Cliarge  in  various  Arms. 


Arm. 

Lock.  . 

Powder, 
A 5- 

Windage. 

Weight 
of  Ball. 

Velocity. 

Ordinary  rifle 

Percussion. 

Grains. 

100 

Inch. 

.015 

Grains. 

219 

Feet. 

2018 

“ 

70 

.015 

219 

1755 

Hall’s  rifle 

Flint. 

70 

.0 

219 

149° 

Hall’s  carbine 

Percussion. 

70 

.0 

219 

1240 

Jenks’s  carbine 

“ 

70 

.0 

219 

1687 

Cadet’s  musket 

Flint. 

70 

•045 

219 

1690 

Pistol 

Percussion. 

35 

.015 

218.5 

947 

Ranges  for  Small  Arms. 

Musket.— With  a ball  of  17  to  pound,  and  a charge  of  no  grains  of  powder,  etc., 
an  elevation  of  36'  is  required  for  a range  of  200  yards;  and  for  a range  of  500 
yards,  an  elevation  of  30  3c/  is  necessary,  and  at  this  distance  a ball  w ill  pass  through 
a pine  board  1 inch  in  thickness. 

Rifle. — With  a charge  of  70  grains,  an  effective  range  of  from  300  to  350  yards  is 
obtained;  but  as  75  grains  can  be  used  without  stripping  the  ball,  it  is  deemed  better 
to  use  it,  to  allow  for  accidental  loss,  deterioration  of  powder,  etc. 

Pistol.—  With  a charge  of  30  grains,  the  ball  is  projected  through  a pine  board 
1 inch  in  thickness  at  a distance  of  80  yards. 


Gunpowder. 

Gunpowder  is  distinguished  as  Musket,  Mortar,  Camion,  Mammoth,  and 
Sporting  powder;  it  is  all  made  in  same . manner,  of  same  proportions  of 
materials,  and  differs  only  in  size  of  its  grain. 

Bursting  or  Explosive  Energy— By  the  experunents  of  Captain  Rodman  U S. 
Ordnance  Corps,  a pressure  of  45  000  lbs.  per  squaie  inch  was  obtained  with  10 
of  powrder,  and  a ball  of  43  lbs. 

Also,  a pressure  of  185000  lbs.  per  sq.  inch  was  obtained  when  the  powder  ivas 
burned  in  its  own  volume , in  a cast-iron  shell  having  diameters  of  3.85  and  12  in.. 

Proof  of  Powder.  ( TJ . £.  Ordnance  Manual.) 

Powder  in  magazines  that  does  not  range  over  180  yards  is  held  to  be  unservice- 
able. 

Good  powder  averages  from  280  to  300  yards;  small  grain,  from  300  to  320  yards. 

Restoring  Unserviceable  Poiuder.-  When  powder  has  been  damaged  by  being 
stored  in  damp  places,  it  loses  its  strength,  and  requires  to  be  worked  o\er.  If 
quantity  of  moisture  absorbed  does  not  exceed  7 per  cent.,  it  is  sufficient  to  diy  it 
to  restore  it  for  service.  This  is  done  by  exposing  it  to  the  sun. 

When  pow'der  has  absorbed  more  than  7 per  cent,  of  w'ater  it  should  be  sent  to  a 
powder  mill  to  be  w’orked  over. 


GUNNERY.  5O3 

3Properti.es  and  IResnlts  of  Gunpowder,  determined  "by 
Experiments.  ( Captain  A.  Mordecai , U.  S.  A.) 


24-PouNDER  Gun. 

Weight  of  ball  and  wad 24.25  lbs. 

“ “ powder 6 “ 

Windage  of  ball 135  inch. 


Musket  Pendulum. 
ball., 
powd 
Windage  of  ball. 


Weight  of  ball 397-5  grains. 

“ “ powder 120  “ 

09  inch. 


Grain. 

C 

Salt- 

petre. 

ompositi 

Char- 

coal. 

on. 

Sul- 

phur. 

Manufacture. 
Where  from. 

Number  of 
Grains  in  io 
Troy  Grains. 

Relative 
Quickness  of 
Burning. 

W ater  ab- 
sorbed by  ex- 
posure to  Air. 

Relative 

Force. 

Cannon,  large... 

“ small 

Musket 

* Dupont’s  Mills, 

77 

569 

ii34 
6174 
5 344 
1 642 
i3J52 
166 
103 
72  808 

295 

2378 

275 

314 

214 

142 

282 

Per  c’t. 
2.77 
3-35 

.677 

.72 

.808 

Rifle 

^76 

TA 

12 

Rifle 

Del. 

3-55 

•9°7 

.728 

•834 

Musket 

Rifle 

Cannon,  uneven. 

“ large... 

Sporting 

75 

12.5 

T 7 

12.5 
IO  ) 

t Dupont’s  Mills, 
Del. 

* Dupont’s  Mills, 
Del. 

Loomis,  Hazard, 
& Co.,  Conn.* 
Waltham  Abbey, 

183 

182 

2.09 
1. 91 

•943 

.788 

•756 

Blasting,  uneven 
Rifle 

/ / 
70 

1 D 

15  . 

X?} 

) 

212 

4.42 

1 

.82 

qqq 

Sporting 

| 76 

15 

9 

204 

Rifle 

T £ 

) 

10  1 

7 

1 1 600 

.OOO 

.865 

/ O 

Aw> 

IO  ) 

England.  * | 

* Glazed.  f Rough. 

Manufacture  of  Powder.  —Powder  of  greatest  force,  whether  for  cannon  or  small 
arms,  is  produced  by  incorporation  in  the  “cylinder  mills.’ ’ 

Effect  of  Size  of  Grain.— Within  limits  of  difference  in  size  of  grain,  which  occurs 
in  ordinary  cannon  powder,  the  granulation  appears  to  exercise  but  little  influence 
upon  force  of  it,  unless  grain  be  exceedingly  dense  and  hard. 

Effect  of  Glazing.  — Glazing  is  favorable  to  production  of  greatest  force,  and  to 
quick  combustion  of  grains,  by  affording  a rapid  transmission  of  flame  through 
mass  of  the  powder.  6 

Effect  of  using  Percussion  Primers.  — Increase  of  force  by  use  of  primers,  which 
nearly  closes  vent , is  constant  and  appreciable  in  amount,  yet  not  of  sufficient  value 
to  authorize  a reduction  of  charge. 

Ratio  of  Relative  Strength  of  different  Powders  for  use  under  water  differ 

but  tittle  from  the  reciprocal  of  the  ratio  between  the  sizes  of  the  grains 

showing  that  the  strength  is  nearly  inversely  proportional  thereto.* 

Mammoth,  .08;  Oliver,  .09;  Cannon,  .18;  Mortar,  1;  Musket,  1.57; 
Sporting  2.61,  and  Safety  Compound  30.62.  ’ 0 / ’ 

IDnalin  is  nitro-glycerine  absorbed  by  Schultze’s  powder. 

For  other  powders  and  explosive  materials  see  Gunnery,  page  443. 

Heat  and.  Explosive  Power.  ( Capt . Noble  and  F.  A.  Abel.) 

One  gram  of  fired  powder  evolves  a mean  temperature  of  730°.  Temper- 
ature of  explosion  3970°.  Volume  of  permanent  gas  (which  is  in  an  in- 
verse ratio  to  units  of  heat  evolved)  at  320  = 250  °. 

The  explosive  power  of  powder,  as  tested  in  Ordnance,  ranges,  for  volumes 
ot  expansion  of  1.5  to  50  times,  from  36  to  17 o foot-tons  per  lb.  burned. 

A charge  of  70  lbs.  gave  to  an  180  lbs.  shot  a velocity  of  1694  feet  per 
second,  equal  to  a total  energy  of  3637  foot-tons,  and  a charge  of  100  lbs. 
0a\e  a velocity  of  2182 feet,  and  an  energy  of  5940 foot-tons. 

aDd  lQVe8tigati0n3  t0  devel°P  a *ystera  of  ^marrae  mines.  Professional 


504 


HEAT. 


HEAT. 

Heat , alike  to  gravity,  is  a universal  force,  and  is  referred  to  both  as 
cause  and  effect. 

Caloric  is  usually  treated  of  as  a material  substance,  though  its  claims 
to  this  distinction  are  not  decided ; the  strongest  argument  in  favor  of 
this  position  is  that  of  its  power  of  radiation.  Upon  touching  a body 
having  a higher  temperature  than  our  own,  caloric  passes  from  it,  and 
excites  the  feeling  of  warmth ; and  when  we  touch  a body  haying  a 
lower  temperature  than  our  own,  caloric  passes  from  our  body  to  it,  and 
thus  arises  the  sensation  of  cold. 

To  avoid  any  ambiguity  that  may  arise  from  use  of  the  same  expres- 
sion, it  is  usual  and  proper  to  employ  the  word  Caloric  to  signify  the 
principle  or  cause  of  sensation  of  heat. 

Heat  U?iit  — For  purpose  of  expressing  and  comparing  quantities  of 
heat,  it  is  convenient  and  customary  to  adopt  a Unit  of  heat  ov  Thermal 
unit]  being  that  quantity  of  heat  which  is  raised  or  lost  in  a defined 
period  of  temperature  in  a defined  weight  of  a particular  substance. 

Thus  a Thermal  unit,  Is  quantity  of  heat  which  corresponds  to  an  interval  of  i°  in 
temperature  of  i lb.  of  pure  liquid  water,  at  and  near  its  temperature  of  greatest 
density. 

Thermal  unit  in  France,  termed  Caloric , Is  quantity  of  heat  which  corresponds 
to  an  interval  of  i°,C.  in  temperature  of  i kilogramme  of  pure  liquid  water , at  and 
near  its  temperature  of  greatest  density. 

Thermal  unit  to  Caloric,  3.96832;  Caloric  to  Thermal  unit,  .251996. 

One  Thermal  unit  or  i°  in  1 lb.  of  water,  772  foot-lbs. 

One  Caloric  or  i°  C.  in  1 kilogramme  of  water,  423.55  kilogrammetres. 

i°  C.  in  1 lb.  water,  1389.6  foot-lbs. 

Ratio  of  Fahrenheit  to  Centigrade,  1.8;  of  Centigrade  to  Fahrenheit,  .555. 

Absolute  Temperature , Is  a temperature  assigned  by  deduction,  as  an 
opportunity  of  observing  it  cannot  occur,  it  being  the  temperature  corre- 
sponding to  entire  absence  of  gaseous  elasticity,  or  when  pressure  and  vol- 
ume =0.  By  Fahrenheit  it  is— 461.2°,  by  Reaumur— 229.20,  and  by  Cen- 
tigrade— 274°. 

Heat  is  termed  Sensible  when  it  diffuses  itself  to  all  surrounding 
bodies ; hence  it  is  free  and  uncombined,  passing  from  one  substance 
to  another,  affecting  the  senses  in  its  passage,  determining  the  height 
of  the  thermometer,  etc. 

Temperature  of  a body,  is  the  quantity  of  sensible  heat  in  it,  present 
at  any  moment. 

Heat  is  developed  by  water  when  it  is  violently  agitated. 

Heat  is  developed  by  percussion  of  a metal,  and  it  is  greatest  at  the  first 
blow. 

Quantities  of  heat  evolved  are  nearly  the  same  for  same  substance,  with- 
out reference  to  temperature  of  its  combustion. 

Mechanical  power  mav  be  expended  in  production  of  heat  either  by  fric- 
tion or  compression,  and  quantity  of  heat  produced  bears  the  same  propor- 
tion to  quantity  of  mechanical  power  expended,  being  1 unit  for  power 
necessary  to  raise  1 lb.  772  feet  in  height.  This  number  of  772  is  termed 
the  mechanical  equivalent  of  heat  (Joules). 


.HEAT. 


505 


Specific  Heat. 

Specific  Heat  of  a body  signifies  its  capacity  for  heat,  or  quantity  re- 
quired to  raise  temperature  of  a body  i°,  or  it  is  that  which  is  ab- 
sorbed by  different  bodies  of  equal  weights  or  volumes  when  their 
temperature  is  equal,  based  upon  the  law,  That  similar  quantities  of 
different  bodies  require  unequal  quantities  of  heat  at  any  given  tempera- 
ture. It  is  also  the  quantity  of  heat  requisite  to  change  the  tempera- 
ture of  a body  any  stated  number  of  degrees  compared  with  that  which 
would  produce  same  effect  upon  water  at  320. 

Quantity  of  heat , therefore,  is  the  quantity  necessary  to  change  the  tem- 
perature of  a body  by  any  given  amount  (as  i°),  divided  by  quantity  of 
heat  necessary  to  change  an  equal  weight  or  volume  of  water  32 0 by  same 
amount. 

Note.— Water  has  greater  specific  heat  than  any  known  body. 

Every  substance  has  a specific  heat  peculiar  to  itself,  whence  a change  of 
composition  will  be  attended  by  a change  of  its  capacity  for  heat. 

Specific  heat  of.  a body  varies  with  its  form.  A solid  has  a less  capacity 
for  heat  than  same  substance  when  in  state  of  a liquid;  specific  heat  of 
water,  for  instance,  being  .5  in  solid  state  (ice),  .622  in  gaseous  (steam), 
and  1 in  liquid. 

Specific  heat  of  equal  weights  of  same  gas  increases  as  density  decreases ; 
exact  rate  of  increase  is  not  known,  but  ratio  is  less  rapid  than  diminution 
in  density. 

Change  of  capacity  for  heat  always  occasions  a change  of  temperature. 
Increase  in  former  is  attended  by  diminution  of  latter,  and  contrariwise. 

Specific  heat  multiplied  by  atomic  weight  of  a substance  will  give 
the  constant  37.5  as  an  average,  which  shows  that  the  atoms  of  all 
substances  have  equal  capacity  for  heat.  This  is  a result  for  which  as 
yet  no  reason  has  been  assigned. 

Thus:  atomic  weights  of  lead  and  copper  are  respectively  1294.5  and  205  7 and 
their  specific  heats  are  .031  and  .095.  Hence  1294.5  X .031  = 40.129,  and  305  7 x 
•095  = 37-  591- 

It  is  important  to  know  the  relative  Specific  Heat  of  bodies.  The  most  conve- 
nient method  of  discovering  it  is  by  mixing  different  substances  together  at  dif- 
ferent temperatures,  and  noting  temperature  of  mixture;  and  by  experiments  it 
appears  that  the  same  quantity  of  heat  imparts  twice  as  high  a temperature  to 
mercury  as  to  an  equal  quantity  of  water;  thus,  when  water  at  ioo°  and  mercury 
at  400  are  mixed  together,  the  mixture  will  be  at  8o°,  the  200  lost  by  the  water 
causing  a rise  of  400  in  the  mercury;  and  when  weights  are  substituted  for  meas- 
ures, the  fact  is  strikingly  illustrated;  for  instance,  on  mixing  a pound  of  mercury 
1 mi4°°  wit,k  a Pounfi  °f  water  at  1600,  a thermometer  placed  in  it  will  fall  to  icSo 

Thus  it  appears  that  same  quantity  of  heat  imparts  twice  as  high  a temperature  to 
mercury  as  to  an  equal  volume  of  water,  and  that  the  heat  which  gives  50  to  water 
will  raise  an  equal  weight  of  mercury  1150,  being  the  ratio  of  1 to  23.  Hence  if 
equal  quantities  of  heat  be  added  to  equal  weights  of  water  and  mercury  their 
temperatures  will  be  expressed  in  relation  to  each  other  by  numbers  1 and  23-  or 
in  order  to  increase  the  temperature  of  equal  weights  of  those  substances  to  the 
same  extent,  the  water  will  require  23  times  as  much  heat  as  the  mercury. 

Capacity  for  Heat  is  relative  power  of  a body  in  receiving  and  re- 
taining heat  in  being  raised  to  any  given  temperature ; while  Specific 
applies  to  actual  quantity  of  heat  so  received  and  retained. 

Specific  Heat  of  ^Air  and.  other  G-ases. 

Specific  heat,  or  capacity  for  heat,  of  permanent  gases  is  sensibly  constant 
1 for  a11  temperatures,  and  for  all  densities.  Capacity  for  heat  of  each  gas  is 

U u 


HEAT, 


506 

same  for  each  degree  of  temperature.  M.  Regnault  proved  that  capacity 
for  heat  for  air  was  uniform  for  temperatures  varying  from  — 220  to 
+437°;  consequently,  specific  heat  for  equal  weights  of  air,  at  constant 
pressure,  averaged  .2377. 


Metals  from,  32  0 to 
212°. 

Antimony. . . .0508 

Bismuth 0308 

Brass °939 

Copper 092 

Cast  iron 1298 

Gold °324 

Lead 0314 

Mercury 

Nickel 1086 

Platinum 0324 


Specific  Heat.  Water  at  32 0 = 1. 

Woods. 

Oak 57  1 

Pear 5 

Pine 65 


Silver 056 

Steel 1165 

Tin 0562 

Wrought  iron  .1138 
Zinc 0955 

Stones. 

Chalk .... 
Limestone 
Masonry. . 

Marble,  gray.  .2694 
“ white.  2158 


. 2149 
. .2174 


Mind  Substances. 

Charcoal 2415 

Coal 2411 

Coke 203 

Glass 1977 

Gypsum 1966 

Phosphorus..  2503 


Sulphur 2026 

Liquids. 

Alcohol 6588 

EtheV 4554 

Linseed  oil  . . .31 

Olive  oil 3096 

Steam ...  .365 

Turpentine  . . -4l6 

Vinegar 72 

Solid. 

Ice 504 


Gases. 


Air. 


•2377 


Hydrogen 2356 

Carbonic  Acid 3308 


Hydrogen 2.4096 

Carbonic  Acid 1714 


Oxygen 2412 

For  Equal  Weights. 

Air 1688 

Oxygen 1559 

Metals  have  least,  ranging  from  Bismuth  .0308  to  Cast  Iron  .1298.  Stones  and 

Mineral  Substances  have  .2  that  of  water,  and  Woods  about  .5.  Liquids,  with  ex- 
ception of  Bromine,  are  less  than  water,  Olive  oil  being  lowest  and  Vinegar  highest. 

Illustration.— If  1 lb.  of  coal  will  heat  1 lb.  of  water  to  ioo°,  — — = — — of  a lb. 

• 033  30.3 

will  heat  1 lb.  of  mercury  to  ioo°. 

To  Compute  Temperature  of’  a Mixture  of*  lilie  Sut>- 
stances. 


W T + w t 


■.t 


w (t'  — t) 


W: 


w ( t ' — t) 


+ f = T.  W representing  weight 


W-fw  ’ T — t'  ’ W 
or  volume  of  a substance  of  temperature  T,  w weight  or  volume  of  a like  substance  of 
temperature  £,  and  t'  temperature  of  mixture  W -j-  w. 

Illustration  i.  — When  5 cube  feet  of  water  (W)  at  a temperature  of  1500  (T)  is 
mixed  with  7.5  cube  feet  (10)  at  500  ( t ),  what  is  the  resultant  temperature  of  the 
mixture? 

5 X 150°  + 7.5  X 500 


5 + 7-5 


1125 
' 12.5 


— 90^. 


2. — How  much  water  at  (T)  ioo°  should  be  mixed  with  30  gallons  (w)  at  6oc,  for 
a required  temperature  of  8o°? 

30(80°  — 6o°)  600  • 

5 — 0-0-  = — = 3°  gallons. 

ioo°  — 8o°  20 


To  Compute  Temperature  of*  a Mixture  of  TJnlilre 
Substances. 


W S T + w .s  t 


— t'; 


10  S (t-r-t') 


— W; 


t'  ( W S + w s)  <x>  w s t 


= T.  W and  w 


W S-fu)i  ’ S (T  — t)  WS 

representing  weights , and  S and  s specific  heat  of  substances. 

Illustration.— To  what  temperature  should  20  lbs.  cast  iron  (W)  be  heated  to 
raise  150  lbs.  ( w ) of  water  to  a temperature  ( t ) of  50°  to  6o°? 


: 1,  and  S = . 1298. 


6o°  (20  X • 1298  + 150  X 1)  'v  150  X 1 X 500  1655.76 


20  X .1298 


2.596 


= 638°. 


HEAT. 


507 


To  Compute  Specific  Heat  at  Constant  Volume. 

S p 

When  Specific  Heat  at  Constant  Pressure  is  known.  ~w  = s-  s represent- 
ing specific  heat  at  constant  pressure,  p proportion  of  heat  absorbed  at  constant  vol- 
ume, H total  heat  absorbed  at  constant  pressure,  and  s specific  heat  at  constant  volume. 


Or  S (t'  Q— -2.742  (V  v)  _ s t and  t>  representing  initial  and  final  tempera- 

ture of  the  gas  and  that  to  which  it  is  raised,  and  V and  v initial  and  final  volumes 
of  the  gas  under  14.7  lbs.  per  sq.  inch,  and  of  it  heated  under  constant  pressure  xn 
cube  feet. 

Illustration. — Assume  1 lb.  air  at  atmospheric  pressure  and  at  320,  doubled  in 
volume  by  heat.  S = . 2377 *,  t - 1'  = 32°  <x,  525°  = 4930  and  V - v = 1 2. 387  * cube 
feet. 

■*377  X 493-(^74^Xi2.387)  = _l688  speclJk  heat 
493 

For  comparative  volumes  of  other  gases,  see  Table,  page  506. 


To  Compute  Specific  Heat  for  Equal  Volume  of  Gas 
and.  Air. 

Rule.— Multiply  specific  heat  of  the  gas  for  equal  weights  of  gas  and  air 
by  specific  gravity  of  gas,  and  product  is  specific  heat  for  equal  volume. 
Example.— What  is  specific  heat  of  air  at  equal  volume  with  hydrogen? 

Specific  heat  of  hydrogen  for  equal  weights  at  constant  volume,  2.4096,  and  speci- 
fic gravity  of  the  gas,  .0692.  (See  Table,  page  506.) 

Then,  2.4096  X .0692  = .1667  specific  heat  for  equal  volumes  at  constant  volume. 
Specific  heat  of  steam,  air  at  unity  = 1.281. 


Capacity  for*  Heat. 

When  a body  has  its  density  increased,  its  capacity  for  heat  is  di- 
minished. The  rapid  reduction  of  air  to  .2  of  its  volume  evolves  heat 
sufficient  to  inflame  tinder,  which  requires  550°. 


Relative  Capacity  for  Heat  of  Various  Bodies.  ( Water  at  320  ==  x.) 


Bodies. 

Equal 

Weights. 

Equal 

Volumes. 

Bodies. 

Equal 

Weights. 

Equal 

Volumes. 

Bodies. 

Equal 

Weights. 

Equal 

Volumes. 

Water. . 

I 

j 

Gold. . . . 

•05 

.966 

Mercury 

.036 

— 

Brass. . . 

.116 

.971 

Ice 

•9 

— 

Silver . . 

.082 

•833 

Copper. . 

.114 

1.027 

Iron 

. 126 

•993 

Tin 

.06 

— 

Glass. . . 

.187 

.448 

Lead . . . 

•°43 

.487 

Zinc. . . . 

. 102 

1 — 

To  Ascertain  Relative  Capacities  of*  Different  Bodies, 
combined  witli  experiment. 

Rule. — Multiply  weight  of  each  body  by  number  of  degrees  of  heat  lost 
or  gained  by  mixture,  and  capacities  of  bodies  will  be  inversely  as  products. 

Or,  if  bodies  be  mingled  in  unequal  quantities,  capacities  of  the  bodies 
will  be  reciprocally  as  quantities  of  matter,  multiplied  into  their  respective 
changes  of  temperature. 

Illustration. — If  1 lb.  of  water  at  156°  is  mixed  with  1 lb.  of  mercury  at  400, 
resultant  temperature  is  1520. 

Thus,  1 x 156°  — 152°  = 40,  and  1X40°^  152°=:  1120.  Hence  capacity  of  water 
for  heat  is  to  capacity  of  mercury  as  1120  to  40,  or  as  28  to  1. 

Sensible  Heat. 

Sensible  heat  or  temperature  to  raise  water  from  320  to  212°  = 180.9°,  or 
heat  units. 


* See  Tables,  pages  506  and  520-21. 


508 


HEAT. 


Latent  Heat. 

Latent  Heat  is  that  which  is  insensible  to  the  touch  of  our  bodies, 
and  is  incapable  of  being  detected  by  a thermometer. 

When  a solid  body  is  exposed  to  heat,  and  ultimately  passes  into  the 
liquid  state  under  its  influence,  its  temperature  rises  until  it  attains  the 
point  of  fusion,  or  melting  point.  The  temperature  of  the  body  at  this 
noint  remains  stationary  until  the  whole  of  it  is  melted ; and  the  heat  mean- 
time absorbed,  without  affecting  the  temperature  or  being  sensible  to  the 
touch  or  to  the  indications  of  a thermometer,  is  said  to  become  latent.  It  is, 
in  fact,  the  latent  heal  of  fusion,  or  the  latent  heat  of  liquidity , and  its  func- 
tion is  to  separate  the  particles  of  the  body,  hitherto  solid,  and  change  their 
condition  into  that  of  a liquid.  When,  on'the  contrary,  a liquid  is  solidified, 
the  latent  heat  is  disengaged. 

If  to  a pound  of  newly-fallen  snow  were  added  a pound  of  water  at  1720, 
the  snow  would  be  melted,  and  320  would  be  resulting  temperature. 

When  a body  is  fusing,  no  rise  in  its  temperature  occurs,  however  great 
the  additional  quantity  of  heat  may  be  imparted  to  it,  as  the  increased  heat 
is  absorbed  in  the  operation  of  fusion.  The  quantity  of  heat  thus  made 
latent  varies  in  different  bodies. 

A pound  of  water,  in  passing  from  a liquid  at  212°  to  steam  at  2120,  re- 
ceives as  much  heat  as  would  be  sufficient  to  raise  it  through  966.6  ther- 
mometric degrees,  if  that  heat,  instead  of  becoming  latent , had  been  sensible . 

If  5 5 lbs.  of  water,  at  temperature  of  320,  be  placed  in  a vessel,  communicating 
with  another  one  (in  which  water  is  kept  constantly  boiling  at  temperature  of  212°), 
until  former  reaches  temperature  of  latter  quantity,  then  let  it  be  weighed,  and 
it  will  be  found  to  weigh  6.5  lbs.,  showing  that  one  lb.  of  water  has  been  received 
in  form  of  steam  through  communication,  and  reconverted  into  water  by  lower 
temperature  in  vessel.  Now  this  pound  of  water,  received  in  the  form  of  steam, 
had  when  in  that  form,  a temperature  of  2120.  It  is  now  converted  into  liquid 
form  and  still  retains  same  temperature  of  2120;  but  it  has  caused  5.5  lbs.  of  water 
to  rise  from  the  temperature  of  320  to  2120,  and  this  without  losing  any  tempera- 
ture of  itself.  Now  this  heat  was  combined  with  the'  steam,  but  as  it  is  not  sensible 
to  a thermometer,  it  is  termed  Latent. 

Quantity  of  heat  necessary  to  enable  ice  to  resume  the  fluid  state  is  equal 
to  that  which  would  raise  temperature  of  same  weight  of  water  140°  ; and  an 
equal  quantity  of  heat  is  set  free  from  water  when  it  assumes  the  solid  form. 

Sum  of  Sensible  and.  Latent  Heats. 

From  Water  at  32°. 


Press- 

ure. 

Latent. 

Sum. 

Press- 

ure. 

Latent. 

Sum. 

Press- 

ure. 

Latent. 

Sum. 

Press- 

ure. 

Latent. 

Sum. 

Lbs. 

0 

0 

Lbs. 

O 

O 

Lbs. 

0 

0 

Lbs. 

0 

0 

14.7 

964-3 

1146.1 

26 

943-7 

II55-3 

55 

912 

1*69 

120 

873-7 

1185.4 

16 

962.1 

1147.4 

27 

942.2 

H55-8 

60 

908 

1170. 7 

130 

869.4 

1187.3 

17 

959- 8 

ii48-3 

28 

940. 8 

1156.4 

65 

904,2 

II72-3 

140 

865.4 

1189 

18 

957-7 

1 149. 2 

29 

939-4 

ii57-i 

70 

900.8 

1173.8 

150 

861.5 

.1190.7 

19 

955-7 

1150. 1 

3° 

937-9 

H57-8 

75 

897-5 

1175.2 

160 

857-9 

1192.2 

20 

952.8 

11509 

32 

935-3 

1 158-9 

80 

894-3 

1176.5 

170 

854-5 

“93-7 

21 

951.3 

ii5i-7 

35 

931.6 

1160.5 

85 

891.4 

II77-9 

180 

85I-3 

II95-1 

22 

949-  9 

1152.5 

37 

929*3 

1161.5 

90 

888.5 

1x79. 1 

190 

848 

1196.5 

23 

948.5 

1153-2 

40 

926 

1162.9 

95 

885.8 

1180.3 

200 

845 

1197.8 

24 

946.9 

II53-9 

45 

920.9 

1164.6 

100 

883. 1 

1181.4 

220 

829.2 

1200.3 

4 

945-3 

ii54-6 

50 

9i6-3 

1167.1 

no 

878.3 

1183.5 

250 

831.2 

1203.7 

Latent  Heat  of  Vaporization , or  Number  of  Degrees  of  Heat  required  to  con- 
vert, following  Substances  from  their  respective  Solidities  to  Vapor  at 
Pressure  of  Atmosphere. 

Alcohol 364°  I 142-6°  I Water c...  966.6° 

Ammonia 860°  Mercury 1570  £inc... — - * • 493q 

Ether  (Sulph.) 163°  I.Oarbonic  Acid.....  298°  I Oil  of  Turpentine. . 124° 


HEAT. 


509 


Latent  Heat  of’  Fusion  of  Solids.  [Person.) 


Substances. 

Melt- 

ing 

Point. 

Specific  Heat. 
Liquid.  Solid, 

In  Heat- 
units  of 
1 lb. 

Substances. 

Melt- 

ing 

Point. 

Specify 

Liquid. 

; Heat. 
Solid. 

In  Heat- 
units  of 
1 lb. 

Tin 

O 

442 

507 

O 

.0637 

.0363 

O 

.0562 

.0308 

25. 6 

Ice 

C 

32 

112 

0 

I 

0 

•504 

.1788 

142.85 

9 

Bismuth. . 

22.7 

Phosphorus 

.2045 

Lead 

Zinc 

617 

773 

.0402 

.0314 

.0956 

9. 86 
50.6 

Spermaceti 

Wax 

120 

142 



148 

175 

Silver 

i873 

— 

•057 

37-9 

Sulphur 

239 

•234 

.2026 

17 

Mercury. . 

39 

•0333 

.0319 

5 

Nitrate  of  soda. . 

591 

•413 

.2782 

ii3 

Cast  iron. . 

3400 

.129 

233 

Nit.  of  potassia . 

642 

•3319 

.2388 

85 

To  Compute  Latent  Heat  of  Fusion  of  a Non-metallic 
Substance. 

C'vc  (£-f-  256°)  = L.  C and  c representing  specific  heats  of  substance  in  solid  and 
liquid  state , t temperature  of  fusion,  and  L latent  heat. 

Illustration.— What  is  latent  heat  of  fusion  of  ice? 


C = -504;  c = i;  and  t = 32°.  

.504  -v  1 X 32  -f-  256  = 142.85°  units. 

Note. — For  Latent  Heat  of  Fusion  of  some  substances,  see  Deschanel’s,  New  York, 
1872,  Heat,  part  2. 

Radiation  of*  Heat. 

Radiation  of  Heat  is  diffusion  of  heat  by  projection  of  it  in  diverging  right 
lines  into  space,  from  a body  having  a higher  temperature  than  space  sur- 
rounding it,  or  body  or  bodies  enveloping  it. 

Radiation  is  affected  by  nature  of  surface  of  body ; thus,  black  and  rough 
surfaces  radiate  and  absorb  more  heat  than  light  and  polished  surfaces. 
Bodies  which  radiate  heat  best  absorb  it  best. 

Radiant  heat  passes  through  moderate  thicknesses  of  air  and  gas  without 
suffering  any  appreciable  loss  or  heating  them.  When  a polished  surface 
receives  a ray  of  heat,  it  absorbs  a portion  of  it  and  reflects  the  rest.  The 
quantity  of  heat  absorbed  by  the  body  from  its  surface  is  the  measure  of 
its  absorbing  power , and  the  heat  reflected,  that  of  its  reflecting  power . 

When  temperature  of  a body  remains  constant  it  is  in  consequence  of 
quantity  of  heat  emitted  being  equal  to  quantity  of  heat  absorbed  by  body. 
Reflecting  powrer  of  a body  is  complement  of  its  absorbing  power ; or,  sum 
of  absorbing  and  reflecting  powers  of  all  bodies  is  the  same. 

Thus,  if  quantity  of  heat  which  strikes  a body=z  ioo,  and  radiating  and  reflecting 
powers  each  90,  the  absorbent  would  be  10. 

Radiating  or  Absorbent  and  Reflecting  Rowers  of 
Substances. 


Substances. 


Lamp  Black 

Water 

! Carbonate  of  Lead 

Lead,  white 

Writing  Paper 

Ivory,  Jet,  Marble 

Resin 

Glass 

India  Ink 

Ice 

Shellac 

Lead 

Cast  Iron, bright  polished 
Platinum,  a little  polish’d 
Mercury 


1 Radiating 
or  Ab- 
sorbing. 

Reflect- 

ing. 

Substances. 

Radiating 
or  Ab- 
sorbing. 

Reflect- 

ing. 

IOO 

_ 

Wrought  Iron,  polished.. 

23 

77 

100 

— 

Lead,  polished 

19 

81 

IOO 

— 

Zinc,  polished 

19 

81 

100 

— 

Steel,  polished 

17 

83 

98 

2 

Platinum,  in  sheet 

17 

83 

93  to  98 

7 to  2 

Tin 

15 

85 

96 

4 

Copper,  varnished 

14 

86 

90 

10 

Brass,  dead  polished 

11 

89 

85 

15 

“ bright  polished. . . 

7 

93 

85 

i5 

Copper,  ham’ered  or  cast 

7 

93 

72 

28 

“ deposited  on  iron 

7 

93 

45 

55 

Gold,  plated 

5 

95 

25 

75 

“ polished 

3 

97 

24 

76 

Silver,  polished 

3 

97 

23 

77 

“ cast,  polished  . . . 

3 

97 

U u* 


HEAT. 


510 


Radiating  and  Absorbing  Rower  of  various  Bodies,  in 
Units  of  Heat  per  Sq.  Root  per  Hour  for  a Difference 
of  1°.  ( Peclet .) 

Unit. 

Iron,  ordinary 5662 

Glass yfTi: 5948 

Iron,  cast . . . . 648 

Wood  sawdust 7225 

Stone,  Brick,  etc 7358 


Unit. 

Silver,  polished 0266 

Copper 0327 

Tin 0439 

Brass,  polished 0491 

Iron,  sheet. 092 


Unit. 

Woollen  stuff 7522 

Oil  paint 7583 

Paper 7706 

Lamp-black 8196 

Water 10853 


To  Compute  Loss  of  Heat  “by  Radiation  per  Sq.  Root. 

1.7  l (T  — Q _ R T representing  temperature  of  pipe,  'which  is  assumed  to  he  .05 
d v 

less  than  that  of  steam ; t temperature  of  air ; l length  of  pipe,  and  v velocity  of  heat 
■in  feet  per  second ; d diameter  in  ins.,  and  R radiation  in  degrees  per  second. 

Illustration. — Assume  temperatures  of  a steam-pipe,  steam,  2120,  2000,  and  air 
6o°,  length  of  pipe  20  feet,  velocity  of  heat  (steam)  15  feet  per  second,  and  diameter 
of  pipe  16  ins. ; what  will  be  loss  of  heat  by  radiation? 

1.7X20  (200-60)  _ 0 

16  X 15 

Reflection. 

Reflection  of  Heat  is  passage  of  beat  from  surface  of  one  substance 
to  another  or  into  space,  and  it  is  the  converse  of  radiation. 

Heat  is  reflected  from  surface  upon  which  its  rays  fall  in  same  manner  as 
light,  angle  of  reflection  being  opposite  and  equal  to  that  of  incidence.  Met- 
als are  the  strongest  reflectors. 

Reflecting  Power  of  various  Substances. 

. .8i 

77 

.6 


Silver 

I Specular  metal 

...  .86  ! 

j Zinc  . 

Gold 

Tin. 

Iron 

Brass — 

1 Steel 

...  .83  ! 

1 Lead 

Communication  and  Transmission  of  Heat. 

Communication  of  Heat  is  passage  of  heat  through  different  bodies 
with  different  degrees  of  velocity.  This  has  led  to  division  of  bodies 
into  Conductors  and  Non-conductors  ; former  includes  such  as  metals, 
which  allow  caloric  to  pass  freely  through  their  substance,  and  latter 
comprise  those  that  do  not  give  an  easy  passage  to  it,  such  as  stones, 
glass,  wood,  charcoal,  etc. 

Velocity  of  cooling,  other  things  being  equal,  increases  with  extent  of  sur- 
face compared  with  volume  of  substance ; and  of  two  bodies  of  same  mate- 
rial, temperature,  and  form,  but  differing  in  volume. 

Transmission  of  Heal  is  passage  of  heat  through  different  bodies  with  dif- 
ferent degrees  of  intensity.  Gaseous  bodies  and  a vacuum  are  highest  in 
order  of  transmitten ts. 

Relative  Power  of  various  Substances  to  Transmit  Heat. 

All  bodies  capable  of  transmitting  heat  are  more  or  less  translucent, 
though  their  powers  of  transmitting  heat  and  light  are  not  in  same  rela- 
tive proportions. 


Flint-glass 67  j Nitric  acid 15 

Gypsum 2 Rock-crystal ..  .62 

Ice 06  I Rape  seed  oil. . .3 


Sulphuric  acid.  .17 

Turpentine 31 

Water 11 


Air 

Alcohol 15 

Crown-glass..  .49  * — ---  , - — * - 

Heat  which  passes  through  one  plate  of  glass  is  less  subject  to  absorption 
in  passing  through,  a second  and  a third  plate.  Of  1000  rays,  451  were  in- 
tercepted by  4 plates  as  follows : 

1st.  381.  2d.  43.  3d.  18.  4th.  9. 


3d.  18. 


HEAT.  5 1 1 


Average  Results  of  Heating  and  Evaporating  W ater  by 
Steam  in  Copper  Pipes  and.  Boilers.  (D.  K.  Clark.) 


Steam  condensed 

Heat  transmitted 

Per  sq.  foot  for  i°  difference  per  1 

hour. 

Heatjng. 

Evaporating. 

Heating. 

Evaporating 

Lbs. 

Lbs. 

Units. 

Units. 

Cast-iron-plate  surface 

.077 

.105 

82 

100 

Copper  plate  surface 

. 248 

•483 

276 

534 

Copper-pipe  surface 

•291 

1.07 

312 

1034 

Whence. — Efficiency  of  copper-plate  surface  for  evaporation  of  water  is 
double  its  efficiency  for  heating;  for  copper-pipe  surface  efficiency  is  more 
than  three  times  as  much ; and  for  cast-iron-plate  surface,  a fourth  more. 

Efficiency  of  pipe  surface  is  a fifth  more  than  that  of  plate  surface  for 
heating,  and  more  than  twice  as  much  for  evaporation. 

Generally,  copper -plate  surface  condenses  .5  lb.  of  steam,  copper-pipe 
1 lb.,  and  cast-iron-plate  surface  .1  lb.  per  sq.  foot  per  i°  of  temperature  per 
hour,  for  evaporation. 

Quantity  of  heat  transmitted  is  at  rate  of  about  1000  units  per  lb.  of  steam 
condensed. 

Transmission  of  Heat  through  Glass  of  different  Colors. 


Direct  =.  100. 


Plate 

1 Blue,  deep 

I Yellow 

Window 

....52  1 

u light 

42 

Orange 

44 

Violet,  deep 

• • • • 53  i 

1 Green 

| Red 

M.  Peclet  defines  law  of  transmission  of  heat  as  : The  flow  of  heat  which 
traverses  an  element  of  a body  in  a unit  of  time  is  proportional  to  its  sur- 
face, and  to  difference  of  temperature  of  the  two  faces  perpendicular  to  direc- 
tion of  flow,  and  is  in  inverse  of  thickness  of  element. 

c 

Or,  (t  — i')  - = H.  t and  f representing  temperatures  of  surfaces , C constant  for 

maternal  1 inch  thick , or  quantity  of  heat  transmitted  per  hour  for  i°  difference  of 
temperature  through  1 unit  of  thickness , T thickness , and  H quantity  of  heat  in  units 
passed  through  plate  per  sq.  foot  per  hour. 

Quantities  of  Heat  transmitted  from  Water  to  Water 
tlirougli  Blates  or  Beds  of  IVIetals  and  otlier  Solid 
Bodies,  1 Inch,  thick,  per  Sq.  Eoot. 

For  i°  Difference  of  Temperature  between  the  two  Faces  per  Hour . 

Selected  from  M.  Peclet’s  tables.  (D.  K.  Clark.) 


Substance. 

C or 
Quantity 
of  Heat. 

Substance. 

C or 
Quantity 
of  Heat. 

Substance. 

C or 

Quantity 
of  Heat. 

Gold 

Units. 

620 

604 

596 

555 

Iron 

Units. 

225 

225 

177 

112 

Marble 

Units. 

24 

2.6 

6.56 

2.16 

Platinum 

Zinc 

Plaster 

Silver 

Tin 

Glass  

Copper 

Lead 

Sand 

The  conditions  are,  that  the  surfaces  of  conducting  material  must  be  per- 
fectly clean,  that  they  be  in  contact  with  water  at  both  faces  of  different 
temperatures,  and  that  the  water  in  contact  with  surfaces  be  thoroughly  and 
constantly  changed.  M.  Peclet  found  that  when  metallic  surfaces  became 
dull,  rate  of  transmission  of  heat  through  all  metals  became  very  nearly 
the  same. 

To  Compute  Units  of  Heat  Transmitted. 

Illustration  i. — If  2000  lbs.  beet  root  juice  at  400  are  contained  in  a copper 
boiler  with  a double  bottom,  and  heated  to  2120,  with  a heating  surface  of  25  sq.  feet, 
and  subjected  to  steam  at  a temperature  of  275°,  for  a period  of  15  minutes,  what 
will  be  the  total  heat,  and  heat  per  degree  of  difference  transmitted  per  sq.  foot  per 
hour? 


512 


HEAT. 


2I2o  _ 40O  x 60  4- 15  = 688°  per  hour,  and  2000  X 688  -f-  25  = 55  040  units  per  sq. 
foot  per  hour. 

(2120  + 400)  -r-  2 = 126°  mean  temperature  of  juice,  and  2750  — 126°  = 1490  mean 
difference  of  temperature. 

Hence,  55040-4-149  = 369.4  units  per  sq.  foot  per  degree  of  difference  per  hour. 

2._  If  48.2  sq.  feet  of  iron  pipe  1.36  ins.  in  diameter,  is  supplied  with  steam  at  2750, 
and  it  raises  temperature  of  882  lbs.  water  from  46°  to  2120  in  4 minutes,  what  will 
be  total  heat  per  sq.  foot  per  hour,  total  heat  per  sq.  foot  per  degree,  and  quantity 
condensed  per  sq.  foot  per  degree  per  hour  ? 

2x2® 46°  X 60  -4-  4 ==  2490°  in  an  hour  ; 46°  -j-  2120  -4-  2 = 1290  mean  temper- 

ature, and  2750  — 129°  = 146°  difference  of  temperature. 


2490°  X 882  _ ^ ^ units  per  hour , 45  563  -4-  x46  = 312. 1 units  per  sq. 

foot  per  degree,  and  total  heat  of  steam  above  1290  ==  1068°. 

Hence  312  - = .292  lbs.  steam  condensed  per  sq.foot  per  degree  per  hour. 

1068 


Evaporation. 


Evaporation  or  Vaporization  is  conversion  of  a fluid  into  vapor,  and 
it  produces  cold  in  consequence  of  heat  being  absorbed  to  form  vapor. 

It  proceeds  only  from  surface  of  fluids,  and  therefore,  other  things  equal , 
must  depend  upon  extent  of  surface  exposed. 

When  a liquid  is  covered  by  a stratum  of  dry  air,  evaporation  is  rapid, 
even  when  temperature  is  low. 

As  a large  quantity  of  heat  passes  from  a sensible  to  a latent  state  during 
formation  of  vapor,  it  follows  that  cold  is  generated  by  evaporation. 

Fluids  evaporate  in  vacuo  at  from  120°  to  1250  below  their  boiling-point. 


Heat  required  to  Evaporate  lib.  Water  at  Temperatures 
below  212°  from  a Vessel  in.  open  air  at  32°, 

( Thomas  Box.) 


< 

£ « 
8 S 
M * 

H 

Water  e vapor’d 
per  sq.  foot  of 
surface  p’r  hour. 

lost  by  radia- 
tion from 
surface. 

lost  in  air. 

X 

M 

> 

to  evaporate  H 
1 lb.  of  wa- 
ter. 

Total  lost 
per  hour. 

Tempera- 

ture. 

Water  evapor’d 
per  sq.  foot  of 
surface  p’r  hour. 

lost  by  radia- 
tion from 
surface. 

HE; 

h 

‘3 

.5 

to  evaporate  H 
1 lb.  of  wa- 
ter. 

O 

Lbs. 

Units. 

Units. 

Units. 

Units. 

0 

Lbs. 

Units. 

Units. 

Units. 

32 

.027 



— 

1091 

29 

132 

.706 

182 

202 

1506 

42 

.04 

270 

424 

1788 

71 

142 

.916 

158 

162 

1445 

52 

.058 

375 

581 

2052 

119 

152 

1.178 

137 

127 

1392 

62 

.083 

4°5 

605 

2110 

i74 

162 

I*5°5 

118 

97 

1346 

72 

.117 

386 

566 

2055 

239 

172 

1895 

106 

72 

1312 

82 

. 162 

358 

504 

1968 

3*9 

182 

2-373 

¥ 

50 

1279 

92 

.223 

3*9 

434 

l862 

4i5 

192 

2.947 

81 

32 

1253 

102 

• 3°3 

280 

366 

1758 

533 

202 

3-633 

ll 

H 

1228 

1 12 

.406 

245 

3°4 

1664 

671 

212 

4.471 

63 

— 

1209 

122 

.528 

211 

250 

1580 

849 

— 

— 

— 

““ 

H e. 

Unit9. 

1068 

1326 

1637 

2039 

2475 

3°45 

3685 

4465 

5397 


To  Compute  Surface  of  a Refrigerator. 

Illustration  of  Table.  — If  it  is  required  to  cool  20  barrels,  of  42  gallons  each,  of 

beer,  from  202°  to  82°  in  an  hour.  4 

Result  to  be  attained  is  to  dissipate  42  X 8.33  (lbs.  U.  S.  gallons)  X 20  X 202  — 82 
= 840000  units  of  heat  per  hour. 

At  202°,  4465  units  are  lost,  and  at  82°,  319,  hence,  average  loss  for  each  temper- 
ature between  oxtremes  ;=  1850  units  per  sq.  foot  per  hour. 

Then  *4ooc??  ^ 4^4  sq.feet  in  a still  air. 

1850 

The  volume  of  air  required  per  hour  in  this  case  would  be  about  100000  cube  feet. 


HEAT. 


513 


To  Compute  Area  of  Grrate  and.  Consumption  of*  Eiiel 
for  Evaporation. 


Illustration  of  Table.— If  it  is  required  to  evaporate  6 Beer  gallons  (282  cube  ins.) 
of  liquid  per  hour,  at  a temperature  not  exceeding  1520. 

6 gallons  = 50  lbs.  At  1520,  water  evaporated  as  per  table  = 1. 178  lbs.  per  hour. 


-- — = 42  sq.feet.  Heat  required  to  effect  this  = 1392  X 50  = 69  600  units. 

1.178 

Assuming  6000  units  as  average  economic  value  of  coals,  then  ^ 6°°  — 11.6  lbs. 


coal,  on  a grate  of  1 sq.foot. 


When  it  is  practicable  to  evaporate  at  a high  temperature,  as  at  or'  above  2120,  it 
is  most  economical. 


Thus,  water  requires  only  1209  units  per  lb.  if  surface  is  exposed,  but  if  enclosed, 
heat  is  reduced  (1209  — 63)  to  1146  units. 

Evaporative  Powers  of  Different  Tubes  per  Degree  of  Heat,  per  Sq.  Foot  of 
Surface. — In  Units. 

Vertical  tube,  230;  Double-bottomed  vessel,  330;  Horizontal  tube  or  Worm,  430. 

To  Compute  Volume  of  Water  Evaporated  in  a given 
Time. 

Illustration.^- What  is  volume  evaporated  at  2120,  in  15  minutes  per  sq.  foot  of 
surface,  in  a double-bottomed  vessel  having  an  area  of  heating  surface  of  17  feet, 
and  subjected  to  steam  at  a pressure  of  25  lbs.  ? 

Temperature  of  steam  at  25  -f  14.7  lbs.  = 269°.  269°  — 2120  = 57°,  and  latent 

heat  927. 

Then  330X57X17X15  = g6  2 m wafer> 

927  x 60 

When  Water  is  at  a Lower  Temperature  than  2120. 

If  120  gallons  or  1000  lbs.  of  water  were  to  be  evaporated  from  420  in  an 
hour,  from  same  vessel  and  under  like  pressure  as  preceding : 

There  would  be  required  1000  X (2120  — 420)  170000  units  of  heat.  Mean  tempera- 
ture of  water  while  being  heated  1=  42°  + 212°  _ Q 

2 ' 

Difference  between  temperature  of  steam  and  water  = 267° 1270  = 1400. 

170000 

Then,  ^ w — • 216  hour  =Z  time  to  raise  water  to  2120 ; hence  1 — . 216  = 

33®  s'  140  17 

.784  hour  left  for  evaporation,  and  quantity  evaporated  = 330  * 57  X 17  X -784_, 
270. 4 lbs. , or  32. 44  gallons. 

Dessiccation. 

Dessiccation , or  the  drying  of  a substance,  is  best  effected  in  a drying 
chamber,  and  it  is  imperative  that  to  attain  greatest  effect  the  hot  air 
should  be  admitted  at  highest  point  of  exposed  substance  and  dis- 
charged at  its  lowest. 

W ood,  submitted  to  an  average  temperature  of  300°  in  an  enclosed  space 
for  a period  of  2.5  days,  will  lose  its  moisture  at  a consumption  of  1 lb.  of 
vood  for  10.5  lbs.  of  wood  dried,  and  evaporating  4 lbs.  of  water,  equal  to 
2.66  lbs.  of  water  per  lb.  of  undried  wood. 

Limit  of.  temperature  for  drying  of  wood  is  340°. 


5 14 


HEAT. 


Evaporation  of  Water  per  Sq.  Eoot  of  Surface  per  Hour. 


Temperature 
of  Water. 

1 

Calm. 

Evaporation 

Light 

Air. 

(Dr.  D 

Brisk 

Wind. 

1 alton . ) 

Temperature 
of  Water. 

1 

Calm. 

Evaporation 

Light 

Air. 

Brisk 

Wind. 

O 

32 

40 

50 

60 

70 

80 

The  rates  ol 

Lbs. 
•0349 
•0459 
•0655 
•°9I7 
•1257 
.1746 
? evaporat 

Lbs. 
.0448 
.0589 
.0841 
•1175 
.1616 
.2241 
:ion  for  tl 

Lbs. 

.055 
.0723 
.1032 
.1441 
.1983 
•2751 
iese  cond 

0 

100 

125 

150 

W5 

200 

212 

itions  of  the  ai 

Lbs. 
.3248 
.6619 
1.296 
2.378 
4.128 
5-239 
r when  pi 

Lbs. 
.4169 
.8494 
1.663 
3- 053 
5.298 
6.724 
erfectly  d 

Lbs. 
.5116 
1-043 
2.043 
3-746 
6. 502 
8.252 
ry  are  as 

i,  1.28,  and  1.57. 

To  Compute  Quantity  of  Water  exposed  to  Air  that  would  he  evaporated  as 
above. — Subtract  tabulated  weight  of  water  corresponding  to  dew-point  from 
weight  of  water  corresponding  to  temperature  of  dry  air,  and  remainder  is 
weight  of  water  that  would  be  evaporated  per  sq.  foot  of  surface  per  hour. 

Distillation. 

Distillation  is  depriving  vapor  of  its  latent  heat,  and,  though  it  may 
be  effected  in  a vacuum  with  very  little  heat,  no  advantage  in  regard  to 
a saving  of  fuel  is  gained,  as  latent  heat  of  vapor  is  increased  propor- 
tionately to  diminution  of  sensible  heat. 

A temperature  of  70°  is  sufficient  for  distillation  of  water  in  a vessel  ex- 
hausted of  air. 

Conduction  or  Convection  of  Heat. 

Air  and  gases  are  very  imperfect  conductors.  Heat  appears  to  be 
transmitted  through  them  almost  entirely  by  conveyance,  the  heated 
portions  of  air  becoming  lighter,  and  diffusing  the  heat  through  the 
mass  in  their  ascent.  Hence,  in  heating  a room  with  air,  the  hot  air 
should  be  introduced  at  lowest  part.  The  advantage  of  double  win- 
dows for  retention  of  heat  depends,  in  a great  measure,  upon  sheet  of  air 
confined  between  them,  through  which  heat  is  very  slowly  transmitted. 

Convection  of  heat  refers  to  transfer  and  diffusion  of  heat  in  a fluid  mass, 
by  means  of  the  motion  of  the  particles  of  the  mass. 

Relative  Internal  Conducting  Rowers  of  Various 
Substances. 


Brass 

. .76 

Cast  Iron 

- -517 

Copper 

. .89 

Cement 

Chalk 

..  .6 

Charcoal 

..  .07 

Slate, 


Metals. 


Gold 1 I 

Porcelain .... 

.012 

Lead 18 

Silver 

•97 

Platinum 98 

1 Terra  Cotta. . 

.011 

Minerals. 

Coal,  anth’cite  1.92 

Fire  brick. . . 

. .61 

“ bitumin.  1.68 

Fire  clay 

. .76 

Coke 1.98 

Glass 

. .96 

Wood  ash. 


Tin 

Wrought  Iron 
Zinc 


Gypsum, 
Lime. . . 
Marble . 
08 


Woods  with  Birch  — .41  with  Silver. 


•3 

•44 


•36 


.2 

.24 

1.22 


Apple 
Ash. . 


Cotton 

Eider  down. . . 


.68  I 
•73  1 

Birch 1 

Chestnut 7 

Ebony 

Elm 

:%\ 

I Oak 

Pine 

Hair  and  Fur  with  Air  = 1. 

-55  I 
•44  1 

I Flannel 2.44 

1 Hemp  Canvas.  .28 

| Hair 

| Hare’s  fur — 

:24sl 

I Silk 

1 Wool 

...  .5 

Liquids  with  Water. 

Alcohol 93  I Proof  spirit 85  I Turpentine 3.1 

Mercury 2.8  | Sulphuric  acid 1.7  | Water z 


HEAT. 


515 


Practical  Deductions  from  preceding  Results. 

Asphalt  is  best  composition  for  resisting  moisture,  and,  being  a slow  con- 
ductor of  heat,  it  is  best  adapted  for  economy  of  heat  and  dryness. 

Slate  is  a very  dry  material,  but,  from  its  quick  conducting  power,  it  is 
not  adapted  for  retention  of  heat. 

Cements.  — Plaster  of  Paris  and  Woods  are  well  adapted  for  lining  of 
rooms,  having  low  conductive  powers,  while  Hair  and  Lime , being  a quick 
conductor,  is  one  of  the  coldest  compositions. 

Fire-brick  absorbs  much  heat,  and  is  well  adapted  for  lining  of  fire-places, 
etc. ; while  Iron , being  a high  conductor  of  heat,  is  one  of  the  worst  of  sub- 
stances for  this  purpose.  Common  brick  is  not  a very  slow  conductor  of  heat. 

CoixLinniaicat  ion. 

Communication  of  Heat  is  passage  of  heat  through  different  bodies 
with  different  degrees  of  velocity.  This  has  led  to  the  division  of 
bodies  into  Conductors  and  Non-conductors  of  caloric;  the  former  in- 
cludes such  as  metals,  which  allow  caloric  to  pass  freely  through  their 
substance,  and  the  latter  comprise  those  that  do  not  give  an  easy  pas- 
sage to  it,  such  as  stones,  glass,  wood,  charcoal,  etc. 

The  velocity  of  cooling,  other  things  being  equal,  increases  with  the  extent 
of  surface  compared  with  volume  of  substance;  and  of  two  bodies  of  same 
material,  temperature,  and  form,  but  differing  in  volume. 

Condensation. 

Tredgold  ascertained  by  experiment  that  steam  at  pressure  (absolute) 
of  17.5  lbs.  per  sq.  inch,  22 1°,  produced  1 cube  foot  of  water  per  hour 
by  condensation  in  182  sq.  feet  of  cast-iron  pipe,  at  a uniform  and  qui- 
escent temperature  of  6o°.  Hence,  condensation  .352  lb.  water  per 
hour,  or  .0022  lbs.  per  degree  of  difference  of  temperature  (221—60). 

From  experiments  of  Mr.  B.  G.  Nichol  in  England,  1875,  it  was  deduced: 

That  rates  of  transmission  of  heat,  between  temperature  of  steam  and 
that  of  water  of  condensation  at  its  exit,  at  the  rate  of  150  feet  per  minute, 
may  be  taken  as  380  units  for  vertical  tubes  and  520  for  horizontal. 


Condensation  of  Steam  in  Cast-iron  IPipes.  (M.  Burnat.) 


Average 
Press,  per 

Temperature. 

Condensation  per  sq 

. foot  of  external  surface  of  pipe 
per  hour. 

Sq. Inch. 

Steam. 

Air. 

Difference. 

Bare. 

Straw. 

Pipe. 

Waste. 

Plaster. 

Lbs. 

O 

0 

0 

Lb. 

Lb. 

Lb. 

Lb. 

Lb. 

22 

233 

36.5 

i96-5 

,.581 

.2 

.229 

.286 

•324 

From  these  data,  following  constants  are  deduced  for  an  absolute  pressure  of 
22  lbs.  per  sq.  inch  of  steam  condensed,  and  heat  passed  off  per  sq.  foot  of  external 
surface  of  pipe  per  hour  of  i°  difference  of  temperature. 


Surface  of  Pipe, 

Steam 
condensed 
per  Sq.  Foot. 

Heat 

passed 

off. 

Surface  of  Pipe. 

I Steam 
condensed 
per  Sq.  Foot. 

Heat 

passed 

off. 

Bare  pipe 

Lb. 

Units. 
2.812 
.968 
1. 108 

Cotton  waste  1 inch. . 
Earth  and  hnir 

Lb. 

.001  46 
.001  65 
.001  56 

Units. 

1.384 

1.S68 

1.486 

Straw  coat 

.003 
.001  02 

Cased  with  clay  pipe. . . 

.001  15 

White  paint 

HEAT. 


516 


Pipes  were  4.72  ins.  diameter,  .25  inch  thick,  and  had  area  of  58.5  sq.  feet, 

Bart rough  surface  as  cast.  Straw  coat— laid  lengthwise  .6  inch  thick  and  bound. 

Pipe— laid  in  clay  pipe  with  an  air  space  between  them,  the  whole  covered  with 
loam  and  straw.  Waste  cotton— 1 inch  thick  and  bound  with  twine.  Plaster— 

laid  in  clay  and  hair  2.36  ins.  thick. 

A wrought-iron  pipe  3.75  ins.  in  external  diameter,  .25  inch  thick,  and  lagged 
with  felt  and  spun  yarn  .5  inch  thick,  condensed  steam  at  2450  at  rate  of  .262  lb. 
per  sq.  foot  per  hour,  in  an  external  temperature  of  6o°. 

Steam  Condensed  per  Sep  Foot  and  per  Degree  per  Hour. 

Mean  Results  of  several  Experiments  with  hare  Cast-iron  Pipes , with  Steam 
at  Absolute  Pressure  of  20  lbs.  per  Sq.  I?ich. 

.4  lb.  per  sq.  foot,  and  .002  39  lb.  per  degree. 

Hence,  to  ascertain  quantity  of  heat  lost  by  condensation  of  .002  39  lb.  = — of  a lb. 

Difference  of  total  and  sensible  heats  of  1 lb.  steam  at  20  lbs.  absolute  pressure  = 
II5Io_j_32o  — 228°  = 955  units,  and  955-4-420  = 2.274  units  = heat  condensed. 

The  loss  of  heat  from  a naked  boiler  in  air  at  62°,  under  an  absolute  pressure  of  50 
lbs.  per  sq.  inch,  was  5.8  units. 

Congelation,  and.  Inqinefaction. 

Freezing  water  gives  out  140°  of  heat.  All  solids  absorb  heat  when 
becoming  fluid. 

Particular  quantity  of  heat  which  renders  a substance  fluid  is  termed 
its  caloric  of  fluidity,  or  latent  heat. 


Temperature  of  Solidification  of  Several  Gases.  [Faraday.) 

Cyanogen 310  I Ammonia 103°  I Sulphuretted  Hydrogen,  123° 

Carbonic  Acid 72°!  Sulphurous  Acid. . . 1050  | Protoxide  of  Nitrogen. . 148 


Frigorific  Mixtures. 


Mixtures. 


I I Fall  of 
Parts.  Temperature. 


Sea  salt 

Nitrate  of  ammonia  . . 
Snow,  or  pounded  ice. 
Muriate  of  ammonia  1 
Nitrate  of  potash  ) 

Snow,  or  pounded  ice. 
Phosphate  of  soda. . . . 
Nitrate  of  ammonia  . . 
Dilute  mixed  acids . . . 

Snow 

Crystallized  muriate  1 

of  lime j 

Snow 

Dilute  sulphuric  acid  . . 

Phosphate  of  soda 

Nitrate  of  ammonia  . . 

Dilute  nitric  acid 

Snow 

Dilute  nitric  acid 


—18  to  —25 


— 5 to  — 18 


-34  to  -5o 


—40  to  —73 


— 68  to  —91 


o to  —34 


o to  — 46 


Nitrate  of  ammonia 

Water 

Snow 

Dilute  sulphuric  acid 
Sulphate  of  soda. . 
Diluted  nitric  acid 
Nitrate  of  ammonia 
Carbonate  of  soda. 

Water ... 

Sulphate  of  soda. . 
Muriate  ofammoni 
Nitrate  of  potash. . 
Dilute  nitric  acid . 
Phosphate  of  soda. 
Dilute  nitric  acid . 

Snow 

Muriate  of  lime. . . 

Potash 

Snow 


Full  of 
Temporature. 


-f5°t0  -j-4 
j — 10  to  — 60 
+50.  to  —3 

+5°  to  — 7 

-{-50  to  — 10 

-j-50  to  — 12 
-{-20  to  — 48 
+32  to  —51 


A Mixture  of  Solid  Carbonic  Acid  and  Sulphuric  Ether,  under  receiver  of  an  air- 
pump,  under  pressures  of  .6  lbs.  to  14  lbs.,  exhibited  a temperature  ranging  no 
I07o  iq  — !66o  which  is  the  most  intense  cold  as  yet  known.  [Faraday.) 


HEAT. 


517 


Melting-points. 


Metals. 


Alloys. 


Aluminium  at  red  heat. . 

Antimony 

Arsenic 

Bismuth 

Bronze 

Calcium  at  red  heat 

Copper 

Gold,  pure 

“ standard 


Iron,  cast . 


2d  melting. 


Wrought 

malleable  forge. 


Lead. . 

Lithium 

Mercury 

Platinum 

Nickel,  highest  forge  heat. 
Potassium 

Silver 

Sodium 

Steel 

Tin 

Zinc 


Alloys. 

Lead  2,  Tin  3,  Bismuth  5. 
“ “ 3?  “ 5- 


810 

365 

476 

1692 

1996 
( 2282 
(2590 
2156 
2000 
2250 
3479* 
2200 
2450 
3700* 
2700 
2912 
3509* 

608 

356 

—39 

3080 

•136 

(1250 

(1873 

194 

2500 

446 

680 


212 

210 


Lead  j 


Tin  4,  Bismuth  5. 

“ 3 

“ 2,  Bismuth  5. 


(solder) 

(soft  solder) . 


Tin 


1,  “ 1,  Bism.  4,  Cadm.  1 
, Bismuth  1 


Zinc  1,  Tin  1. . 


Pnsi'ble  Plugs. 

Lead  2,  Tin  2 

■“  6,  “ 2 


Various  Substances. 

Ambergris 

Beeswax 

Carbonic  acid 

Glass 

Ice 

Lard 

Nitro-Glycerine 

Phosphorus 

Pitch 

Saltpetre 

Spermaceti 

Stearine 

Sulphur 

Tallow 

Wax,  white 


* Rankine. 


240 

334 

199 

552 

475 

360 

368 

155 

286 

336 

392 

399 


372 

383 

388 

410 


145 

151 

— 108 
2 377 
32 
95 
45 
112 

91 
606 
112 
114 

239 

92 
142 


Volume  of  Water,  Antimony,  and  Cast  iron,  in  the  solid  state,  exceeds 
that  of  the  liquid,  as  evidenced  by  the  floating  of  ice  on  water,  and  of  cold 
iron  on  iron  in  a liquid  state. 


IB  oiling-p  oint  s . 

Liquids. 


Alcohol,  s.  g.  813 

Ammonia 

Benzine 

Chloroform 

Ether 

Linseed  oil 

Mercury 

Milk 

Nitric  acid,  s.  g.  1.42 

“ “ “ 1.5 

Oil  of  Turpentine 

Petroleum,  rectified 

Phosphorus 

Sea  water,  average 

Sulphur 

Sulphuric  acid,  s.  g.  1.848 

“ ether  3 


( Under  One  Atmosphere.) 
Liquids. 


W3 

140 

W3 

146 

100 

597 

648 

213 

248 

210 

315 

316 
554 
213.2 
570 
59° 
240 
100 


Turpentine 

Water 

“ in  vacuo 

Whale  oil 

Saturated  Solutions. 

Acetate  of  Soda 

“ “ Potash 

Brine 

Carbonate  of  Soda 

“ “ Potash 

Nitrate  of  Soda 

“ “ Potash 

Salt,  common 

Various  Substances. 
Coal  Tar. 

Naphtha . 


315 

212 

j* 

630 


255-8 

336 

226 

220.3 

275 

250 

240.6 

227.2 


ji8  heat. 

Boiling-points  of  Saturated  -Vapors  under  "Various 
Pressures.  ( Regnault .) 


Temper- 

ature. 

Water. 

Alcohol. 

Ether. 

Chloro- 

form. 

Temper- 

ature. 

Water. 

Alcohol. 

Ether. 

Chloro- 

form. 

0 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

O 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

32 

.089 

.246 

3-53 

— 

212 

14.7 

32.6 

95-17 

45-54 

5° 

.178 

.466 

5-54 

2.52 

230 

20.8 

45-5 

120.9 

58.42 

68 

•337 

• 85i 

8.6 

3.68 

240.8 

25-37 

-rr 

137 

Turp’tine. 

86 

.609 

1-52 

12. 32 

5-34 

248 

29.88 

62.05 

— 

4-97 

104 

1.06 

2-59 

17.67 

7-°4 

266 

39-27 

83.8 

— 

6.71 

122 

1.78 

4.26 

24-53 

10.14 

276.8 

46.87 

— 

— 

— 

140 

2.88 

6.77 

33-47 

14.27 

284 

52.56 

109. 1 

— 

8.94 

158 

4-5i 

10.43 

44.67 

18.88 

302 

69.27 

140.4 

— 

11.7 

176 

6.86 

15.72 

57-CI 

26.46 

3°5-6 

73- °7 

147-3 

— 

194 

10.16 

23.02 

75-4i 

35-°3 

320 

89.97 

” 

13- 1 

Boiling-points  of  Water  corresponding  to  Altitudes  of  Barometer  between 
62  and  31  Ins. 


Barom. 

Boiling-point. 

Barom. 

Boiling-point. 

Barom. 

Boiling-point. 

Barom. 

Boiling-point. 

26 
26.5 

27 

0 

204.91 

205.79 

206.67 

27-5 

28 

28.5 

O 

207. 55 
208.43 
209.31 

29 

29-5 

3° 

O 

210.19 

211.07 

212 

3°-5 

32 

O 

212.88 

213.76 

Boiling-point  of  Salt  water,  213.20.  Water  may  be  heated  in  a Digester 
to  400°  without  boiling. 

Fluids  boil  in  a vacuum  with  less  heat  than  under  pressure  of  atmosphere. 
On  Mont  Blanc  water  boils  at  187°  ; and  in  a vacuum  water  bods  at  98  to 
ioo°)  according  as  it  is  more  or  less  perfect. 

Water  maybe  reduced  to  50  if  confined  in  tubes  of  from  .003  to  .005  inch ^in  diam- 
eter: this  is  in.  consequence  of  adhesion  of  water  to  surface  of  tube,  interfering^ 
a change  in  its  state.  It  may  also  be  reduced  in  its  temperature  below  32  if  it  is 
kept  perfectly  quiescent. 

Effect  upon  Various  Bodies  toy-  Heat. 

Wedgewood’s  zero  is  1077°  (Fahrenheit),  and  each  degree  = 130°. 

In  designation  of  degrees  of  temperature,  symbol  -f-  is  omitted  when  temperature 
is  above  o;  but  when  below  it,  symbol  — must  be  prefixed. 


78 


Degrees 

Acetification  ends 
Acetous  fermen-) 
tation begins. . j 

Air  Furnace 3300 

Ammonia  (liq.)  freezes  — 46 
Blood  (hum. ),  heat  of.  98 
“ freezes.  25 

Brandy  freezes —7 

Charcoal  burns 800 

Cold,  greatest  artific.  —166 
“ “ natural  — 56 

Common  fire 790 

Fire  brick 4000  to  5000 

Gutta-percha  softens. . 145 

Heat,  cherry  red 1500 

11  “ (Daniell)  1141 

“ bright  red i860 

“ red,  visible  by  J IQ77 

“ white 2900 


117 


293 


752 


Degrees 

Highest  natural  tern- 1 
perature,  Egypt . . j 
India-rubber  and  ) 
Gutta-percha  vul-  J 

canize ) 

Iron,  bright  red  in| 

the  dark ) 

Iron,  red  hot  in  twi- ) gg 

light ) 4 

Iron,  wrought,  welds.  .2700 

Ignition  of  bodies 750 

Combustion  of  do. . . 800 
Mercury  volatilizes. . . 680 

Milk  freezes 30 

Nitric  Acid  (sp.grav. ) 

1.424)  freezes J ^ 

Nitrous  Oxide  freezes— 150 

Olive-oil  freezes 36 

Petroleum  boils 306 


Degrees. 

Sea-water  freezes 28 

Snow  and  Salt,  equal)  Q 

parts ) 

Spirits  Turpen.  freezes  14 

Steel,  faint  yellow 43° 

full  “ 470 

purple 53° 

blue 55° 

full  blue 560 

dark  “ 600 

polished,  blue  . . 580 
“ straw  color  460 
Strong  Wines  freeze. . 20 

Sulph.  Acid  (sp.  grav. ) 

1.641)  freezes J ™ 

Sulph.  Ether  freezes.  .—46 

Vinegar  freezes 28 

Vinous  ferment.  ..60  to  77 

Zinc  boils 1872 

Wood,  dried 34° 


Proof  Spirit  freezes. . . —7 
Volume  of*  Several  Biqnids  at  tlieir  Boiling-point. 
Water i i Alcohol. . . . *‘“28  | 1 Ether j . Turpentine 


HEAT. 


519 


TIeiglit  corresponding  to  Boiling-point  of*  Pure  Water. 
Boiling-point  at  Level  of  Sea  — 2120. 


Degree. 

Feet.  | 

Degree. 

Feet. 

Degree. 

| Feet.  J] 

Degree. 

Feet. 

Degree. 

Feet. 

2x1 

52i 

207 

2625 

203 

4761 

199 

6929 

195 

9129 

210 

1044 

206 

3156 

202 

5300 

198 

7476 

194 

9 684 

209 

1569 

205 

3689 

201 

5841 

197 

8025 

193 

10  241 

208 

2096 

204 

4224 

200 

6384  1 

196 

8576 

192 

10800 

Correction  for  temperature  of  air  same  as  given  at  page  428  for  Elevation 
by  a Barometer  by  multiplying  by  C. 

Illustration.— If  water  boils  at  a temperature  of  2000  and  C = 136°, 

Then  6384  X 1.08  = 6894.72  feet. 

Underground  Temperature. 

Mean  increase  of  underground  temperature  per  foot,  from  observations  in 
36  mines  in  various  and  extended  localities,  is  .01565°  = i°  in  64  feet. 

Linear  Expansion  or  Dilatation  of*  a Bar  or  Prism  "by- 
Beat. 


For  i°  in  a Length  of  100  Feet. 


Metals.  Minerals,  etc. 


Inch. 

Antimony 00722 

Bismuth 009  28 

Brass 012  5 

“ yellow 0126 

Brick 001 44 

Cast  Iron 0074 

Cement 009  56 

Copper  from  o°  to  2120 on  5 

u from  320  to  572 0 00418 

Fire  brick 003  3 

Glass 005  74 

“ flint 00541 

“ tube 0612 

Gold— Paris  standard  annealed..  .0101 

“ “ “ unannealed  .0103 

Granite 005  25 

Gun  Metal— 16  copper -}- 1 tin...  .0127 
“ “ 8 copper -j- 1 tin. . . .0121 

Ice 0333 

Iron,  forged 008  14 

11  from  o°  to  2120 007  88 


Inch. 

Iron,  from  320  to  5720 00326 

Iron  wire 00823 

Lead 019 

Marble 005  66 

Palladium 006  67 

Platinum 005  71 

“ from  320  to  5720 00204 

Sandstone 013 

11  008  14 

Silver 012  7 

Speculum  metal 013 

Steel,  rod 007  63 

“ cast 007  2 

“ tempered 00826 

“ not  tempered 007  19 

Tin 0145 

Water 000  222  9 

White  Solder— tin  1 + 2 lead. . .016  7 

Zinc,  forged 0207 

“ sheet 0196 

“ 8 -f- 1 tin 017  9 


Superficial  expansion  is  twice  linear,  and  cubical,  three  times  linear. 


To  Compute  Linear  Expansion  oP  a Substance. 

Divide  1 by  decimal  given  in  above  Table,  and  quotient  will  give  pro- 
portion. 

Illustration  l— A rod  of  copper  100  feet  in  length  will  expand  between  tem- 
peratures of  320  and  2120.  212  — 32  = 180  X -0115  = 2.07  ins. 

2. — A cube  of  cast  iron  of  1 foot  will  expand  in  volume  between  temperatures  of 
62°  and  2120.  212  — 62  = 150,  and  150  X .0074  = i.n,  which -4-  100  for  1 foot  = 

.0111  inch,  and  i2-J-.om  X 3 = 12.0333  ins. 

Some  solids,  as  ice,  cast  iron,  etc.,  have  more  volume  when  near  to  their  melting- 
point  than  when  melted.  This  is  illustrated  in  floating  of  solid  metal  in  the  liquid. 

Expansion,  of  Water. 

Water  expands  from  temperature  of  maximum  density  (see  page  520), 
39. 1 °,  to  46°,  at  which  degree  it  regains  its  initial  volume  of  32°,  and  from 
thence  it  expands  under  one  atmosphere  to  212° ; and  its  cubical  expansion 
is  .0466,  that  is,  its  volume  is  dilated  from  1 at  32°  to  1.0466  at  2120. 

Its  expansion  increases  in  a greater  ratio  than  that  of  temperature. 


HEAT. 


To  Compute  Density  of  Water  at  a given  Temperature, 

62.5  X2 — approximate  density , t representing  temperature  of  water. 


£4-461 


+ 


500 


500  ' 1 4-  461 

Illustration.— What  is  density  . 

of  pure  water  at  298°  ? 2984-46 


62.5  X 2 


+ - 


500 


500  ' 2984-461 

Expansion  of  W a ter*.  (Dalton.) 


= 57.42  lbs.  or  weight  of 
1 cube  foot. 


Temp. 

Expansion. 

Temp. 

Expansion. 

Temp. 

Expansion. 

| Temp. 

Expansion. 

0 

22 

1.0009 

0 

52 

1. 000  21 

0 

112 

1.008  8 

0 

172 

1-02575 

32 

1 

72 

1. 001  8 

132 

1.01367 

192 

1.03265 

*46 

1 

92 

1.00477 

152 

1.01934 

212 

1.0466 

* Greatest  density  39.10. 

Hence,  at  72 °,  water  expands  — ~ = 555- 55^  Part  of  its  original  bulk. 
Expansion  of  Eiqnids  from  32°  to  212°.  Volume  at  320 


Liquids. 

Volume  at  2120. 

Liquids.  I Volume  at  2120. 

7i  i i 

1. 11 
1.08 
1.0154 
1.018433  1 
1. 018  867  9 
1. 11 

Olive  oil 

1.08 

1.06 

1.07 
1.07 
1.046  6 
1.05 

Linseed  oil 

Mercury 

“ 212°  tO  392°.  . . . . 

11  392°  to  57  20 

Nitric  acid 

Sulphuric  acid 

“ ether.. 

Turpentine 

Water 

Water  sat.  with  salt 

Expansion  of  Gases  from  32°  to  212°.  Volume  at  32°=  1. 


Gases. 

Volume 
at  2120. 

Gases. 

Volume 
at  2120. 

Air 1 Atmosphere. . 

3-45  \\ 

Hydrogen 1 

3-35  “ 

Carbonic  acid,  1 

3-32  “ 

1.367  06 
1.369  64 
1.366  13 

1.366  16 
1.37099 
i-384  55 

Nitrous  oxide  . . . 1 Atmosphere. . 
Sulphurous  acid,  1 “ 

1.16  “ 

Carbonic  oxide  ..  1 “ 

Cyanogen 1 “ 

I-3W9 
i-  39°3 
1. 398 

1.3669 

I-3S77 

O’  O"  - - - -t  w 

Expansion  of  Gases  is  uniform  for  all  temperatures. 


Volume  of  One  Pound  of  Various  Gases  at  32  0 under  one  Atmosphere . 

^ 1 _ r i fa/it  Cube 


Cube  feet. 

Air 12.387 

Carbonic  acid 8. 101 

Ether,  vapor 4.777 

Expansion  of  .Adr 


Hydrogen. 
Nitrogen. . 
Olefiant. . . 


Cube  feet. 
78.83 

12.753 

12.58 


Cube  feet. 

Oxygen 12.205 

Mercury 1.776 

Steam 19. 913 


(Dalton.)  ’ 


Temp. 

Expan- 

sion. 

Temp. 

Expan- 

sion. 

Temp. 

Expan- 

Temp. 

Expan- 

sion. 

Temp. 

Expan- 

sion. 

Temp.  | 

0 

0 

0 

0 

0 

0 

32 

1 

4° 

1. 021 

60 

1.066 

80 

1. no 

100 

1. 152 

392 

33 

1.002 

45 

1.032 

65 

1.077 

85 

1.121 

200 

i-354 

482 

34 

1.004 

50 

1.043 

70 

1.089 

90 

1. 132 

212 

i-376 

680 

35 

1.007 

55 

1-055 

75 

1.099 

95 

1 142 

302 

i-558 

772 

i-739 
1. 912 
2.028 
2.312 


To  Compute  Volume  of  a Constant  Weight  of  Adr  or 
Permanent  Gas  for  any  Pressure. 


JT  G at  jlwa  -A-  A — 

When  volume  at  a given  pressure  is  Icnoicn , temperature  remaining  con- 
stant. Rule. — Multiply  given  volume  by  given  pressure  and  divide  by 
new  pressure. 


lib  W J II  Loo  U l C* 

Example. — Pressure  at  212°  = 18.92  lbs.  per  sq.  inch,  and  volume  16.91  cube  feet; 
what  is  volume  at  pressure  of  13.86  lbs. 

16.91  X 13.86-7-18.92  = 12.39  cube  feet. 


HEAT, 


521 


Relative  Densities  of  some  Vapors. 

Water  1.  Alcohol  2.59.  Ether  4.16.  Spirits  of  Turpentine  8.06.  Sulphur  3.59. 

Volume,  Pressure,  and  Density  of  Air  at  Various  Tem- 
peratures. 

Volume  and  Atmospheric  Pressure  at  62°  = 1. 


Volume  of 

Density,  or 

Temper- 

ature. 

1 lb.  of  air  at 

weight  of  one 

atmospheric 
pressure  of 

weight  of 

cube  foot 
of  air  at 

14.7  lbs. 

14.7  lbs. 

O 

Cube  feet. 

Lbs.  per 
Sq. Inch. 

Lbs. 

0 

11583 

12.96 

.086331 

32 

12.387 

13.86 

.080  728 

40 

12.586 

14. 08 

•079  439 

50 

12.84 

14.36 

.077  884 

62 

13.141 

14.7 

.076097 

70 

13-342 

14.92 

•074  95 

80 

i3-  593 

15.21 

•073  565 

90 

13-845 

15-49 

.072  23 

100 

14.096 

15-77 

.070942 

120 

14.592 

16.33 

.068  5 

140 

i5- 1 

16.89 

.066  221 

160 

15-603 

17-5 

.064  088 

180 

16.106 

18.02 

. 062  09 

200 

16.606 

18.58 

.06021 

210 

16.86 

18.86 

•059  313 

212 

16.91 

18.92 

•059135 

220 

17.111 

19.14 

.058  442 

240 

17.612 

19.7 

•056774 

260 

18. 116 

20.27 

•0552 

280 

18.621 

20.83 

•053  71 

300 

19. 121 

21.39 

.052  297 

320 

19.624 

21.95 

.050959 

340 

20.126 

22.51 

.049  686 

Temper- 

ature. 

Volume  of 
1 lb.  of  air  at 
atmospheric 
pressure  of 
14.7  lbs. 

Pressure 
of  a given 
weight  of 

Density,  or 
weight  of  one 
cube  foot 
of  air  at 
14  7 lbs. 

0 

Cube  feet. 

Lbs.  per 
Sq.  Inch. 

Lbs. 

360 

20.63 

23.08 

.048  476 

380 

21. 131 

23.64 

•047  323 

400 

21.634 

24.2 

.046  223 

425 

22.262 

24.9 

.04492 

450 

22.89 

25.61 

.043  686 

475 

23-5l8 

26.31 

•04252 

500 

24. 146 

27.01 

.041 414 

525 

24- 775 

27.71 

■040364 

550 

25-403 

28.42 

•039365 

575 

26.031 

29. 12 

.038415 

600 

26.659 

29.82 

•03751 

650 

27-9I5 

3I-23 

.035  822 

700 

29.171 

32.635 

•034  28 

750 

30.428 

34-04 

.032  865 

800 

31.684 

35-445 

•031  561 

850 

32.941 

36.85 

•030358 

900 

34-197 

38.255 

.029  242 

950 

35-454 

39.66 

.028  206 

1000 

36.811 

41.065 

.027241 

1500 

49-375 

55-H5 

.020  295 

2000 

61.94 

69.165 

.016  172 

2500 

74-565 

83-215 

.013441 

3000 

87-13 

97.265 

.011499 

To  Compute  Volume  of  a Constant  Wei  "-lit  of  Air  or 
other  Permanent  Gras  for  any  other  Pressure  and 
Temperature. 


When  volume  is  known  at  a given  pressure  and  temperature.  Rule. — Mul- 
tiply given  volume  by  given  pressure,  and  by  new  absolute  temperature, 
and  divide  by  new  pressure,  and  by  given  absolute  temperature. 

Example.— Given  volume  16.91  cube  feet,  pressure  13.86  lbs.,  and  temperature 
320;  what  is  volume  at  this  temperature? 

Temperature  for  volume  16.91  =212°. 

16.91  x 13  86  X 32  -f-  461  -j- 13.86  X 212  -j-  461  ==  12.39  feet. 


To  Compute  Pressure  of  a Constant  Weight  of  Air  or 
other  Permanent  Gras  for  any  other  Volume  and 
Temperature. 

When  pressure  is  known  for  a given  volume  and  temperature.  Rule. 

Multiply  given  pressure  by  new  absolute  temperature,  and  divide  by  given 
absolute  temperature. 

Note.— Absolute  temperature  is  found  by  adding  461°  to  temperature. 

Example.— Given  pressure  r3.86  lbs.,  and  temperature  at  this  volume  32°-  what 
is  pressure  at  temperature  of  2120  ? * 


13.86  X 212-1-461  -4-  32  461  = 18.92  lbs. 

X x* 


HEAT. 


522 


To  Compute  Volume  of  a Constant  Weight  of  Air  or 
other  Permanent  Gras  at  any  Temperature. 

When  volume  at  a given  temperature  is  known , pressure  being  constant . 
Rule.— Multiply  given  volume  by  new  absolute  temperature,  and  divide 
by  given  absolute  temperature. 

Absolute  zero-point  by  different  thermometrical  scales  is:  Fahrenheit  — 461.2°; 
Reaumur —219. 20;  Centigrade —2740. 

Example. — Volume  of  1 lb.  air  at  32°  = 12.387  cube  feet;  what  is  its  volume  at 

2120  ? 

12.387  X 212  + 461-7-32  + 461  = 16.91  cube  feet 

To  Compute  Increased.  Volume  of  a Constant  Weight 
of  Air. 

When  initial  volume  at  62°  = 1 under  1 atmosphere . Rule.— To  given 
temperature  add  461,  and  divide  sum  by  523  (32  + 4^1)* 

Example. — Assume  elements  of  preceding  case. 

2I2o  _j_  46 1 -7-  523  = 1.287  comparative  volume  to  1. 


To  Compute  Pressure  of  a Constant  Weight  of  Air  or 
other  G^s  at  62°,  and  at  14.7  lhs.  Pressure  per  Sq.  In., 
with  Constant  Volume,  for  a given  Temperature. 

Rule. — Add  461  to  given  temperature,  and  divide  sum  by  35.58. 
Example.— Temperature  is  2120;  what  is  pressure? 

212  + 461  -7-  35. 58  = 18. 92  lbs. 


To  Compute  Volume,  Pressure,  Temperature,  and 
Density  of  Air. 


^-f-46] 

p 2.71 
P 

2.71 


,__y.  * 4—  — V • V 2.7074P  — 461  — t\  and 

’ 39- 8 v 2.71 

__  — d.  t representing  temperature , p pressure  in  lbs.  per  sq.  inch , V vol- 
' ^ — l-  461  _ . 7 

ume  in  cube  feet,  and  D weight  of  1 cube  foot  at  14.7  lbs.  per  sq.  inch. 

Product  of  volume  and  pressure  of  a constant  weight  of  air,  or  any  other 
permanent  gas,  is  equal  to  product  of  absolute  temperature  and  a coefficient, 

determined  for  each  gas  by  its  density. 

Or,  V » = C £ + 461. 


Coefficients,  as  determined  by  volumes  and  consequent  densities.* 

Air 2.71  1 Hydrogen 1875  | Oxygen 2.99 

Carbonic  acid 4-H  Nitrogen 2.63  Mercury 18.88 

Ether,  vapor. 7.02  | Olefiant 2.67  | Steam i-°o 

* See  D.  K.  Clark,  London,  1877,  page  349. 


From  1 


Decrease  of  Temperature  by  Altitudes. 


to  1 000  feet, 
“ 10000  “ . 
“ 20000  “ 


In  dear  skt/.  With  cloudy  sky. 

1°  in  139  feet i°  in  222  feet. 

1°  11  288  “ i°  “ 33i  “ 

i°  “ 365  “ i°  “ 468 


rp0  Compute  Temperature  to  which  a Substance  of  a 
given  Length  or  Dimension  must  he  Submitted  or 
Reduced,  to  give  it  a Greater  or  Less  Length  or  vol- 
ume by  Expansion  or  Contraction. 

Lineal. — When  Length  is  to  be  increased.  - ^ "H  — T-  L and  l represent- 
ing lengths  of  increased  and  primitive  substance  in  like  denominations , T and  t tem- 
peratures of  L and  l , and  C expansion  of  substance  for  each  degree  of  heat 


HEAT. 


523 


Illustration A copper  rod  at  320  is  100  feet  in  length ; to  what  temperature 

must  it  be  subjected  to  increase  its  length  1.1633  ins-  ? 

Expansion  for  a length  of  100  feet  of  copper  for  i°  = .0115. 


100  X 12-4-1.1633  — 100  X 12  1.1633  , 

^ + 3^  = — + 32  = 133-i60. 


When  Length  is  to  be  reduced. 


L — l 
C 


■T  = t. 


Illustration. — Take  elements  of  preceding  case. 
1201.1633  — 1200 


— 133. 160  = 101. 16  — 133. 16  = 320. 


To  Reduce  Degrees  of  Fahrenheit  to  Reaumur  and.  Cen- 
tigrade, and  Contrariwise. 

Fahrenheit  to  Reaumur.  If  above  zero.  — Multiply  difference 
between  number  of  degrees  and  32  by  4,  and  divide  product  by  9. 

Thus,  2120  — 320  = 1800,  and  1800  X 4-j-  9 = 8o°. 

If  below  zero. — Add  32  to  number  of  degrees ; multiply  sum  by  4,  and 
divide  product  by  9. 

Thus,  — 400  -j-  320  = 720,  and  720  x 4 -r-  9 = — 320. 

Reaumur  to  Fahrenheit.  If  above  freezing-point.  — Multiply 
number  of  degrees  by  9,  divide  by  4,  and  add  32  to  quotient. 

Thus,  8o°  X 9 -r*  4 = 1800,  and  18c0  -j-  32  = 2120. 

If  below  freezing-point . — Multiply  number  of  degrees  by  9,  divide  by  4 
and  subtract  32  from  product.  J ’ 

Thus,  — 320  X9-r4  = 720,  and  72 0 — 32  = — 400. 

Fahrenheit  to  Centigrade.  If  above  zero. — Multiply  difference 
between  number  of  degrees  and  32  by  5,  and  divide  product  by  9. 

Thus,  2120  — 32°  X 5 -4- 9 = 1800  X 5-^9  = ioo°. 

..ff  below  zero. — Add  32  to  number  of  degrees,  multiply  sum  bv  5,  and 
divide  product  by  9. 

Thus,  — 4o°  -f-  320  x 5 -s-  9 = 720  X 5 -5-  9 = — 40° 

Centigrade  to  Fahrenheit.  If  above  freezing-point. — Multiply 

number  of  degrees  by  9,  divide  product  by  5,  and  add  32  to  quotient. 

Thus,  ioo°  X 9 -r-  5 = 1800,  and  1800  -f-  32  = 2120. 

If  beloiv  freezing-point. — Multiply  number  of  degrees  by  9,  divide  product 
by  5,  and  take  difference  between  32  and  quotient. 

Thus,  — io°  X 9 -f-  5 = 180,  and  180  a,  32  = i4o. 

Reaumur  to  Centigrade. — Divide  by  4,  and  add  product. 

Thus,  8o°  -r-  4 = 200,  and  200  + 8o°  = ioo°. 

Centigrade  to  Feanmnr.— Divide  by  5,  and  subtract  product. 

Thus,  ioo°  -r-  5 = 200,  and  ioo°  — 20  = 8o°. 


Corresponding  Degrees  upon  the  Three  Scales. 


Fahr. 

| Cent. 

I Reaum.  jl 

Fahr.  1 

I Cent. 

I Reaum.  [1 

Fahr.  1 

Cent. 

j Reaum. 

212 

| 100 

1 80  (| 

32  | 

1 0 

1 0 11 

—40 

—40 

1 —32 

To  Compute  Expansion  of  Fluids  in  Volume, 
Rule.— Proceed  by  preceding  formulas  for  computing  length  of  a sub- 
stance. Substitute  V and  v for  volume,  instead  of  L and  7,  the  lengths. 


524 


HEAT,  VENTILATION,  BUILDINGS,  ETC. 


Illustration.— A closed  vessel  contains  6 cube  feet  of  water  at  a temperature  of 
4o° ; to  what  height  will  a column  of  it  rise  in  a pipe  1.152  ins.  in  area,  when  it  is 
exposed  to  a temperature  of  1300  ? 

1. 152  ins.  = .008  sq.foot.  C for  water  = .000222  9. 

6 (1  -I-.0002229  (130  — 40))  = 6.125  95,  and  ^ = 15-744  lineal  feet. 


Temperature  by  Agitation. 

Results  of  Experiments  with  Water  enclosed  in  a Vessel  and  violently  Agitated. 
Temperature  of  Air,  60.5°;  of  Water,  59. 50. 


Duration 
of  Agitation. 

Increase 
of  Temperature. 

Duration 
of  Agitation. 

Increase 
of  Temperature. 

Duration 
of  Agitation. 

Increase 
of  Temperature. 

Hour. 

O 

Hours. 

0 

Hours. 

O 

• 5 

10 

2 

19*5 

5 

39-5 

I 

14- 5 

3 

29-5 

6 

42-5 

VENTILATION. 

IB itil clings.  Apartments,  etc. 

In  Ventilation  of  Apartments.— From  3.5  to  5 cube  feet  of  air  are  required 
per  minute  in  winter,  and  5 to  10  feet  in  summer  for  each  occupant.  In 
Hospitals , this  rate  must  be  materially  increased. 

Ventilation  is  attained  by  both  natural  draught  and  artificial  means.  In 
first  case  the  ascensional  force  is  measured  by  difference  in  weight  of  two 
columns  of  air  of  same  height,  the  height  being  determined  by  total  difference 
of  level  between  entrance  for  warm  air  and  its  escape  into  the  atmosphere. 
The  difference  of  weight  is  ascertained  from  difference  of  temperatures  of 
ascending  warm  air  and  the  external  atmosphere,  as  by  table,  page  521?  or 
by  formula,  page  522. 


Volumes  of  Air  discharged  through  a -Ventilator  One 
IPoot  Square  of  Opening,  at  Various  Heights  and 
Temperatures. 


Height  of 
Ventilator 
from 

Base-line. 


Feet. 

10 

15 


25 

30 


Excess  of  Temperature  of  Apartment 
above  that  of  External  Air. 


C.ft. 

116 

142 

164 

184 

201 


C.  ft. 
164 
202 
232 
260 
284 


C.  ft. 
200 
245 
285 
3i8 
347 


C.ft. 

235 

284 

330 

368 

403 


25° 


C.ft. 

260 

318 

368 

410 

450 


3° 


C.ft. 

284 

348 

4°4 

450 

493 


Height  of 
Ventilator 
from 

Base-line. 


Feet. 

35 

40 

45 

50 

55 


Excess  of  Temperature  of  Apartment 
above  that  of  External  Air. 


C.  ft. 
218 
235 
248 
260 
270 


C.ft. 

306 

329 

348 

367 

385 


15^ 


C.  ft. 
376 
403 
427 
450 
472 


C.ft. 

436 

465 

493 

518 

54i 


25^ 


C.  ft. 
486 

518 

55i 

579 

605 


3<^ 

C.  ft. 
531 
57° 
605 
635 
663 


Velocity  of  draft  having  been  ascertained  for  any  particular  case,  together  with 
volume  of  air  to  be  supplied  per  minute,  sectional  area  of  both  air  passages  may  be 
computed  from  these  data. 


Heating  "by  Hot  Water. 

One  sq.  foot  of  plate  or  pipe  surface  at  200°  will  heat  from  40  to  100  cube 
feet  of  enclosed  space  to  70°  where  extreme  depression  of  tempeiature  is 
— io°.  * 

The  range  from  40  to  100  is  to  meet  conditions  of  exposed  or  corner 
buildings,  of  buildings  less  exposed,  as  intermediate  ones  ot  a cluster  or 
block,  and  of  rooms  intermediate  between  the  front  and  rear. 

When  the  air  is  in  constant  course  of  change,  as  required  for  ventilation 
or  occupation  of  space,  these  proportions  are  to  be  very  materially  increased 
as  per  following  rules. 


HEAT,  VENTILATION,  AND  HEATING. 


525 


In  determining  length  of  pipe  for  any  given  space  it  is  proper  to  include 
in  the  computation  the  character  and  occupancy  of  the  space.  Thus,  a 
church,  during  hours  of  service,  or  a dwelling-room,  will  require  less  service 
of  plate  or  length  of  pipe  than  a hallway  or  a public  building. 

Reduction  of  Heat  by  Surfaces  of  Glass  or  Metal.— In  addition  to  the 
volume  of  air  to  be  heated  per  minute  for  each  occupant,  1.25  cube  feet  for 
each  sq.  foot  of  glass  or  metal  the  space  is  enclosed  with  must  be  added. 
The  communicating  power  of  the  glass  and  metal  being  directly  proportion- 
ate to  difference  of  external  and  internal  temperature  of  the  air.  Thus  80 
feet  of  glass  will  reduce  100  feet  of  air  per  minute. 

When  Pipes  are  laid  in  Trenches  in  the  Earth. — The  loss  of  heat  is  es- 
timated by  Mr.  Hood  at  from  5 to  7 per  cent. 

Circulation  of  Water  in  Pipes.— In  consequence  of  the  complex  forms  of 
heating-pipes  and  the  roughness  of  their  internal  surface,  it  is  impracticable 
to  apply  a rule  to  determine  the  velocity  of  circulation,  as  consequent  upon 
difference  of  weights  of  ascending  and  descending  columns  of  the  water. 

For  a difference  of  temperature  in  the  two  columns  of  30°  (ioo°  — 160°) 
and  a height  of  20  feet,  the  velocity  due  to  the  height  would  be  3.74  feet. 
In  practice  not  .3,  and  in  some  cases  but  .1,  would  be  attained. 


Volume  of  Air  Required  per  Hour  for  each  Occupant  in  an  Enclosed  Space. 


To  Compute  Length  of  Iron  ripe  required,  to  Heat  Air 


[General  Morin.) 


1800 
424  to  1060 


Cube  Feet. 


in  an  Enclosed.  Space. 
By  Hot  Water. 


?f  a ™?m  of  a Protected  dwelling  is  4000  cube  feet;  what 
ngin  01  a ms.  nine,  at  onr»o  . . 


526 


HEAT  AND  HEATING. 


Lengths  of  Four-Inch  IPipe  to  Heat  lOOO  Cube  Feet 
of  Air  per  IMUntite.  ( Chas . Hood.) 

Temperature  of  Pipe  200°. 


Temperature 


Temperature  of  Building. 


External  Air. 

45° 

50° 

55° 

| 6o° 

65° 

700  i 

1 75° 

80° 

85° 

I 9°° 

O 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

10 

126 

150 

174 

200 

229 

259 

292 

328 

367 

409 

16 

105 

127 

I5i 

176 

204 

223 

265 

3°° 

337 

378 

20 

91 

112 

135 

160 

187 

216 

247 

281 

318 

358 

26 

69 

90 

112 

136 

162 

190 

220 

253 

288 

327 

30 

54 

75 

97 

120 

145 

173 

202 

234 

269 

307 

36 

32 

52 

73 

96 

120 

147 

175 

206 

239 

276 

40 

18 

37 

58 

80 

104 

129 

157 

187 

220 

255 

50 

— 

— 

19 

40 

62 

86 

112 

140 

171 

204 

Proper  Temperatures  of  Enclosed.  Spaces. 


Temper- 

ature 

required. 


Spaces. 


Temper- 

ature 

required. 


Work-rooms,  manufactories,  etc. 

Churches  and  like  spaces 

Greenhouses 

Schools,  lecture-rooms 

Halls,  shops,  waiting-rooms,  etc. 
Dwelling-rooms 


55 

55 

55 

58 

60 

65 


IB  oiler. 


Dwelling-rooms 

Graperies 

Hothouses 

Drying-rooms,  when  filled 

“ “ for  curing  paper.. 


70 

70 

80 

80 

70 

120 


Boiler  for  steam-heating  should  be  capable  of  evaporating  as  much  water 
as  the  pipes  or  surfaces  will  condense  in  equal  times.  Mr.  Hood  recom- 
mends that  6 sq.  feet  of  direct  heating-surface  of  boiler  should  be  provided 
to  evaporate  a cube  foot  per  hour.  Adopt  mean  weight  of  steam  of  5 lbs. 
above  pressure  of  atmosphere,  or  20  lbs.  absolute  pressure,  condensed  per  sq. 
foot  of  pipe  per  degree  of  difference  of  temperature  per  hour,  viz.,  .002  35  lb. 
(as  given  by  D.  K.  Clark),  the  quantity  of  pipe  or  plate  surface  that  would 
form  a cube  foot  of  condensed  water  per  hour,  weight  of  like  volume  of 
water  62.4  lbs.,  would  be,  per  i°  difference  of  temperature, 

62.4-4.00235  = 26550  sq.feet , and  for  differences  of  1680,  158°,  148°,  and  1080, 
required  surface  would  be  respectively  (26550-4-168  = 158)  158,  168,  179,  and  246 
sq.  feet. 

' Henoe,  assuming,  as  previously  stated,  that  4 sq.  feet  of  direct  and  effec- 
tive heating  boiler-surface,  or  its  equivalent  flue  or  tube  surface,  will  evap- 
orate 1 cube  foot  of  water  per  hour,  158  sq.  feet  of  steam-pipe  or  plate  will 
require  4 sq.  feet  of  direct  surface,  etc.,  for  a temperature  of  6o°,  and  cor- 
respondingly for  other  temperatures. 

Boiler-power.— One  sq.  foot  of  boiler-surface  exposed  to  direct  action  of 
fire,  or  3 sq.  feet  of  flue-surface,  will  suffice,  with  good  coal,  for  heating  50 
sq.  feet  of  4-inch,  66  of  3-inch,  and  100  of  2-inch  pipe.  Mr.  Hood  assigns  the 
proportion  at  40  feet  of  4-inch  pipe  for  all  purposes.  Usual  rate  of  com- 
bustion of  coal  is  10  or  n lbs.  per  sq.  foot  of  grate-surface,  and  at  this  rate, 
20  sq.  ins.  of  grate  suffice  for  heating  40  feet  of  4-inch  pipe. 

Four  sq.  feet  of  direct  heating  boiler-surface,  or  equivalent  flue  or  tube 
surface,  exposed  to  direct  action  of  a good  fire,  are  capable  of  evaporating 
1 cube  foot  of  water  per  hour. 

According  to  M.  Grouvelle,  1 sq.  meter  of  pipe-surface  (10.76  sq.  feet),  heated  to 
6o°  an  ordinary  room  alike  to  a library  or  office,  of  from  90  to  100  cube  meters 
(3178  to  3531  cube  feet). 


HEAT,  WARMING  BUILDINGS,  ETC. 


527 


If  a workshop  to  be  heated  to  a high  temperature,  1 sq.  meter  (10.76  sq.  feet)  of 
surface  is  assigned  to  70  cube  meters  (2472  cube  feet)  = 4.35  sq.  feet  or  5.1 1 lineal 
feet  of  4-inch  pipe  per  1000  cube  feet. 

For  heating  workshops,  having  a transverse  section  of  260  sq.  feet,  with  a window- 
surface  of  one  sixth  total  surface,  it  is  customary  in  France  to  assign  1.33  sq.  feet 
of  iron  pipe  surface  per  lineal  foot  of  shop  = 5.2  sq.  feet  per  1000  cube  feet. 


Illustrations  of  extensive  Heating  by  Steam . ( R . Briggs , M.  I.  C.  E.) 


1.  Total  number  of  rooms,  including  halls  and  vaults 286 

“ Area  of  floor  surface i37  37o  sq.  feet. 

“ Volume  of  rooms 1 923  500  cube  feet. 

“ Number  of  occupants 650 

Maximum  average  of  occupants  at  any  time i3oo 

Volume  per  occupant,  excluding  vaults i443  cube  feet. 


Boilers.—  8 with  173  sq.  feet  of  grate  surface  and  8000  sq.  feet  of  heating  surface. 
Furnishing  steam  in  addition  to  the  above,  to  operate  the  elevators  and  electric 
dynamos,  elevating  water,  and  supplying  steam  to  heat  a distant  building,  requiring 
one  third  of  their  capacity. 

IBy  Steam. 

To  Compute  Length,  of*  Iron  Eipe  required,  to  Heat  Air 
in  an  Enclosed  Space,  with  Steam  at  5 lhs.  per  Sq. 
Inch  above  Pressure  of*  Atmosphere. 

# Rule.-— Multiply  volume  of  air  in  cube  feet  to  be  heated  per  minute,  by 
difference  of  temperature  in  space  and  external  air,  divide  product  by  coeffi- 
cients in  preceding  table,  and  quotient  will  give  length  of  4-inch  pipe  in 
lineal  feet,  or  area  of  plate-surface  in  sq.  feet. 


Temperature  of  steam  at  5 lbs.-f-  pressure  = 228°.  Hence,  if  temperature  of  space 
required  is  6o°,  700,  8o°,  or  1200,  the  differences  will  be  1680,  158°,  148°,  and  io8u, 
which  for  a coefficient  of  .5,  as  given  in  rule  for  hot  water,  would  be  336,  316,  296, 
and  216,  for  a pipe  4 ins.  in  diameter,  and  for 


6o° 

7°° 

8o° 

120° 

237 

222 

l62 

.168 

158 

148 

108 

. 84 

79 

74 

54 

Illustration. — Volume  of  combined  spaces  of  a factory  is  50000  cube  feet;  what 
surface  of  wrought-iron  plate  at  200°  is  necessary  to  maintain  a temperature  of  500 
■when  external  air  is  at  o°  ? 


50000  X 5° — o x 6666  square  feet. 
200  — 50 


Coal  Consumed  per  Hour  to  IT  eat  IOO  Feet  of  IPipe. 
( Chas . Hood.) 

Diam.  of  Difference  of  Temperature  of  Pipe  and  Air  in  Space,  in  Degrees. 


Pipe. 

150 

i45 

140 

135 

130 

125 

120 

ii5 

IIO 

105 

IOO 

95 

90 

8S 

80 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

I 

1. 1 

1. 1 

1. 1 

1 

1 

•9 

•9 

•9 

.8 

.8 

•7 

•7 

•7 

.6 

.6 

2 

2-3 

2.2 

2.2 

2.1 

2 

i.g 

1.8 

1.8 

1 -7 

1.6 

1.4 

1.4 

i-3 

1.2 

3 

3-5 

3-4 

3-3 

3-i 

3 

2.9 

2.8 

2.7 

2-5 

2.4 

2-3 

2.2 

2. 1 

2 

1.8 

4 

4-7 

4-5 

4.4 

4.2 

4.1 

3-9 

3-7 

3.6 

3-4 

3-2 

3-i 

2.9 

2.8 

2.6 

2.5 

To  warm  a fhctory,  according  to  M.  Claudel,  43  feet  in  width  by  10.5  high,  a single 
line  of  hot-water  pipe  6.25  ins.  in  diameter  per  foot  of  length  of  room,  appears  to  be 
sufficient,  temperature  in  pipe  being  from  1700  to  1800.  Also,  water  being  at  1800, 
and  air  at  6o°,  making  a difference  of  1200,  it  is  convenient  to  estimate  from  1.5 
to  1.75  sq.  feet  of  water-heated  surface  as  equivalent  to  one  sq.  foot  of  steam-heated 
surface,  and  to  allow  from  8 to  9 sq.  feet  of  hot- water  pipe-surface  per  1000  cube 
feet  of  room. 

M.  Grouyelle  states  that  4 sq.  feet  of  cast-iron  pipe-surface,  whether  heated  by 
steam  or  by  water  at  176°  to  1940,  will  warm  1000  cube  feet  of  workshop,  main- 
taining a temperature  of  6o°.  Steam  is  condensed  at  rate  of  .328  lb.  per  sq.  foot 
per  hour. 


HEAT,  WARMING  BUILDINGS,  ETC. 


528 


2 (22  L.  Greene.)  Length  of  fronts  of  buildings 

Total  volume  of  rooms 

Radiating  surfaces,  direct,  10804. 

indirect,  23  296, 

Boilers Grate-surface 

Heating  surface 


...  2 000  lineal  feet. 
2 574  084  cube  feet. 

| 34 100  sq.  feet. 

...  180  “ 

...5863  “ 


Volume  of  Air  Heated  by  Radiators  ; Consumption  of 
Coal  ; Areas  of  Grate  and  Heating-surface  of  toiler. 

( RoVt  Briggs.) 

Per  100  Sq.  Feet  of  Warming-surface  of  Radiator . 

Pressure  of  steam  per  sq.  inch  + 1 

atmosphere  in  lbs 1 

Heat  from  radiators  per  minute  \ 

in  units 

Volume  of  air  heated  i°  per  min 

ute  in  cube  feet - 

Efficiency  of  radiators  in  ratio . 

Coal  consumed  per  hour  in  lbs . 

Area  of  grate  consuming  8 lbs 

coal  per  hour  in  sq.feet 

do.  12  lbs 

Heating  surface  of  boiler  ; coal  1 
consumed  per  hour  X2.8  in  sq.feet ) 

8 lbs.  X 2.8 

12  lbs.  X 2.8 


— 

3 

10 

30 

456 

486 

537 

642 

25  IIO 

26  772 

29  570 

35  352 

I 

1.066 

1.178 

1.408 

3-c4 

3-24 

3-58 

4.28 

•38 

•405 

.448 

— 

— 

— 

.298 

•357 

8.512 

9.072 

10.02 

11.98 

22.4 

22.4 

22.4 

— 

— 

— 

33-6 

33-6 

60 
74i 
40  803 
1.625 
4.94 

.412 

13-83 

33-6 


By  Hot-Air  Furnaces  or  Stoves. 

A square  foot  of  heating  surface  in  a hot-air  furnace  or  stove  is  held  to 
be  equivalent  to  7 sq.  feet  of  hot  water  pipe. 

M.  Peclet  deduced  that  when  the  flue-pipe  of  a stove  radiated  its  heat 
directly  to  air  of  a space,  the  heat  radiated  per  sq.  foot  per  hour,  for  1 
difference  of  temperature,  were,  for:  Cast  iron,  3.65  units;  Wrought  iron,  1.45 
units,  and  Terra  cotta  .4  inch  thick,  1.42  units. 

In  ordinary  practice,  1 sq.  foot  of  cast  iron  is  assigned  to  328  cube  feet 
of  space. 


Open  ZETires. 


According  to  M.  Claudel,  the  quantity  of  heat  radiated  into  an  apart- 
ment from  an  ordinary  fireplace  is  .25  of  total  heat  radiated  by  combustible. 

For  wood  the  heat  utilized  is  but  from  6 to  7 per  cent.,  and  for  coal  13  per 
cent. 

In  combustion  of  wood,  chimney  of  an  ordinary  open  fireplace  draws 
from  1000  to  1600  cube  feet  of  air  per  pound  of  fuel,  and  a sectional  area 
of  from  50  to  60  sq.  ins.  is  sufficient  for  an  ordinary  apartment. 

Proportions  of  fuel  required  to  heat  an  apartment  are:  For  ordinary  fire- 
places, 100 ; metal  stoves,  63  \ and  open  fires,  13  to  16. 

Furnaces. 

By  D.  K.  Clark,  from  investigations  of  Mr.  J.  Lothian  Bell 

Cupola. — M.  Peclet  estimates  that  in  melting  pig-iron  by  combustion 
of  30  per  cent,  of  its  weight  of  coke,  14  per  cent,  only  of  the  heat  of  combus- 
tion is  utilized. 

IVIetalHirgical. — According  to  Dr.  Siemens,  1 ton  of  coal  is  consumed 
in  heating  1.66  tons  of  wrought  iron  to  welding-point  of  2700°,  and  a ton 
of  coal  is  capable  of  heating  up  39  tons  of  iron;  from  which  it  appears  that 
only  4.5  per  cent,  of  whole  heat  is  appropriated  by  the  iron. . Similarly,  lie 
estimates  1.5  per  cent,  of  whole  heat  generated  is  utilized  in  melting  pot 


HEAT  AND  HEATING. HYDRAULICS. 


529 


steel  in  ordinary  furnaces,  whilst,  in  his  regenerative  furnace,  1 ton  of  steel 
is  melted  by  combustion  of  1344  lbs.  of  small  coal,  showing  that  6 per  cent, 
of  the  heat  is  utilized. 

Blast-furnace.— Mr.  Bell  has  formed  detailed  estimates  of  appro- 
priation of  the  heat  of  Durham  coke  in  a blast-furnace;  from  which  is  de- 
duced following  abstract : 

Durham  coke  consists  of  92.5  per  cent,  of  carbon,  2.5  of  water,  and  5 of 
a«li  and  sulphur.  To  produce  1 ton  of  pig-iron,  there  are  required  1232  lbs. 
of  limestone,  and  5388  lbs.  of  calcined  iron-stone ; the  iron-stone  consists  of 
2083  lbs.  of  iron,  1008  lbs.  of  oxygen,  and  2509  lbs.  of  earths.  There  is 
formed  813  lbs.  of  slag,  of  which  123  lbs.  is  formed  with  ash  of  the  coke, 
ami  690  lbs.  with  the  limestone.  There  are  2397  lbs.  of  earths  from  the  iron- 
stone, less  93  lbs.  of  bases  taken  up  by  the  pig-irou  and  dissipated  m fume, 
say  2314  lbs.  Total  of  slag  and  earths,  3 12  7 16s. 

Mr.  Bell  assumes  that  30.4  per  cent,  of  the  carbon  of  the  fuel,  which  es- 
capes in  a gaseous  form,  is  carbonic  acid;  and  that,  therefore,  only  5X*27 
per  cent,  of  heating  power  of  fuel  is  developed,  and  remaining  48.73  per 
cent,  leaves  tunnel-head  undeveloped.  He  adopts,  as  a unit  of  heat,  the 
heat  required  to  raise  the  temperature  of  112  lbs.  of  water  33*8  . 


HYDRAULICS. 

Descending  Fluids  are  actuated  by  same  laws  as  Falling  Bodies. 

A Fluid  will  fall  through  1 foot  in  .25  of  a second,  4 feet  in  .5  of  a 
second,  and  through  9 feet  in  .75  of  a second,  and  so  on. 

Velocity  of  a fluid,  flowing  through  an  aperture  in  side  of  a vessel, 
reservoir,  or  bulkhead,  is  same  that  a heavy  body  would  acquire  by  fall- 
ing freely  from  a height  equal  to  that  between  surface  of  fluid  and 
middle  of  aperture. 

Velocity  of  a fluid  flowing  out  of  an  aperture  is  as  square  root  of 
height  of  head  of  fluid.  Theoretical  velocity,  therefore,  in  feet  per  sec- 
ond, is  as  square  root  of  product  of  space  fallen  through  in  feet  and 
64.333  = V 2 gh\  consequently,  for  one  foot  it  is  V 64.333  = 8.02  feet. 
Mean  velocity,  however,  of  a number  of  experiments  gives  5.4  feet, 
or  .673. 

In  short  ajutages  accurately  rounded,  and  of  form  of  contracted  vein, 
( vena  contracta),  coefficient  of  discharge  = .974  of  theoretical. 

Fluids  subside  to  a natural  level,  or  curve  similar  to  Earth’s  convexity;  apparent 
level,  or  level  taken  by  any  instrument  for  that  purpose,  is  only  a tangent  to  Earth’s 
circumference;  hence,  in  leveling  for  canals,  etc.,  difference  caused  by  Earth’s  cur- 
vature must  be  deducted  from  apparent  level,  to  obtain  true  level. 

Deductions  from  Experiments  on  Discharge  of  Fluids 
from  Reservqirs. 

1.  That  volumes  of  a fluid  discharged  in  equal  times  by  same  apertures 
from  same  head  are  nearly  as  areas  of  apertures. 

2.  That  volumes  of  a fluid  discharged  in  equal  times  by  same  apertures, 
under  different  heads,  are  nearly  as  square  roots  of  corresponding  heights 
of  fluid  above  surface  of  apertures. 

3.  That,  on  account  of  friction,  small-lipped  or  thin  orifices  discharge  pro- 
portionally more  fluid  than  those  which  are  larger  and  of  similar  figure, 
under  same  height  of  fluid. 

Y Y 


53°  HYDRAULICS. 


4.  That  in  consequence  of  a slight  augmentation  which  contraction  of  the 
fluid  vein  undergoes,  in  proportion  as  the  height  of  a fluid  increases,  the  flow 
is  a little  diminished. 

5.  That  if  a cylindrical  horizontal  tube  is  of  greater  length  than  its  di- 
ameter, discharge  of  a fluid  is  much  increased,  and  may  be  increased  with 
advantage,  up  to  a length  of  tube  of  four  times  diameter  of  aperture. 

6.  That  discharge  of  a fluid  by  a vertical  pipe  is  augmented,  on  the  prin- 
ciple of  gravitation  of  falling  bodies ; consequently,  greater  the  length  of  a 
pipe,  greater  the  discharge  of  the  fluid. 

7.  That  discharge  of  a fluid  is  inversely  as  square  root  of  its  density. 

8.  That  velocity  of  a fluid  line  passing  from  a reservoir  at  any  point  is 
equal  to  ordinate  of  a parabola,  of  which  twice  the  action  of  gravity  (2  q ) 
is  parameter,  the  distance  of  this  point  below  surface  of  reservoir  being  the 
abscissa*  Or,  velocity  of  a jet  being  ascertained,  its  curve  is  a parabola 
parameter  of  which  = 4 h,  due  to  velocity  of  projection^ 

9*  Volume  of  water  discharged  through  ah  aperture  from  a prismatic 
vessel  which  empties  itself,  is  only  half  of  what  it  would  have  been  during 
the  time  of  emptying,  if  flow  had  taken  place  constantly  under  same  head 
and  corresponding  velocity  as  at  commencement  of  discharge ; consequently 
the  time  in  which  such  a vessel  empties  itself  is  double  the  time  in  which 
all  its  fluid  would  have  run  out  if  the  head  had  remained  uniform. 

10.  Mean  velocity  of  a fluid  flowing  from  a rectangular  slit  in  side  of  a 
reservoir  is  two  thirds  of  that  due  to  velocity  at  sill  or  lowest  point,  or  it  is 
that  due  to  a point  four  ninths  of  whole  height  from  surface  of  reservoir. 

11.  When  a fluid  issues  through  a short  tube,  the  vein  is  less  contracted 
than  in  preceding  case,  in  proportion  of  16  to  13 ; and  if  it  issues  through 
an  aperture  which  is  alike  to  frustum  of  a cone,  base  of  which  is  the  aper- 
ture, the  height  of  frustum  half  diameter  of  aperture,  and  area  of  small  end 
to  area  of  large  end  as  10  to  16,  there  will  be  no  contraction  of  the  vein. 
Hence  this  form  of  aperture  will  give  greatest  attainable  discharge  of  a fluid. 

12.  Velocity  of  efflux  increases  as  square  root  of  pressure  on  surface  of  a 
fluid. 

13.  In  efflux  under  water,  difference  of  levels  between  the  surfaces  must  ; 
be  taken  as  head  of  the  flowing  water. 

14.  To  attain  greatest  mechanical  effect,  or  vis  viva,  of  water  flowing 
through  an  opening,  it  should  flow  through  a circular  aperture  in  a thin 
plate,  as  it  has  less  frictional  surface. 

From  Cond.viits  or  IPipes.  ( Bossut .) 

1.  Less  diameter  of  pipe,  the  less  is  proportional  discharge  of  fluid. 

2.  Discharges  made  in  equal  times  by  horizontal  pipes  of  different  lengths, 
but  of  same  diameter,  and  under  same  altitude  of  fluid,  are  to  one  another 
in  inverse  ratio  of  sq.  roots  of  their  lengths. 

3.  In  order  to  have  a perceptible  and  continuous  discharge  of  fluid,  the 
altitude  of  it  in  a reservoir,  above  plane  of  conduit  pipe,  must  not  be  less 
than  .082  ins.  for  every  100  feet  of  length  of  pipe. 

4.  I11  construction  of  hydraulic  machines,  it  is  not  enough  that  elbow's  and  ) 
contractions  be  avoided,  but  also  any  intermediate  enlargements,  the  in- 
jurious effects  of  which  are  proportionate,  as  in  following  Table,  for  like 
volumes  of  fluid,  under  like  heads  in  pipes,  having  a different  number  of  \ 
enlarged  parts. 


No. 

of  Parts. 

Velocity. 

II  No. 

of  Parts. 

Velocity. 

1 No.  1 

of  Parts. 

Velocity. 

I No. 
of  Parts. 

Velocity. 

0 

1 

II  1 

.741 

1 3 1 

•569 

1 5 

•454 

* See  D’Aubuisson,  pnge  66.  t Humber,  page  57. 


HYDRAULICS. 


531 


Friction. 

Flowing  of  liquids  through  pipes  or  in  natural  channels  is  materially  af- 
fected by  friction. 

If  equal  volumes  of  water  were  to  be  discharged  through  pipes  of  equal 
diameters  and  lengths,  but  of  following  figures : 


Discharges  from  Compound  or  Divided.  Reservoirs. 

Velocity  in  each  may  be  considered  as  generated  by  difference  of  heights 
in  contiguous  reservoirs ; consequently,  square  root  of  difference  will  rep- 
resent velocities,  which,  if  there  are  several  apertures,  must  be  inversely  as 
their  respective  areas. 

Note.— When  water  flows  into  a vacuum,  32.166  feet  must  be  added  to  height  of 
it  ; and  when  into  a rarefied  space  only,  height  due  to  difference  of  external  and 
internal  pressure  must  be  added. 

TELOCITY  OF  WATER  OR  OF  FLUIDS. 

Coefficients  of  Discharge. 

Coefficient  of  Discharge  or  Efflux  is  product  of  coefficients  of  Contraction 
and  Velocity. 

It  is  ascertained  in  practice  that  water  issuing  from  a Circular  Aperture 
in  a thin  plate  contracts  its  section  at  a distance  of  .5  its  diameter  from 
aperture  to  very  nearly  .8  diameter  of  aperture,  so  as  to  reduce  its  area 
from  1 to  about  .61.*  Velocity  at  this  point  is  also  ascertained  to  be  about 
.974  times  theoretical  velocity  due  to  a body  falling  from  a height  equal 
to  head  of  water.  Mean  velocity  in  aperture  is  therefore  .974,  which,  X 
.61  = .594,  theoretical  discharge ; and  in  this  case  .594  becomes  coefficient  of 
discharge , which,  if  expressed  generally  by  C,  will  give  for  discharge  itself 

C1V2  g hxC  — V.  a representing  area  of  aperture,  and  V volume  discharged  per 
second.  Or,  4.95  a y/h  z=z  V.  Or,  3.91  d2  y/h  — V.  d representing  diameter  in  feet. 

Hence,  for  cube  feet  per  second,  4.95  a y/h , or  3.91  d2  y/h. 

Illustration. — Assume  head  of  water  10  feet,  diameter  of  opening  1.127  feet, 
area  1 sq.  foot,  and  C ==  .62. 

Then  1V2  g 10X  .62  = 15.72  cube  feet.  4.97  X 1 X y/io  = 15.72  cube  feet , and 
3.91  X 1.1272  X y/ 10  =.  15.7  cube  feet. 

For  square  aperture  it  is  .615,  and  for  rectangular  .621. 

Volume  of  water  or  a fluid  discharged  in  a given  time  from  an  aperture 
of  a given  area  depends  on  head,  form  of  aperture,  and  nature  of  approaches. 

V2 

64-333  h = v2,  and  ^ ^ ==  h.  h representing  height  to  centre  of  opening  in  feet. 

Note.  — Head , or  height , h,  may  be  measured  from  surface  of  water  to  centre  of 
aperture  without  practical  error,  for  it  has  been  proved  by  Mr.  Neville  that  for  cir- 
cular apertures,  having  their  centre  at  the  depth  of  their  radius  below  the  surface, 
and  therefore  circumference  touching  the  surface,  the  error  cannot  exceed  4 per 
cent,  in  excess  of  the  true  theoretical  discharge,  and  that  for  depths  exceeding  three 


Bayer,  .61.  Observed  discharges  of  water  coincide  nearer  to  unit  of  Bayer  than  that  of  all  others. 


532 


HYDRAULICS. 


times  the  diameter,  the  error  is  practically  immaterial.  For  rectangular  apertures 
it  is  also  shown  that,  when  their  upper  side  is  at  surface  of  the  water,  as  in  notches 
the  extreme  error  cannot  exceed  4.17  per  cent,  in  excess;  and  when  the  upper  is 
three  times  depth  of  aperture  below  the  surface,  the  excess  is  inappreciable. 

For  notches , weirs , slits,  etc.,  however,  it  is  usual  to  take  full  depth  for  head,  when 
.666  only  of  above  equation  must  be  taken  to  ascertain  the  discharge. 

Experiments  show  that  coefficient  for  similar  apertures  in  thin  plates,  for 
small  apertures  and  low  velocities,  is  greater  than  for  large  apertures  and 
high  velocities,  and  that  for  elongated  and  small  apertures  it  is  greater  than 
for  apertures  which  have  a regular  form,  and  which  approximate  to  the 
circle. 

When  Discharge  of  a Fluid  is  under  the  Surface  of  another  body  of  a 
lilce  Fluid. — The  difference  of  levels  between  the  two  surfaces  must  be  taken 
as  the  head  of  the  fluid. 

Or,  V2  g (h  — hf)  = v. 

When  Outer  Side  of  opening  of  a discharging  Vessel  is  pressed  by  a Force . 
— The  difference  of  height  of  head  of  fluid  and  quotient  of  pressures  on  two 
sides  of  vessel,  divided  by  density  of  fluid,  must  be  taken  as  heads  of  fluid. 

j ( 2)  ‘ X 144A 

Or,  2 g l h J =v.  S representing  density  of  fluid. 

Illustration.— Assume  head  of  water  in  open  reservoir  is  12  feet  above  water- 
line in  boiler,  and  pressures  of  atmosphere  and  steam  are  14.7  and  19.7  lbs. 

Then  f g X ^.333  x („  = 5.56^ 

When  Water  flows  into  a rarefied  Space , as  into  Condenser  of  a Steam- 
engine , and  is  either  pressed  upon  or  open  to  Atmosphere. — The  height  due  to 
mean  pressure  of  atmosphere  within  condenser,  added  to  height  of  water 
above  internal  surface  of  it,  must  be  taken  as  head  of  the  water. 

Of,  V2  g{h-\-  hf)  = v. 

Illustration. — Assume  head  of  water  external  to  condenser  of  a steam-engine  to 
be  3 feet,  vacuum  gauge  to  indicate  a column  of  mercury  of  26.467  ins.  (=  13  lbs.), 
and  a column  of  water  of  13  lbs.  = 29. 9 feet. 

Then  V2  g (3  + 29.9)  = V 64.333  X 32-9  = V21 16- 57  = 46  feet. 

Relative  "Velocity-  of  Discharge  of  Water  through.  differ- 
ent Apertures  and.  under  lihe  Heads. 

Velocity  that  would  result  from  direct , unretarded  action  of  the  column  of 


water  tuhich  produces  it,  being  a constant . or 1 

Through  a cylindrical  aperture  in  a thin  plate 625 

A tube  from  2 to  3 diameters  in  length,  projecting  outward 8125 

A tube  of  the  same  length,  projecting  inward 6812 

A conical  tube  of  form  of  contracted  vein 974 

Wide  opening,  bottom  of  which  is  on  a level  with  that  of  reservoir; 

sluice  with  walls  in  a line  with  orifice;  or  bridge  with  pointed  piers .96 

Narrow  opening,  bottom  of  which  is  on  a level  with  that  of  reservoir; 

abrupt  projections  and  square  piers  of  bridges 86 

Sluice  without  side  walls 63 


Discharge  or  Efflux  of*  "Water  for  various  Openings  and 
Apertures. 

Rectangular  'W'eir. 

Weirs  are  designated  Perfect  when  their  sill  is  above  surface  of  natural 
stream,  and  Imperfect , Submerged , or  Drowned  when  it  is  below  that  surface. 


HYDRAULICS. 


533 


Height  measured  from  Surface  of  Water  to  Sill.  {Jas.  B.  Francis.) 


Mean  Head. 

I Length  of  Opening. 

I Mean  Discharge  per  Second. 

I Mean  Coefficient. 

.62  to  1.55  feet. 

| 10  feet. 

32.9  cube  feet. 

1 *623 

Principal  causes  for  variation  in  coefficients  derived  from  most  experi- 
ments giving  discharge  of  water  over  weirs  arises  from, 

j Depth  being  taken  from  only  one  part  of  surface,  for  it  has  been  proved 
that  heads  on,  at,  and  above  a weir  should  be  taken  in  order  to  determine 
true  discharge. 

2.  Nature  of  the  approaches,  including  ratio  of  the  water-way  in  channel 
above,  to  water-way  on  weir. 

When  a weir  extends  from  side  to  side  of  a channel,  the  contraction  is 
less  than  when  it  forms  a notch,  or  Poncelet  weir,  and  coefficient  sometimes 
rises  as  high  as  .667. 

When  weir  or  notch  extends  only  one  fourth,  or  a less  portion  of  width, 
coefficient  has  been  found  to  vary  from  .584  to  .6. 

When  wing-boards  are  added  at  an  angle  of  about  64°,  coefficient  is  greater 
than  even  when  head  is  less. 


Computation  of*  "Vol  vim  e of  Discharge. 

Mean  velocity  of  a fluid  issuing  through  a rectangular  opening  in 
side  of  a vessel  is  two  thirds  of  that  due  to  velocity  at  sill  or  lower 
edo-e  of  opening,  or  it  is  that  due  to  a point  four  ninths  of  whole  height 
from  surface  of  fluid. 


Height  measured  from  Surface  of  Head  of  Water  to  Sill  of  Opening. 

Rule.— Multiply  square  root  of  product  of  64.333  and  height  or  whole 
depth  of  the  fluid  in  feet,  by  area  in  feet,  and  by  coefficient  for  opening,  and 
two  thirds  of  product  will  give  volume  in  cube  feet  per  second. 

Or,f6  *^0  = V;  = 

t representing  time  in  seconds  and  V volume  in  cube  feet. 

Example.— Sill  of  a weir  is  1 foot  below  surface  of  water,  and  its  breadth  is  10 
feet;  what  volume  of  water  will  it  discharge  in  one  second? 

C = .623,  V64.33  X 1 X ioX^  - - 80.2,  and  % 80.2  X .623  = 33.32  cube  feet. 

Note. Mean  coefficient  of  discharge  of  weirs,  breadth  of  which  is  no  more  than 

third  part  of  breadth  of  stream,  is  two  thirds  of  .6  = .4;  and  for  weirs  which  extend 
whole  width  of  stream  it  is  two  thirds  of . 66 £ = -444- 

Or,  214  y/h?  = V in  cube  feet  per  minute.  When  h is  in  ins. , put  5. 15  for  214. 

Or,  G bhy/TgTi  — V.  C for  a depth  .1  of  lengths  .417,  and  for  .33  of  length  = .4. 


Or,  by  formula  of  Jas.  B.  Francis:  3.33  (L-.mH)  H^-V. 

L representing  length  of  weir  and  H depth  of  water  in  canal,  sufficiently  fai  fiorn 
weir  to  be  unaffected  by  depression  caused  by  the  current , both  in  feet , and  n number 
of  end  contractions. 

Note.— When  contraction  exists  at  each  end  of  weir,  n — 2 ; and  when  weir  is  of 
width  of  canal  or  conduit,  end  contraction  does  not  exist,  and  n = o. 

This  formula  is  applicable  only  to  rectangular  and  horizontal  weirs  in  side  of  a 
dam,  vertical  on  water-side,  with  sharp  edges  to  current;  for  if  bevelled  or  rounded 
off  in  any  perceptible  degree,  a material  effect  will  be  produced  in  the  discharge; 
it  is  essential  also  that  the  stream,  after  passing  the  edges,  should  in  nowise  be 
restricted  in  its  flow  and  descent. 


Y Y* 


534 


HYDRAULICS. 


In  cases  in  which  depth  exceeds  one  third  of  length  of  weir,  this  formula  is  not 
applicable.  In  the  observations  from  which  it  was  deduced,  the  depth  varied  from 
7 to  nearly  19  ins.  • 

With  end  contraction,  a distance  from  side  of  canal  to  weir  equal  to  depth  on 
weir  is  least  admissible,  in  order  that  formula  may  apply  correctly. 

Depth  of  water  in  canal  should  not  be  less  than  three  times  that  on  weir  for  ac- 
curate computation  of  flow. 


Illustration.—  If  an  overfull  weir  has  a length  of  7.94  feet  and  a depth  of  .986 
(as  determined  by  a hook  gauge),  what  volume  will  it  discharge  in  24  hours? 


= -522444 


3-33  (7-94  — -2  X .986)  .986^  = 3.33  X 7.94  — .1972  X .97907  = 3.33  X 7-7428  X 
.97907  = 25.243  875,  which  X 60  X 60  X 24  = 2 181 061  cube  feet. 

By  Logarithms.— Log.  3. 33 
7.7428 

3.  _ 

.986s  = 1.993  877 

3 

2)  1.981  631 

1.990815  = 1.990815 
1.402  157 

Log.  24  hours  = 86  400  seconds.  4. 936  514 

6.338  671 

Log.  6.33867  = 2 181 073  cube  feet.  C in  this  case  = .615. 

Or.  2i4\/H3  and  5.15V&3  — V,  if  stream  above  the  sill  is  not  in  motion.  H 
representing  height  of  surface  of  water  above  sill  in  feet , h in  inches ; and 
214VH3-I-.035  v2  H3  — V,  if  in  motion,  v representing  velocity  of  approach  of 
water  in  feet  per  second , and  V volume  in  cube  feet  discharged  over  each  lineal  foot 
of  sill  per  minute. 

In  gauging,  waste-board  must  have  a thin  edge.  Height  measured  to  level  of  sur- 
face not  affected  by  the  current  of  overfall.  ( Molesworth .) 

To  Compute  Depth,  of  Flow  over  a,  Sill  that  will  Dis- 
charge a given  Volume  of  Water. 


/ 3 V 

\2  C b a/  2 g 


-7c  = cl.  h = — representing  height  due  to  velocity  (v)  as  it 


, 3 \ 2 

- -f-  lc2  \3 

fows  to  the  weir. 

Note.— When  back-water  is  raised  considerably,  say  2 feet,  velocity  of  water  ap- 
proaching weir  (7c)  may  be  neglected. 


Rectangular  Notches,  or  ‘Vertical  Apertures  or  Slits. 

A Notch  is  an  opening',  either  vertical  or  oblique,  in  side  of  a vessel,  reser- 
voir, etc.,  alike  to  a narrow  and  deep  weir. 

Vertical  Apertures  or  Slits  are  narrow  notches  or  weirs,  running  to  or 
near  to  bottom  of  vessel  or  reservoir. 

Coefficient  for  opening,  8 ins.  by  5,  mean  .606  {Poncelet  and  Lesbros). 

Coefficient  increases  as  deptli  decreases,  or  as  ratio  of  length  of  notch  to 
its  depth  increases. 

When  sides  and  under  edge  of  a notch  increase  in  thickness,  so  as  to  be  converted 
into  a short  open  channel,  coefficients  reduce  considerably,  and  to  an  extent  beyond 
what  increased  resistance  from  friction,  particularly  for  small  depths,  indicates. 

Poncelet  and  Lesbros  found,  for  apertures  8x8  ins.,  that  addition  of  a horizontal 
shoot  21  ins.  long  reduced  coefficient  from  .604  to  .601,  with  a head  of  about  4 feet; 
but  for  a head  of  4.5  ins.  coefficient  fell  from  .572  to  .483. 

For  Rule  and  Formulas,  see  preceding  page. 


HYDRAULICS. 


535 


.Rectangular  Openings  or  Sluices,  or  Horizontal  Slits. 

Height  measured  from  Surface  of  Head  of  Water  to  Upper  Side  and  to  Sill 
of  Opening. 

c inch  by  i inch.  Head,  7 to  23  feet.  = .621. 


Coefficient  for 


{Opening, 


23 


= .614. 

3 feet  11  1 foot.  “ 1 “ 2 11  = .641. 

Poncelet  and  Lesbros  deduced  that  coefficient  of  discharge  increases  with  small 
and  very  oblong  apertures  as  they  approach  the  surface,  and  decreases  with  large 
and  square  apertures  under  like  circumstances. 

Coefficients  ranged,  in  square  apertures  of  8 by  8 ins.,  under  a head  of  6 ins.  to 
rectangular  apertures,  8 by  4 ins. ; under  a head  of  10  feet,  from  .572  to  .745. 

In  a Thin  Plate , C = .616  (Bossut) ; C = .'61  ( Michelotti ). 

To  Compute  Discharge. 

Rule. — Multiply  square  root  of  64.333  and  breadth  of  opening  in  feet,  by 
coefficient  for  opening,  and  by  difference  of  products  of  heights  of  water  and 
their  square  roots,  and  two  thirds  of  whole  product  will  give  discharge  in 
cube  feet  per  second. 

Or,  — b\f2g  (fiT/h  — h'y/h')  C = V;  —=z — 2 -~t\  and 

3 f 6 V 2 g (h  y/h  — h'  y/h')  C 

V 

— v.  h and  h'  representing  depth  to  sill  and  opening  in  feet,  and  v velocity 

b ( k — k') 

in  feet  per  second. 

Example.— Sill  of  a rectangular  sluice,  6 feet  in  width  by  5 feet  in  depth,  is  9 feet 
below  surface  of  water;  what  is  discharge  in  cube  feet  per  second? 

C rz  .625,  9 — 5 = 4,  and  V2gX6x62$X(9V9— 4X  V4)  = 38o*95  cube  feet. 
Or,  V 2 g d a C ==  V.  d representing  depth  to  centre  of  opening  in  feet. 
d = 9 — 2.5  = 6.5,  a = 6 X 5 = 30?  and  V64.33  X 6.5  X 3°  X .625  = 383.44  cube  ft. 


Sluice  'W'eirs  or  Sluices. 

Discharge  of  water  by  Sluices  occurs  under  three  forms— viz.,  Unimpeded , 
Impeded , or  Partly  Unimpeded. 

To  Compute  Discharge  when  Unimpeded. 

C d b V2  g h — V.  d representing  depth  of  opening  and  li  taken  from  centre  of 
opening  to  surface  of  water. 

If  velocity,  /j,  with  which  water  flows  to  sluice  is  considered, 

1 / V \2  v V 

— (ftt)  — =d;  and  — d. 

*9\CdbJ  O6V20/1  „ , / d \ 

CV2 9\h  ~T) 

h'  representing  height  to  which  water  is  raised  by  dam  above  sill. 

Illustration.  — How  high  must  the  gate  of  a sluice  weir  be  raised,  to  discharge 
250  cube  feet  of  water  per  second,  its  breadth  being  24  feet  and  height,  hf,  5 feet? 

C by  experiment  = .6.  d approximately  = 1. 

2 5°  25°  r. 

— - = 1.0204  jeez. 


.6X24  V 64-33(5— i)  I4‘ 


250 
4 X 17.014 


To  Compute  Discharge  when  Impeded. 


C d b yj  2 g h = V,  and 


-~d. 


C b V 2 g h 

h representing  difference  of  level  between  supply  and  back-water.  ’ 


536 


HYDRAULICS. 


To  Compute  Discharge  when  partly  Impeded. 

C 6 V 2 g yj h — ~ -f*  <*V*)  — y*  d'  representing  depth  of  back-water  above 

upper  edge  of  sill. 

Illustration.— Dimensions  of  a sluice  are  18  feet  in  breadth  by  .5  in  depth- 
height  of  opening  above  surface  of  water  .7  feet,  and  difference  between  levels  of 
supply  and  surface  water  is  2 feet;  what  is  discharge  per  second? 

6 X 18  X 8. 02  ^.7  yj  2 — — -f- . 5 y/2j  = 86. 62  X • 896  -f- . 707  = 138. 85  cube  feet. 

Coefficients  of  Circnlar  Openings  or  Slnices. 

Height  measured  from  Surface  of  Head  of  Water  to  Centre  of  Opening. 

Contraction  of  section  from  1 to  .633,  and  reduction  of  velocity  to  .074;  hence 
• 633  X -974  = .617  (Neville). 

In  a Thin  Plate , C = .666  {Bossut)\  .631  (Venturi)  ] .64  (Eytelwein). 

Cylindrical  Ajutages , or  Additional  Tubes,  give  a greater  discharge  than 
apertures  in  a thin  side,  head  and  area  of  opening  being  the  same ; but  it 
is  necessary  that  the  flowing  water  should  entirely  fill  mouth  of  ajutage. 

Mean  coefficient,  as  deduced  by  Castel,  Bossut,  and  Eytelwein,  is  .82. 

Short  Tubes,  Month. -pieces,  and.  Cylindrical  Prolonga- 
tions or  Ajutages. 

Fig*  4*  If  an  aperture  be  placed  in  side  of  a 

vessel  of  from  1.5  to  2.5  diameters  in 
thickness,  it  is  converted  thereby  into  a 
short  tube,  and  coefficient,  instead  of  being 
reduced  by  increased  friction,  is  increased 
from  mean  value  up  to  about  .815,  when 
opening  is  cylindrical,  as  in  Fig.  4 ; and 
when  junction  is  rounded,  as  in  Fig.  5,  to  form  of  contracted  vein,  coefficient 
increases  to  .958,  .959,  and  .975  for  heads  of  1, 10,  and  15  feet. 

Conically  Convergent  and  Divergent  Tubes. 

In  conically  divergent  tube,  Fig.  6,  coeffi- 
cient of  discharge  is  greater  than  for  same 
tube  placed  convergent,  fluid  filling  in  both 
cases,  and  the  smaller  diameters,  or  those  at 
same  distance  from  centres,  O 0,  being  used 
in  the  computations. 

A tube,  angle  of  convergence,  O,  of  which 
is  50  nearly,  with  a head  of  from  1 to  10 
feet,  axial  length  of  which  is  3.5  ins.,  small 
diameter  1 inch,  and  large  diameter  1.3  ins., 
gives,  when  placed  as  at  Fig.  6,  .921  for  co- 
efficient ; but  when  placed  as  at  Fig.  7,  co- 
efficient increases  up  to  .948.  Coefficient  of  velocity  is,  however,  larger  for 
Fig.  6 than  for  Fig.  7,  and  discharging  jet  has  greater  amplitude  in  falling. 
If  a prismatic  tube  project  beyond  sides  into  a vessel,  coefficient  will  be  re- 
duced to  .715  nearly. 

Form  of  tube  which  gives  greatest  discharge  is  that  of  a truncated  cone, 
lesser  base  being  fitted  to  reservoir,  Fig.  7.  Venturi  concluded  from  his  ex- 


HYDRAULICS. 


537 


periments  that  tube  of  greatest  discharge  has  a length  9 times  diameter  of 
lesser  opening  base,  and  a diverging  angle  of  50  6' — discharge  being  2.5 
greater  than  that  through  a thin  plate,  1.9  times  greater  than  through  a 
short  cylindrical  tube,  and  1.46  greater  than  theoretic  discharge. 

Compound  IMovitli-pieces  and.  Ajutages. 


Coefficients  for  Mouth-pieces,  Short  Tubes,  and  Cyl- 
indrical Prolongations. 

Computed  and  reduced  by  Mr.  Neville,  from  Venturi's  Experiments. 


Description  of  Aperture,  Mouthpiece,  or  Tube. 

C.  for 
Diam.  ab. 

C.  for 
Diam.  o 

1.  An  aperture  1.5  ins.  diameter,  in  a thin  plate 

2.  Tube  1.5  ins.  diameter,  and  4.5  ins.  long,  Fig.  4 

.823 

•974 

.823 

3.  Tube,  Fig.  5,  having  junction  rounded  to  form  of  contracted 
vein 

.611 

•956 

•934 

4.  Short  conical  convergent  mouth-piece,  Fig.  6 

.607 

5.  Like  tube  divergent,  with  smaller  diameter  at  junction  with 

reservoir;  length  3.5  ins.,  or  = 1 in.,  and  ab  = 1.3  ins.  . . . 

6.  Double  conical  tube,  a 0,  S T,  r b,  Fig.  9,  when  ab  — S T = 1. 5 

ins.,  or  — 1.21  ins.,  a 0 — .92  in.,  and  0 S — 4.1  ins 

7.  Like  tube  wrhen,  as  in  Fig.  8,  a o rb  = o S Tr,  and  aoS— ' 

1.84  ins 

.561 

.948 

.928 

1.428 

.823 

.823 

1.266 

8.  Like  tube  when  ST — 1.46  ins.,  and  0 S — 2.17  ins 

1.266 

9.  Like  tube  when  S T—  3 ins.,  and  0 S'—  9.5  ins 

.911 

1.4 

10.  Like  tube  when  0 S — 6. 5 ins. , atid  S T — 1. 92  ins 

1.02 

1.569 

11.  Like  tube  when  ST  — 2.25  ins.,  and  0 S — 12.125  ms 

1-215 

i-855 

12.  A tube,  Fig.  10,  when  os  = r t — 3 ins., -dr  t=zSt  ~ 1.21  ins., 
and  tube  oSTr,  as  in  No.  6,  ST  = 1.5  ins.,  and  sS=4. 1 ins. 

•895 

i-377 

Mean  of  various  experiments  with  tubes  of  .5  to  3 ins.  in  diameter,  and 
with  a head  of  fluid  of  from  3 to  20  feet,  gave  a coefficient  of  .813 ; and  as 
mean  for  circular  apertures  in  a thin  plate  is  .63,  it  follows  that  under 
similar  circumstances,  .813  -4-  .63=  1.29  times  as  milch  fluid  flows  through 
a tube  as  through  a like  aperture  in  a thin  plate. 

Preceding  Table  gives  coefficients  of  discharge  for  figures  given,  and  it 
will  be  found  of  great  value,  as  coefficients  are  calculated  for  large  as  well 
as  small  diameters,  and  the  necessity  for  taking  into  consideration  form  of 
junction  of  a pipe  with  a reservoir  will  be  understood  from  the  results. 


Circular  Slnices,  etc. 


To  Compute  Discharge. 

Height  measured  from  Surface  of  Head  of  Water  to  Centre  of  Opening. 

Rule. — Multiply  square  root  of  product  of  64.333  and  depth  of  centre  of 
opening  from  surface  of  water,  by  area  of  opening  in  square  feet,  and  this 
product  by  coefficient  for  the  opening,  and  whole  product  will  give  discharge 
in  cube  feet  per  second. 

Or,  V2  g d,  a C = V.  a representing  area  in  sq.  feet , and  d depth  of  surface  of 
Jluid  f'om  centre  of  opening  in  feet. 


HYDRAULICS. 


538 


Example.— Diameter  of  a circular  sluice  is  1 foot,  and  its  centre  is  1.5  feet  below 
surface  of  the  water;  what  is  discharge  in  cube  feet  per  second? 

Area  of  1 foot  = . 7854 ; C = . 64,  and  V64. 333X1.5  X . 7854  X . 64  = 4. 938  cube  feet 

When  Circumference  reaches  Surface  of  Water.  3/2  g r,  .9604  aC  = V. 
r representing  radius  of  circle  in  feet. 

Illustration.— In  what  time  will  800  cube  feet  of  water  be  discharged  through  a 
circular  opening  of  .025  sq.  foot,  centre  of  which  is  8 feet  below  surface  of  water? 


C = .63. 


800  800 

3/2  gd  X .025  X .63  2268  X .025  X .63 


= 2239.58  = 37  min.  19.6  sec. 


Note.— For  circular  orifices,  the  formula  Vzgd  aC  = V is  sufficiently  exact  for 
all  depths  exceeding  3 times  diameter;  the  finish  of  openings  being  of  more  effect 
than  extreme  accuracy  in  coefficient. 


Semicircular  Slnices. 

When  Diameter  is  either  Upward  or  Downward.  V2  gd  a C = V.  d repre- 
senting depth  of  centre  of  gravity  of figure  from  surface. 


When  Diameter  as  above  is  at  Depth  d,  below  Surface.  V2  g d 1. 188  a C = V. 


Circular,  Semicircular,  Triangular,  Trapezoidal,  3?ris- 
matic  Wedges,  Sluices,  Slits,  etc. 

See  Neville , London , i860 ,pp.  51-63,  and  Weisbach , vol.  \.p.  456. 

For  greater  number  of  apertures  at  any  depth  below  surface  of  water, 
product  of  area,  and  velocity  of  depth  of  centre,  or  centre  of  gravity, 
if  practicable  to  obtain  it,  will  give  discharge  with  sufficient  accuracy.* 

Discharge  from  "Vessels  not  Receiving  any-  Supply. 

For  prismatic  vessels  the  general  law  applies,  that  twice  as  much  "would 
be  discharged  from  like  apertures  if  the  vessels  were  kept  full  during  the 
time  which  is  required  for  emptying  them. 


To  Compute  Time. 


2 A fh  2 Ah 
Qay/zg  v 


Illustration — A rectangular  cistern  has  a transverse  horizontal  section  of  14 
feet,  a depth  of  4 feet,  and  a circular  opening  in  its  bottom  of  2 ins.  in  diameter;  in 
what  time  will  it  discharge  its  volume  of  water,  when  supply  to  it  is  cut  off  and 
cistern  allowed  to  be  emptied  of  its  contents? 
h = \feet,  a ==  22  X .7854-4-  144  = . 0218,  0=3.613,  and  V2  gh  x a X C = .2143 

2 ^ 14-  X 4 

cube  foot  per  second.  Then - = 522.6  seconds. 

.2143 


To  Compute  Time  and  Fall. 

Depression  or  subsidence  of  surface  of  water  in  a vessel,  corresponding  to 
a given  time  of  efflux,  is  h — h'.  h'  representing  lesser  depth. 

— 2 (fh  — hf)  ■=.  t.  Inversely,  (fh  — — av^2gf  tY  = h'. 

C a V2  <7  \ 2 A / 

Illustration.  — In  what  time  will  the  water  in  cistern,  as  given  in  preceding 
case,  subside  1.6  feet,  and  how  much  will  it  subside  in  that  time? 

A =3  14,  C = .6,  a = . 0218,  f 2 g — 8.02,  h = 4,  h' = 4 — 1.63=2.4. 

.6  X.  0^8X8.02  x (V4 — V*.  • ♦)  = ^ x (*  - 1. 55)  = 120.1  seconds, 
f . .6  X .0218  X 8.02  \2  . , t 

^ v 4 2 x X120.1  j =2—  .45  =32.4/^;  hence,  4 - 2.4  =3 1.6  feet. 

When  Supply  is  maintained. — Divide  result  obtained  as  preceding  by  2. 


HYDRAULICS. 


539 


Discharge,  wlien  Form  and.  Dimensions  of  Vessel  of 
Efflux  are  not  known. 

Volume  discharged  may  be  estimated  by  observing  heads  of  the  water  at 
equal  intervals  of  time;  and  at  end  ot  half  time  of  discharge,  head  of  water 
will  be  .25  of  whole  height  from  surface  to  delivery.  ^ ^ ^ 

When  t = such  interval.  For  openings  in  bottom  or  side , Vaty/ig  ^ 1 j 

= V,  for  I depth  ; Cat  fTy  + + = V for  2 depths  ; and 

GaW~y  ^ + + + = v for  4 depths. 

Note. — At  end  of  half  time  of  discharge,  head  of  water  will  be  . 25  of  whole  height 
from  surface  to  delivery. 

Weirs  or  Notclies. 

- Cb  t V2 ~g  {y/hs  -f-  4 y/h^  + y/h3  2)  = V.  b representing  breadth  in  feet. 

9 

Illustration. — A prismatic  reservoir  9 feet  in  depth  is  discharged  through  a 
notch  2.222  feet  wide,  surface  subsiding  6.75  feet  in  935  seconds;  what  is  volume 
discharged? 

C ==  .6,  = 9 — 6.75  — 2. 25  feet,  and  ^ 6 X 2.222  X 935  X 8.02  (-/93  + 4 

V2.253-}-  Vo3)  = 2221.6  X 40.5  = 89974.8  cube  feet. 

When  there  is  an  Influx  and  Efflux. 

If  a reservoir  during  an  efflux  from  it  has  an  influx  into  it,  determination 
of  time  in  which  surface  of  water  rises  or  falls  a certain  height  becomes  so 
complicated  that  an  approximate  determination  is  here  alone  essayed. 

A state  of  permanency  or  constant  height  occurs  whenever  head  of  water  is  in- 

r decreased  by  — — fc.  I representing  influx  in  cube  feet  per  second. 

2 g \C  a) 

Ai  v 

Time  ( t ) in  which  variable  head  » increases  by  volume  (v)  — 1 _c  a > 

and  time  in  which  it  sinks  height,  7c,  by  — fl_=. — — Time  of  efflux,  in  which 

Ca V2 gx  — I 

subsiding  surface  falls  from  A to  Ax,  etc.,  and  head  of  water  from  k to  At,  when 

k is  represented  by is 

C a V2  g 

h — hA  / A 4 Ai  ■ 2 A2  4A3  A4  \ t 

12  C a VPg  V^3  V*  V/(7 

Illustration.— In  what  time  will  surface  of  water  in  a pond,  as  in  a previous 
example,  fall  6 feet,  if  there  is  an  influx  into  it  of  3.0444  cube  feet  per  second? 

— 3-°4t  r-,— .8.  C . 537  and  a = . 8836. 

V 537  X. 8836X8.02  537 

4 X 495  o°°  1 2 X 41QOQQ  . 4 X 325  000 

8'  4.301— .8  4.123-  8 _t~  3-937  — -8 


creased  or  i 


-537  X .8836  X 8.02 
20  — 14  / 600000 


12  X -537  X .8836  X 8.02 
265000  \ 6 


X I 


\4-472  " 


— ^jvoo_\  _ — 6 — x j 480201  = 194486  seconds  = 54  h.,  1 min.,  26  sec. 
3.742  — .8/  45-665 

Prismatic  Vessels. 

If  vessel  has  a uniform  transverse  section,  A. 


Then 


2 A — [ y/h  - Vhz  + Vk  X hyp.  log.  (^.!l 1 ~t- time  in  which 

' a Vz  g L \vtli  — V «7  J 


head  of  water  flows  from  h to  ht. 


540 


HYDRAULICS. 


Illustration.— A reservoir  has  a surface  of  500000  sq.  feet,  a depth  of  20  feet;  it 
is  fed  by  a stream  affording  a supply  of  3.0444  cube  feet  per  second,  and  outlet  has 
an  area  of  .8836  sq.  foot;  in  what  time  will  it  subside  6 feet? 


y/lc,  as  before,  = . 8,  C = 
X2'3°3]  = 238 


. 2 X 500  000 

537,  and  — n±- 

. C 7 


< 50 X ly/zo  — -/mP- 8 Xhyp.log. 
av 2 0 L 


To  Compute  Fall  in  a given  Time. 

This  is  determining  head  hx  at  end  of  that  time,  and  it  should  be  sub- 
tracted from  head  h at  commencement  of  discharge.  Put  into  preceding 
equation  several  values  of  hi,  until  one  is  found  to  meet  the  condition. 

Illustration. — Take  a prismatic  pond  having  a surface  of  38750  sq.  feet,  a depth 
to  centre  of  opening  of  sluice  of  10.5  feet,  a supply  of  33.6  cube  feet,  and  a discharge 
of  40  cube  feet  per  second. 

y/Jc  — .84. 

Putting  these  numerical  values  into  the  equation,  and  assuming  different  values 
for  hi'  a value  which  nearly  satisfies  the  equation  is  4.  Consequently,  10.5 — 4 — 
6-5  feet,  fall. 


A Ic 
3 


a 


hyp.  lo 


„ ht+Vhik+Tc 

WK  -y/k) 


+ V»2  arc  (tang.  = 2Vfc^^j  ] = t; 


3 L V “7  \ V I V "1/  J 

(- — p = &;  arc  (tang.  = y,  arc  tangent  of  which  = y,  and  I as  preceding. 

f C by/2g/ 


According  as  1c  is  ^ h , and  influx  of  water,  I^fC  ly/agh3,  there  is  a rise  or  fall 

of  fluid  surface,  the  condition  of  permanency  occurring  when  hz  ==Jc,  and  time  cor- 
responding becomes  co. 

Illustration.— In  what  time  will  water  in  a rectangular  tank,  12  feet  in  length 
by  6 feet  in  breadth,  rise  from  sill  of  a weir  or  notch,  6 inches  broad,  to  2 feet 
above  it,  when  5 cube  feet  of  water  flow  into  the  tank  per  second  ? 

hi  — 2,  7i  — o,  A = 12  X 6 = 72,  1 = 5,  & = -5j  C = .6. 

5 


Jc  = 


( i \t= 

\f  .6  X -5  X 8.02/ 


= 3. 1172  = 2.1338. 


[.6  X -5  X 8.< 

! f.  , ...  2 -f- V2  X 2. 13384-2.1338  . , A 

[^hyp.  logarithm arc  (tang.  = 


>men7?„X«-*33,8| 

3 X 5 

— ~^3nX  2 ) 1 = 10.2423  X hyp.  log.  6,1  3-4641  X arc  (tang.  -)  = 

2V2-I338  + v2/ J .002162  \ 4-3356/ 

10.2423  x [7-961  — (3-461  X arc,  tangent  of  which  ===  .56497,  or  290  28'  = 29.466, 
length  of  which  = .5143)  = 1-781]  = 10.2423  — 7-961  — 1.781  = 10.2423  X 6.18  = 
63.297  seconds. 


Discharge  of  'Water  under  Variable  Pressures. 


To  Compute  Time,  Tiise  and.  Fall,  and  Volume. 

- y/  2 g x = v.  x representing  variable  head , A and  a areas  of  transverse  horizon - 
tal  section  of  vessel  and  discharge , and  v theoretical  velocity  of  efflux. 

To  Compute  Volume. 

A y — Y.  y representing  extent  of  fall , and  V volume  of  water  discharged , as 
h — h'. 

Illustration.— Assume  elements  of  preceding  case. 

A = 14.  y — 4 feet.  Then  56X4  = 224  cube  feet. 


HYDRAULICS, 


541 


Discharge  from  Vessels  of  Communication. 
When  Reservoir  of  Supply  is  maintained  at  a uniform  Height. — Fig.  11. 

2 A fh 


To  Compute  Tiipe. 


-=  t. 


Ca  f 2 g 

Illustration  i. — In  what  time  will  level  of  water  in  a receiving  vessel  having  a 
section  of  14  sq.  feet  attain  height  of  that  in  supply,  through  a pipe  2 ins.  in  diam- 
eter, placed  4 feet  below  level  of  supply  ? 

~ ^ 2 X 14  X a/4  56  , 

C = .613.  -7 ■ n — — — — = 522.3  seconds. 

**  .613  X .0218  X 8.02  .1072 


_A  Fjg-  «• 


2 Assume  C,  vessel,  Fig.  11,  to  be  a cylinder  18 

ins.  in  diameter,  head  of  water  in  A = 4 feet,  at  A' 
1 foot,  and  2 feet  below  outlet  o;  in  what  time  will 
water  in  vessel  run  out  and  over  at  o through  a pipe, 
a,  1.5  ins.  diameter? 

h — h'  = 4 — 1 — 2=1  foot.  C . 8. 

Hsr— 

; v 288  

t/i)  — X 1. 73  — 1 = 32-  73  seconds. 

0.424 

When  Vessel  of  Supply  has  no  Influx,  and  is  not  indefinitely  great  compared 
with  Receiving  Vessel . 

2 A A 'y/h 

■ - ±=t.  A representing  section  of  receiving  vessel , t time  in  which 


A 

h 

i 

! A' 

iss 

a 

■ - 

1 

2 A A'  {y/h  — fh') 


C a (A  -f-  A')  f 2 g 

the  two  surfaces  of  water  attain  same  level ; and 

C a ( A -f-  A')  sTTh 

which  level  falls  from  h to  h'. 

Illustration. — Section  of  a cistern  from  which  wa' 
feet,  and  section  of  receiving  cistern  is  4 sq.  feet;  initial 
and  diameter  of  communicating  pipe  is  1 inch,  in  what 
in  both  vessels  attain  like  levels? 


= t.  ' • 


= .82. 


i"=-7854- 


2 X 10  X 4 3/3 


.82  X .7854  X - — X 8.02 
144 


— 276  seconds. 


Discharge 


from  a Notch.*  in 
3 A 


Side  of  a Vessel. 


■==  ( —77 77" ) —t.b  breadth  of  notch  in  feet. 

2 q VA  fh) 


When  it  has  no  Influx.  , , 

C b x V2  g 

Illustration.— If  a reservoir  of  water,  no  feet  in  length  by  40  in  breadth,  has  a 
notch  in  end  of  9 ins.  in  width;  in  what  time  will  head  of  wrater  of  15  ins.  fall  to  6? 
C = .6.  9"  = .75  foot.  h'L 

3 X no  X 40  x /_ 1 1 \ _ 13200  > 

8.02  \y/ .5  fl.25/  ~ 


h = 


X.25. 


- X 1.414  — -894  = 1901  seconds. 


•6  X .75  X 8.02  \ * 5 3/1-25/  3.61 

Note. -For  discharge  of  vessels  in  motion,  see  Weisbach,  vol.  1,  pp.  394-396. 
Reservoirs  or  Cisterns. 

To  Compute  Time  of*  billing  and  of  Emptying  a Reser- 
voir under  Operation  of  Doth  Supply  and  Discharge. 

V Y 

g £>  — T>  and  j)\_g  — fc  V representing  volume  of  vessel , S supply  of  water , 

and  D discharge  of  water , both  per  minute , and  in  cube  feet.  T time  of  filling  vessel 
and  t time  of  discharging  it,  both  in  minutes. 


indefinite”  €xtends  to  tlle  bottom  o{  tbe  reservoir,  etc.,  the  time  for  the  water  to  run  out  is 

Z z 


542 


HYDRAULICS. 


Irreguilar-Sliaped  Vessels,  as  a Pond,  Lake,  etc. 
To  Compute  Time  and.  Volume  Discharged. 
Operation.— Divide  whole  mass  of  u’ater  into  four  or  six  strata  of  equal 

depths.  , 

* 2 a2  4 a3  a 4\  , 

. + VT2+VA3+VAi)“’’’ 


h — hA 
2 C ay/ : 


(a  ,4  ar 

= X VVA+VA"1" 


Then,  /or  4 Strata , 

etc.,  representing  depths  of  strata  at  a,  ai,  etc.,  commencing  at  surface;  a*,  as, 

ft  — 

etc.,  toeing  areas  of  first,  second , etc.,  transverse  sections  of  pond . etc.  ; and  — — — 
Xa-f-4at-{-2a2-J-4a3-{-a4:=:V. 

Fig.  12.  A 


Illustration. — In  what  time 
?C  will  depth  of  water  in  a lake, 
A 6 C,  Fig.  12,  subside  6 feet,  sur- 
faces of  its  strata  having  follow- 
ing areas,  outline  of  sluice  being 
a semicircle,  18  ins.  wide,  9 deep, 
and  60  feet  in  length? 


a at  20  feet  (h  ) depth  of  water  = area  of  600000  sq.  feet, 
ai  “ 18.5  “ (7m)  “ “ = “ 495000  “ 

a2  “ 17  “ (h?)  “ “ = “ 410000  “ 

a3  “ 15.5  “ (^3)  “ = “ 325000  “ 

a4  u 14  “ (/i4)  “ = “ 265000  “ 

a = area  of  18 -4- 2 = .8836  sq.feet;  € = .537. 


Then 


-14 


265  ooo\ 
3-742  > 


12  X-537  X .8836  X 8.02 


/600000  . 4X495000  , 2 X 4ioo°o  4X325000 

x {-rzr+  — -+ 


4.472 


4.3OI 


4.123 


3-937 


: X 1 194  431  = 156  938  sec.  = 43  h. , 35  min.  38  sec. 


”45-665 

And  discharge  ir'-—  X (600  000  + 4 X 495  000  -f  2X  410  000  + 4 X 325  000  -f-  265  000) 
12 

= .5X4  965  000  = 2 482  500  cw&e  feet. 

For  6 Strata,  put  2a4,  instead  of  a4,  and  4 as  and  a6  additional,  and  divide  by 
18  instead  of  12. 

Flow  of  Water  in  Beds. 

Flow  of  water  in  beds  is  either  Uniform  or  Variable.  It  is  uniform  when 
mean  velocit}r  at  all  transverse  sections  is  the  same,  and  consequently  when 
areas  of  sections  are  equal ; it  is  variable  when  mean  velocities,  and  there- 
fore areas  of  sections,  vary. 


To  Compute  Ball  of  Blow. 

C — x — = h.  C representing  coefficient  of  friction , l length  of  flow,  p perimeter 
a . 2 g 

of  sides  and  bottom  of  bed,  and  hfall  in  feet. 

Illustration. — A canal  2600  feet  in  length  has  breadths  of  3 and  7 feet,  a depth 
of  3 feet,  with  a flow  of  40  cube  feet  per  second;  what  is  its  fall ? 

C = as  per  table  below  .007  565 ; p — X 2 + 3 = 10.2;  a = 15;  and 

2600X10.2  2.662 

v — 40  -4- 15  = 2.66.  Hence  .007565  X X 7 = 1.4 7 feet- 

* 3 15  64.33 

/a 

q f I p 2 gli  — v. 

Illustration. — A canal  5800  feet  in  length  has  breadths  of  4 and  12  feet,  a depth 
of  5,  and  a fall  of  3 ; what  is  velocity  and  volume  of  flow? 

P = a/5^  + 42  X 2 + 4 = 16.8,  and  a = 40. 

Then  A / ^ — — =r  X 64. 33  X 3 = V.0542  X 193  = 3-23  feet-  Hence 

V .007  565  X 5800  X 10.8 
volume  = 40  x 3. 23  = 129. 2 cube  feet. 


HYDRAULICS 


543 


Coefficients  of  Friction  of  Flow  of  Water  in  Beds,  as 
in  Rivers,  Canals,  Streams,  etc. 


Forms  of  Transverse  Sections  of  Canals,  etc. 

Resistance  or  friction  which  bed  of  a stream,  etc.,  opposes  to  flow  of  water, 
in  consequence  of  its  adhesion  or  viscosity,  increases  with  surface  of  contact 
between  bed  and  water,  and  therefore  with  the  perimeter  of  water  profile,  or 
of  that  portion  of  transverse  section  which  comprises  the  bed. 

Friction  of  flow  of  water  in  a bed  is  inversely  as  area  of  it. 

Of  all  regular  figures,  that  which  has  greatest  number  of  sides  has  for 
same  area  least  perimeter ; hence,  for  enclosed  conduits,  nearer  its  trans- 
verse profile  approaches  to  a regular  figure,  less  the  coefficient  of  its  friction ; 
consequently,  a circle  has  the  profile  which  presents  minimum  of  friction. 

When  a canal  is  cut  in  earth  or  sand  and  not  walled  up,  the  slope  of  its 
sides  should  not  exceed  45  °. 


Variable  motion  of  water  in  beds  of  rivers  or  streams  may  be  reduced  to 
rules  of  uniform  motion  when  resistance  of  friction  for  an  observed  length 
of  river  can  be  taken  as  constant. 

To  Compute  Volume  of  Water  flowing  in  a River. 


Illustration. — A stream  having  a mean  perimeter  of  water  profile  of  40  feet  for 
a length  of  300  feet  has  a fall  of  9.6  ins. ; area  of  its  upper  section  is  70  sq.  feet,  and 
of  its  lower  60;  what  is  volume  of  its  discharge? 


Friction  in  flow  of  water  through  pipes,  etc.,  of  a uniform  diameter  is  in- 
dependent of  pressure,  and  increases  directly  as  length,  very  nearly  as  square 
of  velocity  of  flow,  and  inversely  as  diameter  of  pipe. 

With  wooden  pipes  friction  is  1.75  times  greater  than  in  metallic. 

Time  occupied  in  flowing  of  an  equal  quantity  of  water  through  Pipes  or 
Sewers  of  equal  lengths,  and  with  equal  heads,  is  proportionally  as  follows : 

In  a Right  Line  as  90,  in  a True  Curve  as  100,  and  in  a Right  Angle  as  140. 


In  Feet  per  Second. 


Velocity.  I C. 


C.  Velocity.  C.  Velocity. 


C.  Velocity.  C. 


.3  .008 
.4  .007 
.5  .007 
.6  .007 


.00815  .7  .00773  i-5 

.00797  .8  .00769  2 

.00785  .9  .00766  2.5 

.00778  1 .00763  3 


2-5 

3 


2 


•007  59 
• 007  52 
.007  51 
.007  49 


5 .00745 
8 .00744 
10  .00743 
12  .00742 


"V" ai*iaL>le  VEotion. 


V2  gh 


- = V.  A and  Ax  representing  areas  of  upper 


and  lower  transverse  sections  of 
flow. 


To  obtain  C for  velocity  due  to  this  case,  92.35 
coefficient  for  which,  see  Table  above,  = .007  44. 


70  -f-  60  X — 
12 


V 64. 33  x (9.6^- 12) 


==  394. 6 cube  feet ; 


and  mean  velocity  = 


394.6  X 2 
70-j-  60 


= 6.0 7 feet,  C for  which  is  .007  45. 


FRICTION  IN  PIPES  AND  SEWERS. 


HYDRAULICS. 


To  Compute  Head,  necessary'  to 'overcome  Friction  of 
Dipe.  ( Weisbach. ) 

( ols,  I 'Q17  46\  A x — = h'.  h'  representing  head  to  overcome  friction  of 

V Vv  ) d 5-4  . 

flow  in  pipe,  l length  of  pipe,  and  v velocity  of  water  per  second,  all  in  feet , and  d 
internal  diameter  of  pipe  in  ins. 

Illustration. — Length  of  a conduit-pipe  is  1000  feet,  its  diameter  3 ins.,  and  the 
required  velocity  of  its  discharge  4 feet  per  second;  what  is  required  head  of  water 
to  overcome  friction  of  flow  in  pipe? 

(.0144  -f  X X ~ = .023  13  X 333-333  X 2.963  = 22.845/eeZ. 

Head  here  deduced  is  height  necessary  to  overcome  friction  of  water  in 
pipe  alone. 

Whole  or  entire  head  or  fall  includes,  in  addition  to  above,  height  between 
surface  of  supply  and  centre  of  opening  of  pipe  at  its  upper  end.  Conse- 
quently, it  is  whole  height  or  vertical  distance  between  supply  and  centre 
of  outlet. 


To  Compute  wliole  Head,  or  Heiglit  from  Surface  of 
Supply  to  Centre  of  Discharge. 

(C><f  + ,.5)X^  = ». 

1.5  is  taken  as  a mean,  and  is  coefficient  of  friction  for  interior  orifice,  or  that  of 
upper  portion  of  pipe. 

/ .017  46\  ^ 

To  obtain  C or  coefficient.  ^0144  d — J = C. 

For  facilitating  computation,  following  Table  of  coefficients  of  resistance 
is  introduced,  being  a reduction  of  preceding  formula : 


Coefficients  of  Friction  of  YV ater. 
In  IPipes  at  Different  Velocities. 


V. 

C. 

V. 

C. 

V. 

C. 

V. 

C. 

V. 

C 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

4 

•0443 

2 8 

.025 

5 

.0221 

7 4 

.0208 

n 6 

.0195 

8 

.0356 

3 

.0244 

. 5 4 

.0219 

7 8 

.0206 

12 

.0194 

z 

.0317 

3 4 

.0239 

5 8 

.0217 

8 

.0205 

12  6 

.0193 

1 4 

.0294 

3 8 

.0234 

6 

.0215 

8 6 

.0204 

13 

.0191 

1 8 

.0278 

4 

.0231 

6 4 

.0213 

9 

.0202 

14 

.0189 

2 

.0266 

4 4 

.0227 

6 8 

.0211 

10 

.0199 

15 

.0188 

2 4 

.0257 

4 8 

.0224 

7 

.0209 

11 

.0196 

16 

.0187 

Illustration  1.— Coefficient  due  to  a velocity  of  4 feet  per  second  is  .0231. 


2.— Take  elements  of  preceding  case. 


1000  X 12  , . 

(.0231  X b i-5)  X 


' — 93-9  X • 


- = 23.35  feet. 


Note. — In  preceding  formula  Z was  taken  in  feet,  as  the  multiplier  of  12  for  ins. 
was  cancelled  by  taking  5.4  for  2 g,  but  in  above  formula  it  is  necessary  to  restore 
this  multiplier. 


Ftadii  of  Curvatures. 

When  Pipes  branch  off  from  Mains,  or  when  they  are  deflected  at  right 
angles,  radius  of  curvature  should  be  proportionate  to  their  diameter.  Thus, 


Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Diameter 

2 to  3 

3 to  4 

6 

8 

10 

Radius 

18 

20 

30 

42 

60 

HYDRAULICS, 


545 


Curves  and.  Bends. 


Resistance  or  loss  of  head  due  to  curves  and  bends,  alike  to  that  of  friction, 
increases  as  square  of  velocity ; when,  however,  curves  have  a long  radius 
and  bends  are  obtuse,  the  loss  is  small. 


Curved  Circular  Pipe.  ( Weisbach).  x £.131  + 1-847  2 J X ^ = h- 

presenting  angle  of  curve , d diameter  of  pipe , r radius  of  curve,  and  h height 
friction  or  resistance  of  curve,  all  in  feet. 

facility  of  computations,  following  values  of  .131  -f  1.847  are  intro- 


For 

duced. 


Coefficients  of  Resistance. 

In  Curved  Pipes  with  Section  of  a Circle. 


A 

2 r 


( -1  1 

•I31  II 

•25  I 

•145  II 

•4 

| .206  | 

1 -6  ! 

•44  I. 

1 -75  | 

.806  1] 

1 *9  I 

1 -15 

•3 

- 1 58 

•45 

•244 

•65 

•54 

.8 

•977 

•95 

l -2  1 

.138  II 

•35  1 

.178  || 

•5  ! 

1 -294  1 

1 -7  1 

.661  | 

1 -85  1 

i-i77  1 

| 1 

1.408 
1.674 
1.978 

Illustration.  — If  in  a pipe  18  ins.  in  diameter  and  1 mile  in  length  there  is  a 
right-angled  curve  of  5 feet  radius,  what  additional  head  of  flow  should  be  given  to 
attain  velocity  due  to  a head  of  20  feet? 

a = 90°,  v for  such  a pipe  and  head  = 4 feet  per  second;  18  = 1.5  and  • 1,5 
— .15,  and  . 15  by  table  1= . 133. 

— X -133  X = .5  X .133  X = -016  53  foot. 

a.  „„  64.33 


2X5 


Hence, 


180  64.33 

Note.— If  angle  is  greater  than  900,  head  should  be  proportionately  increased. 

Bent  or  Angular  Circular  Pipes. 

Coefficient  for  angle  of  bend  = .9457  sin.2  x -}-  2.047  sin.4  x.  Hence, 


X 

I 10° 

20°  I 

| 3°° 

1 4°° 

45° 

50°  | 

| 55°  | 

6o° 

65° 

0 

0 

C 

1 • 046  I 

•139 

•364 

•74  1 

1 -984 

1 1-26  j 

i-556  | 

| 1. 861 

1 2.158  | 

2.431 

and  — x C = h.  x representing  half  angle  of  bend. 

2 9 

Illustration.  — Assume  v = 4 feet,  and  angle  =:  900 ; x 


90'-' 


Then  - 


64-33 


X -984  = .2447  foot  additional  head  required. 


In  Valve  Grates  or  Slide  Valves. 
In  Rectangular  Pipes. 


r 

1 1 1 

•9 

.8 

■7  | 

.6  | 

•5 

| .4  | .3  | .2  | .1 

C 

1 .0  I 

.09 

•39  1 

•95  1 

2.08 

I 4.02 

| 8.12  | 17.8  | 44.5  1 193 

r — ratio  of  cross  section. 

In  Cylindrical  Pipes. 


li 

0 

.125 

•25 

•375 

•5 

.625 

•75 

•875 

r 

1 

•948 

.856 

•74 

.609 

.466 

•3i5 

•*59 

C 

.0 

.07 

.26 

.81 

2.06 

5-52 

17 

97.8 

h — relative  height  of  opening. 

In  a Throttle  Valve.  In  Cylindrical  Pipes. 


A 

5° 

IO° 

15° 

20°  1 25° 

300 

35° 

40° 

45° 

0 

0 

IT) 

6o° 

700 

r 

•9i3 

.826 

.741 

•658  I .577 

•5 

.426 

•357 

•293 

•234 

•i34 

.06 

C 

.24 

•52 

•9 

1.54  1 2-51 

3-91 

6.22 

10.8 

18.7 

32.6 

118 

75i 

A = angle  of  position. 


Z z* 


546 


HYDRAULICS. 


In  a Clack  or  Trap  "Valve. 


Angle  of  opening 

I5° 

2°°  1 

25° 

3°° 

35° 

40°  I 

45°  I 

5o° 

55°  1 

6o° 

65° 

700 

C 

90 

1 62  j 

42 

30  1 

20  | 

14 

9-5 

| 6.6 

| 4.6 

1 3-2 

2-3 

i*7 

In  a Cock.  In  Cylindrical  Pipes. 


A 

5° 

IO° 

I5° 

20° 

25°  I 30° 

35° 

40° 

45° 

50° 

55° 

6o° 

65° 

r 

.926 

.85 

.772 

.692 

•6i3  -535 

•458 

.385 

•3X5 

•25 

•x9 

•x37 

.091 

C 

•05 

.29 

•75 

1.56 

3-i  15-47 

9.68 

*7-3 

31.2 

52.6 

106 

206 

486 

In  a Conical  "Valve.  ( 1-645  ^ — 1 ^ = C.  a and  a ' = areas  of  pipe 
and  opening. 


(CL  \ 

- — 1 j = C.  c = a factor,  rang- 
ing from  .624  for  = -1  t°  1for  ^7  = 1,  bein9  greater  the  greater  the  ratio. 

Illustration. — If  a slide  valve  is  set  in  a cylindrical  pipe  3 ins.  in  diameter  and 
500  feet  in  length,  is  opened  to  .375  of  diameter  of  pipe  (hence,  .625  diameter  closed), 
what  volume  of  water  will  it  discharge  under  a head  of  100  feet,  coefficient  of  en- 
trance of  pipe  assumed  at  .5  ? 


C,  by  table,  p.  54 5,  pipe  being  .625  closed  = 5.52. 


V 2 gfh 


V^-s+c+c^) 

C =from  table,  p.  5445/0?'  an  assumed  velocity  of  11  feet  6 ins.  = .0195. 


Then 


V 64.33  x V 100 


; 8.03X10  803  an,,5M 

V(7-°2-f-39)  6-78 


//  . . 500  X I2\ 

^i-  5 + 5-  52  + -OI95 ) 

Hence,  area  of  3 ins.  =7.07,  and  7.07  X 12  X 11-85  = 1005.4  cube  feet  per  second. 


Valves.  ( Conical , Spherical,  or  Flap.) 


Conical  or  Splierical  Valve  IPnppet. 

v 2 

Height  due  to  resistance  or  loss  of  head  of  water—  11  — . v representing 
velocity  of  water  in  f ull  dia  meter  of  pipe  or  vessel. 

(—7—, 1^  = C.  A and  A'  representing  transverse  areas  of  vessel  and  of  valve 

opening , and  (1.645  ~ — 1^  = C of  contraction  in  general. 

Illustration. — If  A'  = .5  of  vessel,  C = (1.645  X ~ — = 2.292  = 5-24- 


Clack  or  Trap  Valve. — C decreases  with  diameter  of  vessel. 

Illustration. If  a single-acting  force  pump,  6 ins.  in  diameter,  delivers  at  each 

stroke  5 cube  feet  of  water  in  4 seconds,  diameter  of  valve  seat  3.5  ins.,  and  of  valve 
4.5;  what  resistance  has  water  in  its  passage,  and  what  is  loss  of  mechanical  effect? 

a — . 196.  (=^  = . 34  ratio  of  transverse  area  of  opening.  1 — (~^j  — • 44  ratio 

of  annular  contraction  to  transverse  area  of  vessel. 

Hence,  .39  mean  ratio,  and  coefficient  of  resistance  corresponding 

2 

thereto  = — iV  = 3. 22  2 =10.37.  5 -—  = 6.3 7 velocity  per  second. 

V -39  / 4 X -190 


HYDRAULICS. 


547 


--372-  = .63  height  due  to  velocity.  Consequently,  10.37  X .63  = 6.53  height  due  to 
64-33 

resistance  of  valve,  and  - X 62. 5 X 6. 53  — 510.15  lbs.  mechanical  effect  lost. 


Discharge  of*  YVater  in.  Pipes. 

For  any  Length  and  Head,  and  for  Diameters  from 
1 Inch,  to  IO  Feet.  In  Cube  Feet  per  Minute.  (Beardmore.) 


Diam. 

Tab.  No. 

I Diam. 

Tab.  No. 

Diam. 

Ins. 

Ft.  Ins. 

Ft.  Ins. 

I 

4.71 

9 

1 147.6 

I 

II 

1.25 

8.48 

10 

1493-5 

2 

i-5 

13.02 

11 

1894.9 

2 

I 

i-75 

19- 

1 

2 356 

2 

2 

2 

26.69 

I 

1 

2 876.7 

2 

3 

2-5 

46.67 

I 

2 

3 463-3 

2 

4 

3 

73-5 

I 

3 

4 II5-9 

2 

5 

3-5 

108. 14 

I 

4 

4836.9 

2 

6 

4 

151.02 

I 

5 

5628.5 

2 

7 

4-5 

194.84 

I 

6 

6 493- 1 

2 

8 

5 

263.87 

I 

7 

7 433 

2 

9 

6 

416.54 

I 

8 

8449 

2 

10 

7 

612.32 

I 

9 

9 544 

2 

11 

8 

854.99 

1 

10 

10722 

3 

Tab.  No. 

Diam. 

Tab.  No. 

Diam. 

Tab.  No. 

11 983 

Ft.  Ins. 
3 4 

39  329 

Ft.  Ins. 
4 9 

115  854 

13328 

3 2 

42  040 

5 

131  7°3 

14  758 

3 3 

44  863 

5 3 

148791 

16  278 

3 4 

47  794 

5 6 

167  139 

17  889 

3 5 

5o835 

5 9 

186786 

19  592 

3 6 

53995 

6 

207  754 

21  390 

3 7 

57265 

6 6 

253  781 

23  282 

3 8 

60  648 

7 

305  437 

25  270 

3 9 

64156 

7 6 

362  935 

27  358 

3 10 

67  782 

8 

426  481 

29  547 

3 11 

71526 

8 6 

496  275 

31834 
34  228 

4 

4 3 

75  392 
87730 

S 6 

572  508 
655  369 

36725 

4 6 

101  207 

IO 

745  038 

This  Table  is  applicable  to  Sewers  and  Drains  by  taking  same  proportion 
of  tabular  numbers  that  area  of  cross-section  of  water  in  sewer  or  drain 
bears  to  whole  area  of  sewer  or  drain. 


Formula  upon  which  the  table  is  constructed  is,  2356 y/i  xds  = V in 

cube  feet  per  minute , and  39.27  yj  ~ X d^  — V in  cube  feet  per  second,  h represent- 
ing height  of  fall  of  water  and  d diameter  of  pipe  and  l length , all  in  feet. 

To  Compute  Discharge. 

( Eytelwein .)  4-7T  ~ ^ and  . 538  = d.  d=z  diameter  of  pipe  in 

ins.,  I length  of  pipe  and  h head  of  water,  both  in  feet. 

( Hawksley .)  ~~  = d,  and  -y — ~ — G.  G — number  of  Imperial 

gallons  per  hour,  and  l length  of  pipes  in  yards. 

{Neville.)  140  Vr  s — n vr  s = v in  feet  per  second,  r — hydraulic  mean  depth 
in  feet,  and  s sine  of  the  inclination  or  total  fall  divided  by  total  length. 

v 47. 124  d2  — V,  and  v 293.7286  d2  = Imperial  gallons  per  minute,  d = diameter 
of  pipe  in  feet. 


To  Compute  Volume  discharged. 

When  Length  of  Pipe,  Height  or  Fall,  and  Diameter  are  given . Rule. 
— Divide  tabular  number,  opposite  to  diameter  of  tube,  by  square  root  of 
rate  of  inclination,  and  quotient  will  give  volume  required  in  cube  feet  per 
minute. 

Example. — A pipe  has  a diameter  of  9 ins.,  and  a length  of  4750  feet;  what  is 
its  discharge  per  minute  under  a head  of  17.5  feet? 

Tab.  No.  9 ins.  = 1x47.6,  and  ^Z-— - — — = 69.67  cube  feet. 

/475Q  16.47 

V *7-5 


548 


HYDRAULICS. 


To  Compute  Diameter. 

When  Length , Head,  and  Volume  are  given.  Rule. — Multiply  discharge 
per  minute  by  square  root  of  ratio  of  inclination ; take  nearest  corresponding 
number  in  Table,  and  opposite  to  it  is  diameter  required. 

Example. — Take  elements  of  preceding  case. 


69.67  x < 


' = 1147.61,  and  opposite  to  this  is  q ins. 

I7-5 


/ v l 

°r’  V i542  k ~ d infeeL  v representing  velocity  in  feet  per  second  and  l length 
in  feet. 

To  Compute  Head. 

When  Length , Discharge , and  Diameter  are  given . Rule.  — Divide 

tabular  number  for  diameter  by  discharge  per  minute,  square  quotient,  and 
divide  length  of  pipe  by  it ; quotient  will  give  head  necessary  to  force  given 
volume  of  water  through  pipe  in  one  minute. 

Example.— Take  elements  of  preceding  cases. 

1147.61 


69. 67 


16.47;  I6-472-- 271.3;  4750-5-271.2  = 17.5  feet. 


To  Compute  whole  Head  necessary  to  furnish  requisite 
Discharge. 

See  Formula  and  Illustration,  page  544. 

To  Compute  Velocity. 

When  Volume  and  Diameter  alone  are  given.  Rule.  — Divide  volume 
when  in  feet  per  minute  by  area  in  feet,  and  quotient,  divided  by  60,  will 
give  velocity  in  feet  per  second. 

Example. — Take  elements  of  preceding  case. 

— 6^6?- 5-  60  = 2.63  feet. 

•752  X *7^54 

When  Volume  is  not  given.  Rule.  — Multiply  square  root  of  product  of 
height  of  pipe  by  diameter  in  feet,  divided  by  length  in  feet,  by  50,  and 
product  will  give  velocity  in  feet  per  second.  (Beardmore.) 

To  Compute  Inclination  of  a Dipe. 

When  Volume , Diameter , and  Length  are  given. 

Illustration. — Take  elements  of  preceding  case. 

(6q.67\2  i 17.5 

7-]  X — 7 — .000874  X 4.214  = .003  68,  and  = . 003 68,  or  4750  x .00368 

235°  / -75°  475° 

= 17.49  feet  head. 

To  Compute  Elements  of  Dong 
V 4 V 


/ V \ 2 1 h 

I2356/  d5  l' 


A 3. 1416  X d2 


= ..2732^=,;  (i+«  + oi).£=»; 


Dipes. 

V2  g h 

Vi+«+<4 


— v; 


and  .4787 


5Q5Xd-J-c  l)~ 


= d in  ins. 


This  latter  formula  will  only  give  an  approximate  dimension  in  consequence  of 

4 V 

unknown  element  d,  and  also  of  C,  as  v = 

’ ’ 3.1416  xd2 

For  Illustration,  see  Miscellaneous  Illustration,  page  556. 


HYDRAULICS. 


549 


To  Compute  Vertical  Height  of  a Stream  projected  from 
IPipe  of  a Fire-engine  or  IPnmp. 

Rule.— Ascertain  velocity  of  stream  by  computing  volume  of  water  run- 
ning or  forced  through  opening  in  a second;  then  by  Rule  in  Gravi  ation, 
pagf  488,  ascertain  height  to  which  stream  would  be  elevated  if  wholly  un- 
obstructed, which  multiply  by  a coefficient  for  particular  case. 

In  great  heights  and  with  small  apertures,  coefficients  should  be  reduced. 
In  consequence  of  the  varying  elements  and  conditions  of  operation  of  fne- 
engines,  it  is  difficult  to  assign  a coefficient  for  them.  Difference  between 
actual  discharge  and  that  as  computed  by  capacity  and  stroke  of  cylinder, 
as  ascertained  by  Mr.  Larned,  1859,  was  18  per  cent.  = a coefficient  of  .82. 

\ steam  fire-engine  of  the  Portland  Company,  discharging  a stream  1.125  ms  in 
diameter  through  100  feet  2.5  inch  hose,  gave  a theoretical  head,  computed  from 
actual  discharge,  of  225  feet.  Lid  stream  vertically  projected  was  200  feet;  hence 
coefficient  in  this  case  was  .88.  . ^ , 

Example.— If  a fire  engine  discharges  14  cube  feet  of  water  vertically  through  a 
pipe  .75  inch  in  diameter  in  one  minute,  how  high  will  the  water  be  projected. 

x < x 1728  --  .4417  area  of  pipe,  -4- 12  ins.  in  a foot,  -4-  60  seconds  = 76  07  /^  ve- 
locity; and  as  coefficient  of  such  a stream  = at  .85,  then  114.1  X .85  — 96.98  feet 

Or  H — '°°22  H — h.  H representing  head  at  nozzle , and  d height  of  jet , both  in 
d 

feet,  and  d diameter  of  nozzle  in  ins.  (R.  F.  Hartford.) 

Illustration. — Assume  head  of  no  feet  and  diameter  of  nozzle  .75  inch. 

IIO  1IO  _ 35.5  = 74.5./M. 

•75 

Note.  — The  loss  of  head  is  greater  with  ring  than  with  smooth  nozzles.  E.  B. 

Weston,  Am.  Soc.  C.  E.,  puts  the  difference  at  .000  171  v2. 

The  loss  of  head  increases  with  the  absolute  height  of  the  jet,  and  is  less  with  an 
increase  of  its  diameter.  This  loss  increases  nearly  in  ratio  of  square  of  height  ot 
jet,  and  varies  nearly  in  inverse  ratio  to  its  diameter. 

Cylindrical  Ajutage. 

Mean  coefficient  as  determined  by  Mariotte  and  B°ssut  = .oo3°66  square 
of  effective  head  for  cylindrical  ajutages;  hence,  for  conical,  alike  to  that  ot 
an  engine  pipe,  coefficient  ranges  from  .72  to  .9,  or  a mean  of  .81. 

By  formula  of  D’Aubuisson,  .003047  h2  = h'. 

Effective  head,  or  h,  in  preceding  example  = 114.1.  Then  ”4-i  — *003047  X 
ii4.i2  = ii4-i  — 39-67  = 74-43  feet  height  of  jet. 

Hence,  fora  conical  or  engine  pipe,  74.43  X .81  = 60.29  feet,  or  a coefficient  of  .535. 


To  Coin 


Fig-  13- 


p„te  Distance  a.  Jet  of  Water  will  Toe  projected 

A • _ Z „ 


B C Fig  13  is  equal  to  twice  square  root  of  A 0 X 0 B. 

* is  4 times  as  deep  below  A as^a  is,  s will  discharge 


If 


twice  volume  of  water  that  will  flow  from  a in  same  time, 
as  2 is  yj  of  A s and  1 is  yj  of  A a. 

Note  —Water  will  spout  farthest  when  0 is  equidistant 

from  A and  B ; and  if  vessel  is  raised  above  a plane,  B must 

be  taken  upon  plane. 

C B Volumes  of  water  passing  through  equal  apertures  in 

same  time  are  as  square  roots  of  their  depths  from  surface. 

Rule. — Multiply  square  root  of  product  of  distance  of  opening  from  sur- 
face  of  wrater,  and  its  height  from  plane  upon  which  water  flows,  in  feet  by 
2,  and  product  will  give  distance  in  feet. 

Example.— A vessel  20  feet  deep  is  raised  5 feet  above  a plane;  how  far  will  a jet 
reach  that  is  5 feet  from  bottom  of  vessel? 

20  — 5 X 5 + 5 = I5°>  and  y/l$°  X 2 = 24.495  feet. 


550 


HYDRAULICS. 


Velocity  of  a jet  of  water  flowing  from  a cylindrical  tube  is  determined  to 
be  .974  to  .98  of  actual  to  theoretic  velocity,  or  = .82  of  that  due  to  height 
of  reservoir.  Hence  volume  of  discharge  through  a cylindrical  opening 


Jets  d’Eau.  (Fig.  14.) 

That  a jet  may  ascend  to  greatest  practicable  height, 
communication  with  supply  should  be  perfectly  free. 

Short  tubes  shaped  alike  to  contracted  fluid  vein,  and 
conically  convergent  pipes,  are  those  wrhich  give  greatest 
velocities  of  efflux.  Hence,  to  attain  greatest  effect,  as  in 
fire-engines,  long  and  slightly  conically  convergent  tubes 
or  pipes  should  be  applied. 

In  order  to  diminish  resistance  of  descending  water,  a 
jet  must  be  directed  with  a slight  inclination  from  vertical. 

Effect  of  combined  causes  which  diminish  height  of  a jet  from  that  due 
to  elevation  of  its  supply  can  only  be  determined  by  experiments.  Great 
jets  rise  higher  than  small  ones. 

With  cylindrical  tubes,  velocity  being  reduced  in  ratio  of  1 to  .82,  and  as 
heights  of  jets  are  as  squares  of  these  coefficients  or  ratios,  or  as  1 to  .67, 
height  of  a jet  through  a cylindrical  tube  is  two  thirds  that  of  head  of 
water  from  which  it  flows. 

H C = h.  H representing  head  of  water , C coefficient , and  h height  of  jet.  ( Moles - 
worth. ) 


When  d = H -r*  300, 

C = .96. 

When  d = H -f-  1500, 

C = 

.8. 

“ “=  11  rr-  45°: 

“ = •93- 

“ “ = 11  -r-  1800, 

u _ 

•7- 

U U _ U _A_  (500? 

‘•  = .9. 

u “ = “ -i-  2800, 

u 

.6. 

“ 800, 

‘‘=.87. 

“ “=“-^-3500, 

11  = 

•5- 

U C<  _ CC  _L_  IOOO} 

“ = .85. 

“ “=“-f-45oo, 

u 

.25. 

FLOW  OF  WATER  IN  RIYERS,  CANALS,  AND  STREAMS. 

Running  Water. — Water  flows  either  in  a natural  or  artificial  bed  , 
or  course.  In  first  case  it  forms  Streams,  Brooks,  and  Bivers  ; in 
second,  Drains,  Cuts,  and  Canals. 

Bed  of  a water-course  is  formed  of  a Bottom  and  two  Banks  or  Shores. 

Transverse  Section  is  a vertical  plane  at  right  angles  to  course  of  the  ; 
flowing  water ; Perimeter  is  length  of  this  section  in  its  bed. 

Longitudinal  Section  or  Profile  is  a vertical  plane  in  the  course  or  thread 
of  current  of  flowing  water. 

Slope  or  Declivity  is  the  mean  angle  of  inclination  of  surface  of  the  water 
to  the  horizon. 

Fall  is  vertical  distance  of  the  two  extreme  points  of  a defined  length  of 
the  flowing  course,  measured  upon  a horizontal  plane,  and  this  fall  assigns  \ 
angle  for  defined  length  of  the  course. 

Line  or  Thread  of  Current  is  the  point  where  flowing  water  attains  its 
maximum  velocity. 

Mid-channel  is  deepest  point  of  the  bed  in  thread  of  current.  Velocity  is  ; 
greatest  at  surface  and  in  middle  of  current ; and  surface  of  flowing  wrater 
is  highest  in  current,  and  lowest  at  banks  or  shore. 

A River,  Canal,  etc.,  is  in  a state  of  pemnanency  wThen  an  equal  quantity 
of  water  flows  through  each  of  its  transverse  sections  in  an  equal  time,  or 
when  V,  product  of  area  of  section , and  mean  velocity  through  whole  extent 
of  the  stream , is  a constant  number . 


= .82  aV 2 g h. 
Fig.  14. 


hydraulics. 

To  Compute  Mean  Depth  of  Flowing  Water. 

bv  number  of  divisions,  and  quotient  is  the  mean  depth. 

To  Compute  Mean  Area  of  Flowing  Water. 

Rule  i.-Multiply  breadth  or  breadths  of  the  stream,  etc.,  by  the  mean 
depth  or  depths,  and  product  is  the  area. 

2. Divide  the  volume  flowing  in  cube  feet  per  second  by  mean  velocity 

in  feet  per  second,  and  quotient  is  area  m sq.  iee  . 

To  Compute  Volume  of  Flowing  Water. 

Rule. — Multiply  area  of  the  stream,  etc.,  in  sq  feet,  by  the  mean  velocity 
of  its  flow  in  feet,  and  product  is  volume  in  cube  feet. 

To  Compute  Mean  Velocity  of  Flowing  Water. 

steamrefcfand6 q^tenCSued^y  coefficienT  oTvelocity^lgive 

rolydtoSes'ftom  ii^e  of  current  toward  banks,  and,  toobtainmean  superficial 
velocity,  ui  + P2  + ^3_  : hence, 

n 

To  Compute  Mean  Velocity  in  whole  Profile  of  a Navi- 
gamble  River,  etc., 

V+7 - 2 V V = tel ocity  at  bottom,  and  V+T5 - V V = mean  velocity. 

In  rivers  of  low  velocities  multiply  mean  velocity  by  .8. 

Obstruction  in  Fivers.  ( Molesworth .) 

58.6 


q_0-x(  — ) — 1 = R.  v representing  velocity  in  ins.  per  second  previous 


to  obstruction,  Xand  a areas  of  river  unobstructed  and  at  obstruction  in  sq.feet,  and 
R rise  in  feet  nTwtrnrted  flow  of  a river  is  6 feet  per  second,  and 

- a“d  5°  «*  wbat 

be  rise  in  feet?  . 2 

JL + oSx  V-i  = -664  X.332  = .i54 feet- 
58.6^  3 \9°/ 

FloW  of  Water  in  Lined  Channels.  (Bazin.) 

VC  D _ v 1 n d representing  mean  hydraulic  depth  in  feet,  F 

p ’ / 1 \ fall  or  length  of  channel  to  fall  of  i,x  and 

jcfy-hpJ  J no 


jail,  VI  bc/tt/n/n  wj  VI 

y as  per  table , and  C as  per  table  p.  543 


Plastered. 
Cut  Stone 


. .0000045 
.000013 


y 

10. 16 
4-354 


X I 

V 

.00006  1 

1. 219 

.000  35  I 

.214 

p 2 D — u 

A = 

it  — u » . - 

For  Sections  of  Uniform  Area,  as  Canals,  Sewers,  etc.  f 
area  of  flow  in  sq.  feet,  P wet  perimeter  of  section,  and  D fall  of 

“iS^TKATiON.-Area  of  transverse  section  of  a sewer  is  50  sq.  feet,  its  wet  perim- 
eter 20  feet,  and  its  fall  5 feet  per  mile.  — — 

7 (§2  x 2 x 5)  = V25  = 5 feet  For  Sections  of  Rivers.  12  JB  p = v. 
Illcstratioh.— Assume  area  500  sq.  feet,  wet  perimeter  200,  and  fall  5 feet  per  mile. 

12  ./5X— = n Vi2-5  = 42-4  feet. 

V 200 


552 


HYDRAULICS. 


Hydraulic  Fadius  oi  Me  an  Depth  is  obtained  bv  dividing  qtcsl  of  trans- 
verse  section  by  wet  perimeter,  both  in  feet. 


To  Compute  Fall  per  Mile  for  a required  Mean  Velocity. 

/VXl2\2 *  _ 

y I2  - I -r*  2 r = D.  r representing  hydraulic  radius  in  ins. 


Upper  surface  of  flowing  water  is  not  exactly  horizontal,  as  water  at  its  surface 
flows  with  different  velocities  with  respect  to  each  other,  and  consequently  exert 
on  each  other  different  pressures. 


If  v and  v i are  velocities  at  line  of  current  and  bank  of  a stream,  the  difference 
^.2 

of  the  two  levels  rs — h. 

2 9 

2 

Illustration.—  If  v = 5feet,  and  ux  9i>;  then  *2~~'9  X 5 _ ±75_  = 8 foof 

2 9 64.33 

A velocity  of  7 to  8 ins.  per  second  is  necessary  to  prevent  deposit  of  slime  and 
growth  of  grass,  and  15  ins.  is  necessary  to  prevent  deposit  of  sand. 

Maximum  velocity  of  water  in  a canal  should  depend  on  character  of  bed  of  the 
channel. 


Thus,  Mean  Velocity  should  not  exceed  per  second  over 


Fine  clay 6 ins.  I River  sand. .. 

A slimy  bed 8 “ Small  gravel. 

Common  clay 6 “ | Large  shingle. 


1 ft. 
1 “ 

3 “ 


Broken  stones. 

Stones 

Loose  rocks. . . 


4 ft. 

6 “ 
10  “ 


To  Compute  Velocity  of*  Flow  or  Discharge  of  Water  in 
Streams,  Pipes,  Canals,  etc. 

1.  When  Volume  discharged  per  Minute  is  given  in  Cube  Feet , and  Area  of 
Canal , etc.,  in  Sq.  Feet.  Rule. — Divide  volume  by  area,  and  quotient,  di- 
vided by  60,  will  give  velocity  in  feet  per  second. 

2.  When  Volume  is  given  in  Cube  Feet , and  Area  in  Sq.  Ins.  Rule.— Di- 
vide volume  by  area ; multiply  quotient  by  144,  and  divide  product  by  60. 

3-  When  Volume  is  given  in  Cube  Ins.,  and  Area  in  Sq.  Ins.  Rule. — Di- 
vide volume  by  area,  and  again  by  12  and  by  60. 

To  Compute  Flow  or  Volume  of  Discharge. 

1.  When  Area  is  given  in  Sq.  Feet.  Rule.— Multiply  area  of  flow  by  its 
velocity  in  feet  per  second,  and  product,  multiplied  by  60,  will  give  volume 
in  cube  feet  per  minute. 

2.  When  Area  is  given  in  Sq.  Ins.  Rule. — Multiply  area  by  its  velocity, 
and  again  by  60,  and  divide  product  by  144. 

Note  i. — Velocities  find  discharges  here  deduced  are  theoretical,  actual  results  de- 
pending upon  coefficient  of  efflux  used.  Mean  velocity,  however,  as  before  given, 
page  529,  may  be  taken  at  y/Yg  .673  = 5.4  feet,  instead  of  8.02  feet. 

2. — As  a rule,  with  large  bodies,  as  vessels,  etc.,  their  floating  velocity  is  some- 
what greater  than  that  of  flow  of  water,  not  only  because  in  floating  they  descend 
an  inclined  plane,  formed  by  surface  of  the  water,  but  because  they  are  but  slightly 
affected  by  the  irregular  intimate  motion  of  water:  the  variation  for  small  bodies 
is  so  slight  that  it  may  be  neglected. 


To  Coinpnte  ITeight  of  Head,  of  Flowing  Water. 

When  Volume  and  Area  of  Flow  are  given  in  Feet.  Rule. — Divide  vol- 
ume in  feet  per  second  by  product  of  area,  and  $ coefficient  for  opening,  and 
square  of  quotient,  divided  by  64.33,  will  give  height  in  feet. 

Example. — Assume  volume  266.48  cube  feet,  area  40  sq.  feet,  and  C = -623. 


Then  ( 266  ^ .Vh- 64.33  = 2^  = 4/^ 

\4°  X £ -623/  4 33  64.33  4 


HYDRAULICS. 


553 


Su."bxxierged  or  Drowned  Orifices  and.  Weirs. 

When  wholly  submerged  (Fig.  1 5). —Available  pressure  at  any  point  in  depth 
of  orifice  is  equal  to  difference  of  pressure  on 

Fig.  15-  « each  side.  

Whence,  C V2  g h = v,  and  C a V 2 g h = V. 
a representing  area  of  sluice  in  sq.feet. 
Illustration. — Assume  opening  3 feet  by  5, 

Then,  .5  X 3X5  V64.33  X 4 = 7-5  X 16.04  = 

' — 120. 3 cube  feet  per  second. 

When  partly  submerged  (Fig.  16).  h'-h  = d = submerged  depth,  and  h - 
P “ h"  — d'  — remaining  portion  of  depth;  whence 

Fig-  l6-  m. d'  -f-  d = entire  depth,  and 

C l V2 ~g  (d  y/h  -f  f h y/h  — h"  y/h")  = Y. 
Illustration.  — Assume  opening  as  above,  h = 
4 feet , fc'  = 6,  h"  ==  3,  and  C = .5.  Then  d = 6 — 

w \ ^ 2 fee^‘  o 

I — 1 I rri  Then  . ^ X «;  X 8.02  (2  a/4  + X 4 V4  — 3 V3) 
' — 1 — i k = 20.05  X 5-869  = 117.67  cube  feet  per  second. 

Fig.  17. 


When  drowned  (Fig.  17). 

ClVz  gh  (d-\-%h)  = V. 
Illustration.  — Assume  opening  as  above, 
h — 4 feet,  d = 2,  and  C = -52. 

Then,  .52  X 5 X V64.33  X 4 X (2  + f 4)  ==  2.6 
IRD  X 16.04  X 4-66  = 194.34  cube  feet  per  second. 

CANAL  LOCKS. 

Single  Locks. 

When  a fluid  passes  from  one  level  or  reservoir  to  another,  through  an 
aperture  covered  by  the  fluid  in  the  latter,  effective  head  on  each  point  of 
aperture,  and  consequently  head  due  to  velocity  of  efflux  at  each  instant,  is 
the  difference  of  levels  of  the  two  reservoirs  at  that  instant. 

Hence  C a -JTght  = V per  second.  K representing  difference  of  levels. 

To  Compute  Time  of  Filling  and  Discharging  a Single 
Lock.-Fig.  IS. 


When  Sluice  in  Upper  Gate  is  entirely  under  Water , and  above  Lower  Level. 


A It' 


h — time  of  filling  up  to  centre  of  sluice. 


C a V 2 gh  . . ^r, 

h representing  height  of  centre  of  sluice  in  upper 
gate  from  surface  of  canal  or  reservoir,  and  h height  gj 
of  centre  of  sluice  in  upper  gate  from  lower  sur-  ^ 
face , or  water  in  the  lode  or  river,  all  in  feet , ana 

2 A h — time  of  filling  the  remaining  space, 

C a V2  g h 

where  a gradual  diminution  of  head  of  water  occurs. 

Consequently,  - — ^ — — t time  of  filling  a single  loclc. 

C a V2  gh 

When  Aperture  or  Sluice  in  Lower  Gate  is  entirely  under  Water , and  above 

Lower  Level.  2 A __  ^me  0f  emptying  or  discharging  it.  a ' representing 

C a'  VTg 

area  of  lower  sluice. 

3 A 


554 


HYDRAULICS. 


Illustration. — Mean  dimensions  of  a lock,  Fig.  18,  are  200  feet  in  length  by  24 
in  breadth;  height  of  centre  of  aperture  of  sluice  from  upper  and  lower  surfaces  is 
5 feet;  breadth  of  both  upper  and  lower  sluices  is  2.5  feet;  height  of  upper  is  4 feet, 
and  of  lower— entirely  under  water— 5 feet;  required  the  times  of  filling  and  dis* 
charging. 

h = 5,  h'  = 5,  A = 200  X 24  = 4800,  C = .545,  a = 4 X 2. 5 = 10,  a'  = 5 X 2.5  = 12.5. 
4800  X 5 24  opo 

— ■ = = 245. 59  seconds  = time  of  filling  lock  up  to  centre  of 

.545X10  xVTJh  97-72 

, . A 2 X4800X  5 48000  _ , 

sluice;  and  :==■ = 491.18  seconds  — time  of fillingremain- 

.545X10XV2  gh  97-72 

ing  space , or  lock  above  centre  of  sluice , and  245. 59  -f-  491.18  = 736.77  seconds , whole 
time . 


Or  (5  + 2 X 5)  X 4800  _ 72  000  __ 


736.77  sec.  = time  of  filling. 


.545  X 12.5  X VTg 


.545X10XV2  gh  97-72 
3°  358.08 

= = 554.9  seconds  — time  of  discharging. 

54-7 

When  Aperture  or  Sluice  in  Upper  Gate  is  entirely  under  Water  and  below 

Lower  Level.  r — — time  of  filling  lock. 

Cay/zg 

When  Sluice  in  the  Lower  Gate  is  in  part  above  Surface  of  Lower  Level 

i • 1 1 * 2A(/l-f/0  , . . , . 

and  in  part  below  it.  7 — = time  of  dis 


Cby/zg  (ft^h  + h'  — ^ + d'  y/h  -j-  h'^j 


charging,  d and  d'  representing  distances  of  part  of  aperture  above  and  of  below 
surface  of  lower  water , b breadth  of  aperture,  and  h and  h ' as  before. 

Illustration. — Assume  sluice  in  preceding  example  to  be  1 foot  above  lower 
level  of  water,  or  that  of  lower  canal;  what  is  time  of  discharge  of  lock,  distance 
of  part  of  aperture  1 foot  and  of  that  below  surface  of  water  4 feet? 

2X4800(5  + 5)  96000 


• 545  X 2.5  X 
96000 


171-95 


02  [1  X V5  + 5 — (i-i-2)  + 4X  V5  + 5] 

558.3  seconds. 


‘ 10.93  X (3.082  + 12.65) " 


Double  Lock.  (J.  D.  Van  Buren,  Jr.) 

A double  lock  is  not  a duplication  of  a single  lock  in  its  operation,  for  in 
lower  chamber  supply  of  water  w 
is  from  upper  one,  having  no  % 5 

influx,  instead  of  a uniform  sup-  » o Uppgr 

ply  flowing  directly  from  sur-  M wi  A 

face  level  of  canal  or  feeder. 

Operation,  therefore,  of  a 
double  lock  is  complex,  addition 
to  formula  for  a single  lock  be- 
ing that  of  discharging  of  water 
in  upper  lock  to  All  lower,  the 
head  of  water  gradually  decreas- 
ing in  the  chamber,  which  is 


closed  from  upper  reach  during  discharge  into  lower. 

To  Compute  Time  required  for  Water  to  Fall  from 
Upper  to  Uniform  Water  Level. 

1.  -r— — (v//-f  v/2 Tit—-  y/z  h — 2 d)  — t,  A representing  horizontal  area  of  lock, 
C a y/  g 

and  a area  of  sluice  opening,  both  in  sq.  feet,  C coefficient  of  discharge  = .545  for 
openings  with  square  arrises , g acceleration  of  gravity,  f depth  of  centre  of  sluice 


HYDRAULICS. 


555 


below  uniform  level , h depth  of  centre  sluice  opening  belowuppei  water  level,  and  d 
height  of  centre  of  sluice  above  lower  water  level , alt  in  feet , and  t time  Jo)  water  to 
fall  from  upper  to  uniform  water  level , in  seconds. 

a — 5 ; /=  6 ; = 14;  and  d = 


•545; 


Illustration.  — A ==  2000  sq.  feet ; C 
2 feet.  (Fig.  19.) 

Then, — — — = X 7~74I=*9  = 367-6  seconds. 

.545X5X5-67  I5-45 

Tf  7 A V/  _t.  = _ 366. 34  seconds. 

2-Ifd  = °>  r.nTTn-1'  ^^XO^.67  15-45 


CaV0  ri  -545X5X5-67 

Note  -/is  never  greater  than  Z (lift  in  feet);  it  is  equal  to l when id  = o;  f>  is 
equal  to  l when  A ^o,  never  greater.  In  each  case  it  is  the  unbalanced  head  above 
sluice,  however  far  below  the  lowest  water  le\el  the  sluice  is. 

To  Fill  Upper  Lock  or  Empty  Lower. 

To  fill  upper  lock  or  empty  lower,  when  the  sluice  is  below  the  lowest  water-line, 
in  either  case,  takes  the  same  time;  for  the  head  diminishes  at  the  same  rate,  one 
from  the  upper  surface,  the  other  from  the  bottom. 

Aa/z /__£  Her e,f  being  below  lowest  water  level  of  lock  ~ Z feet,  as  d = o, 

Ca^/g  ' ’ 

2000  y/  2 X 8 8000  7 

and  f— whole  = ^ = — = 5-7-8  seconds. 

To  Discharge  a like  Volume  under  a Constant  Head. 


Ay If  A //m.-  _ 

C aV7g  c*VsJ..-v 


Tl 


545  X 5 V 64.33 


-258.9  seconds, 


Or,  one  half  the  time  given  by  preceding  case. 

The  times  deduced  by  preceding  formulas  are  in  the  following  proportions  in 

/ a/ 2 / 1 

order,  as  i : V 2 : — , or  i : v 2 : -~r  • 

’ 2 V2 

If  sluice  of  upper  lock,  through  which  it  is  filled,  is  above  lowest  water  level, 
then,  by  combining  formulas  3 and  4,  the  time  is  thus  deduced. 

To  fill  from  Lowest  Water  Level  of  said  Lock  to  Level  of  Centre  of  Sluice. 

5 A /'  representing  height  of  centre  of  sluice  above  said  lowest  water 

Vay/^g  level. 

To  fill  remaining  Portion  of  Lock  above  Sluice. 

6.  2 A ^ ^ - — t".  f"  representing  depth  below  upper  water  level  of  centre  of 
Cay/2  g 

A 

sluice  or  remaining  portion  of  lift.  Hence,  t'  -j-  Z"  ==  77-— 7—  iff'  + 2 V f")  =F.  & 

V ft  V 2 3 

To  fill  Lower  Lock  under  Constant  Head  from  Upper  Canal  Level. 

C a^/—g\+h  V*  ) ^ 

8.  Tf  both  lifts  are  the  same,  h — f=l.  and  — - ^ — 2*  / — ) = ^ 

If  lower  lock  is  filled  from  upper  one  under  a constant  head,  when  latter  is  drawn 
down  to  lowest  level,  formula  7 will  apply  by  making  h —fi  and 

A — (2  V/+  -77);  which  is  identical  with  7,  for/ =/2  and  d —fi  the  cases 

Cay/2  g \ v/ / 

being  the  same. 


556 


HYDRAULICS. 


MISCELLANEOUS  ILLUSTRATIONS. 

1.  If  external  height  of  fresh  water,  at  6o°  above  injection  opening  in  condenser 
of  a steam-engine,  is  3 feet,  and  the  indicated  vacuum  at  23  ins.,  velocity  of  water 
llowing  into  condenser  is  thus  determined.  [Formula page  532.) 

v = V 2 g [h  + h').  lif  representing  height  of  a column  of  water  equivalent  to  press- 
ure of  atmosphere  within  condenser. 

Assuming  mean  pressure  of  atmosphere  = 14.7  lbs.  per  sq.  inch,  height  of  a column 
of  fresh  water  equivalent  thereto  = 33.95  feet. 

Then,  if  1 inch  = .4912  lbs.,  23  ins.  = 11.3  lbs.;  and  if  14.7  lbs.  = 33.95  feet,  11.3 
lbs.  — 26. 1 feet. 

Hence  v = V2  g (3  + 26. 1)  = 43.27  feet,  less  retardation  due  to  coefficient  of  both 
influx  and  efflux. 

2.  What  breadth  must  be  given  to  a rectangular  weir,  to  admit  of  a flow  of  6 cube 
feet  of  water,  under  a head  of  8 ins.  ? [Formula  page  533.) 

6 6 r / 

' 1 •== 77-7 — = 2.21  Jeet. 

^X. 625 V20  66  .417X6.55 

3.  It  being  required  to  ascertain  volume  of  water  flowing  in  a stream,  a tem- 
porary dam  is  raised  across  it,  with  a notch  in  it  2 feet  in  breadth  by  1 in  depth, 
which  so  arrests  flow  that  it  raises  to  a head  of  1.75  feet  above  sill  of  notch;  what 
is  volume  of  flow  per  second?  [Formula  page  533. ) 


C = .635.  - X -635  X2X1.75  V2^X  J-75  ==  1-481  X 10.6  = 15.7  cube  feet. 

3 

4.  A rectangular  sluice  6 feet  in  breadth  by  5 in  depth,  has  a depth  of  9 feet  of 
water  over  its  sill,  and  discharges,  as  per  example  page  535,  380.95  cube  feet  per 
second ; what  is  velocity  of  flow  ? [Formula  page  535. ) 

380.9s  = 38095  ^ 

6X  (9  — 4)  30 

2 ^/h  3 ■Jli'3 

If  volume  was  not  given : — C v 2 g X y—. — = v.  C = . 625. 

3 ti — ti 

Then  — x .625  X 8.02  X ^729 — 3.341  X 3-8  = 12.7  feet. 

3 9 — 4 

5.  If  a river  has  an  inclination  of  1.5  feet  per  mile,  is  40  feet  in  breadth  with  nearly 
vertical  banks,  and  3 feet  depth;  what  is  volume  of  its  discharge  ? [Formula p.  542.) 

Perimeter  40  -f-  2 X 3 = 46  feet;  hydraulic  mean  depth  — 2.61  feet; 


a = 120  feet; 
Then. 


46 

C per  table , page  543,  for  assumed  velocity  of  2.5  feet  — .0075. 


■ X 64.33  X 1.5  = V.0659  X 96.5  = 2. 52  feet  velocity. 


.0075  X 5280  X 46  ' 

Hence  120  X 2.52  = 302.4  cube  feet. 

6.  What  is  head  of  water  necessary  to  give  a discharge  of  25  cube  feet  of  water 
per  minute,  through  a pipe  5 ins.  in  diam.  and  150  feet  in  length  ? [Formula p.  548.) 

Tabular  number  for  diameter  5 ins.,  page  547,  = 263.87. 

< 2 

Then  263.87 -r- 25  = 111.3,  and  150-4-  111.3  = I-35  feet- 

If  this  pipe  had  2 rectangular  knees  or  bends,  what  then  would  be  head  of  water 
required?  [Formula page  545.) 

C,  page  545,  for  — = .984,  area  of  5 ins.  — . 136  feet,  and  -4-  60  = 3.06  feet 

velocity.  Then  — X -984  X 2 = .2863,  which,  added  to  1.35  = 1.64  feet. 

04-33 

By  formulas  foot  of  page  548,  C = .o24,  and  c .505  velocity  — 3.06  feet ; head  — 
1.49  feet,  and  volume  26.38  cube  feet. 

7.  Tf  a stream  of  water  has  a mean  velocity  of  2.25  feet  per  second  at  a breadth 
of  560  feet,  and  a mean  depth  of  9 feet,  what  will  be  its  mean  velocity  when  it  has 
a breadth  of  320  feet,  and  a mean  depth  of  7.5  feet?  [Rule page  548.) 

s6oX9Xa.25  = M34Q=  ^ 

320  X 7-  5 2400 


HYDRAULICS. 


557 


8 What  volume  will  a pipe  48  feet  in  length  and  2 ins.  in  diameter,  under  a head 
of  5 feet,  deliver  per  second  ? ( Formula  page  547.) 


Tabular  number  far  diameter  2 ins.,  page  547,  .=  26.69. 

, 2^,69;—  8.61,  "which  -f-  60  = . 143  cube  feet. 


V 


748 


*=  31* 


Then 


If  this  pipe  had  5 curves  of  9O0,  with  radii  — = - = .5;  wliat  vvould  be  its  dis' 
charge  per  second  ? ^ ^ 

V = .i43;  a = 2-4- 144  = .0139;  Gper  table  = ~ = .294;  v --^^  = 10.29  feet. 

Then  .294  X^X  = - M7.X  1-64  ff  241,  w/itc/i  X 5 /or  5 curves  = 1.2  = 

* 180P  04.33 

/lClyt£  due  to  resistance  of  curves,  h =.  5 — 1. 2 = 3. 8. 

Hence,  if  V 2 g 5 = .143;  V 2 g 3.8  = .125  cube  feet. 

a If  a slide  stop  valve,  set  in  a cylindrical  conduit  500  feet  in  length  and  3 ins.  in 
diameter,  is  raised  so  as  to  close  .625  of  conduit;  what  volume  will  it  discharge 
under  a head  of  4 feet  ? ( Formula  page  546. ) 

C for  conduit  = . 5,  for  friction  . 025 r and  for  slide  valve  .375  open , table,  page  545, 
5.52,  d = .25,  and  a = 7.07  sq.  ins. 

Then  - , •»  *9-- — : ■■■■■•::  l6;°^  - - = 2.13  feet  velocity , and 

,/  500  \ V(7-°2  + 5°) 

f ('+.5+5-5-  + ^S  — •) 

3.13  x 12X7-07  = 180.71  cube  ins. 

10.  If  a single  lock  chamber  is  200  feet  in  length  by  24  in  breadth,  with  a depth 
of  10  feet,  centre  of  upper  gate,  which  is  4 feet  in  depth  by  2.5  in  breadth,  is  at 
middle  of  depth  of  chamber,  lower  gate,  5 feet  in  depth  by  2.5  in  breadth  and  wholly 
immersed;  what  is  time  required  for  filling  and  discharging  it?  (Formula p.  553.) 

C = .6i5,  7t  = 5,  h'  = 5,  A = 200  X 24 •=  4800,  a = 4X2.5  — 10,  and  a = 5 
X 2.5  = 12.5 

(2X5  + 5)  48o^_  = 7^  = 652.8  seconds  time  of  filling. 

• 615X10^64.33.*  5 .,  110-27 

2 x 4800  X V 5 + 5. 3^83^  _ 491.4  seconds  time  of  emptying. 

, ..-p=zr  — 61.73 

.615  x 12.5  V 2 g 

1,.  In  a moderately  direct  and  uniform  course  of  a river,  the  depths  and  velocities 
are  as  follows;  what  is  the  volume  of  its  flow  and  what,  its  mean  velocity?  (p.  551.) 
Feet.  Feet.  Feet.  Feet.  Feet. 


Area  of  profiles  = 5 X 3 + 

12  X 6 -f-  20  X n + 15  X 8 -f- 

7 X 4 = 455  sq.feet. 


Distances 5 12  20  *5  7 

Depths 3 6 11  8 4 

Mean  velocity 1.9  2.3  2.8  2.4  2.1  , - - . — . 

1S  x 1.9  ■ + 72  X 2.3  -f  220  X 2.8  + 120  x 2.4  + 28  X 2. 1 = 1156.9  cube  feet  volume, 

and  115— 2 — 3.54  feet  velocity. 

455 

IVtiner’s  Inch. 

A “ Miner’s  inch  ” is  a measure  for  flow  of  water,  and  is  an  opening  one 
inch  square  through  a plank  two  inches  in  thickness,  under  a head  ox  six 
inches  of  water  to  upper  edge  of  opening. 

It  will  discharge  11.625  U.  S.  gallons  water  in  one  minute. 

Theoretical  HP  under  different  Heads. 


18°  I 

7°'  I 

60  | 

5°  | 

4° 

30  1 20  1 15  |io  | 5I 

3| 

1 4-°6| 

1 4-641 

1 5-4i 

1 6.5 

| 8.12 

10.8I16.2I21.6I32.5I65I 

io8|; 

Water  Inch  ( Pouce  d eau).- —Circular  opening  of  1 inch  in  a thin  plate  is 
equal  to  a discharge  of  19.1953  cube  meters  per  24  hours. 

3 A* 


558 


HYDRODYNAMICS. 


HYDRODYNAMICS. 


Hydrodynamics  treats  of  the  force  of  action  of  Liquids  or  Inelastic 
Fluids,  and  it  embraces  Hydraulics  and  Hydrostatics:  the  former  of 
which  treats  of  liquids  in  motion,  as  flow  of  water  in  pipes,  etc.,  and 
latter  of  pressure,  weight,  and  equilibrium  of  liquids  in  a state  of  rest. 

Fluids  are  of  two  kinds,  aeriform  and  liquid,  or  elastic  and  inelastic, 
and  they  press  equally  in  all  directions,  and  any  pressure  communicated 
to  a fluid  at  rest  is  equally  transmitted  throughout  the  whole  fluid. 

Pressure  of  a fluid  at  any  depth  is  as  depth  or  vertical  height,  and 
pressure  upon  bottom  of  a containing  vessel  is  as  base  and  perpendicu- 
lar height,  whatever  may  be  the  figure  of  vessel.  Pressure,  therefore, 
of  a fluid,  upon  any  surface , whether  Vertical , Oblique , or  Horizontal , is 
equal  to  weight  of  a column  of  the  fluid,  base  of  which  is  equal  to  sur- 
face pressed,  and  height  equal  to  distance  of  centre  of  gravity  of  sur- 
face pressed,  below  surface  of  the  fluid. 

Side  of  any  vessel  sustains  a pressure  equal  to  its  area,  multiplied  by 
half  depth  of  fluid,  and  whole  pressure  upon  bottom  and  against  sides 
of  a vessel  is  equal  to  three  times  weight  of  fluid. 

Pressure  upon  a number  of  surfaces  is  ascertained  by  multiplying 
sum  of  surfaces  into  depth  of  their  common  centre  of  gravity,  below 
surface  of  fluid. 

When  a body  is  partly  or  wholly  immersed  in  a fluid,  vertical  press- 
ure of  the  fluid  tends  to  raise  the  body  with  a force  equal  to  weight  of 
fluid  displaced ; hence  weight  of  any  quantity  of  a fluid  displaced  by  a 
buoyant  body  equals  weight  of  that  body. 

Centre  of  Pressure  is  that  point  of  a surface  against  which  any  fluid 
presses,  to  which,  if  a force  equal  to  whole  pressure  were  applied,  it 
would  keep  surface  at  rest.  Hence  distance  of  centre  of  pressure  of 
any  given  surface  from  surface  of  fluid  is  same  as  Centre  of  Percussion. 

Centres  of*  Pressure. 

Parallelogram , Side,  Base , Tangent , or  Vertex  of  Figure  at  Surface  of  Fluid,  is  at  ;i 
.66  of  line  (measuring  downward)  that  joins  centres  of  two  horizontal  sides. 

Triangle , Base  uppermost,  is  at  centre  of  a line  raised  vertically  from  lower  apex, 
and  joining  it  with  centre  of  base;  aud  Vertex  uppermost , it  is  at  .75  of  a line  let 
fall  perpendicularly  from  vertex,  and  joining  it  with  centre  of  base. 


Right-angled  Triangle , Base  uppermost , is  at  intersection  of  a line  extended  from 
centre  of  base  to  extremity  of  triangle  by  a line  running  horizontally  from  centre 
of  side  of  triangle.  Vertex  or  Extremity  uppermost , is  at  intersection  of  a line  ex- 
tended from  the  centre  of  the  base  to  the  vertex,  by  a line  running  horizontally  from 
. 375  of  side  of  triangle,  measured  from  base. 


Trapezoid , either  of  parallel  Sides  at  Surface , 


6 + 35' 


- X a = d.  b and  b'  repre- 


2 b -{-  4 b 

senting  breadths  of  figure,  d distance  from  surface  of  fluid,  and  a length  of  line  join- 
ing opposite  sides. 


Circle , at  1.25  of  its  radius,  measured  from  upper  edge. 

3 p r 

Semicircle,  Diameter  at  Surface  of  Fluid,  — = d.  r representing  radius  of  circle 

16 

1 5 » r — 32  r 

and  p = 3. 1416.  Diam.  downward, = d. 

J i2i>  — 16 


HYDRODYNAMICS. 


559 


Side,  Base,  or  Tangent  of  Figure  below  Surface  of 
Blviid. 


Rectangle  or  Parallelog'm. 


h'3  — hs 
X ....  ^ = d ; 


3 m o -f-  wi2 


= d 


and  — = d". 

30 


3 h' * — /i2  ~ 3 0 

h and  Ji'  representing  depths  of  upper  and  lower  surfaces  of  figure  and  d depth , 
both  from  surface  of  fluid,  m half  depth  of  figure,  o depth  of  centre  of  gravity  of 
figure  from  surface  of  fluid,  d'  distance  from  upper  side  of  figure,  and  d distance 
from  centre  of  gravity. 

1-*  10  o*  . 


Triangle.  — Vertex  Uppermost. 


--d\ 


- ==  d'.  Base  Uppermost. 


Z2+I8°2_.  1 representing  depth  of  figure,  d distance  from  surface  of  fluid  upon 

a line  from  vertex  to  centre  of  base,  and  d'  distance  from  centre  of  gravity  of  figure. 

Circle  4 °~  + y2  _ d or  :L_  = distance  from  centre  of  circle. 

•4°  40  *2  *6Z2  - 

Semicircle.  —Diam.  Horizontal  and  Upward  or  Downward.  -—  — -^-Q  + 0 = a 5 

3 P l ~~  4 1 . 4 l __  an(j  i — c.  d representing  distance  from 

op  ’3J3  ’ 4 o 9 p o 

surface  of  fluid,  d'  distance  of  centre  of  gravity  from  centre  of  arc,  d ' distance  of 
centre  of  gravity  from  diameter  when  it  is  uppermost,  and  c centre  of  pressure. 


Pressure. 

To  Compute  Pressure  of  a FlnicL  upon  Bottom  of  its 
Containing  Vessel. 

Rule —Multiply  area  of  base  by  height  of  fluid  in  feet,  and  product  by 
weight  of  a cube  foot  of  fluid. 

To  Compnte  Pressure  of  a Blnid  upon  a V ertical.  In- 
clined, Curved,  or  any  Surface. 

Rule.— Multiply  area  of  surface  by  height  of  centre  of  gravity  of  fluid 
in  feet,  and  product  by  weight  of  a cube  foot  of  fluid. 

Example  i. — What  is  pressure  upon  a sloping  side  of  a pond  of  fresh  water  10  feet 
square  and  8 feet  in  depth  ? 

Centre  of  gravity,  8-4-2  = 4 fed  from  surface.  Then  102  X 4 X 62. 5 = 25  000  lbs. 

2.  —What  is  pressure  upon  staves  of  a cylindrical  reservoir  when  filled  with  fresh 
water,  depth  beiDg  6 feet,  and  diameter  of  base  5 feet? 

5 x 3-1416=  15-708  feet  curved  surface  of  reservoir,  which  is  considered  as  a plane. 

15.708  X 6 X 6-4-2  = 282.744,  which  X 62.5  = 17671.5  lbs. 

3,  a rectangular  flood-gate  in  fresh  water  is  25  feet  in  length  by  12  feet  deep; 

what  is  pressure  upon  it? 

25  X 12  X 12-4-2  = 1800,  which  X 62.5  = 112  500  lbs. 

When  water  presses  against  both  sides  of  a plane  surface,  there  arises^  from 
resultant  forces,  corresponding  to  the  two  sides,  a new  resultant,  which  is 
obtained  by  subtraction  of  former,  as  they  are  opposed  to  each  other. 

Illustration  -Depth  of  water  in  a canal  is  7 feet;  in  its  adjoining  lock  it  is  4 
feet,  and  breadtn  of  gates  is  15  feet;  what  mean  pressure  have  they  to  sustain,  and 
what  is  depth  of  point  of  its  application  below  surface  ? 

7 x 15=  105,  and  4 X 15  =60  sq.feet.  (105  X —60  X 2)  X 62.5  = 1546.875  lbs., 
mean  pressure. 

Then  1546.875-5-62.5  = 247.5  =cube  feet  pressing  upon  gates  upon  high  side,  and 
247. 5 -4- 15  X 7 = 2. 35  feet  = depth  of  centre  of  gravity  of  mean  pressure. 

To  Compute  Pressure  on  a Slaice. 

Awd  = P,  and  C P = P'.  A representing  area  of  sluice  in  sq.  feet,  w weight  of 
water  per  cube  foot,  d mean  depth  of  sluice  below  surface , in  feet,  P pressure  on  sluice , 
and  P'  power  required  to  operate  it,  both  in  lbs. 

C = .68  when  sluice  is  of  wood,  and  .31  when  of  iron. 


HYDRODYNAMICS. 


560 


Example.— What  is  pressure  on  a sluice-gate  3 feet  square,  its  centre  of  gravity 
being  30  feet  below  surface  of  a pond  of  fresh  water? 


3 X 3 X 30  = 270,  which  X 62.5  = 16  875  lbs. 

To  Compute  Pressure  of  a,  Column  of  a pTliaid.  per 
Sq.  Inch. 

Rule. — Multiply  height  of  column  in  feet  by  weight  of  a cube  foot  of 
fluid,  and  divide  product  by  144 ; quotient  will  give  weight  or  pressure  per 
sq.  inch  in  lbs. 

Note. —When  height  is  given  in  ins.,  omit  division  by  144. 

PIPES. 

To  Compute  required  Thickness  oP  a ripe. 

Rule.— Multiply  pressure  in  lbs.  per  sq.  inch  by  diameter  of  pipe  in  ins., 
and  divide  product  by  twice  assumed  tensile  resistance  or  value  of  a sq! 
inch  of  material  of  which  pipe  is  constructed. 

By  experiment,  it  has  been  found  that  a cast-iron  pipe  15  ins.  in  diameter,  and 
.75  of  an  inch  thick,  will  support  a head  of  water  of  600  feet;  and  that  one  of’ oak, 
of  same  diameter,  and  2 ins.  thick,  will  support  a head  of  180  feet  ? 

Example  1.— Pressure  upon  a cast-iron  pipe  15  ins.  in  diameter  is  300  lbs.  per  sq. 
inch;  what  is  required  thickness  of  metal? 

300  X 15  = 4500,  which  -r-  3000  X 2 . 7 5 inch. 

Note. — Here  3000  is  taken  as  value  of  tensile  strength  of  cast  iron  in  ordinary 
small  water-pipes.  This  is  in  consequence  of  liability  of  such  castings  to  be  im- 
perfect from  honey-combs,  springing  of  core,  etc. 

2. — Pressure  upon  a lead  pipe  1 inch  in  diameter  is  150  lbs.  per  sq.  inch;  what  is 
required  thickness  of  metal  ? 

Here  500  is  taken  as  value  of  tensile  strength. 


150  X 1 = 150,  which  rf-  500  X 2 = . 15  inch. 


Cast-iron  IPipes. 

To  Compute  Thickness,  etc.,  of*  Flanged  IPipes. 


For  75  lbs.  Pressure. 

.025  D -f-  .25  =c  T 

.03  D-f  .3  = t 

.05  D-f- 1. 15  —l 

•°3  R-f  -35  —f 

1.05  D -f-  4.25  d -f-  1.25  = o 

1.05  D -f-  2 X d -}-  1 — o' 

A xp~ 


. 7 D -f-  2. 2 = n ; 


For  100  lbs.  Pressure. 
•03  D+  -3  - 

•035  D + -45  = 

I>+  1.15 

6- 


4000 


•05 

.04  D + _ 

11  D + 5 X d-\-  1.5 

11  D + 2.5  X d-f-  1.4 
and 


— T 

==  t 

— I 

= f 

=z  O 


— 0 


\/-78S4 


+■■■0  = d. 


D representing  diam.  of  pipe , T thickness  of  metal,  t thickness  and  l length  of  boss, 
f thickness  of  flange,  o diam.  of  flange,  o'  diam.  of  centres  at  bolt  holes,  and  d diam. 
of  bolts,  all  in  ins.;  A area  of  pipe  and  a area  of  bolt  at  base  of  its  thread,  in  sq.  ins., 
p pressure  in  lbs.  per  sq.  inch , and  C a coefficient  due  to  diam.  of  bolt. 

Thus,  diam. . 125 -f- .032,  .25 -h .064,  .5 -{-•  107,  i-f-.i6,  1.5 -f-. 214,  and  2 -[-.285. 

Illustration.— What  should  be  dimensions  of  a flanged  pipe,  10  ins.  in  diameter, 
for  a pressure  of  100  lbs.  per  sq.  inch? 

.7  X 10  -f-  2. 2 = 9. 2 = 10  number  of  bolts,  and  diam.  10  ins.  = 78. 54  ins.  area  = A. 


78.54  X .00^0 y and  J.J96 35  + c = v. 25  = ,5  ; hence,  .5  + . ro7  = 

4000  v .7054 

.607  = .625  lbs.  diameter  of  bolts  ; .03  X 10 +.3  = .6  = thickness  of  metal;  .035  X 10 
— }- . 45  = . 8 — thickness  of  flange;  .05  X 10  -f- 1. 15  = 1.65  = length  of  boss;  .04X10 
-f- . 6 = 1 s=  thickness  of  flange  ; i.iXio-f-5X.625-f-i.5  = i5.625  = diameter  of 
flange;  and  1.1  X 10  -f  2.5  X .625  -f  1.4  — 13.9625  diameter  of  bolt  holes. 


For  Tables  of  Cast-iron  Pipes,  see  page  132. 


i 

\ 


HYDRODYN  AMICS. 


56l 


To  Compute  Elements  of  Water-pipes. 

000 124  c;  P d 4-  C = t:  or,  .000054  H d -J-  C = t\  -4336H==P;  and 

142 d2  v 2.  A.11  = W.  P representing  pressure  of  water  in  lbs.  per  sq.  inch,  D andd 

external  anil  internal  diameters  of  pipe,  and  t thickness  of  metal,  all  in  ins.,  C coeffi- 
cient for  diameter  of  pipe,  and  H head  of  water  in  feet. 

C = .37  for  pipes  less  than  12  ins.  in  diameter,  .5  from  12  to  30,  and  .6  from  30  to  50. 


To  Compute  Weight  of  Pipes. 

To  Diameter  add  thickness  of  metal,  multiply  sum  by  10  times  thickness, 
and  product  will  give  weight  in  lbs.  per  foot  of  length. 

Weight  of  Faucet  end  is  equal  to  8 ins.  of  length  of  pipe. 


Hydrostatic  Press. 

To  Compute  Elements  of  a Hydrostatic  Press. 

PEA  _ w . W E a — . W E a __  p P A E __  a p represcnting  power  or  press- 
l*  a 1 P l * E A W l 

ure  applied , W weight  or  resistance  in  lbs.,  I and  V lengths  of  lever  and  fulcrum  in 
ins.  or  feet,  and  A and  a areas  of  ram  and  piston  in  sq.  ins. 

Illustration. — Areas  of  a ram  and  piston  are  86.6  and  1 sq.  ins.,  lengths  of  lever 
and  fulcrum  4 feet  and  9 ins.,  and  power  applied  20  lbs. ; what  is  weight  that  may 
he  sustained? 

*0X4  X12X86.6  = 83136  = 3 m 

9X1  9 

To  Compute  Thickness  of  HVTetal  to  Resist  a given 
Pressure. 

Rule.— Multiply  pressure  per  sq.  inch  in  lbs.  by  diameter  of  cylinder  in 
ins.,  and  divide  product  by  twice  estimated  tensile  resistance  or  value  of 
metal  in  lbs.  per  sq.  inch,  and  quotient  will  give  thickness  of  metal  required. 

Example.— Pressure  required  is  9000  lbs.  per  sq.  inch,  and  diameter  of  cylinder  is 
5.3  ins. ; what  is  required  thickness  of  metal  of  cast  iron? 

0000  X 5-  3 47  7°° 

Value  of  metal  is  taken  at  6000.  - z — — - — = 3-975  xns' 

6000  X 2 12  000 

Values  of  Different  Metals  in  Tons.  ( Molesworth . ) 

Cast  iron 41  | Gun  metal 22  | Wrought  iron..  .14  | Steel 06 

Hydraulic  Ram. 

Useful  effect  of  an  Hydraulic  Ram,  as  determined  by  Eytelwein,  varied 
from  .9  to  .18  of  power  expended.  When  height  to  which  water  is  raised 
compared  to  fall  is  low,  effect  is  greater  than  with  any  other  machine ; but 
it  diminishes  as  height  increases. 

Length  of  supply  pipe  should  not  be  less  than  .75  of  height  to  which 
water  is  to  be  raised,  or  5 times  height  of  supply ; it  may  be  much  longer. 

To  Compute  Elements. 

.00113  VA  = EP;  ^^  = v;  i.45VV  = d;  -75  = ^ 5 and|xV^  = 

efficiency.  V and  v representing  volumes  expended  and  raised,  in  cube  feet  per 
minute,  li  and  h'  heights  from  which  water  is  drawn  and  elevated  in  feet,  D and  d 
diameters  of  supply  and  discharging  pipes  in  ins.,  and  IP  effective  horsepower. 

Illustration. — Heights  of  a fall  and  of  elevation  are  10  and  26.3  feet,  and  vol- 
umes expended  and  raised  per  minute  are  1.71  and  .543  cube  feet. 

.001 13  X 1. 71  X 10  = .0193  IP  *,  881  ^o°19j  = 1-71  cubefeto;  1.45^/1.71  = 1.89 

ins. ; . 75  Ji. 71  = . 975  ins. ; and  §■  X ~543  X — . 696  efficiency. 

' J v ' * ' 6 1. 71  X 10 


HYDRODYNAMICS. 


62 


Results  of  Operations  of  Hydraulic  Rams. 


Strokes 
per  M. 

| Fall. 

Eleva- 

tion. 

Wai 

Expen'd. 

ter 

Raised. 

Useful 

Effect. 

j Strokes 
j per  M. 

Fall. 

Eleva- 

tion. 

Wai 

Expen’d. 

ter 

Raised. 

Useful 

Effect. 

No. 

66 

50 

36 

3i 

Feet. 

10.06 

9-93 

6.05 

5.06 

Feet. 

26.3 

38.6 

38.6 

38.6 

C.  Ft. 
1.71 
1-93 
i-43 
1.29 

C.  Ft. 
•543 
.421 
.169 

•113 

•9 

.85 

•75 

.67 

No. 

15 

10 

Feet. 

3.22 

*•97 

22.8 

8-5 

Feet. 

38.6 

38.6 
196.8 

52.7 

C.  Ft. 

1.98 

1.58 

.38 

2 

C.  Ft. 
.058 
.014 
.029 
.186 

.67 

•57 

V online  01  air  vessel  — volume  of  delivery  pipe.  One  seventh  of  watei 
may  be  raised  to  about  4 times  head  of  fall,  or  one  fourteenth  8 times  or  one  twenty 
eighth  16  times.  ’ * 


WATER  POWER. 

Water  acts  as  a moving  power,  either  by  its  weight  or  by  its  vis  viva , and 
m latter  case  it  acts  either  by  Pressure  or  by  Impact. 

Natural  Effect  or  Power  of  a fall  of  water  is  equal  to  weight  of  its  volume 
and  vertical  height  of  its  fall. 

If  water  is  made  to  impinge  upon  a machine,  the  velocity  with  which  it 
impinges  may  be  estimated  in  the  effect  of  the  machine.  Result  or  effect 
however,  is  in  nowise  altered ; for  in  first  case  P = Vto  h,  and  in  latter  = 

~ V w.  Y representing  volume  in  cube  feet,  tv  weight  in  lbs.,  and  v velocity 
of  flow  in  f eet  per  second. 

62.5  V h — P,  and  3.2*  a fh  — V.  P representing  pressure  in. lbs.,  a area  of  open- 
ing in  sq.  feet,  and  h height  of  flow  in  feet  per  second. 


To  Compute  Rower  of  a Rail  of  Water. 

Rule. — Multiply  volume  of  flowing  water  in  cube  feet  per  minute  bv 
62.5,  and  this  product  by  vertical  height  of  fall  in  feet. 

Note.  — When  Flow  is  over  a Weir  or  Notch , height  is  measured  from  surface  of 
tail-race  to  a point  four  ninths  of  height  of  weir,  or  to  centre  of  velocity  or  pressure 
of  opening  of  flow. 

When  Flow  is  through  a Sluice  or  Horizontal  Slit,  height  is  measured  from  sur- 
face of  tail-race  to  centre  of  pressure  of  opening. 

Example.— What  is  power  of  a stream  of  water  when  flowing  over  a weir  5 feet 
in  breadth  by  1 in  depth,  and  having  a fall  of  20  feet  from  centre  of  pressure  of  flow? 

By  Rule,  page  533,  — 5 X 1 V2  ^ 1 X -625  = 16.68  cube  feet  per  second. 

16.68  X 60  X 62. 5 X 20  = r 251 000  lbs. , which  -4-  33000  = 37.91  horses 1 power. 

Or,  .1135  V h = theoretical  IP.  h representing  height  from  race  in  feel. 

Illustration.— If  flow  of  a stream  is  17.9  cube  feet  per  second,  to  what  height 
and  area  of  flow  of  1 foot  in  depth  should  it  be  dammed  to  attain  a power  of  10 
horses. 


33  000  X 10  ssoo  pc 

= 5500  lbs.  per  second,  and  = 88  cube  feet  per  second.  : 

02.5  179 


60 


4. 92  feet  height.  Hence,  — . 6 V2  g X 1 = 3. 2,  and  17. 9 -4-  3. 2 = 5. 59  sq.  feet. 


' Water  sometimes  acts  by  its  weight  and  vis  viva  simultaneously,  by  com- 
bining effect  of  an  acquired  velocity  with  fall  through  which  it  flows  upon 
wheel  or  instrument. 


In  this  case 


V x 62. 5 = mechanical  effect. 


* As  determined  by  — C. 


HYDRODYNAMICS. 


563 


WATER-WHEELS. 

Water-wheels  are  divided  into  two  classes,  Vertical  and  Horizontal. 
Vertical  comprises  Overshot , Breast , and  Undershot ; and  Horizontal, 
Turbine , Impact , or  Reaction  wheels. 

Vertical  wheels  are  limited  by  construction  to  falls  of  less  than  60  feet. 
Turbines  are  applicable  to  falls  of  any  height  from  1 foot  upward. 

Vertical  wheels  applied  to  a fall  of  from  20  to  40  feet  give  a greater 
effect  than  a Turbine,  and  for  very  low  falls  Turbines  give  a greater  effect. 

Sluices. — Methods  of  admitting  water  to  an  Overshot  or  Breast 
Wheel  are  various,  consisting  of  Overfall , Guide-bucket,  and  Penstock. 

An  Overfall  Sluice  is  a saddle-beam  with  a curved  surface,  so  as  to  direct  the 
current  of  water  tangentially  to  buckets;  a Guide-bucket  is  an  apron  by  which 
water  is  guided  in  a course  tangential  to  buckets;  and  a Penstock  is  sluice-board  or 
^ate  placed  as  close  to  wheel  as  practicable,  and  of  such  thickness  at  its  lower  edge 
as  to  avoid  a contraction  of  current.  Bottom  surface  of  penstock  is  formed  with  a 
parabolic  lip. 

SL.roud.iiag  of  a wheel  consists  of  plates' at*its  periphery,  which 
form  the  sides  of  the  bucket. 

Height  of  fall  of  a water-wheel  is  measured  between  surfaces  of  water  in  penstock 
and  in  tail-race , and,  ordinarily,  two  thirds  of  height  between  level  of  reservoir  and 
point  at  which  water  strikes  a wheel  is  lost  for  all  effective  operation. 

Velocity  of  a wheel  at  centre  of  percussion  of  fluid  should  be  from  .5  to  .6  that 
of  flow  of  the  water. 

Total  effect  in  a fall  of  water  is  expressed  by  product  of  its  weight 
and  height  of  its  fall. 

Hatio  of  Effective  Power  of  Water  Motors. 
Overshot  and  high}  from  t 6 to  _ Undershot,  Poncelet’s,  from  .6  to. 4 to  1 

breast j lrom  t0  t0  1 Undershot “ .27  to  .45  to  1 

Turbine u .6  to  .8  to  1 Impact  and  Reac- 1 u 0 to. 5 to  1 

Breast “ .45  to  .65  to  1 tion j 

Hydraulic  Ram .6  to  1 Water-pressure  engine  .8  to  1 


Overshot-wheel, 

Overshot- wheel. — The  flow  of  water  acts  in  some  degree  by  impact, 
but  chiefly  by  its  weight. 

Lower  the  speed  of  wheel  at  its  circumference,  the  greater  will  be  mechan- 
ical effect  of  the  water,  in  some  cases  rising  to  80  per  cent, ; with  velocities 
of  from  3 to  6.5  feet,  efficiency  ranges  from  70  to  75  per  cent.  Proper  ve- 
locity is  about  5 feet  per  second. 

Humber  of  buckets  should  be  as  great,  and  should  retain  water  as  long,  as 
practicable.  Maximum  effect  is  attained  when  the  buckets  are  so  numerous 
and  close  that  water  surface  in  the  bucket  commencing  to  be  emptied  should 
come  in  contact  with  the  under  side  of  the  bucket  next  above  it.  Moles- 
worth  gives  12  ins.  apart. 

Curved  buckets  give  greatest  effect,  and  Radial  give  but  .78  of  effect  of 
Elbow  buckets.  Wheel  40  feet  in  diameter  should  have  152  buckets. 

Small  wheels  give  a less  effect  than  large,  in  consequence  of  their  greater 
centrifugal  action,  and  discharging  water  from  the  buckets  at  an  earlier 
period  than  with  larger  wheels,  or  when  their  velocity  is  lower. 

When  head  of  water  bears  to  fall  or  height  of  wheel  a proportion  as  great 
as  1 to  4 or  5,  ratio  of  effect  to  power  is  reduced.  The  general  law  there- 
fore is,  that  ratio  of  effect  to  power  decreases  as  proportion  of  head  to  total 
head  and  f all  increases. 


564 


HYDRODYNAMICS. 


Wheel  with  shallow  Shrouding  acts  more  efficiently  than  one  where  it  is 
deep,  and  depth  is  usually  made  10  or  12  ins.,  but  in  some  cases  it  has  been 
increased  to  15. 

Breadth  of  a wheel  depends  upon  capacity  necessary  to  ffive  the  buckets 
to  receive  required  volume  of  water. 

Form  of  Buckets.  Radial  buckets — that  is,  when  the  bottom  is  a right  line in- 

volve so  great  a loss  of  mechanical  effect  as  to  render  their  use  incompatible  with 
economy;  and  when  a bucket  is  formed  of  two  pieces,  lowrer  or  inner  piece  is 
termed  bottom  or  floor,  and  outer  piece  arm  or  wrist.  Former  is  usuallv  placed  in 
a line  with  radius  of  wheel.  J 1 

Line  of  a circle  passing  through  elbow , made  by  junction  of  floor  and  arm  is 
termed  division  circle,  or  bucket  pitch,  and  it  is  usual  to  put  this  at  one  half  depth 
of  shrouding.  F 

When  arm  of  a bucket  is  included  in  division  angle  of  buckets,  that  is,  n 

representing  number  of  buckets,  the  cells  are  not  sufficiently  covered,  except  for  verv 
shadow  shrouding;  hence  it  is  best  to  extend  arm  of  a bucket  over  1.2  of  division 
angle,  so  as  to  cover  or  overlap  elbow  of  bucket  next  in  advance  of  it. 

Construction  of  Buckets  (Fig.  1).— Capacity  of  bucket  should  be  3 times  volume 
of  water. 


Fig. 


Fairbairn  gives  area  of  opening  of  a bucket  in  a 
wheel  of  great  diameter,  compared  to  the  volume  of  it 
as  5 to  24. 

Buckets  having  a bottom  of  two  planes,  that  is,  with 
two  bottoms,  and  two  division  circles  or  bucket  pitches 
and  an  arm,  give  a greater  effect  than  with  one  bottom. 
When  an  opening  is  made  in  base  of  buckets  so  as 
D l to  afford  an  escape  of  air  contained  within,  without  a 
! : loss  of  water  admitted,  the  buckets  are  termed  ven- 
tilated, and  effective  power  of  wheel  is  much  greater 
than  with  closed  buckets. 

D — distance  apart  at  periphery  = d%  d depth  of 
shrouding,  s length  of  radial  start  =.33'd,  l length  of 
bucket  curves  1.25  a in  large  wheels,  and  1 in  wheels 
under  25  feet,  a angle  of  radius  of  curve  of  bucket 
with  radial  line  of  wheel  at  points  of  bucket  ~ 1^0’ 
(Molesworth.) 


To  Compute  Radius  and  Revolutions  of*  an  Overshot- 
wheel,  and  Height  of  Fall  of  Water. 

h — h' 

- = 


When  whole  Fall  and  Velocity  of  Flow , etc.,  are  given . 
hc  v2  ..  o.i4i6 


- = w. 


- 1. 1 — h'. 


and 


1 -j-  cos.  a ~ 
h representing  height  of  ivhole 


3.1416  r 7 2 g 3o 

{non  hwM^LhtfWnen  ^ CTKe  °fOravity  of  discharge  and  half  depth  of  bucket 
cntZZfni-  wajerfi°wsf  velocity  of  flow  in  feet  per  second,  a angle  which  point  of 
nil  7 a,buIcJceIt  makes  with  summit  of  wheel,  n number  of  revolutions 

per  minute,  c velocity  of  wheel  at  its  circumference  per  second,  and  r its  radius. 

iinSwWM1  °f  whole  fall  is  distance  between  surface  of  water  in  flume  and 
h °Wer  buckets  are  emptied  of  water,  and  as  a proportion  of  velocity 
ot  flow  is  lost,  it  is  proper  to  assume  height  h'  as  above  given.  * 

fal1  °tf  w,ater  is  30  feet,  velocity  of  its  flow'  is  16  feet  per  second, 
JJSna  7?  ,aiPact  «Pf>n  buckets  is  120,  and  required  velocity  of  w heel  is  8 feet  per 
wheel?  lat  1S  required  rad,us,  number  of  revolutions,  and  height  of  fall  upon 

X 1.1=4.38  feet;  cos.  i2°  = .978;  = ^ = 12.95  fed  radius; 

24O 

40.68 


2 g 

30  x s ^ 

3.1416  X 12.95 


= 5.9,  revolutions. 


HYDRODYNAMICS. 


565 


When  Number  of  Revolutions  and  Ratio  between  Velocities  of  Flow  and  at 
Circumference  of  Wheel  are  given. 

^.000772  (»w)8AHr(r-H*>s,  0)2  — i-t-cos.ig)  _r  x__  v ^ 3.1416  nr 
.000386  (xn) 2 ’ c’  30 

Illustration. — If  number  of  revolutions  are  5,  x = 2,  and  fall,  etc. , as  in  previous 
case;  what  is  radius  of  wheel,  velocity  of  flow,  and  height  of  fall? 

y.000772  (2  X 5)2  X 3°  + (*-978)2  — 1-978  _ ^ 

.000386  (2  X 5)2  .0386 

3.1416  X 5 X 13-4*.  __  ~'0?feet . Hence  7.03  X 2 = 14.06  velocity  of  flow,  and  -1-4- 6 - 
30  04.33 

X i.i  = 3-37  /«*■ 

To  Compute  "Width,  of  an  Overshot-wheel. 

C V 

— - — w.  C representing  a coefficient  = 3,  when  buckets  are  filled  to  an  excess,  and 

5 when  they  are  deficiently  filled , V volume  of  water  in  cube  feet  per  second , s depth 
of  shrouding,  w luidth  of  buckets,  both  in  feet,  and  c'  velocity  of  wheel  at  centre  of 
shrouding , in  feet  per  second. 

Illustration. — A wheel  is  to  be  31  feet  in  diameter,  with  a depth  of  shrouding  of 
1 foot,  and  is  required  to  make  5 revolutions  per  minute  under  a discharge  of  10 
cube  feet  of  water  per  second ; what  should  be  width  of  buckets  ? 

Assume  C = 4,  and  c'  = — — * X 3*4l6  X 5 _ y Then  4X  ^Q^feet. 

60  1 x 7*054 

To  Compute  Number  of  Buckets. 

7(1  + -!^  -4-  12  = d,  and  — ^ = n.  D representing  diameter  of  wheel,  d dis- 
\ • 83/  d 

tance  between  centres  of  buckets,  in  feet,  and  n number  of  buckets. 

Illustration. — Take  elements  of  preceding  case. 

31  — 1 X 3- 1416  X 


Then 


' ^1  -j-  = 7X2.24-12  = 1.283,  an(l 


360® 

buckets  ; hence  — — = 50,  angle  of  subdivision  of  buckets. 
72 


1.283 


1 = 73*4,  ^7  72 


To  Compute  Effect  of  an  Overshot- wheel. 

TV"~(ST"+/) 


V h w 


- = P.  w representing  weight  of  cube  foot  of  water  in  lbs., 

v'  velocity  of  it  discharged  at  tail  of  wheel,  in  feet  per  second,  V volume  of  flow  in 
cube  feet,  and  f friction  of  wheel  in  lbs. 

Illustration.  — A volume  of  12  cube  feet  per  second  has  a fall  of  10  feet.,  wheel 
using  but  8.5  feet  of  it,  and  velocity  of  water  discharged  is  9 feet  per  second;  what 
is  effect  of  fall  ? 

Friction  of  wheel  is  assumed  to  be  750  lbs. 

12  X 3.5  X 62.5— — X 12  X 62.5  + 750^  M 

. V64.33 / _ 6373  — (1.26  X 750  + 750  _ 4680  _ 

12X10X62.5  — 7500  7500 

.624  = ratio  of  effect  to  power  ; and  4680  X 60  seconds  -H  33  000  = 8.51  IP. 

To  Compute  Power  of  an  Overshot- wheel. 

Pule.  — Multiply  weight  of  water  in  lbs.  discharged  upon  wheel  in  one 
minute  by  height  or  distance  in  feet  from  centre  of  opening  in  gate  to  sur- 
face of  tail-race;  divide  product  by  33000,  and  multiply  quotient  by  as- 
sumed or  determined  ratio  of  effect  to  power.  Or,  for  general  purposes, 
divide  product  by  50  000,  and  quotient  is  IP. 

Or,  .0852  Vh  = IP,  and  - - 7 — — V per  second ; or,  = V Per  minute. 

3B 


HYDRODYNAMICS. 


566 


Mechanical  Effect  of  water  is  product  of  its  weight  into  height  from  which 
it  falls. 

Example.— Volume  of  water  discharged  upon  an  overshot- wheel  is  640  cube  feet 
per  minute,  and  effective  height  of  fall  is  22  feet;  what  is  H?? 

640  X 62. 5 X 22  __  2g>6  which,  X -75  = assumed  ratio  of  effect  to  power  = 20  IP. 

33000 

TJ seful  Effect  of  an  OversRot- wheel. 

With  a large  wheel  running  in  most  advantageous  manner,  .84  of  power 
may  be  taken  for  effect. 

Velocity  of  a wheel  bears  a constant  ratio,  for  maximum  effects,  to  that 
of  the  flowing  water,  and  this  ratio  is  at  a mean  .55. 

Ratio  of  effect  to  power  with  radial-buckets  is  .78  that  of  elbow-buckets. 
Ratio  of  effect  decreases  as  proportion  of  head  to  total  head  and  fall  increases. 
Thus,  a wheel  10  feet  in  diameter  gave,  with  heads  of  water  above  gate, 
ranging  from  .25  to  3.75  feet,  a ratio  of  effect  decreasing  from  .82  to  .67  of 
power. 

Higher  an  overshot-wheel  is,  in  proportion  to  whole  descent  of  water, 
greater  will  be  its  effect.  Effect  is  as  product  of  volume  of  water  and  its 
perpendicular  height. 

Weight  of  arch  of  loaded  buckets  in  lbs.  is  ascertained  by  multiplying 
.444  of  their  number  by  number  of  cube  feet  in  each,  and  that  product  by  40. 


IT  ndershot-wheel. 

Undershot-wheel  is  usually  set  in  a curb,  with  as  little  clearance  for 
escape  of  water  as  practicable ; hence  a curb  concentric  to  this  wheel  is  more 
effective  than  one  set  straight  or  tangential  to  it. 

Computations  for  an  undershot-wheel  and  rules  for  construction  are  near- 
ly identical  with  those  for  a breast-wheel. 

Buckets  are  usually  set  radially,  but  they  may  be  inclined  upward,  so  as 
to  be  more  effectively  relieved  of  water  upon  their  return  side,  and  they  are 
usually  filled  from  .5  to  .6  of  their  volume.  Depth  of  shrouding  should  be 
from  15  to  18  ins.,  in  order  to  prevent  overflow  of  water  within  the  wheel, 
which  would  retard  it. 

Velocity  of  periphery  should  equal  theoretical  velocity  due  to  head  of 
water  X .57. 

Note.— When  constructed  without  shrouding,  as  in  a current- wheel,  etc.,  buckets 
become  blades. 

Sluice-gate  should  be  set  at  an  inclination  to  plane  of  curb,  or  tangential 
to  wheel,  in  order  that  its  aperture  may  be  as  close  to  wheel  as  practicable ; 
and  in  order  to  prevent  partial  contraction  of  flow  of  water,  lower  edge  ot 
sluice  should  be  rounded. 

Effect  of  an  undershot-wheel  is  less  than  that  of  a breast-wheel,  as  the 
fall  available  as  weight  is  less  than  with  latter. 


4 


To  Compute  Power  of  an  TJndersliot-wlieel. 

Proceed  as  per  rule  for  an  overshot-wheel,  using  93  750  for  50  000,  and  .4 
for  .75. 

Or,  V h .000 66  = IP ; or,  — = V.  V representing  volume  of  water  in  cube 
feet  per  minute , and  h head  of  water  in  feet. 


HYDRODYNAMICS. 


567 


TPoncelet’s  Wheel. 

Ponce  let’s  Wheel. — Buckets  are  curved,  so  that  flow  of  water  is  in 
course  of  their  concave  side,  pressing  upon  them  without  impact ; and  effect 
is  greater  than  when  water  impinges  at  nearly  right  angles  to  a plane  sur- 
face or  blade. 

This  wheel  is  advantageous  for  application  to  falls  under  6 feet,  as  its 
effect  is  greater  than  that  of  other  undershot  wheels  with  a curb,  and  for 
falls  from  3 to  6 feet  its  effect  is  equal  to  that  of  a Turbine. 

For  falls  of  4 feet  and  less,  efficiency  is  65  per  cent.,  for  4.25  to  5 feet,  60 
per  cent.,  and  from  6 to  6.5  feet,  55  to  50  per  cent. 

In  its  arrangement,  aperture  of  sluice  should  be  brought  close  to  face  of 
wheel.  First  part  of  course  should  be  inclined  from  40  to  6°  ; remainder  of 
course,  which  should  cover  or  embrace  at  least  three  buckets,  should  be  car- 
ried concentric  to  wheel,  and  at  end  of  it  a quick  fall  of  6 ins.  made,  to  guard 
against  effect  of  back-water.  Sluice  should  not  be  opened  over  1 foot  in  any 
case,  and  6 ins.  is  a suitable  height  for  falls  of  5 and  6 feet. 

Distance  between  two  buckets  should  not  exceed  8 or  10  ins.,  and  radius 
of  wheel  should  not  be  less  than  40  ins.,  or  more  than  8 feet. 

Plane  of  stream  or  head  of  water  should  meet  periphery  of  wheel  at  an 
angle  of  from  240  to  30°.  Space  between  wheel  and  its  curb  should  not  ex- 
ceed .4  of  an  inch. 

Depth  of  shrouding  should  be  at  least  .25  depth  of  head  of  water,  or  such 
as  to  prevent  water  from  flowing  through  it  and  over  the  buckets,  and  width 
of  wheel  should  be  equal  to  that  of  stream  of  impinging  water. 

Effect  of  this  wheel  increases  with  depth  of  water  flow,  and,  therefore, 
other  elements  being  equal,  as  filling  of  buckets,  to  obtain  maximum  effect, 
water  should  flow  to  buckets  without  impact,  and  velocity  of  wheel  should 
be  only  a little  less  than  half  that  of  velocity  of  water  flowing  upon  wheel. 

To  Compute  Ti’oportions  of  a,  IPoncelet  'Wlieel. 

Note. — As  it  is  impracticable  to  arrive  at  the  results  by  a direct  formula,  they 
must  be  obtained  by  gradual  approximation. 

Example. — Height  of  fall  is  4.5  feet;  volume  of  water  40  cube  feet  per  second, 
radius  of  wheel  = 2 h,  or  9 feet ; depth  of  the  stream  = . 75  feel ; and  C assumed  at  .9. 

V representing  volume  of  water  in  cube  feet  per  second , h height  of  fall , d depth  of 

shrouding  = — . — — ]-  d' ; d'  opening  of  and  e width  of  sluice,  r radius  of  curva- 
4 2 g 

ture  of  buckets  ==  - ^ --- , and  a of  wheel , all  in  feet ; n number  of  revolutions  = 

J cos.  2’  J , J > ^ pa 

per  minute  ; c velocity  of  circumference  of  wheel  and  v velocity  of  water,  both  in  feet 
per  second  ; C coefficient  of  resistance  of  flow  of  water ; x angle  between  plane  of 

flowing  water  and  that  of  circumference  of  wheel  at  point  of  contact,  sin.  of  - — 

2 

Vcos.  z ; z angle  made  by  circumference  of  wheel  with  end  of  buckets  — 2 tang,  y ; 
and  y angle  of  direction  of  water  from  circumference  of  wheel  — 


Then  v — .9  yj 2 g (h  — — j = .9  x 16.29  = 14-66  ftet  velocity  of  wheel , being 


568 


HYDRODYNAMICS. 


14.66  — .66  . . 

less  than  half  velocity  of  water  ; c = = 7 Jeet  / 


4 


.25,  angle  corresponding  to  which  = 140  30';  n = 


to  which  = 140  30' ; n = 3°^~-  = 7-  43  revolutions ; 


— 7-  43  revolutions  ; 


sin.  of  700  34'  .'.  x — 1410  8'.  Effect  is  a maximum  when  c = .5  v cos.  y. 


back  from  perpendicular  line  wnicn  passes  mrougu 

wheel,  the  breast  should  then  incline  1 in  10,  or  1 in  15  towards 

sluice. 

After  passing  axis  of  wheel  in  tail-race,  curb  should  make  a 
'*  sudden  dip  of  6 ins. 


Number  of  buckets  1.6  D -f  1.6,  D = diameter  of  wheel  in  feet.  Shrouding  . 33  to 
,5  depth  of  head  of  water,  and  D 2 7t,  and  not  less  than  7 or  more  than  16  feet. 


Breast-wheel  is  designed  for  falls  of  water  varying  from  5 to  15  feet, 
and  for  flows  of  from  5 to  80  cube  feet  per  second.  It  is  constructed  with 
either  ordinary  buckets  or  with  blades  confined  by  a Curb. 

Enclosure  within  which  water  flows  to  a breast-wheel,  as  it  leaves  the  sluice 
is  termed  a Curb  or  Mantle. 

When  blades  are  enclosed  in  a curb , they  are  not  required  to  hold  water ; 
hence  thev  may  be  set  radial , and  they  should  be  numerous,  as  the.  loss  of 
water  escaping  between  the  wheel  and  the  curb  is  less  the  greater  their  num- 
ber ; and  that  they  may  not  lift  or  carry  up  water  with  them  from  tail-race, 
it  is  proper  to  give  them  such  a plane  that  it  may  leave  the  water  as  nearly 
vertical  as  may  be  practicable. 

Distance  between  two  buckets  or  blades  should  be  from  1.3  to  1.5  times 
head  over  gate  for  low  velocity  of  wheel  and  more  for  a high  velocity,  or 
equal  to  depth  of  shrouding,  or  at  from  10  to  15  ins. 

It  is  essential  that  there  should  be  air-holes  in  floor  of  buckets,  to  prevent 
air  from  impeding  flow  of  water  into  them,  as  the  -water  admitted  is  nearly 
as  deep  as  the  interval  between  them  ; and  velocity  of  wheel  should  be  such 
that  buckets  should  be  filled  to  .5  or  .625  of  their  volume. 

When  wheels  are  constructed  of  iron,  and  are  accurately  set  in  masonry, 
a clearance  of  .5  of  an  inch  is  sufficient. 


Fig.  2. 


Construction  of  Buckets  (Fig.  2).  ( Molesworth .) 


' / cle,  drawn  at  a’ distance’  of  s X 1.17  from  periphery  of  wheel,  is 
'jj  centre  from  which  bucket  is  struck  with  radius,  b a.  Radius  of 
^ wheel  should  not  be  less  than  7,  or  more  than  16  feet. 


From  point  of  bucket,  a,  draw  a line,  a 6,  at  an  angle  of  26° 
, with  radial  line,  point  6,  where  this  line  cuts  an  imaginary  cir- 


curb  should  lit  wheel  accurately  for  18  or  20  ins.,  measured 


'*  sudden  dip  of  6 ins. 

To  Compute  3?ower  of  a,  Poncelet  Wheel. 


V h .001 13  = IP,  and  880  = V.  V = velocity  of  theoretical  -periphery  — . 55.* 


33reast-wlieel. 


in  feet  per  second. 


HYDRODYNAMICS. 


569 


High  Breast-wheel  is  used  when  level  of  water  in  tail-race  and  penstock 
or  forebay  are  subject  to  variation  of  heights,  as  wheel  revolves  in  direction 
in  which  water  flows  from  blades,  and  back-ivater  is  therefore  less  disad- 
vantageous, added  to  which,  penstocks  can  be  so  constructed  as  to  admit  of 
an  adjustable  point  of  opening  for  the  water  to  flow  upon  the  wheel. 

Effect  of  this  wheel  is  equal  to  that  of  the  overshot,  and  in  some  instances, 
from  the  advantageous  manner  in  which  water  is  admitted  to  it,  it  is  greater 
when  both  wheels  have  same  general  proportions. 

Under  circumstances  of  a variable  supply  of  water,  Breast-wheel  is  better 
designed  for  effective  duty  than  Overshot , as  it  can  be  made  of  a greater 
diameter;  whereby  it  affords  an  increased  facility  for  reception  of  water 
into  its  buckets,  also  for  its  discharge  at  bottom ; and  further,  its  buckets 
more  easily  overcome  retardation  of  back-water,  enabling  it  to  be  worked 
for  a longer  period  in  back-water  consequent  upon  a flood. 

In  a well-constructed  wheel  an  efficiency  of  93  per  cent,  was  observed  by  M. 
Morin,  and  Sir  Wm.  Fairbairn  gives,  at  a velocity  of  circumference  of  wheel  of 
5 feet,  an  efficiency  of  75  per  cent.  Velocity  usually  adopted  by  him  was  from  4 
to  6 feet  per  second,  both  for  high  and  low  falls;  a minimum  of  3.5  feet  for  a fall  of 
40  and  a maximum  of  7 feet  for  a fall  of  5 to  6 feet. 


When  water  flows  at  from  io°  to  120  above  horizontal  centre  of  wheel,  Fairbairn 
gives  area  of  opening  of  buckets,  compared  with  their  volume,  as  8 to  24. 

The  capacity  between  two  buckets  or  blades  should  be  very  nearly  double  that  of 
volume  of  water  expended. 

To  Compute  [Proportions  and.  Effect  of  a Breast-wheel. 

Illustration. — Flow  of  water  is  15  cube  feet  per  second;  height  of  fall,  measured 
from  centre  of  pressure  of  opening  to  tail-race,  is -8. 5' feet;  velocity  of  circumference 
of  wheel  5 feet  per  second;  and  depth  of  buckets  or  blades  1 foot,  filled  to  .5  of  their 
volume. 

Width  of  wheel  -=  — ,d  representing  depth , and  v velocity  of  buckets  ; — -5  ■■  — 3; 

and  as  buckets  are  but  .5  filled,  3-=-  .5  ‘=±  6 feet.  Assume  water  is  to  flow  with  double 
velocity  of  circumference  of  wheel;  v = $x  2 = 10  feet;  and  fall  required  to  gen- 
erate this  velocity  = — X 1.1  = hf—  Xn  = 1.71  feet. 

2 g 64.33 

Deducting  this  height  from  total  fall,  there  remains  for  height  of  curb  or  shroud- 
ing, or  fall  during  which  weight  of  water  alone  acts,  h — h'  = 8.5  — ■ 1.71  =6.79  feet 

Making  radius  of  wheel  12  feet,  and  radius  of  bucket  circle  n feet,  whole  mechan- 
ical effect  of  flow  of  water  = 15  x 62. 5 x 8.5  = 7968.75  lbs.,  from  which  is  to  be  de- 
ducted from  10  to  15  per  cent,  for  loss  of  water  by  escape. 

Theoretical  effect,  as  determined  by  M.  Morin,  velocity  of  circumference  about 
.5  of  that  of  water,  and  within  velocities  of  1.66  to  6 feet. 

((y  cos.  a — v)  v \ 

(-  h J V 62.5.  a representing  angle  of  direction  of  velocity  with 

which  water  flows  to  wheel  at  centre  of  thread  of  flow  and  direction  of  velocity  of 
wheel  at  this  Line , and  h"  h — h'  in  feet. 

a is  here  assumed  at  200.  See  Weisbacli,  London,  1848,  vol.  ii.  page  197,  and  for 
the  necessarily  small  value  of  a,  its  cosine  may  be  taken  at  1.  Cos.  20°  = . 94. 

Then  ^ + 6.79^  X 15  X 62.5  = 7.474  X 15  X 62.5  = 7006.9  Z6s.,  which 

is  to  be  reduced  by  a coefficient  of  .77  for  a penstock  sluice,  and  .8  for  an  overfall 
sluice. 

Theoretical  effect,  as  determined  by  Weisbach,  7273  lbs.,  from  which  are 
to  be  deducted  losses,  which  he  computes  as  follows : 

Loss  by  escape  of  water  between  wheel  and  curb = 916 

Loss  by  escape  at  sides  of  wheel  and  curb — 

Friction  and  resistance  of  water  = 2. 5 per  cent — 160 

1256  lbs. 

3B* 


570 


HYDRODYNAMICS. 


Friction  of  wheel  as  per  formula,  page  571,  =Wr  nC  .0086; 
'16500  . 5X6o 


and  n = 5 ^ w --  = 4 revolutions. 

12  X 2 X 3-i4l6 

Then  16  500  X 2. 18  X 4 X .08  X .0086  = 98.99  lbs. 


.o4s^= 

C = .o8. 


.048^^  = 4-36  ins.; 
r = 4.  36  -r-  2 — 2. 

whence  7oo6-9  I256  + 9; 9 __  efficiency,  upon  assumption  of  losses  as  com- 

’ 7968-75 

puted  by  Weisbach. 

To  Compute  Power  of  a Breast-wheel. 

Rule. — Proceed  as  per  rule  for  an  overshot-wheel,  using  55000  and  .65 
with  a high  breast,  and  62  500  and  .6  for  a low  breast. 

Or,  High  breast,  .0612  V h = H>,  and  = V;  and  Low  breast  .0546  V h = 

EP,  andit|^  = V. 

..  I5X62.5X8.5X60 

Illustration.— Assume  elements  of  preceding  case.  Then 

— 14.49,  which  x -7  = 10. 14  horses. 


Or, 


7006.9-1256+102-6X60  = ja27  horses 


33000 


Openinqs  of  Buckets  or  Blades.-High  Breast,  .33  sq.  foot,  and  Lou,  Breast,  .2  sq 
foot  for  each  cube  foot  of  their  volume,  or  generally  6 to  8 in  opening  in  a high 
breast  and  9 to  12  in  a low  breast. 

Forms  of  Buckets.-Two  Part.  d=D,  s = .5  d,  l 1.25  d in  large  wheels,  and  =d 
in  wheels ‘less  than  25  feet  in  diameter. 

Three  Part  Buckets.— d divided  into  3 equal  parts ; Z = . 25  d,  d = D,  s = • 3?  d,l  — 
d in  large  wheels,  and  .75  d in  wheels  less  than  25  feet  in  diameter. 

Ventilating  Buckets  (. Fairbairn's ).  Spaces  are  about  1 inch  in  width. 

Notes  —A  Committee  of  the  Franklin  Institute  ascertained  that,  with  a high 
breast- wheel  20  feet  in  diameter,  water  admitted  under  a head  of  9 ins.,  and  at  17 
feet  above  bottom  of  wheel,  elbow-buckets  gave  a ratio  of  effect  to  power  of  .731  at 
a maximum,  and  radial-blades  .653.  With  water  admitted  at  a height  01  33  feet 
8 ins.,  elbow-buckets  gave  .658,  and  radial  blades  .628. 

At  10.96  feet  above  bottom  of  wheel,  with  a head  of  4.29  feet,  elbow-buckets  gave 
.544,  and  blades  .329. 

At  7 feet  above  bottom  of  wheel,  and  a head  of  2 feet,  a low  breast  gave  for 
elbow-buckets  .62,  and  for  blades  .531. 

At  3 feet  8 ins.  above  bottom  of  wheel,  and  a head  of  1 foot,  elbow-buckets  gave 
•555;  antl  b^des  .533. 

Cmrrent-wlieel. 

Current-wheel. — D.  K.  Clark  assigns  the  most  suitable  ratio  of  veloc- 
ity of  blades  to  that  of  current  as  40  per  cent. 

Depth  of  blades  should  be  from  .25  to  .2  of  radius;  it  should  not  be  less 
than  12  or  14  ins.  Diameter  is  usually  from  13  to  16.5  feet,  with  12  blades ; 
but  it  is  thought  that  there  might  be  an  advantage  in  applying  18  or  even 
24.  The  blades  should  be  completely  submerged  at  lower  side,  but  not  more 
than  2 ins.  under  water,  and  not  less  than  2 at  one  time. 

a s (u_s)2  — ip.  a representing  area  of  vertical  section  of  immersed  blades  in 
sq.  feet , s velocity  of  wheel  at  circumference,  and  v of  stream,  both  in  feet  per  second. 

Or  .38  — V 62. 5 = useful  effect.  Hence,  efficiency  = . 38. 

’ a g 


HYDRODYNAMICS. 


571 


Fritter- wheel. 

Flutter  or  Saw-mill  Wheel — Is  a small,  low  breast-wheel  operating  under 
a high  head  of  water ; the  design  of  its  construction,  water  being  plenty,  is 
the  attainment  of  a simple  application  to  high-speed  connections,  as  a gang 
or  circular  saw.  In  effect  it  is  from  .6  to  .7  that  of  an  overshot-wheel  of 
like  head  of  fall. 

Y $ 

— ( v — s)  — IP.  v and  s as  preceding. 

150 

Friction,  of  Journals  or  Gudgeons. 

A very  considerable  portion  of  mechanical  effect  of  a wheel  is  lost  in  ef- 
fect absorbed  by  friction  of  its  gudgeons. 

To  Compute  Friction,  of  Journals  or  Gudgeons  of  a 
W ater- wheel. 

WrnC  .008  6=f  W representing  weight  of  wheel  in  lbs.,  r radius  of  gudgeon  in 
ins.,  and  n number  of  revolutions  of  wheel  per  minute. 

For  well-turned  surfaces  and  good  bearings,  C = .c>75  with  oil  or  tallow;  when 
best  of  oil  is  well  supplied  = .054;  and,  as  in  ordinary  circumstances,  when  a black- 
lead  unguent  is  alone  applied  = .n. 

Illustration.— A wheel  weighing  25000  lbs.  has  gudgeons  6 ins.  in  diameter,  and 
makes  6 revolutions  per  minute;  what  is  loss  of  effect? 

Assume  C = . 08.  Then  25000  X - X 6 X .08  X .0086  = 309.6  lbs. 

'Weights. — Iron  wheels  of  18  to  20  feet  in  diameter  will  weigh  from  800  to 
1000  lbs.  per  IP 

Wood  wheels  of  30  feet  in  diameter,  2000  to  2500  lbs.  per  IP. 


To  Compute  Diameter  and  Journals  of*  a Shaft,  Stress 
laid,  uniformly  along  its  Length. 


l 

Cast  Iron, — = d.  Wood,  6.  ] 

9-6  . ^ 

lbs. , l length  of  shaft  between  journals  in  feet , and  d diameter  of  shaft  in  its  body 
in  ins. 


'/F- 


d.  W representing  weight  or  load  in 


Journals  or  Gudgeons. — Cast  Iron , .048  d. 


When  Shaft  has  to  resist  both  Lateral  and  Torsional  Stress. — Ascertain 
the  diameter  for  each  stress,  and  cube  root  of  sum  of  their  cubes  will  give 
diameter. 

To  Compute  Dimensions  of  Arms. 

Cast  Iron,  = w-  d representing  diameter  of  shaft,  and  w width  of  arm,  both 

• w 

in  ms. , n number  of  arms , — = t,  and  t thickness  of  arm. 

When  Arm  is  of  Oak,  w should  be  1.4  times  that  of  iron,  and  thickness  .7  that 
of  width. 

Memoranda. 


A volume  of  water  of  17.5  cube  feet  per  second,  with  a fall  of  25  ffeet,  applied  to  an 
undershot- wheel,  will  drive  a hammer  of  1500  lbs.  in  weight  from  100  to  120  blows 
per  minute,  with  a lift  of  from  1 to  1.5  feet.* 

A volume  of  water  of  21.5  cube  feet  per  second,  with  a fall  of  12.5  feet,  applied  to 
a wheel  having  a great  height  of  water  above  its  summit,  being  7.75  feet  in  diame- 
ter, will  drive  a hammer  of  500  lbs.  in  weight  100  blows  per  minute,  with  a lift  of  2 
feet  10  ins.  Estimate  of  power  31.5  horses. 


* Volume  of  water  required  for  a hammer  increases  in  a much  greater  ratio  than  velocity  to  be  given 
to  it,  it  being  nearly  as  cube  of  velocity. 


HYDRODYNAMICS. 


572 


A Stream  and  Overshot  Wheel  of  following  dimensions— viz.,  height  of  head  to 
centre  of  opening,  24.875  ins. ; opening,  1.75  t>y  8°  ins.  ; wheel,  22  feet  in  diameter 
by  8 feet  face;  52  buckets,  each  1 foot  in  depth,  making  3.5  revolutions  per  minute 
—drove  3 run  of  4.5  feet  stones  130  revolutions  per  minute,  with  all  attendant  ma- 
chinery, and  ground  and  dressed  25  bushels  of  wheat  per  hour. 

4.5  bushels  Southern  and  5 bushels  Northern  wheat  are  required  to  make  1 bar- 
rel of  flour. 

A Breast-wheel  and  Stream  of  following  dimensions— viz.,  head,  20  feet;  height 
of  water  upon  wheel,  16  feet;  opening,  18  feet  by  2 ins. ; diameter  of  wheel,  26  feet 
4 ins. ; face  of  wheel,  20  feet  9 ins. ; depth  of  buckets,  15.75  ins. ; number  of  buck- 
ets, 70;  revolutions,  4.5  per  minute  — drove  6144  self-acting  mule  spindles;  160 
looms,  weaving  printing-cloths  27  ins.  wide  of  No.  33  yarn  (33  hanks  to  a lb.),  and 
producing  24000  hanks  in  a day  of  n hours. 

Horizontal  "Wlieels. 

In  horizontal  water-wheels,  water  produces  its  effect  either  by  Impact , 
Pressure , or  Reaction , but  never  directly  by  its  weight. 

These  wheels  are  therefore  classed  as  Impact,  Pressure,  and  Reaction,  but 
are  now  designated  by  the  generic  term  of  Turbine. 

Turbines. 

Turbines,  being  operated  at  a higher  number  of  revolutions  than  Ver- 
tical Wheels,  are  more  generally  applicable  to  mechanical  purposes  ; but 
in  operations  requiring  low  velocities,  Vertical  Wheel  is  preferred. 

For  variable  resistances,  as  rolling-mills,  etc.,  Vertical  Wheel  is  far 
preferable,  as  its  mass  serves  to  regulate  motion  better  than  a small 
wheel. 

In  economy  of  construction  there  is  no  essential  difference  between 
a Vertical  Wheel  and  a Turbine.  When,  however,  fall  of  water  and 
volume  of  it  are  great,  the  Turbine  is  least  expensive.  Variations  in 
supply  of  water  affect  vertical  wheels  less  than  Turbines. 

Durability  of  a Turbine  is  less  than  that  of  a Vertical  Wheel;  and  it  is 
indispensable  to  its  operation  that  the  water  should  be  free  from  sand,  silt, 
branches,  leaves,  etc. 

With  Overshot  and  Breast  Wheels,  when  only  a small  quantity  of  'water  is 
available,  or  when  it  is  required  or  becomes  necessary  to  produce  only  a por- 
tion of  the  power  of  the  fall,  their  efficiency  is  relatively  increased,  from  the 
blades  being  but  proportionately  filled;  but  with  Turbines  the  effect  is  con- 
trary, as  when  the  sluice  is  lowered  or  supply  decreased  water  enters  the 
wheel  under  circumstances  involving  greater  loss  of  effect.  To  produce 
maximum  effect  of  a stream  of  water  upon  a wheel,  it  must  flow  without  im- 
pact upon  it,  and  leave  it  without  velocity;  and  distance  between. point  at 
which  the  water  flows  upon  a wheel  and  level  of  water  in  reservoir  should 
be  as  short  as  practicable. 

Small  wheels  give  less  effect  than  large,  in  consequence  of  their  making  a 
greater  number  of  revolutions  and  having  a smaller  water  arc. 

In  Ilwh-pressure  Turbines  reservoir  (of  wheel)  is  enclosed  at  top,  and  water 
is  admitted  through  a pipe  at  its  side.  In  Low-pressure,  water  flows  into  res- 
ervoir, which  is  open. 

In  Turbines  working  under  water,  height  is  measured  from  surface  of 
water  in  supply  to  surface  of  discharged  water  or  race ; and  when  they  work 
in  air,  height  is  measured  from  surface  in  supply  to  centre  of  wheel. 

In  order  to  obtain  maximum  effect  from  water,  velocity  of  it,  whep.  lead- 
ing a Turbine,  should  be  the  least  practicable. 


HYDRODYNAMICS. 


573 


Efficiency  is  greater  when  sluice  or  supply  is  wide  open,  and  it  is  less,  af- 
fected by  head  than  by  variations  in  supply  of  water.  It  varies  but  little 
with  velocity,  as  it  was  ascertained  by  experiment  that  when  35  revolutions 
gave  an  effect  of  .64,  55  gave  but  .66. 

When  Turbines  operate  under  water,  the  flow  is  always  full  through  them  ; 
hence  they  become  Reaction-wheels , which  are  the  most  efficient. 

Experiments  of  Morin  gave  efficiency  of  Turbines  as  high  as  .75  of  power. 

Angle  of  plane  of  water  entering  a Turbine,  with  inner  periphery  of  it, 
should  be  greater  than  90°,  and  angle  which  plane  of  water  leaving  reservoir 
makes  with  inner  circumference  of  Turbine  should  be  less  than  90°. 

When  Turbines  are  constructed  without  a guide  curve *,  angle  of  plane  of 
flowing  water  and  inner  circumference  of  wheel  = 90°. 

Great  curvature  involves  greater  resistance  to  efflux  of  water ; and  hence 
it  is  advisable  to  make  angle  of  plane  of  entering  water  rather  obtuse  than 
acute,  say  ioo° ; angle  of  plane  of  water  leaving,  then,  should  be  50°,  if  in- 
ternal pressure  is  to  balance  the  external ; and  if  wheel  operates  free  of 
water,  it  may  be  reduced  to  25°  and  30°. 

If  blades  are  given  increased  length,  and  formed  to  such  a hollow  curve 
that  the  water  leaves  wheel  in  nearly  a horizontal  direction,  water  then  both 
impinges  on  blades  and  exerts  a pressure  upon  them ; therefore  effect  is 
greater  than  with  an  impact-wheel  alone. 

Turbines  are  of  three  descriptions : Outward,  Downward,  and  Inward  flow. 

Outward-flow  Turbines. 

Fourneyron  Turbine,  as  recently  constructed,  may  be  considered  as  one 
of  the  most  perfect  of  horizontal  wheels;  it  operates  both  in  and  out  of 
back-water,  is  applicable  to  high  or  low  falls,  and  is  either  a high  or  low 
pressure  turbine. 

In  high-pressure,  the  reservoir  is  closed  at  top  and  the  water  is  led  to  it 
through  a pipe.  In  low-pressure,  the  water  flows  directly  into  an  open  res- 
ervoir. Pressure  upon  the  step  is  confined  to  weight  of  wheel  alone. 

Fourneyron  makes  angle  of  plane  of  water  entering  =go°,  and  angle  of 
plane  of  water  leaving  = 30°. 

Efficiency  is  reduced  in  proportion  as  sluice  is  lowered,  for  action  of  water 
on  wheel  is  less  favorably  exerted.  M.  Morin  tested  a Fourneyron  turbine 
6.56  feet  in  diameter,  and  he  found  that  efficiency  varied  from  a minimum 
of  24,  to  79  per  cent.,  when  supply  of  water  was  reduced  to  .25  of  full  supply. 
In  practice,  radial  length  of  blades  of  wheel  is  .25  of  radius,  for  falls  not  ex- 
ceeding 6.5  feet,  .3  for  falls  of  from  6.5  to  19  feet,  and  .66  for  higher  falls. 

To  Compute  Elements  and  Results. 

High  Pressure,  6.6  y/h  — v;  -=rA;  V'1'77 V—  Df;  12.6  ^ = and 

v y/h  ' h ’ 

,079  V h = IP.  h representing  head  of  water , v velocity  of  turbine  at  periphery  per 
minute , and  D internal  diameter  of  turbine,  all  in  feet,  V volume  of  water  in  cube  feet 
per  second , A sum  of  area  of  orifices  in  sq.  feet,  and  IP  effective  horse-power. 

1.2  D = external  diameter  of  turbine  in  feet,  when  it  is  more  than  6 feet,  and  1.4 
when  it  is  less  than  6 feet.  Number  of  guides  = number  of  blades  t when  less  than 
24,  and  number -i-  3 when  greater  than  24.  Area  of  section  of  supply-pipe  = .4  V. 

For  construction  of  blades  and  guides,  see  Molesworth,  London,  1882,  page  540. 


* Guide  curves  are  plates  upon  centre  body  of  a Turbine,  which  give  direction  to  flowing  water, 
or  to  blades  of  wheel  wnich  surround  them, 
t In  extreme  cases  of  very  high  falls  diameter  given  by  this  formula  may  be  increased. 

$ Fourneyron’s  rule  for  the  number  of  blades  is  constant  number  36,  irrespective  of  size  of  turbine. 


574 


HYDRODY N AM  ICS. 


Operation  of*  Higli-Freasvire  Turbines. 

I 120 


h 

I 3° 

I 4°  1-  5° 

60 

1 70  | 

1 80 

V 

4.2 

3- 1 2-5  1 

2. 1 

1.8 

1.6 

V 

[36 

1 42  1 47 

[51 

1 55 

159 

1 x4°  1 

0 

00 

0 

VO 

•9! 

00 

1 78 

1 84  | 89  | 

94 


— hp.a.a  oj  waier  in  jtti,  v uvluiuo  m/u.c,c ■>  v 

and  v velocity  of  periphery  of  turbine  in  feet  per  second. 


Boydkn  Turbine.  — Mr. Boyden,  of  Massachusetts,  designed  an 

outward-flow  turbine  of  75  BP,  which  realized  an  efficiency  of  88  per  cent. 
Peculiar  features,  as  compared  with  a Fourneyron  turbine,  are,  1st,  and  most 
important,  the  conduction  of  the  water  to  turbine  through  a vertical  tiun- 
cated  cone,  concentric  with  the  shaft.  The  water,  as  it  descends,  acquires  a 
gradually  increasing  velocity,  together  with  a spiral  movement  in  direction 
of  motion  of  wheel.  The  spiral  movement  is,  in  fact,  a continuation  of  the 
motion  of  the  water  as  it  enters  cone.— 2d.  Guide-plates  at  base  are  inclined, 
so  as  to  meet  tangentially  the  approaching  water— 3d.  A “ diffuser  ” or  annu- 
lar chamber  surrounding  wheel,  into  which  water  Irom  wheel  is  discharged. 
This  chamber  expands  outwardly,  and,  thus  escaping  velocity  of  water,  is 
eased  off  and  reduced  to  a fourth  when  outside  of  diffuser  is  reached.  Effect 
of  diffuser  is  to  accelerate  velocity  of  water  through  machine ; and  gain  of 
efficiency  is  3 per  cent.  Diffuser  must  be  entirely  submerged.  (D.  K.  Clark.) 

Poncelet  Turbine.— This  wheel  is  alike  to  one  of  his  under  shot- wheels 
set  horizontally,  and  it  is  the  most  simple  of  all  horizontal  wheels. 


To  Compute  IClerneiits  of  CTeiieral  IProportion  and. 
Ttesnlts.  (Lt.  F.  A.  Mahan , TJ.  S.  A.) 

5D2V/i=t;  .iD  = H;  4.49^  = **, 
d 


.0425  D2  h jh  — BP;  4-85  sj  ifjl  — 13 ; 


D 4 D 

3 (I)  -p  10)  = N ; ~ ™ ; — 


= W;  D- 


V' 


and  C coefficient  for  V'  in  terms  of  V = — . 


^ = .5  N to  .75  N = n ; - =W; 

f 71 

D and  d representing  exterior  and  in- 


terior diameters  of  wheel,  H and  h heights  of  orifices  of  discharge  at  outer  circum- 
ference and  of  fall  acting  on  wheel , w and  w'  shortest  distances  between  two  adjacent 
blades  and  two  adjacent  guides , all  in  feet,  V,  V',  and  v velocities  due  to  fall  of  watei 
passing  through  narrowest  section  of  wheel,  and  of  interior  circumference  of  wheel, 
all  in  feet  per  second , N and  n numbers  of  blades  and  guides,  and  BP  actual  horse- 
power. 

For  falls  of  from  5 feet  to  40,  and  diameters  not  less  than  2 feet,  n w should  be 
equal  to  diameter  of  wheel.  H equal  to.iD  ,nw'  = d,  and  4 w = width  of  crown. 
For  falls  exceeding  this,  H should  be  smaller,  in  proportion  to  diameter  ot  wheel. 

Downward-flow  Turbines. 

In  turbines  with  downward  flow,  wheel  is  placed  below  an  annular  series 
of  guide-blades,  by  which  water  is  conducted  to  wheel,  llie  water  strikes 
curved  blades,  and  falls  vertically,  or  nearly  so,  into  tail-race;  consequently, 
centrifugal  action  is  avoided,  and  downward  flow  is  more  compact. 

Fontaine  Turbine  vields  an  efficiency  of  70  per  cent.,  when  fully 
charged.  When  supply  of  water  is  shut  off  to  .75,  by  sluice,  efficiency  is 
cn  per  cent.  Best  velocity  at  mean  circumference  of  wheel  is  equal  to  55 
per  cent,  of  that  due  to  height  of  fall.  It  may  vary  .25  of  this  either  way, 
without  materially  affecting  efficiency. 

In  operation  the  water  in  race  is  in  immediate  contact  with  wheel,  and  its 
efficiencv  is  greatest  when  sluice  is  fully  opened.  Its  efficiency,  also,  is  less 
affected  by  variations  of  head  of  flow  than  in  volume  of  water  supplied; 
hence  they  are  adapted  for  Tide-mills . 


HYDRODYNAMICS. 


575 


Jonval  Turbine. — This  wheel  is  essentially  alike  in  its  principal  propor- 
tions to  Fontaine’s,  and  in  principle  of  operation  it  is  the  same.  Water  in 
race  must  be  at  a certain  depth  below  wheel. 

For  convenience,  it  is  placed  at  some  height  above  level  of  tail-race,  within 
an  air-tight  cylinder,  or  “ draft-tube,”  so  that  a partial  vacuum  or  reduction 
of  pressure  is  induced  under  wheel,  and  effect  of  wheel  is  by  so  much  in- 
creased. Resulting  efficiency  is  same  as  if  wheel  was  placed  at  level  of  tail- 
race  ; and  thus,  while  it  may  be  placed  at  any  level,  advantage  is  taken  of 
whole  height  of  fall,  and  its  efficiency  decreases  as  volume  of  water  is  di- 
minished or  as  sluice  is  contracted. 

To  Coinpixte  Elements  ancL  Ftesnlts. 

Low  Pressure. — For  falls  of  30  feet  and  less. 

V 1/ 1.77  V IP 

6 y/h  = v ; = — = A;  — --- — = D*;  12.7  — r:Y;  and  .079  V h = IP. 

h representing  head  of  water,  v velocity  of  turbine  at  periphery  per  minute , and  D 
internal  diameter  of  turbine,  all  in  feet,  V volume  of  water  in  cube  feet  per  second , 
A sum  of  area  of  orifices  in  sq.  feet,  and  IP  effective  horse-power. 

1.2  D = external  diameter  of  turbine  in  feet,  when  it  is  more  than  6 feet,  and  1.4 
when  it  is  less  than  6 feet.  Number  of  guides  = number  of  bladesf  when  less  than 
24,  and  number  -4-  3 when  greater  than  24.  Area  of  section  of  supply-pipe  = .4  V. 

For  construction  of  blades  and  guides,  see  Mples worth,  London,  1882.  page  540. 


Low-Pressure  Turbines.  ( Molesworth .) 


•0 

5 BP 

10  IP 

15  BP 

20 

BP 

30  BP 

40  IP 

50  IP 

V 

V 

R 

V 

R 

V 

R 

Y 

R 

V 

R 

V 

R 

V 

R 

2.5 

9.48 

25 

34 

50 

24 

75 

20 

100 

*7 

5 

13-38 

12.5 

81 

25 

57 

38 

47 

5o 

41 

75 

33 

100 

28 

126 

26 

7-5 

16.38 

8.5 

136 

17 

97 

25 

79 

33 

68 

5i 

56 

68 

48 

85 

43 

10 

18.96 

6-3 

180 

12.6 

128 

19  , 

105 

25 

90 

38 

75 

5o 

64 

63 

58 

15 

23.22 

4.2 

3i9 

8.4 

226 

12.6 

185 

W 

160 

25 

!3i 

33 

113 

42 

100 

20 

26.82 

— 

— 

6-3 

329 

9-3 

273 

12.6 

232 

18.9 

194 

25 

164 

31 

148 

25 

39 

— 

— 

— 

— 

7-5 

358 

10 

310 

i5 

253 

20 

220 

25 

196 

30 

32.88 

8.4 

380 

12.6 

310 

17 

268 

21 

240 

v representing  velocity  of  centre  of  blades  in  feet  and  V volume  of  water,  in  cube 
feet,  both  per  second , R revolutions  per  minute , and  IP  effective  horse-power. 

Vertical  Shaft.  3/— ' — diameter  of  shaft  in  ins. 


Inward-flow  Tiarbiiie. 

Inward-flow  Turbine.  — Inward-flow  or  vortex  wheel  is  made  with 
radiating  blades,  and  is  surrounded  by  an  annular  case,  closed  externally, 
and  open  internally  to  wheel,  having  its  inner  circumference  fitted  with  four 
curved  guide-passages.  The  water  is  admitted  by  one  or  more  pipes  to  the 
case,  and  it  issues  centripetally  through  the  guide-passages  upon  circum- 
ference of  wheel.  The  water  acting  against  the  curved  blades,  wheel  is 
driven  at  a velocity  dependent  on  height  of  fall,  and  water  having  expended 
its  force,  passes  out  at  centre.  This  wheel  has  realized  an  efficiency  as  high 
as  77.5  per  cent.  It  was  originally  designed  by  Prof.  James  Thomson. 

Swain  Turbine. — Combines  an  inward  and  a downward  discharge.  Re- 
ceiving edges  of  buckets  of  wheel  are  vertical  opposite  guide-blades,  and 
lower  portions  of  the  edges  are  bent  into  form  of  a quadrant.  Each  bucket 
thus  forms,  with  the  surface  of  adjoining  bucket,  an  outlet  which  combines 
an  inward  and  a downward  discharge.  One,  72  ins.  in  diameter,  was  tested 


* In  extreme  cases  of  very  High  falls  diameter  given  by  this  formula  may  be  increased, 
t Fourneyron’s  rule  for  the  number  of  blades  is  constant  number  36,  irrespective  of  size  of  turbine. 


HYDRODYNAMICS. 


576 

bv  Mr  J.  B.  Francis,  for  several  heights  of  gate  or  sluice,  from  2 to  13.08 
ins.,  and  circumferential  velocities  of  wheel  ranging  from  60  to  80  per  cent, 
of  respective  velocities  due  to  heads  acting  on  wheel. 

For  a velocity  of  60  per  cent.,  and  for  heights  of  gate  varying  within  limits  al- 
ready stated,  efficiency  ranged  from  47.5  to  76.5  per  cent.,  and  for  a velocity  of  80 
r>er  cent  it  ranged  from  37.5  to  83  per  cent.  Maximum  efficiency  attained  was  84 
per  cent.,  with  a 12-inch  gate  and  a velocity-ratio  of  76  per  cent. ; but  from  9- inch 
to  iq-inch  gate,  or  from  .66  gate  to  full  gate,  maximum  efficiency  varied  within 
very  narrow  limits— from  83  to  84  per  cent.,— velocity-ratios  being  72  per  cent,  for 
o-inch  gate,  and  76.5  per  cent,  for  full  gate.  At  half-gate,  maximum  efficiency  was 
78  per  cent.,  when  velocity-ratio  was  68  per  cent.  At  quarter-gate,  maximum  effi- 
ciency was  61  per  cent.,  and  velocity-ratio  66  per  cent. 

Tremor t Turbine,  as  observed  by  Mr.  Francis,  in  his  experiments  at 
Lowell,  Mass.,  gave  a ratio  of  effect  to^power  as  .793  to  1. 

Victor  Turbine  is  alleged  to  have  given  an  effect  of  .88  per  cent,  under 
a head  of  18.34  feet,  with  a discharge  of  977  cube  feet  of  water  per  minute, 
and  with  343.5  revolutions. 

Tangential  Wlieel. 

Wheels  to  which  water  is  applied  at  a portion  only  of  the  circumference 
are  termed  tangential.  They  are  suited  for  very  high  falls,  -where  diameter 
and  high  tangential  velocity  mav  be  combined  with  moderate  revolutions. 
The  Girard  turbine  belongs  to  this  class.  It  is  employed  at  Goeschenen 
station  for  St.  Gothard  tunnel , it  operates  under  a head  of  279  feet.  _ The 
wheels  are  7 feet  10.5  ins.  in  diam.,  having  80  blades,  and  their  speed  is  160 
revolutions  per  minute,  with  a maximum  charge  of  water  of  67  gallons  per 
second.  An  efficiency  of  87  per  cent,  is  claimed  for  them  at  the  I aris 
water-works ; ordinarily  it  is  from  75  to  80  per  cent.  (D.  A.  Clark.) 

Impact  and.  Reaction  "Wlieel. 

Impact-wheel. — Impact  Turbine  is  most  simple  but  least  efficient  form 
of  impact-wheel.  It  consists  of  a series  of  rectangular  buckets  or  blades, 
set  upon  a wheel  at  an  angle  of  50°  to  70°  to  horizon;  the  water  flows  to 
blades  through  a pyramidal  trough  set  at  an  angle  of  20°  to  40  , so  that 
the  water  impinges  nearly  at  right  angles  to  blades.  Effect  is  .5  entire  me- 
chanical effect,  which  is  increased  by  enclosing  blades  in  a border  or  frame. 

If  buckets  are  given  increased  length,  and  formed  to  such  a hollow  curve 
that  the  water  leaves  wheel  in  nearly  a horizontal  direction,  the  water  then 
impinges  on  buckets  and  exerts  a pressure  upon  them ; effect  therefore  is 
greater  than  with  the  force  of  impact  alone. 

By  deductions  of  Weisbach  it  appears  that  effect  of  impact  is  only  half 
available  effect  under  most  favorable  circumstances. 

Reaction-wheel.— Reaction  of  water  issuing  from  an  orifice  of  less 
capacity  than  section  of  vessel  of  supply,  is  equal  to  weight  of  a column  of 
water , basis  of  which  is  area  of  orifice  or  of  stream,  and  height  of  which,  is 
twice  height  due  to  velocity  of  water  discharged. 

Hence  the  expression  is  2.  — - a u>  = R.  w representing  weight  of  a cube  foot  of 
2 g 

water  in  tbs.,  and  a area  of  opening  in  sq.feet. 

Wiiitelaw’s  is  a modification  of  Barker’s;  the  arms  taper  from  centre 
towards  circumference  and  are  curved  in  such  a manner  as  to  enable  the 
water  to  pass  from  central  openings  to  orifices  in  a line  nearly  right  and 
radial,  when  instrument  is  operating  at  a proper  velocity ; in  order  that  very 
little  centrifugal  force  may  be  imparted  to  the  water  by  the  revolution  ot 
the  arms,  and  consequently  a minimum  of  frictional  resistance  is  opposed 
to  course  of  the  water.  , 


HYDRODYNAMICS. 


577 

A Turbine  9.55  feet  in  diameter,  with  orifices  4.944  ins.  in  diameter,  oper- 
ated by  a fall  of  25  feet,  gave  an  efficiency  of  75  per  cent.,  including  friction 
of  gearing  of  an  inclined  plane. 

When  a reaction  wheel  is  loaded,  so  that  height  due  to  velocity,  corresponding  to 
velocity  of  rotation  v,  is  equal  to  fall,  or  — 7t,  or  v = f2  gh,  there  is  a loss  of  17 

v 2 

per  cent,  of  available  effect;  and  when  — =12/1,  there  is  a loss  of  but  10  per  cent. ; 
v2 

and  when  — = 4 h,  there  is  a loss  of  but  6 per  cent.  Consequently,  for  moderate 

falls,  and  when  a velocity  of  rotation  exceeding  velocity  due  to  height  of  fall  may 
be  adopted,  this  wheel  works  very  effectively. 

Efficiency  of  wheel  is  but  one  half  that  of  an  undershot-wheel. 

When  sluice  is  lowered,  so  that  only  a portion  of  wheel  is  opened,  efficiency 
of  a Reaction-wheel  is  less  than  that  of  a Pressure  Turbine. 

Ratio  of  Effect  to  Power  of  several  Turbines  is  as  follows : 

Poncelet 65  to  .75  to  1 I Jonval 6 to  7 to  1 

Fourneyron 6 to  .75  to  1 | Fontaine .6  to  .7  to  1 

Barker’s  Mill. — Effect  of  this  mill  is  considerably  greater  than  that 
which  same  quantity  of  water  would  produce  if  applied  to  an  undershot- 
wheel,  but  less  than  that  which  it  would  produce  if  properly  applied  to  an 
overshot-wheel. 

For  a description  of  it,  see  Grier’s  Mechanics’  Calculator , page  234;  and  for  its 
formulas,  see  London  Artisan , 1845,  page  229. 

IMPULSE  AND  RESISTANCE  OF  FLUIDS. 

Impulse  and  Resistance  of  "Water. — Water  or  any  other  fluid, 
when  flowing  against  a body,  imparts  a force  to  it  by  which  its  condition  of 
motion  is  altered.  Resistance  which  a fluid  opposes  to  motion  of  a body 
does  not  essentially  differ  from  Impulse. 

Impulse  of  one  and  same  mass  of  fluid  under  otherwise  similar  circum- 
stances is  proportional  to  relative  velocities  c- p r of  fluid. 

For  an  equal  transverse  section  of  a stream,  the  impulse  against  a surface 
at  rest  increases  as  square  of  velocity  of  water. 

Impulse  against  Plane  Surfaces. — The  impulse  of  a stream  of  water  de- 
pends principally  upon  angle  under  which,  after  impulse,  it  leaves  the  water ; 
it  is  nothing  if  the  angle  is  o,  and  a maximum  if  it  is  deflected  back  in  a 

line  parallel  to  that  of  its  flow,  or  180°,  2 ~ V ic  — p* 

9 

When  Surface  of  Resistance  is  a Plane , and  = 90°,  then  S-  V w = P,  and 
for  a surface  at  rest , 2 a h w = P.  a representing  area  of  opening  in  sq.feet . 

. ^~2  vj.  if  an<^  v representing  velocities  of  water  and  of  surf  ace  upon  which  it 
impinges  in  feet  per  second , w weight  of  fluid  per  cube  foot  in  lbs. , A transverse  section 
of  stream,  in  sq.  ins.,  and  cip  v relative  moiions  of  waiter  and  surface. 

.Normal  impulse  of  water  against  a plane  surface  is  equivalent  to  weight 
of  a column  which  has  for  its  base  transverse  section  of  stream,  and  for 

altitude  twice  height  due  to  its  velocity,  2 li  ==  2 — . 

2 9 

Resistance  of  a fluid  to  a body  in  motion  is  same  as  impulse  of  a fluid 
moving  with  same  velocity  against  a body  at  rest. 


Weisbach,  New  York,  1870,  vol.  i.  page  1008. 

3C 


HYDRODYNAMICS. 


578 

Maximum  Effect  of  Impulse.  — Effect  of  impulse  depends  principally  on 
velocity  v of  impinged  surface.  It  is,  for  example,  o,  both  when  v = c and 
v — o ; hence  there  is  a velocity  for  which  effect  of  impulse  is  a maximum 

= (c  — v)  v\  that  &,  v=  ^ , and  maximum  effect  of  impulse  of  water  is  ob- 
tained when  surface  impinged  moves  from  it  with  half  velocity  of  water. 

Illustration.— A stream  of  water  having  a transverse  section  of  40  sq.  ins.,  dis- 
charges 5 cube  feet  per  second  against  a plane  surface,  and  flows  off  with  a velocity 

of  12  feet  per  second;  effect  of  its  impulse,  then, is  Vw,==p;  c— 5 *J44==i8; 


g = 32-16;  10  = 62.5; 


18- 


32.16 


x 5 X 62.5  = 58.28  lbs. 


Hence  mechanical  effect  upon  surface  = P v = 58.28  X 12  = 699.36  lbs. 
Maximum  effect  would  be  v = - = 


X 5 X~—=gfeet,  and  - 


40 


X^-X  5X62.5 
20 


— - X 5.036  X 312.5  = 786.87  lbs.;  and  hydraulic  pressure  _ = 87. 44  lbs. 

2 9 


When  Surface  is  a Plane  and  at  an  Angle,  then  (1  — cos.  a)  — 


V«?  = P. 


Illustration. — A stream  of  water,  having  a transverse  section  of  64  sq.  ins.,  dis- 
charges 17.778  cube  feet  per  second  against  a fixed  cone,  having  an  angle  of  con- 
vergence from  flow  of  stream  of  500,  hydraulic  pressure  in  direction  of  stream ; 


40;  cos.  50°=  .642  79.  (1  — .642  79)  32  i£  X 17- 77s  X 62.5  = 


the„  c = 

64—I44 

.357  21  X 1382.2  = 494.26  lbs. 

When  Surf  ace  of  Resistance  is  a Plane  at  go°,  and  has  Borders  added  to 
its  Perimeter , effect  will  be  greater,  depending  upon  height  of  border  and 
ratio  of  transverse  section  between  stream  and  part  confined. 

Oblique  Impulse  — In  oblique  impulse  against  a plane,  the  stream  may  flow 
in  one,  two,  or  in  all  directions  over  plane. 

When  Stream  is  confined  at  Three  Sides , (1  cos.  a)  ~ V w = V. 

When  Stream  is  confined  at  Two  Sides , C—~  sin.  a2  V w — P. 

Normal  impulse  of  a stream  increases  as  sine  of  angle  of  incidence ; par- 
allel impulse  as  square  of  sine  of  angle ; and  lateral  impulse  as  double  the 
angle. 

When  an  Inclined  Surface  is  not  Bordered,  then  stream  can  spread  over 
it  in  all  directions,  and  impulse  is  greater,  because  of  all  the  angles  by 
which  the  water  is  deflected,  a is  least ; hence  each  particle  that  does  not 
move  in  normal  plane  exerts  a greater  pressure  than  particle  in  that  plane, 


, 2 sin.  a2  c — v 
and  1 -|- sinTa5  X ~g~ 


v«?  = p. 


Impulse  and  Resistance  against  Surfaces. 

Coefficient  of  resistance,  C,  or  number  with  which  height  due  to  velocity  is  to  be 
multiplied,  to  obtain  height  of  a column  of  water  measuring  this  hydraulic  press- 
ure, varies  for  bodies  of  different  figures,  and  only  for  surfaces  which  are  at  right 
angles  to  direction  of  motion  is  it  nearly  a definite  quantity. 

According  to  experiments  of  Du  Buat  and  Thibault,  0 = 1.85  for  impulse  of  air 
or  water  against  a plane  surface  at  rest,  and  for  resistance  of  air  or  water  against  a 
surface  in  motion,  C = 1.4.  In  each  case  about  .66  of  effect  is  expended  upon  front 
surface,  and  .34  upon  rear. 


hydrodynamics. 


579 


Comparison  between  Xu-iToines  and.  otlier*  "Water- wheels. 

Turbines  are  applicable  to  falls  of  water  at  any  height,  from  i to  500  feet. 

Their  efficiency  for  very  high  falls  is  less  than  for  smaller,  in  consequence 
of  the  hydraulic  resistances  involved,  and  which  increase  as  the  square  of 
the  velocity  of  the  water.  They  can  only  be  operated  in  clear  water. 

With  Fourneyron’s,  the  stress  and  pressure  on  the  step  is  that  of  the  wheel 
in  motion ; with  Fontaine’s,  the  whole  weight  of  the  water  is  added  to  that 
of  the  wheel;  they  are  well  adapted,  however,  for  tide-mills.  Experiments 
on  Jouval’s  gave  equal  results  with  Fontaine’s. 

Vertical  Water-wheels  are  limited  in  their  application  to  falls  under  60 
feet  in  height. 

For  falls  of  from  40  to  20  feet  they  give  a greater  effect  than  any  turbine ; 
for  falls  of  from  20  to  10  feet,  they  are  equal  to  them ; and  for  very  low 
falls,  they  have  much  less  efficiency. 

Variations  in  the  supply  of  water  effect  them  less  than  turbines. 

Water-pressure  Engine. 

By  experiments  of  M.  Jordan,  he  ascertained  that  a mean  useful  effect  of 
.84  was  attainable. 

Weisbach,  London,  1848,  vol.  ii.  page  349. 


PERCUSSION  OF  FLUIDS. 

When  a stream  strikes  a plane  perpendicular  to  its  action,  force  with 
-which  it  strikes  is  estimated  by  product  of  area  of  plane,  density  of  fluid, 
and  square  of  its  velocity. 

Or,  A d v2  = P.  A representing  area  in  sq.  feet,  d weight  of  fluid  in  lbs.,  and  v 
velocity  in  feet  per  seeond. 

If  plane  is  itself  in  motion,  then  force  becomes  Ad  (v  — v')2  = P.  v'  representing 
velocity  of  plane. 

If  C represent  a coefficient  to  be  determined  by  experiment,  and  h height 
due  to  velocity  v , then  v2  = 2 g k,  and  expression  for  force  becomes 

A C 2 g h = P. 

CENTRIFUGAL  PUMPS.  (D.  K.  Clark.) 

Appold  Pump,  made  with  curved  receding  blades,  is  the  form  of 
centrifugal  pump  most  widely  known  and  accepted.  M.  Morin  tested  three 
kinds  of  centrifugal  or  revolving  pumps : 

1st,  on  model  of  Appold  pump ; 2d,  one  having  straight  receding  blades 
inclined  at  an  angle  of  45 0 with  the  radius,  and  3d,  one  having  radial  blades. 
They  were  12  ins.  in  diameter  and  3.125  ins.  in  length,  and  had  central  open- 
ings of  6 ins.  Their  efficiencies  were  as  follows : 

1.  Curved  blades. . 48  to  68  per  cent.  | 2.  Inclined  blades. . 40  to  43  per  cent. 

3.  Radial  blades 24  per  cent. 

Height  to  which  water  ascends  in  a pipe,  by  action  of  a centrifugal  pump, 
would,  if  there  were  no  other  resistances,  be  that  due  to  velocity  of  circum- 
v2 

ference  of  revolving  wheel,  or  to  — . Results  of  experiments  made  by  the 

author  on  two  pumps,  in  1862,  yielded  following  data,  showing  height  to 
which  water  was  raised,  without  any  discharge : 


Gwyknk’s  Pump 
(blades  partly  radial, 
curved  at  ends). 


Diameter  of  pump- wheel 4 feet. 

Revolutions  per  minute 177 

Velocity  of  circumference  per  second. . . 37.05  feet. 

Head  due  to  the  velocity 21.45  “ 

Actual  head 18.21  “ 

Do.  do.  in  parts  of  head  due  to  velocity,  85  per  cent. 


Appold  Pump 
(blades,  curved). 


7 ins. 


580  HYDRODYNAMICS. IMPACT  OR  COLLISION. 

Mr.  David  Thomson  made  similar  experiments  with  Appold  pumps  of  from  1.25 
to  1. 71  feet  in  diameter,  the  results  of  which  showed  that  the  actual  head  was  about 
90  per  cent,  of  the  head  due  to  the  velocity. 

M.  Tresca,  in  1861,  tested  two  centrifugal  pumps,  18  ins.  in  diameter,  with  a cen- 
tral opening  of  9 ins.  at  each  side.  The  blades  were  six  in  number,  of  which  three 
sprung  from  centre,  where  they  were  .5  inch  thick;  the  alternate  three  only  sprung 
at  a distance  equal  to  radius  of  opening  from  centre.  They  were  radial,  except  at 
ends,  where  they  were  curved  backward,  to  a radius  of  about  2.25  ins. ; and  they 
joined  the  circumference  nearly  at  a tangent.  Width  of  blades  was  taper,  and  they 
were  5.75  ins.  wide  at  nave,  and  only  2.625  ins-  at  ends:  so  designed  that  section  of 
outflowing  water  should  be  nearly  constant. 

M.  Tresca  deduced  from  his  experiments  that,  in  making  from  630  to  700  revolu- 
tions per  minute,  efficiency  of  the  pump,  or  actual  duty  in  raising  water,  through  a 
height  of  31.16  feet,  amounted  to  from  34  to  54  per  cent,  of  work  applied  to  shaft; 
or  that,  in  the  conditions  of  the  experiment,  the  pump  could  raise  upward  of  16200 
cube  feet  of  water  per  hour,  through  a height  of  33  feet,  with  about  30  IP  applied 
to  shaft,  and  an  efficiency  of  45  per  cent. 

According  to  Mr.  Thomson,  maximum  duty  of  a centrifugal  pump  worked  by  a 
steam-engine  varies  from  55  per  cent,  for  smaller  pumps  to  70  per  cent,  for  larger 
pumps.  They  may  be  most  effectively  used  for  low  or  for  moderately  high  lifts,  of 
from  15  to  20  feet;  and,  in  such  conditions,  they  are  as  efficient  as  any  pumps  that 
can  be  made.  For  lifts  of  4 or  5 feet  they  are  even  more  efficient  than  others. 

At  same  time,  larger  the  pump  higher  lift  it  may  work  against.  Thus,  an  18-inch 
pump  works  well  at  20-feet  lift,  and  a 3-feet  pump  at  30-feet  lift.  A 21-inch  wheel 
at  40-feet  lift  has  not  given  good  results:  high  lifts  demand  very  high  velocities. 

Efficiency  is  influenced  by  form  of  casing  of  pump.  Hon.  R.  C.  Parsons  made  exper- 
iments with  two  14-inch  wheels  on  Appold’s  and  on  Rankine’s  forms.  In  Rankine’s 
wheel  blades  are  curved  backwards,  like  those  of  Appold’s,  for  half  their  length; 
and  curved  forwards,  reversely,  for  outer  half  of  their  length.  Deducing  results  of 
performance  arrived  at,  following  are  the  several  amounts  of  work  done  per  lb.  of 
water  evaporated  from  boiler  : 

Work  done  per  lb.  of 
water  evaporated. 

Foot-lbs.  Ratio. 


Appold  wheel,  in  concentric  circular  casing 11385  1.06 

u “ in  spiral  casing 15996  1.5 

Rankine  wheel,  in  concentric  circular  casing 10748  1 

“ “ in  spiral  casing 12954  1.2 


These  data  prove: — 1st,  that  spiral  casing  was  better  than  concentric  casing;  2d, 
that  Appold’s  wheel  was  more  efficient  than  Rankine’s  wheel. 


IMPACT  OR  COLLISION. 

Impact  is  Direct  or  Oblique.  Bodies  are  Elastic  or  Inelastic.  The 
division  of  them  into  hard  and  elastic  is  wholly  at  variance  with  these 
properties ; as,  for  instance,  glass  and  steel,  which  are  among  hardest 
of  bodies,  are  most  elastic  of  all. 

Product  of  mass  and  velocity  of  a body  is  the  Momentum  of  the  body. 

Principle  upon  which  motions  of  bodies  from  percussion  or  collision  are 
determined  belongs  both  to  elastic  and  inelastic  bodies ; thus  there  exists  in 
bodies  the  same  momentum  or  quantity  of  motion,  estimated  in  any  one  and 
same  direction,  both  before  collision  and  after  it. 

Action  and  reaction  are  always  equal  and  contrary.  If  a body  impinge 
obliquely  upon  a plane,  force  of  blow  is  as  the  sine  of  angle  of  incidence.  ( 

When  a body  impinges  upon  a plane  surface,  it  rebounds  at  an  angle  equal 
to  that  at  which  it  impinged  the  plane,  that  is,  angle  of  reflection  is  equal  to 
that  of  incidence. 

Effect  of  a blow  of  an  elastic  body  upon  a plane  is  double  that  of  an  in- 
elastic one,  velocity  and  mass  being  equal  in  each ; for  the  force  of  blow 


IMPACT  OK  COLLISION. 


58i 


from  inelastic  body  is  as  its  mass  and  velocity,  which  is  only  destroyed  by 
resistance  of  the  plane ; but  in  an  elastic  body  that  force  is  not  only  destroyed, 
being  sustained  by  plane,  but  another,  also  equal  to  it,  is  sustained  by  plane, 
in  consequence  of  the  restoring  force,  and  by  which  the  body  is  repelled  with 
an  equal  velocity ; hence  intensity  of  the  blow  is  doubled. 

If  two  perfectly  elastic  bodies  impinge  on  one  another,  their  relative  ve- 
locities will  be  same,  both  before  and  after  impact;  that  is,  they  will  recede 
from  each  other  with  same  velocity  with  which  they  approached  and  met. 

If  two  bodies  are  imperfectly  elastic,  sum  of  their  moments  will  be  same, 
both  before  and  after  collision,  but  velocities  after  will  be  less  than  in  case 
of  perfect  elasticity,  in  ratio  of  imperfection. 

Effect  of  collision  of  two  bodies,  as  B and  b , velocities  of  which  are  differ- 
ent, as  v and  v\  is  given  in  following  formulas,  in  which  B is  assumed  to 
have  greatest  momentum  before  impact. 

If  bodies  move  in  same  direction  before  and  after  impact,  sum  of  their 
moments  before  impact  will  be  equal  to  their  sum  after. 

If  bodies  move  in  same  direction  before,  and  in  opposite  direction  after 
impact,  sum  of  their  moments  before  impact  will  be  equal  to  difference  of  their 
sums  after. 

If  bodies  move  in  opposite  directions  before,  and  in  same  direction  after 
impact,  difference  of  their  moments  before  impact  will  be  equal  to  their 
sum  after . 

If  bodies  move  in  opposite  directions  before,  and  in  opposite  directions 
after  impact,  difference  of  their  moments  before  impact  will  be  equal  to  their 
difference  after . 


To  Compute  Velocities  of  Inelastic  Bodies  after  Impact. 


When  Impelled  in  Same  Direction.  ^ v __  n b and  b representing 

weights  of  the  two  bodies , V and  v their  velocities  before  impact , and  r velocity  of  bodies 
after  impact , all  in  feet. 


Consequently 


V — v 
’B  + 6 


X 


b =.  velocity  lost  by  B,  and 


V — v 

B + 6 


X B = velocity  gained  by  b. 


Note. — In  these  formulas  it  is  assumed  that  V>v.  If  V<+  the  result  will  be 
negative,  but  may  be  read  as  positive  if  lost  and  gained  are  reversed  in  places. 


Illustration.— An  inelastic  body,  6,  weighing  30  lbs.,  having  a velocity  of  3 feet, 
is  struck  by  another  body,  B,  of  50  lbs.,  having  a velocity  of  7 feet;  the  velocity  of 

b after  impact  will  be  , 

5°  X 7 + 3°  X 3 __  44°  _ , , 

50  + 30  80  S'  5 / • 

When  Impelled  in  Opposite  Directions.  — ^ ~ ? V = r. 


B + 6 

Illustration. — Assume  elements  of  preceding  case. 

50  X 7 — 30  X 3 _ 260 
50  + 3°  _ 80 

B V 

When  One  Body  is  at  Rest.  - = r. 

B + 6 

Illustration.— Assume  elements  as  preceding. 

35o 
80 


;=  3.2s  feet. 


50  X 7 35o  - , 

= a7  = 4-75  M- 


50+30" 

When  Bodies  are  inelastic,  their  velocities  after  impact  will  be  alike 

AC* 


5 82  IMPACT  OR  COLLISION. 

To  Compute  Velocities  of  Elastic  Bodies*  after  Impact. 

B — bV  - \-2bv  2BV — B — bv 

When  Impelled  in  One  Direction.  and — r- 


B + 6 

Illustration.— Assume  elements  as  preceding. 


B -j-  6 


50  — 30  X 7 + 2 X 3°X  3 _32°. 


' 80 


: 4 /eei,  and 


2X  50X7  — 5Q— 3°X  3 _64° 


50  + 30 


= ^ = 8 ** 


50+30 

Or,  V — ~~~-r  V — v = velocity  of  A,  and  v + 2 jr  V — v = velocity  of  r. 

B o B -J-  0 

IFAen  Impelled  in  Opposite  Directions. 

B — b V (\)  2b  v 2 BV  — B — b v 

B++  -R?an  B + fc  -• 

Illustration. — Assume  elements  as  preceding. 

50  — 30X7  ^2X30  X 3 _ 140 x8o ^ an^  ? X 5° X 7 + 5(3  30  X 3 ' — 

80  ’ 


50  + 30 


5o  + 3o_ 

< 3°  X 7 _ 

50  -J-  30  80 


7°°,+jg  _ 9-  5 feet.  Or,  2 & - = velocity  lost  by  B.  Ag  2 X 30  X 7 + 3 

80  B + b 

= 7-5  feet.  

V B — 6 „ _ 2 B V 

One  Body  is  at  Rest.  B & — R>  and  — r- 

Illustration. — Assume  elements  as  preceding. 

7*5°- 3° _n2  = I /«*,  and  2X?oX7  = Z22  = 8.75 /<*(. 

5O+3O  80  ’ 50+30  80 

To  Compute  Velocities  of  Imperfect  Elastic  Bodies  after 
Impact. 

Effect,  of  Collision  is  increased  over  that  of  perfectly  inelastic  bodies,  but 
not  doubled,  as  in  case  of  perfectly  elastic  bodies ; it  must  be  multiplied  by 


id or 


m + ™,  when  - represents  degree  of  elasticity  relative  to  both  per- 
fect inelasticity  and  elasticity. 

in  -1-  n B m -j-  n 

Moving  in  same  Direction.  V X (V  — v)  = R ; and  v -J 

X B (V  _ V)  — r.  m and  n representing  ratio  of  perfect  to  imperfect  elasticity. 

t and  n = 2 and  1. 

7 x — ^ — X V~~~3  = 7 — i'5X^X  4 = 7 — 2*25  = 4*75  feet,  and  3 + 


Illustration.— Assume  elements  as  preceding. 

7~ 

2+1 


' 50  + 3°  ' 


-X 


50 

50  + 30 


X 7 — 3 = 3 + 3-75  = 6-75/ee*- 


When  Moving  in  Opposite  Directions 

, m + n ~ 

md  

m 

(b--&) 

V m / 


_ m + n b ( V+ j _ R apd  A-X(T  + «)-»=i«-. 

m * B-f&  m B + *> 


BV 


When  One  Body  is  at  Rest.  ■ — — — R>  and 
Illustration. — Assume  elements  of  preceding  case. 

1 * (5°- ; * 3°) 


‘■(■+3 


— r. 


50  + 30 

3|2XA5i=6.s625/«t 


l XS°~^  = 3.6625/^, 


80 


and 


B + b 

50X7X  (1  + j) 
50  + 30 


LIGHT. 


583 


LIGHT. 

Light  is  similar  to  Heat  in  many  of  its  qualities,  being  emitted  in 
form  of  rays,  and  subject  to  same  laws  of  reflection. 

It  is  of  two  kinds,  Natural  and  Artificial ; one  proceeding  from  Sun 
and  Stars,  the  other  from  heated  bodies. 

Solids  shine  in  dark  only  at  a temperature  from  6oo°  to  700°,  and  in 
daylight  at  iooo°. 

Intensity  of  Light  is  inversely  as  square  of  distance  from  luminous 
body. 

Velocity  of  Light  of  Sun  is  185  000  miles  per  second. 

Standard  of  Intensity  or  of  comparison  of  light  between  different  methods 
of  Illumination  is  a Sperm  Candle  “short  6,”  burning  120  grains  per  hour. 


Candles. 

A Spermaceti  candle  .85  of  a inch  in  diameter  consumes  an  inch  in  length 
in  1 hour. 

Decomposition  of*  JLaglit. 


Colors. 

Maximum 

Ray. 

< 

Primary. 

Contrasts. 

Second’y. 

Tertiary. 

C 

Primary. 

'ombinations. 

Secondary. 

Tertiary. 

Violet. . . . 

Chemical. 

__ 



_ 

Blue. . . 1 

Indigo 

— 

— 

. 

Brown. 

Yellow.  } 

ureen. . ; 

1 

Dark. 

Blue 

Electrical. 

Blue. 

— 

— 

Blue. . . ) 

Purple.  ' 

( 

Green. 

Green 

— 

— 

Green. 

Green. 

Red j 

Orange.  ] 

1 

Yellow . . . 

Light. 

Yellow. 

— 

— 

Green. . j 

[ 

Gray. 

Orange . . . 

— 

— 

Orange. 

Broken. 

Yellow.  1 

Purple.  ] 

Red 

Heat. 

Red. 

Purple. 

Green. 

Red. ...  j 

Orange.  J 

f 

Brown. 

All  colors  of  spectrum,  when  combined,  are  white. 


Consumption  and  Comparative  Intensity-  of*  Liglit 
of  Candles. 


Candle. 

No.  in  a 
Lb. 

Diameter. 

Length. 

Consumption 
per  Hour. 

Light  comp’d 
with  Carcel. 

Wax 

Inch. 

Ins. 

Grains. 

3 

Q.h 

15 

i 135 

.09 

Spermaceti 

3 

•075 

j 

u 

3 

.1 

a. 

*5 
i3-5 
a c 

.09 

a 

4 

6 

Tallow 

.04 

5 

12.5 

) 

\ 

3 

1 

| 204 

•°7 

U 

3 

4 

•9 

.8. 

*5 

13-75 

Compared  with  1000  Cube  Feet  of  Gas. 


Candle. 

Gas  = i. 

Con- 

sump- 

tion. 

Light. 

Con- 
sumption 
for  equal 
Light. 

Candle. 

Gas  = i. 

Con- 

sump- 

tion. 

Light. 

Con- 
sumption 
for  equal 
Light. 

Paraffine. 
Sperm . . . 

.098 

•095 

Lbs. 

3-5 

3-9 

Lbs. 
35-5 
41. 1 

103 

120 

Adamantine. 
Tallow 

.108 

.074 

Lbs. 

5-i 

5-i 

Lbs. 

47.2 

53-8 

137 

155 

In  combustion  of  oil  in  an  ordinary  lamp,  a straight  or  horizontally  cut  wick 
gives  great  economy  over  one  irregularly  cut. 


584  LIGHT. 

Relative  Intensity,  Consumption,  Illumination,  and 
Cost  of  various  Modes  of  Illumination. 

Oil  at  11  cents,  Tallow,  at  14  cents,  Wax  at  52  cents,  and  Stearine  at  32  cents  per 
lb.  100  cube  feet  coal  gas  at  14  cents,  and  100  cube  feet  of  oil  gas  at  52  cents. 


Illuminator. 

Illumi- 
nation. 
CftTcel 
Lamp 
= 100. 

Actual 

Cost 

t/our. 

Cost  for 
equal 
Inten- 
sity. 

Illuminator. 

Illumi- 
nation. 
Carcel 
Lamp 
= 100. 

Actual 

Cost 

per 

Hour. 

Cost  for 
equal 
Inten- 
sity. 

Carcel  Lamp 

100 

Cents. 

.87 

Per  H’r. 

.87 

Stearine  Candle  5 to  lb. 

66.6 

Cents. 

•59 

Per  H’r. 
4-J3 

Lamp  with  in- 1 
verted  reserv’r.  J 

57-8 

.89 

•99 

Tallow  “ 6 “ 

Sperm  “ 6 “ 

54 

67-5 

•25 

.89 

2-34 

S’7* 

Astral  Lamp 

48.7 

•56 

1.78 

Coal  Gas 

— 

1.22 

.96 

Wax  Candle  6 to  lb. 

61.6 

.92 

6.31 

Oil  Gas 

— 

1.25 

.98 

1000  cube  feet  of  13-candle  coal  gas  is  equal  to  7.5  gallons  sperm  oil,  52.9  lbs.  mold, 
and  44.6  lbs.  sperm  candles. 

Candles,  I^amps,  Rlmids,  and  Gas. 

Comparison  of  several  Varieties  of  Candles,  Lamps,  and  Fluids , with  Coal * Gas , de- 
duced from  Reports  of  Com.  of  Franklin  Institute , and  of  A.  Frye , M.  D.,  etc. 


Candle. 

Intensity 
of  ' 

Light. t 

Light 

at  Equal  ( 
Costs. 

Cost  com- 
pared with 
Gas  for 
Equal  Light. 

Candle. 

Intensity 

of 

Light.f 

Light 
at  Equal 
Costs. 

Cost  com- 
pared with 
Gas  for 
Equal  Light. 

Diaphane 

•7 

.8 

•58 

•5 

•54 

.85 

15. 1 

16.2 

7-5 

. Tallow,  short  6’s,  1 
double  wick  . . j 

Wax,  short  6’s 

Palm  oil 

1 

.8 
1 7 

1 

.61 

•77 

7- 1 

14.4 

10.5 

Spermaceti,  short  6’s. 
Tallow,  short  6’s, ) 
single  wick  . . . } 

* City  of  Philadelphia  \ Compared  with  a fish-tail  jet  of  Edinburgh  gas,  containing  12  per  cent, 
of  condensable  matter  and  consuming  1 cube  toot  per  hour. 


Lamp  and  Fluid. 

Inten- 
sity of 
Light. 

Light 

at 

Equal 

Cost. 

Time  of 
Burning 
1 Pint 
of  Oil. 

Lamp  and  Fluid. 

Inten- 
sity of 
Light. 

Light 

at 

Equal 

Cost. 

Time  of 
Burning 
1 Pint 
of  Oil. 

Carcel. 

Sperm  oil,  max’m 
‘ ‘ mean. 

11  min'm 

Lard  oil 

1.8 

I-35 

1.2 

•97 

Hours. 

6.32 

9.87 

14.6 

XI*3 

Has 

1 

I 

Hours. 

2.15 

1.22 

.69 

•77 

Semi-solar,  Sperm  oil 

Solar,  Sperm  oil 

Camphene 

i-*5 

1.76 

i-75 

•93 

i-55 

3.08 

&75 

8.42 

9-31 

Loss  of  Light  by  Use  of  Glass  Globes . 

Clear  Glass,  12  per  cent.  | Half  ground,  35  per  cent.  | Full  ground,  40  per  cent. 

Refraction. 

Relative  Index  of  Refraction- Is.  Ratio  of  sine  of  angle  of  incidence  to  sine  of 
angle  of  refraction,  when  a ray  of  light  passes  from  one  medium  iuto  another. 

Absolute  Index  or  Index  of  Refraction— Is,  When  a ray  passes  from  a vacuum  into 
any  medium,  the  ratio  is  greater  than  unity. 

Relative  index  of  refraction  from  any  medium,  as  A,  into  another,  as  B,  is  always 
oqual  to  absolute  index  of  .B,  divided  by  absolute  index  of  A. 

Absolute  index  of  air  is  so  small,  that  it  may  be  neglected  when  compared  with 
liauids  or  solids;  strictly,  however,  relative  index  for  a ray  passing  from  air  into  a 
given  substance  must  be  multiplied  by  absolute  index  for  air,  in  order  to  obtain 
like  index  of  refraction  for  the  substance. 

Mean  Indices  of  Refraction. 

Humors  of  eye 1.34 

Salt,  rock 1.55 

Water,  fresh 1.34 

“ sea 1.34 — 


Air  at  320. 

Alcohol 

Canada  balsam . , . . 

••  1 -.54 

Crystalline  lens... 

..  1.34 

Glass,  fluid 

X i-s8 
“J  1-64 

“ ernwa 

...1  ‘-53 

J x-50. 

LIGHT. 


585 


Gras. 

Retort. — A retort  produces  about  600  cube  feet  of  gas  in  5 hours  with  a 
charge  of  about  1.5  cwt.  of  coal,  or  2800  cube  feet  in  24  hours. 

In  estimating  number  of  retorts  required,  one  fourth  should  be  added  for 
being  under  repairs,  etc. 

Pressure  with  which  gas  is  forced  through  pipes  should  seldom  exceed  2.5 
ins.  of  water  at  the  Works,  or  leakage  will  exceed  advantages  to  be  obtained 
from  increased  pressure. 

The  average  mean  pressure  in  street  mains  is  equal  to  that  of  1 inch  of 
water. 

When  pipes  are  laid  at  an  inclination  either  above  or  below  horizon,  a cor- 
rection will  have,  to  be  made  in  estimating  supply,  by  adding  or  deducting 
.01  inch  from  initial  pressure  for  every  foot  of  rise  or  fall  in  the  length  of  pipe. 

It  is  customary  to  locate  a governor  at  each  change  of  level  of  30  feet. 

Illuminating  power  of  coal-gas  varies  from  1.6  to  4.4  times  that  of  a tallow 
candle  6 to  a lb. ; consumption  being  from  1.5  to  2.3  cube  feet  per  hour,  and 
specific  gravity  from  .42  to  .58. 

Higher  the  flame  from  a burner  greater  the  intensity  of  the  light,  the 
most  effective  height  being  5 ins. 

Standard  of  gas  burning  is  a 15-hole  Argand  lamp,  internal  diameter  .44 
inch,  chimney  7 ins.  in  height,  and  consumption  5 cube  feet  per  hour,  giving 
a light  from  ordinary  •coal-gas  of  from  10  to  12  candles,  with  Cannel  coal 
from  20  to  24  candles,  and  with  rich  coals  of  Virginia  and  Pennsylvania  of 
from  14  to  16  candles. 

In  Philadelphia,  with  a fish-tail  burner,  consuming  4.26  cube  feet  per  hour, 
illuminating  power  was  equal  to  17.9  candles,  and  with  an  Argand  burner, 
consuming  5.28  cube  feet  per  hour,  illuminating  power  was  20.4  candles. 

Gas,  which  at  level  of  sea  would  have  a Value  of  100,  would  have  but  60 
in  city  of  Mexico. 

Internal  lights  require  4 cube  feet,  and  external  lights  about  5 per  hour. 
When  large  or  Argand  burners  are  used,  from  6 to  10  are  required. 

An  ordinary  single-jet  house  burner  consumes  5 to  6 cube  feet  per  hour. 

# Street-lamps  in  city  of  New  York  consume  3 cube  feet  per  hour.  In  some 
cities  4 and  5 cube  feet  are  consumed.  Fish-tail  burners  for  ordinary  coal 
gas  consume  from  4 to  5 cube  feet  of  gas  per  hour. 

A cube  foot  of  good  gas,  from  a jet  .033  inch  in  diameter  and  height  of 
flame  of  4 ins.,  will  bum  for  65  minutes. 

Resin  Gas. — Jet  .033,  flame  5 ins.,  1.25  cube  feet  per  hour. 

Purifiers.— Wet  purifiers  require  1 bushel  of  lime  mixed  with  48  bushels 
of  water  for  10  000  cube  feet  of  gas. 

Dry  purifiers  require  1 bushel  of  lime  to  10000  cube  feet  of  gas,  and  1 
superficial  foot  for  every  400  cube  feet  of  gas. 


Intensity-  of  Light  with  Equal  Volumes  of  Gras  from 
different  Burners. 

Equal  to  Spermaceti  Candle  burning  120  Grains  per  Hour. 


Burners. 

El 

1 

enditu 
'eet  pe 
2 

ire  in  ( 
r Hou 
3 

Cube 

r. 

4 

Burners. 

I 

Single -jet,  1 foot 

Fish-tail  No.  3 

Bat’s  wing 

2.6 

3-5 

3 

4 

4.1 

4.2 

4-3 

4-5 

Argand,  16  holes 

Argand,  24  holes 

Argand,  28  holes 

•32 

•33 

•34 

3.8 
5-3 

5.8 


586 


LIGHT, 


Material. 

Cube 

Feet. 

Material. 

Cube 

Feet, 

Material. 

Cube 

Feet. 

Boghead  Cannel 
Wigan  Cannel. . 

Cannel 

Cape  Breton,  ] 
uCow  Bay,” 
etc , 

■{ 

I- 

13  334 
15426 
8960 
15  000 

9500 

Cumberland 

English,  mean 

Newcastle j 

Oil  and  Grease 

Pictou  and  Sidney. . 
Pine  wood 

9 8 (DO 
11  000 
9500 
10000 
23000 
8000 
11  800 

Pittsburgh 

Resin 

Scotch | 

Virginia 

“ West’n. 

Walls-end 

| 

9520 

15600 

10300 

15000 

8960 

9500 

12000 

i Chaldron  Newcastle  coal,  3136  lbs.,  will  furnish  8600  cube  feet  of  gas  at 
a specific  gravity  of  .4,  1454  lbs.  coke,  14. 1 gallons  tar,  and  15  gallons  am- 
moniacal  liquor. 

Australian  coal  is  superior  to  Welsh  in  producing  of  gas. 

Wigan  Cannel,  1 ton,  has  produced  coke,  1326  lbs. ; gas,  338  lbs. ; tar, 
250  lbs. ; loss,  326  lbs. 

Peat , 1 lb.  will  produce  gas  for  a light  of  one  hour. 

Fuel,  required  for  a retort  18  lbs.  per  100  lbs.  of  coal. 

In  distilling  56  lbs.  of  coal,  volume  of  gas  produced  in  cube  feet  when 
distillation  was  effected  in  3 hours  was  41.3,  in  7,  37.5,  in  20,  33.5,  and  111 
25,  31-7- 

Flow  of  Gras  ill  Pipes. 

Flow  of  Gas  is  determined  by  same  rules  as  govern  that  of  flow  of  water. 
Pressure  applied  is  indicated  and  estimated  in  inches  of  water,  usually  from 
.5  to  1 inch. 

Volumes  of  gases  of  like  specific  gravities  discharged  in  equal  times  by  a 
horizontal  pipe,  under  same  pressure  and  for  different  lengths,  are  inversely 
as  square  roots  of  lengths. 

Velocity  of  gases  of  different  specific  gravities,  under  like  pressure,  are  in- 
versely as  square  roots  of  their  gravities. 

By  experiment,  30000  cube  feet  of  gas,  specific  gravity  of 
charged  in  an  hour  through  a main  6 ms.  in  diameter  and  22.5  feet  in  lengtli. 

Loss  of  volume  of  discharge  by  friction,  in  a pipe  6 ins.  in  diameter  and  1 
mile  in  length,  is  estimated  at  95  per  cent. 

Diameter  and  Lengtli  of  Gas-pipes  to  transmit  given 
Volumes  of  GJ-as  to  Branch-pipes.  [Vi.  Uie.) 


Volume 
per  Hour. 

Diameter. 

Length. 

Volume 
per  Hour. 

Diameter. 

Length. 

Volume 
per  Hour. 

Diameter. 

Length. 

Cube  Feet. 

50 

250 

500 

700 

Ins. 

•4 

1 

1.97 

2.65 

Feet. 

100 

200 

600 

1000 

Cube  Feet. 
1000 
1500 
2000 
2000 

Ins. 

3.16 

3-87 

5-32 

6.33 

Feet. 

1000 

1000 

2000 

4000 

Cube  Feet. 
2000 
6000 
6000 
8000 

Ins. 

7 

7- 75 
9.21 

8- 95 

Feet. 

6000 

1000 

2000 

1000 

Regulation  of  Diameter  and  Extreme  Length  °f  Tub- 
ing,  and  Number  of  Burners  permitted. 


Diameter 

of 

Tubing. 

Length. 

Capacity 

of 

Meters. 

Burners. 

Diameter 

of 

Tubing. 

Length. 

Capacity 

of 

Meters. 

Burners. 

Ins. 

•25 

•375 

•5 

.625 

Feet. 

6 

20 

30 

40 

Light. 

3 

5 

10 

20 

No. 

9 

i5 

30 

60 

Ins. 

•75 

1 

1.25 

i-5 

Feet. 

50 

70 

100 

150 

Light. 

30 

45 

60 

100 

No. 

90 

135 

180 

300 

LIGHT. 


587 

Temperature  of  Gases.  Combustion  of  a cube  foot  of  common  eras  will 
heat  650  lbs.  of  water  i°.  & 

Services  for  Lamps. 


Lamps. 

Length 
from  Main. 

Diameter 
of  Pipe. 

Lamps. 

Length 
from  Main. 

Diameter 
of  Pipe. 

Lamps. 

Length 
from  Main. 

Diameter 
of  Pipe. 

No. 

2 

6 I 

Feet. 

40 

40 

50 

Ins. 

•375 

•5 

.625 

No. 

10 

15 

1 20 

Feet. 

100 

130 

150 

Ins. 

•75 

1 

1.25 

No. 

25 

30 

Feet. 

180 

200 

Ins. 

i-5 

I-75 

Volumes  of  Has  Discharged  per  Hour  under  a Pressure 
of  Half  an  Inch,  of  Water. 

Specific  Gravity  .42. 


Diam.  of 
Opening. 

Volume. 

Diam.  of 
Opening. 

Volume. 

1 Diam.  of 
Opening. 

Volume. 

I Diam.  of 
Opening. 

Volume. 

Ins. 

•25 

•5 

Cube  Feet. 
80 
321 

Ins. 

•75 

1 

Cube  Feet.  | 
723 

1287  . 1 

| Ins. 

1-125 
1 1-25 

Cube  Feet. 
1625 
2010  | 

Ins. 

*•5 
1 5 

Cube  Feet. 
2885 
46  150 

To  Compute  -Volume  ofGas  Discharged  through  a Dipe. 

1000  \J G l ~ and  0(53  \f~~ti  =d'  d representing  diameter  of  pipe , and 

h height  of  water  in  ins.,  denoting  pressure  upon  gas , l length  of  vine  in  yards  G 
specif  c gravity  of  gas,  and  V volume  in  cube  feet  per  hour.  P V 1 

G may  be  assumed  for  ordinary  computation  at  .42,  and  h .5  to  1 inch, 
of ^ pTi^ard^' ~AssumQ  diameter  of  Pipe  1 inch,  pressure  1.68  ins.,  and  length 
w A x 1.68  /1.68 

1°°°  X “2  x x - IQOO  X = 2000  cube  feet, 

»nd  nd-r  v c /4000000  X -42  X i , /168000000 

.063  X -g-. =!/—. L68  =I.°S  ins. 

Note.— For  tables  deduced  by  above  formulas  see  Molesworth,  1878,  page  226. 

Dimensions  of  Mains,  with  Weight  of  One  Length. 
Diameter  in  ins. 


Length  in  feet 

Thickness  in  ins.  . . . 
Weight  in  lbs 


4 

6 

8 

9 

i°  1 14 

18 

9 

9 

9 

9 

9 9 

9 

ss'375 

•375 

•5 

•5 

•5  -625 

•75 

288 

224 

400 

454 

489  1 868 

13^ 

n ',0 

1484 

GAS  ENGINES. 

*n8the  iLenoir  Znnh\e’ the  best  proportions  of  air  and  gas  are,  for  common 
gas,  8 volumes  of  air  to  i of  gas,  and  for  cannel  gas,  n of  air  to  i of  gas. 
The  time  of  explosion  is  about  the  27th  part  of  a second. 

An  engine,  havinga  cylinder  4.625  ins.  in  diameter  and  8.75  ins.  stroke  of 
piston,  making  185  revolutions  per  minute,  develops  a half  horse-power. 

Distribution  of  Heat  Generated  in  the  Cylinder.  (. M ; Tresca.) 

Dissipated  by  the  water  and  prcS^^'l  Losses 

ucts  of  combustion 6Q 

Converted  into  work ’ * * ^ | 

Hence  efficiency  as  determined  by  the  brake  — 4 per  cent. 

Atmospheric  Gas  Engine. 

opcdS1^6eTP  and  ?he  ins'  in  diameter>  making  81  strokes  per  minute,  devel- 
flaming  2 rabe  feet  ( jf  ***  ^ CyUnder  “ CUbe  aDd  for  in‘ 


Per  eent. 
•••  27 

ICO 


5 88  LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES. 

LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES. 

Essentially  from  a Treatise  by  Brig -Gen' l Q.  A.  Gillmore,  U.S.A.* 
Lime. 

Calcination  of  marble  or  any  pure  limestone  produces  lime  (quick- 
lime) Pure  limestones  burn  white,  and  give  richest  limes. 

Finest  calcareous  minerals  are  rhombohedral  pnsrns  of  calcareous 
spar,  the  transparent  double-reflecting  Iceland  spar,  and  white  or  statu- 

^ Property  of  hardening  under  water,  or  when  excluded  from  air,  con- 
ferredPupon  a paste  of  lime,  is  effected  by  presence  of  foreign  sub- 
stanees— as  silicum,  alumina,  iron,  etc.— when  their  aggregate  presence 
amounts  to  .1  of  whole. 

Times  are  classed:  i.  Common  or  Fat  limes, which  do  not  set  in  water. 

2 Poor  or  Meagre,  mixed  with  sand,  which  does  not  alter  its  condition. 

Hydraulic  lime,  containing  8 to  12  per  cent,  of  silica,  alumina,  iron, 
pto  set  slowly  in  water.  4.  Hydraulic,  containing  12  to  20  per  cent,  of 
similar1  ingreiUents'sets  in  wate/ in  6 or  8 days.  5.  Eminently  Hydraulic 
enntainimr  20  to  to  per  cent,  of  similar  ingredients,  sets  in  water  in  2 to  4 
days.  6.gHydraulic  Cement,  containing  30  to  50  per  cent.  °*  yS1’’  ^ 
few  minutes,  and  attains  the  hardness  of  stone  m a few  ““"terras  Artnes 
Pozzuolanas  including  pozzuolana  properly  so  called,  Tra&s  01  ienas,  Arenes, 
Ochreous^rths,  Basaltic  sands,  and  a variety  of  similar  substances. 

Indications  of  Limestones.  They  dissolve  wholly  or  partly  in  weak  acids 
with  brisk  effervescence,  and  are  nearly  insoluble  in  water. 

Kick  Limes  are  fully  dissolved  in  water  frequently  renewed  and  they 

the  action  of  air.3  They  are  rendered  Hydraulic  by  admixture  of  pozzuolana 

or  trass.  , , - . 

Rich,  fat,  or  common  Limes  usually  contain  less  than  10  per  cent,  of  im- 

purities. 

Hydraulic  Limestones  are  those  which  contain  iron  and  clay,  so  as  to  en- 
able them  to  produce  cements  which  become  solid  when  under  watei.  _ 
Poor  Limes  have  all  the  defects  of  rich  limes,  and  increase  tut  slightly  in 
1 . 11.  ti.„  Doorer  limes  are  invariably  basis  of  the  most  rapidly  - setting 
and  most  durable  cements  and  mortars,  and  tliev  ™a,“thef2j“ 
which  have  the  property,  when  in  combination  with  silica,  etc.,  ot  lndurat  in*, 
under  water,  and  are  therefore  applicable  for  admixture  of  hydraulic  cements 
f c Alike  to  rich  limes  tliev  will  not  harden  if  in  a state  of  paste 

underwater  0?£V?t  0 orTeSded  from  contact  with  the  atmosphere 
o aH  n ac  d gL.  They  should  be  employed  for  mortar  only  when  it  is 
ini  practicable0  to  procure  common  or  hydraulic  lime  or  cement,  m which  case 
it  is  recommended  to  reduce  them  to  powder  by  grinding. 

Hydraulic  Limes  are  those  which  readily  Harden  under  water  T, he  most 
valuable  or  eminently  hydraulic  set  from  the  ^dtothe^h  day  . 

ro^lrft^  ^eacTp°^e 

They  absorb  less  water  than  pure  limes,  and  only  increase  m bulk  from  1.75 
to  2.5  times  their  original  volume. _ 


. See  .’.o  hie  Treat!...  on  Lime.,  Hydraulic  Cement.,  and  Mortar.,  in  Paper,  on  Practical  Engineer, 
ig,  Engineer  Department,  U.  S.  A. 


LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES.  589 

Inferior  grades,  or  moderately  hydraulic , require  a period  of  from  15 
to  20  days’  immersion,  and  continue  to  harden  for  a period  of  6 months. 

Resistance  of  hydraulic  limes  increase  if  sand  is  mixed  in  proportion 
of  50  to  180  per  cent,  of  the  part  in  volume ; from  thence  it  decreases. 

M.  Vicat  declares  that  lime  is  rendered  hydraulic  by  admixture  with  it  of  from 
33  w>  4°  per  cent,  of  clay  and  silica,  and  that  a lime  is  obtained  which  does  not 
slake,  and  which  quickly  sets  under  water. 

Artificial  Hydraulic  Limes  do  not  attain,  even  under  favorable  circum- 
stances, the  same  degree  of  hardness  and  power  of  resistance  to  compression 
as  natural  limes  of  same  class. 

Close-grained  and  densest  limestones  furnish  best  limes. 

Hydraulic  limes  lose  or  depreciate  in  value  by  exposure  to  the  air. 

Pastes  of  fat  limes  shrink,  in  hardening,  to  such  a degree  that  they  can- 
not be  used  as  mortar  without  a large  proportion  of  sand. 

Arenes  is  a species  of  ochreous  sand.  It  is  found  in  France.  On  account 
or  the  large  proportion  of  clay  it  contains,  sometimes  as  great  as  .7,  it  can  be 
made  into  a paste  with  water  without  any  addition  of  lime;  hence  it  is  some- 
times used  m that  state  for  walls  constructed  en  pise,  as  well  as  for  mortar. 
Mixed  with  rich  lime  it  gives  excellent  mortar,  which  attains  great  hardness 
under  water,  and  possesses  great  hydraulic  energy. 

i?°fvolcaniJ  or[Sh}- , Xt  comprises  Trass  or  Terras,  the  Arenes, 
some  of  the  ochreous  earths,  and  the  sand  of  certain  graywackes,  granites, 
schists,  and  basalts;  their  principal  elements  are  silica’' and  alumina,  the 
former  preponderating.  None  contain  more  than  10  per  cent,  of  lime. 

nf^er  fiDe,y  l3ulverized>  without  previous  calcination,  and  combined  with  paste 
°££at  m proportions  suitable  to  supply  its  deficiency  in  that  element  itpos 
sesses  hydraulic  energy  to  a valuable  degree.  It  is  used  in  combination  with  X 

bKion ^.r^temperattreVr^o  Ca'CiU,“g  day  aDd  driviD«  oflrthe  wat»r  ' ■»»- 

Bnclc  or  Tile  Dust  combined  with  rich  lime  possesses  hydraulic  energy. 

?r  VrrOSy  ls-a  blue"black  and  is  also  of  volcanic  origin,  ‘it 
requires  to  be  pulverized  and  combined  with  rich  lime  to  render  it  fit  for 
use,  and  to  develop  any  of  its  hydraulic  properties. 

assaaasar’-- " *~ 6 * * 

JUlriZ£  Slll<r?  jUn\e.d  }vith  rich  lime  Produces  hydraulic  lime  of  ex- 
cellent quality.  Hydraulic  limes  are  injured  by  air-slaking  in  a ratio  vary- 
ing directly  with  their  hydraulicity,  and  they  deteriorate  by  age. 

cWeirSK11  “ d“mp  S°U  °r  CXp°SU,e- 1,ydraulic  limea  ■»«»  be  ex- 

Hydraulic  Lime  ofTeil  is  a silicious  hydraulic  lime;  it  is  slow  in  setting 
requn  ing  a period  of  from  18  to  24  hours. 


Cerrients. 

*?*$&%* ■ CeTnts  c?.ntain  a .Iarger  proportion  of  silica,  alumina,  magnesia, 
tion  U nrf  °f  Pr?cedlnS  varieties  of  lime ; they  do  not  slake  after  oflcina- 

set  under  waterPfr‘°r  t0  a ‘®  YCry  best  °f  h-vdra,l!ic  limes-  as  s°me  of  them 
set  under  water  at  a moderate  temperature  (65°)  in  from  3 to  4 minutes- 

an  ZSppuqU!re  as  many  hours.  They  do  not  shrink  in  hardening,  and  make 
an  excellent  mortar  without  any  admixture  of  sand. 

i D 


590  LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES. 


When  exposed  to  air,  they  absorb  moisture  and  carbonic  acid  gas,  and  are 

^tt’abont  .33  strength  of  Portland, and  is  not  adapted  for  use  with  sand. 
Kosendale  Cement  is  from  Rosendale,  New  York.  , 

Portland  Cement  is  made  in 

hours  from  its  ft, 

^Property  of  setting  s,ow  may 

cement,  as  the  Boulogne,  when  required  f uctions  having  to  be  executed  un- 
immediate  causes  of  dest™ctl0*b  J other  hand  a quick-setting  cement  is  always 
der  water  and  between  tides.  On  the ‘ , active  supervision.  A slow- 

difficult  of  use  ; it  requires  special  w°Ame  „ossesses  the  advantage  of  being 

- 


Conclusions  derived  from  Mr.  Grant's  Experiments. 

, Portland  cement  improves  by  age,  if  kept  from  moisture, 
j ^nd  of  a «S‘  sand  is  about  .„  strength  of  neat  cement; 


5.  Strong  cement-  is  ueav,,; u ^ e 


w.  Less  water  used  in  mixing  cement  in  • n wetted  before  use. 

7.  Bricks,  stones,  etc  used  with ‘ “mem  should  be  v ^ ,f  kept  dry 

8'  SrfSSnd wnent  in  a few  months  are  equal  to  Blue  bncks, 
Bramley-Fall  stone,  or  Yorkshire  landings.  ^ picked  stock  bri cUs. 

usIdfa^nTofVater  Sill  wash  away  the  cement. 


Artificial  Cement  is  made  by  a combination  of  slaked  lime  with  unburned 
clay  in  suitable  proportions.  , - „ . t0  a slight  calcination. 


Mortar.  . 

Lime  or  Cement  Paste. is  l1 1 that^Su^e^ 
portion  should  be  determine  y , ~ ^ ov  SpaCes  in  sand  or  coarse 

iZSiiiZTSft SZ£  Ss  M to  -***•  ] 

of  the  mass.  . . . fm.mpd  into  a paste  after  having  ! 

JSfStXS  - >“  - “■  i 

'ZtZSZSrTsx,  ttfssa%iss»  a : 

not  very  great,  if  the  sea-water, 

mortars,  however,  is  consid  > 0f  finely-ground  cement  and  clean 

Pointing  Mortar  is  composed  of  » t?ie  Volume  of  cement  paste  is 

slig^ltlv'in^xce^'of  the^olume  h voids  or  spaces  in  the  sand.  The  volume 


LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES.  [Jgi 

of  sand  varies  from  2.5  to  2.75  that  of  the  cement  paste,  or  by  weight,  1 of 
cement  powder  to  3 to  3.33  of  sand.  The  mixture  should  be  made  under 
shelter,  and  in  small  quantities. 

All  mortars  are  much  improved  by  being  worked  or  manipulated;  and  as  rich 
imes  gain  somewhat  by,  exposure  to  the  air,  it  is  advisable  to  work  mortar  in 
then  render  it  fit  for  use  by  a second  manipulation 
White  lime  will  take  a larger  proportion  of  sand  than  brown  lime. 

Use  of  salt-water  in  the  composition  of  mortar  injures  adhesion  of  it. 

. Wh,en  a small  quantity  of  water  is  mixed  with  slaked  lime,  a stiff  paste 
is  made,  which,  upon  becoming  dry  or  hard,  has  but  very  little  tenacity,  but 
by  being  mixed  with  sand  or  like  substance,  it  acquires  the  properties  of  a 
cement  or  mortar.  t 1 

Proportion  of  sand  that  can  be  incorporated  with  mortar  depends  partly 
upon  the  decree  of  fineness  of  the  sand  itself,  and  partly  upon  character  of 
the  lime.  I'or  rich  limes,  the  resistance  is  increased  if  the  sand  is  in  pro- 
portions varying  from  50  to  240  per  cent,  of  the  paste  in  volume;  beyond 
tins  proportion  the  resistance  decreases. 

1,  clean  sharp  sand,  2.5.  An  excess  of  water  in  slaking  the  lime 
swells  the  mortar,  which  remains  light  and  porous,  or  shrinks  in  drying-  an 
excess  of  sand  destroys  the  cohesive  properties  of  the  mass.  ’ 

It  is  indispensable  that  the  sand  should  be  sharp  and  clean. 

Stone  Mortar.- 8 parts  cement,  3 parts  lime,  and  31  parts  of  sand;  or  1 
cask  cement  325  lbs.,  .5  cask  0:  lime,  120  lbs.,  and  14.7  cube  feet  of  sand== 
10.5  cube  feet  of  mortar. 

Parts  cement,  3 parts  lime,  and  27  parts  of  sand;  or  1 
cask  cement,  325  lbs.,  .5  cask  of  lime,  120  lbs.,  and  12  cube  feet  of  sand=r 
10  cube  feet  of  mortar. 

Brown  Mortar, —Lime  1 part,  sand  2 parts,  and  a small  quantity  of  hair. 
together.and  ^ cement  and  sand> lessen  about  -33  in  volume  when  mixed 

Calcareous  Mortar , being  composed  of  one  or  more  of  the  varieties  of  lime 
whb" T 75-  r artlficiaJ’  mixed  with  sand,  will  vary  in  its  properties 
method  of  .Lnipuladom  * “ "Sed’  **  natU,'e  and  (lua,ity  of  sand> 

Turkish.  Plaster,  or  Hydraulic  Cement. 

1 00  lbs.  fresh  lime  reduced  to  powder,  10  quarts  linseed-oil,  and  i to  2 
ounces  cotton.  Manipulate  the  lime,  gradually  mixing  the  oil  and  cotton,  in 
a wooden  vessel,  until  mixture  becomes  of  the  consistency  of  bread-dough 
,narm’  an?  wh(!u  required  for  use,  mix  with  linseed-oil  to  the  ■consistency  of  nas’te 
reslsfthe  ifffectVniumWity'to^  motal,  joined  or  coated  with  it,’ 

Stucco. 

Stucco  or  Exterior  Plaster  is  term  given  to  a certain  mortar  designed  for 
exterior  plastering;  it  is  sometimes  manipulated  to  resemble  variegated 
marble,  and  consists  of  i volume  of  cement  powder  to  2 volumes  of  dry  sand. 

Li  India,  to  water  for  mixing  the  plaster  is  added  1 lb.  of  sugar  or  molas- 
ses to  8 Imperial  gallons  of  water,  for  the  first  coat ; and  for  second  or  finish- 
ing, 1 lb.  sugar  to  2 gallons  of  water. 

Powdered  slaked  lime  and  Smith’s  forge  scales,  mixed  with  blood  in  suit- 
able proportions,  make  a moderate  hydraulic  mortar,  which  adheres  well  to 
masonry  previously  coated  with  boiled  oil. 


592  LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES. 

that  of  the  cement  paste. 

Khorassar,  or  Turkish.  Mortar, 

sistency,  and  lay  between  the  courses  of  brick  or  stones. 

Mortars. 

Mortars  used  for  inside  plastering  are  termed  Coarse,  Fine,  Gauge  or  bard 
finish,  and  Stucco. 

mcttick. 1 w\,S  for  7o  upon 
“when  full  time  for  hardening  cannot  be  allowed  substitute  from  J5  to  so  per 

be  *** 

coarse  Staff.  — Common  lime  mortar,  as  made  for  brick  masonry, 
with  a small  quantity  of  hair ; or  by  volumes,  lime  paste  (30  lbs.  lime) 
part,  sand  2 to  2.25  parts,  hair  .16  part. 

/lime  nutty). — Lump  lime  slaked  to  a paste  with  a mod- 
eraTe  volume  of water!  and  afterward  diluted  to  consistency  of  cream,  and 
then  to  harden  by  evaporation  to  required  consistency  for  w orking. 

In  this  state  it  is  used  for  a slipped  coat,  and  when  mixed  with  sand  or  plaster  of 
Paris,  it  is  used  for  finishing  coat. 

t.,1  TTivii<=;'h  is  composed  of  from  3 to  4 volumes  fine 

each,  fine  stuff  and  plaster. 

Scratch  Coat—  First  of  three  coats  when  laid  upon  laths,  and  is  fiom  .25 
^7?  of  an  inch  in  thickness. 

One-coat  IForifc.-Plastering  in  one  coat  without  finish,  either  on  masonry 

or  laths— that  is,  rendered  or  laid.  . 

Two-coat  Work.— Plastering  in  two  coats  is  done  either  in  a laid  coa 
and  set,  or  in  a screed  coat  and  set. 

Screed  coat  is  also  termed  a Floated  coat.  Laid ! first  coat  in  two^oat 
work  is  resorted  to  in  common  work  instead  of  screeding,  when  finished l sur 
Tee  is  not  required  to  be  exact  to  a straight-edge.  It  is  laid  m a coat  of 
about  .5  inch  in  thickness. 

Laid  coat,  except  for  very  common  work,  should  be  hand-floated. 

Firmness  and  tenacity  of  plastering  is  very  much  increased  by  band- flea  mg. 
Screeds  are  strips  of  mortar  6 to  8 inches  in  width,  and  of  weired  thick- 
ness of  first  coat,  applied  to  the  angles  of  a room,  ”S°  whe!  thesehave 
become  of  a straight-edge,  the  inter- 

spaces  between  the  screeds  are  filled  out  flush  with  them. 

ativelv  even  surface. 

This  finish  answers  when  the  surface  is  to  be  nnislied  in  distemper,  or  paper. 


LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES.  593 


Concrete  or  Beton 

Is  a mixture  of  mortar  (generally  hydraulic)  with  coarse  materials,  as 
gravel,  pebbles,  stones,  shells,  broken  bricks,  etc.  Two  or  more  of  these 
materials,  or  all  of  them,  may  be  used  together.  As  lime  or  cement  paste  is 
the  cementing  substance  in  mortar,  so  is  mortar  the  cementing  substance  in 
concrete  or  beton.  The  original  distinction  between  cement  and  beton  was 
that  latter  possessed  hydraulic  energy,  while  former  did  not. 

Hydraulic.  — 1.5  parts  unslaked  hydraulic  lime,  1.5  parts  sand,  1 part 
gravel,  and  2 parts  of  a hard  broken  limestone.  1 

This  mass  contracts  one  fifth  in  volume.  Fat  lime  may  be  mixed  with  concrete 
without  serious  prejudice  to  its  hydraulic  energy.  concrete, 

"Various  Compositions  of  Concrete. 

Hydraulic.  308  lbs.  cement  = 3.65  to  3.7  cube  feet  of  stiff  paste.  12  cube 
feet  of  loose  sand  = 9.75  cube  feet  of  dense. 

For  Superstructure.— n.75  cube  feet  of  mortar  as  above,  and  16  cube  feet 
of  stone  fragments. 

Sea  Wall.  Boston  Harbor.  Hydraulic. — 308  lbs.  cement,  8 cube  feet  of 
sand,  and  30  cube  feet  of  gravel.  Whole  producing  32.3  cube  feet. 

lbs.  cement,  80  lbs.  lime,  and  14.6  cube  feet  dense 
sands.  Whole  producing  12.825  cube  feet. 

is  ,mad«  ?f  clay  or  earth  rammed  in  layers  of  from  3 to  4 ins.  in  depth  In 
Tms  Znnfr  wljfa  rcoiroTmtS.1601  eXte™al  SUrfa0e  °fa  wa“  Oracled 
Asphalt  Composition. 

1.  Mineral  pitch  i part,  bitumen  ii,  powdered  stone,  or  wood  ashes  7 parts. 

^ Parts’ clay  3 parts,  and  sand  1 part,  mixed  with  a little  oil  makes  a 
very  fine  and  durable  cement,  suitable  for  external  use.  ’ 

Flooring.— % lbs.  of  composition  will  cover  1 sup.  foot,  .75  inch  thick 
Asphaltum  55  lbs.  and  gravel  28.7  lbs.  will  cover  an  area  of  10.75  sq.  feet. 

~£u!verized  burnt  clay  93  parts,  litharge,  ground  very  fine  7 narfs 
mixed  with  a sufficient  quantity  of  pure  linseed  oil.  5 7 P j 

pa?tsSbyCwe'ightand  pulverized  calcareous  stone  ,4,  litharge  2,  and  linseed  oil  4 

appUecfmlmt'be^sa^urared  with  od.an  ^ SUrfaCe  Up0“  which  il  is  “>  bo 

an4d  parts’ oil  o^esin  6.25  parts, 

Artificial  Mastic.—  Composition  of  i square  yard  .9  inch  thick: 

Mineral  tar ao5  cube  ins.  | Gravel.  .'275  cube  ins. 

«“  ii  Slakedlime •• 

^ • 1249  cube  ins. 

nfSnMiioi  morescence.— White  alkaline  efflorescence  upon  the  surface 
XMortom  rad  tKtei"/  natural  hydraulic  hme  or  cement  is  the  basis, 
its  formation  * h * f 1 m the  ProPortl<>n  of  .025  of  its  weight  will  prevent 

■ CiystaHiratnm  of  these  salts  within  ‘he  pores  of  bricks,  into  which  thev  have 

been  absorbed  from  the  mortar,  causes  disintegration.  ’ y a\e 

Distemper  is  term  for  all  coloring  mixed  with  water  and  size. 

Grmtmg.^lQYtar  composed  of  lime  and  fine  sand,  in  a semi-fluid  state 
poured  into  the  upper  beds  and  internal  joints  of  masonry  * 

aJ)* 


594  limes,  cements,  mortars,  and  concretes. 

slaking. 

Slaked  Lime  is  a hydrate  of  lime,  and  it  absorbs  a mean  of  2.5  times  its 
volume  and  2.25  times  its  weight  of  water.  „ 

Lime  (quicklime)  must  be  slaked  before  it  can  be  used  as  a matrix  for 

^SS£&,a3SaprS^ 

gSSSSSifSS 

consequent  upon  its  reduction.  . p^npcsive  Quantity,  is  termed 

rawaffitsas 

"aSliXjUnfc. i>  il'<  "'.■“Ijf  "*■?? tt'b.fcS'SSaS 

without  essential  deterioration. 

P Urn,  mt  W-A  Cask  * Lin«  = »o  «*,»■"  — »-  7'8  “ 

8. 1 5 cube  feet  of  stiff  paste.  - » 

A Cask  of  Cement  = 300*  lbs.,  will  make  from  3.7  to  3.75  c 

A Cask  of  Portland  Cement  = 4 bushels  or  5 cube  feet  = 420  lbs. 

A Cask  of  Roman  Cement  = 3 bushels  or  3.75  cube  feet  — 3 4 s- 


.5  inch. 

. 2.25  yards 


.75  inch. 
1.5  yards 


* inch. 

1. 14  yards. 


A Husliel  of  cement  will  cover. 

From  experiments  of  General  Totten,  it  appeared  that  r 

1 volume  oflime  slaked  with. 33  its  volume  of  water  gave  2.27  u u 

i “ “ “ -66  t.  u u “ 

' 0.:  cube  foot  of  dry  cement,  mixed  with  .33  cube  foot  of  water,  will  make  .63  to 
635  cube  foot  of  stiff  paste. 

one  volume  or  mass,  for  use  as  required. ; — ; — 

" ~3oo  lba.  net  is  standard ; it  usually  overruns  8 lbs. 


LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES.  595 


Mortar,  Cement,  See.  ( Molesworth .) 

Mortar.— -1  of  lime  to  2 to  3 of  sharp  river  sand. 

Or,  1 of  lime  to  2 sand  and  1 blacksmith’s  ashes,  or  coarsely  ground  coke. 
Coarse  Mortar. — 1 of  lime  to  4 of  coarse  gravelly  sand. 

Concrete.— 1 of  lime  to  4 of  gravel  and  2 of  sand. 

Hydraulic  Mortar—  1 of  blue  lias  lime  to  2.5  of  burnt  clay,  ground  to- 
gether. J & 

Or,  1 of  blue  lias  lime  to  6 of  sharp  sand,  1 of  pozzuolana  and  1 of  calcined 
ironstone. 


Beton. — 1 of  hydraulic  mortar  to  1.5  of  angular  stones. 

Cement.  — 1 of  sand  to  1 of  cement.— If  great  tenacity  is  required,  the  ce- 
ment should  be  used  without  sand. 


Portland  Cement 

Is  composed  of  clayey  mud  and  chalk  ground  together,  and  afterwards  cal- 
cined at  a high  temperature— after  calcining  it  is  ground  to  a fine  powder. 

Strength,  of  Mortars,  Cements,  and.  Concretes. 
Deduced  from  Experiments  of  Vicat,  Paisley , Treussart,  and  Voisin. 

Tensile 


Weight  or  Power  required  to  Tear  asunder  One  Sq.  Inch. 

Cement  Mortar.  (42  days  old.) 

Proportion  of  Sand  to  1 of  Cement. 


Roman . . , 
Portland. , 


0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

284 

284 

*99 

166 

142 

128 

116 

106 

99 

92 

95  lbs. 

142 

142 

1L3 

92 

79 

67 

57 

42 

35 

25 

— “ 

Bricls,  Stone,  and.  Grranite  Masonry.  (320  days  old.) 
Experiments  of  General  Gillmore , U.  S.  A. 

Cement  on  Bricks. 


Pure,  average  . 


Sand  1 ) 

Cement  1 j 
Sand  1 j 
Cement  2 j 
Sand  1 ) 

Cement  3 j 


Lbs. 

30.8 

*5-7 

12.3 

6.8 


Pure. , 


Cement  on  Granite. 

Lbs. 


Sand  1 1 

Cement  1 j 
Sand  1 j 
Cement  2 j 
Sand  1 1 

Cement  3 j 


27-5 

20.8 

12.6 

9.2 


Delafeld  and  Baxter.  Lbs. 

Pure  cement 68 

Cement  4) 

Sand  1 4} 68 

Cement  8 ) 

Siftings  1 j 80 

Cement  1 ) 

Siftings  1 j 82 

Cement  1 ) 

Siftings  2 J 74 

Lawrence  Cement  Co. 

Pure  cement 87 


Sand  1 1 

Cement  4 j 
Water  1 ) 
Cement  2 J 
Water  .42 ) 
Cement  1 j 
Water  .33 ) 
Cement  1 j 


Lbs. 

7-9 

20.5 

37-25 

29.15 


James  River.  Lbs. 

Pure  cement 87 

Cement  4 ) 

Sand  1 / 62 

Newark  Lime  and  Cement 
Co. 

Pure  cement 04 

Cement  1 ) 

Sand  2 j 4° 

Newark  and  Rosendale. 

Pure  cement 75 

Cement  1 ) 

Sand  1 j 


„ Lbs. 

Newark  and  Rosendale. 
Cement  1 ) 

Sand  3 j 7 

Pure,  without) 
mortar,  mean)  45 

Mortar. 

Lime  paste  1,  sand  2.5,  6 


cement  paste  5 u 


596  LIMES,  CEMENTS,  MORTARS,  AND  CONCRETES. 


Pure  Cement. 


Lbs. 

Portland,  in  sea- water,  45  days 266 

“ English,  6 months 424 

Roman  “Septaria,”  1 year 191 

“ masonry,  5 months 77 

Rosendale,  9 months 700 

Lawrence  Cement  Co 1210 


Lbs. 

Boulogne  100,  water  50 112 

Portland,  natural,  1 year 675 

“ artificial,  Eng.,  1 year...  462 

“ English,  320  days 1152 

“ “ 1 month 393 

Newark  and  Rosendale 339 

Transverse. 

Reduced  to  a uniform  Measure  of  One  Inch  Square  and  One  Foot  in  Length . 
Supported  at  Both  Ends. 

Experiments  of  General  Grillmore. 

Formed  in  molds  under  a pressure  of  32  lbs.  per  sq.  inch,  applied  until  mortar 
had  set.  Exposed  to  moisture  for  24  hours,  and  then  immersed  in  sea-water. 
Prisms  2 by  2 by  8 ins.  between  supports. 

Reduced  by  Formula  2 JJL.  _ “ = C.  C coefficient  of  rupture,  and  a weight  of 
3 4 v cl  2 

portion  of  prism  l. 

Cement . Mortar. 


James  River. 

Thick  cream 

Thin  paste 

Stiff  paste 

Rosendale  “ Hoffman.  ” 

Thin  paste 

Stiff  paste 

“ Delafield  and  Baxter.” 

Thin  paste 

Stiff  paste 

English. 

Portland,  pure 

Stiff  paste 

Cumberland,  Md.,  pure  . 
High  Falls,  U1-) 
ster  Co. , N.  Y. ) 
Complete  calcination. 


Age. 

Pure. 

Days. 

Lbs. 

59 

3-9 

320 

5-8 

59 

6.9 

320 

9 

320 

8.9 

, 320 

8.5 

, 320 

12 

■ 320 

16 

. 320 

13 

. 320 

13.2 

• 95 

8.4 

. 95 

4.2  1 

Portland,  Eng.,  stiff  paste 
Roman,  “ “ “ 


Cumberland,  Md 

Akron,  N.  Y 

James  River,  Ya 

Pulverized  and  re- ) 
mixed  after  set — j 

Fresh 

Kingston  and  Rosendale. 
High  Falls,  Ul- ) 

sterCo.,N.Y. ) 

Fresh  water  to  a stiff) 

paste ) 

Sea-water  to  a stiff  paste 
Lawrence  Cement  Co. 
Fresh 


Days. 

320 

20 

100 

320 

320 

320 


Ocfi 

Lbs. 


7.8 

8.4 

8.8 


Crusliixig. 

Cements,  Stones,  etc.  ( Crystal  Palace , London.) 
Reduced  to  a uniform  Measure  of  One  Sq.  Inch. 

Material. 


Portl’d  cem’t,  area  1,  height  1. 
“ cement ) 

“ sand. . . j 

“ stone  ■ 


Destructive 

Pressure. 

Material. 

Lbs. 

1680 

Portland  cement  1 ) 

“ sand  4 ) 

1244 

“ cement  1 ) 

1144 

“ sand  7 ) 

Roman  cement,  pure 

Lbs. 

[3 

2- 5 
6 

12.8 
8.8 
8.6 

3- 6 
9 

7.6  6.6 

3-2 
4.4 
2.6 

I10.2  I — 


Destructive 

Pressure. 


Lbs. 

1244 

692 

342 


General  deductions. 

, Particles  of  unwound  cement  exceeding  .0125  of  an  inch  in  diameter  may  be 
allowed  in  cement  paste  without  sand,  to  extent  of  50  per  cent,  of  whole,  without 
detriment  toTts  properties,  while  a corresponding  proportion  of  sand  tnjures  the 
strength  of  mortar  about  40  per  cent. 


LIMES,  CEMENTS,  MORTARS,  ETC. — MASONRY.  597 

2.  When  these  unground  particles  exist  in  cement  paste  to  extent  of  66  per  cent 
of  whole,  adhesive  strength  is  diminished  about  28  per  cent.  For  a corresponding 
proportion  of  sand  the  diminution  is  68  per  cent. 

3-  ^d5^iori  siftings  exercises  a less  injurious  effect  upon  the  cohesive  than  upon 
the  adhesive  property  of  cement.  The  converse  is  true  when  sand,  instead  of  sift- 
ings, is  used.  ’ 

4.  In  all  mixtures  with  siftings,  even  when  the  latter  amounted  to  66  per  cent  of 
whole,  cohesive  strength  of  mortars  exceeded  their  adhesion  to  bricks.  Same're- 
sults  appear  to  exist  when  siftings  are  replaced  by  sand,  until  volume  of  the  latter 
exceeds  20  per  cent,  of  whole,  after  which  adhesion  exceeds  cohesion. 

5.  At  age  of  320  days  (and  perhaps  considerably  within  that  period)  cohesive 
strength  of  pure  cement  mortar  exceeds  that  of  Croton  front  bricks.  The  converse 
is  true  when  the  mortar  contains  50  per  cent,  or  more  of  sand. 

6.  When  cement  is  to  be  used  without  sand,  as  may  be  the  case  when  qroulina  is 
resorted  to,  or  when  old  walls  are  to  be  repaired  by  injections  of  thin  paste,  there  is 
no  advantage  in  having  it  ground  to  an  impalpable  powder. 

7.  For  economy  it  is  customary  to  add  lime  to  cement  mortars,  and  this  may  be 

done  to  a considerable  extent  when  in  positions  where  hydraulic  activity  and 
strength  are  not  required  in  an  eminent  degree.  y 

8.  Hamming  of  concrete  under  water  is  held  to  be  injurious. 

9.  Mortars  of  common  lime,  when  suitably  made,  set  in  a very  few  days  and  with 
such  rapidity  that  there  is  no  need  of  awaiting  its  hardening  in  the  prosecution  of 

tnfar®  Olay.  The  fusibility  of  clay  arises  from  the  presence  of  impurities, 
^uch  as  lime,  iron,  and  manganese.  These  may  be  removed  by  steeping  the  clay  in 

bfiSfa  to  maMerWaBhlDS  “ WitU  Wat°r-  °rucibles  from  common 

Votes  by  General  GiUmore,  U.  S.  A. -Recent  experiments  have  developed  that 
most  American  cements  will  sustain,  without  any  great  loss  of  strength  a dose  of 
lime  paste  equal  to  that  of  the  cement  paste,  while  l dose  equal  to  5 fo  ’7t  the  vol 
ume  of  cement  paste  may  be  safely  added  to  any  Rosendale  cement  without  nro 
ducing  any  essential  deterioration  of  the  quality  of  the mortar NeYtois  the 
hydraulic  activity  of  the  mortars  so  far  impaired  by  this  limited  addlto  of  lime 
paste  as  to  render  them  unsuited  for  concrete  under  water  or  other  submarine 
economy  ^ ^ US6  °f  iS  secured  tlle  d°uble  advantages  of  slow  setting  and 

Notes  by  General  Totten , TJ.  S.  A.—  240  lbs.  limez 
8. 15  cube  feet  of  stiff  paste. 

cube^foot  when^packe^'as  at  ^manufaemry1"6^  WheQ  l0°Se>  WlU  meaSUre  *8  *°  '8 

cherbourg’  D°ver- Aide™^ 


t cask,  will  make  from  7.8  to 


MASONRY. 

Brickwork. 

eof°ofhiS  “ T™?ment  °f  ,bricks  or  stones> laid  aside  of  and  above 
dno»  ^ er,.s0.  la  )he  vertical  joint  between  any  two  bricks  or  stones 
does  not  coincide  with  that  between  any  other  two. 

This  is  termed  “breaking  joints.” 

Header  is  a brick  or  stone  laid  with  an  end  to  face  of  wall. 

Stretcher  is  a brick  or  stone  laid  parallel  to  face  of  wall. 

Header  Course  or  Bond  is  a course  or  courses  of  headers  alone. 

Stretcher  Course  or  Bond  is  a course  or  courses  of  stretchers  alone. 

Closers  are  pieces  of  bricks  inserted  in  alternate  courses,  in  order  to  obtain 
1 bond  by  preventing  two  headers  from  being  exactly  over  a stretcher 
Ene/ksh  Bond  is  laying  of  headers  and  stretchers  in  alternates  courses. 


MASONRY. 


598 

Flemish  Bond  is  laying  of  headers  and  stretchers  alternately  in  each  course. 
Gauged  Work.— Bricks  cut  and  rubbed  to  exact  shape  required. 

String  Course  is  a horizontal  and  projecting  course  around  a building. 
Corbelling  is  projection  of  some  courses  of  a wall  beyond  its  face,  in  order 
to  support  wall-plates  or  floor-beams,  etc. 

Wood  Bricks , Pallets , Plugs,  or  Slips  are  pieces  of  wood  laid  in  a wall  in 
order  the  better  to  secure  any  woodwork  that  it  may  be  necessary  to  fasten 
to  it. 

Reveals  are  portions  of  sides  of  an  opening  in  a wall  in  front  of  the  recesses 
for  a door  or  window  frame. 

Brick  Ashlar. — Walls  with  ashlar-facing  backed  with  brick. 

Grouting  is  pouring  liquid  mortar  over  last  course  for  the  purpose  of  filling 
all  vacuities. 

Larrying  is  filling  in  of  interior  of  thick  walls  or  piers,  after  exteiior  faces 
are  laid,  with  a bed  of  soft  mortar  and  floating  bricks  or  spawls  in  it. 

Rendering  (Eng.)  is  application  of  first  coat  on  masonry,  Laying  if  one 
or  two  coats  on  laths,  and  u Pricking  up  ” if  three-coat  work  on  laths. 

Bricks  should  be  well  wetted  before  use.  Sea  sand  should  not  be  used  in  the 
composition  of  mortar,  as  it  contains  salt  and  its  grains  are  round,  being  worn  by 
attrition,  and  consequently  having  less  tenacity  than  sharp- edged  giains. 

A common  burned  brick  will  absorb  1 pint  or  about  one  sixth  of  its  weight  ot 
water  to  saturate  it.  The  volume  of  water  a brick  will  absorb  is  inversely  a test  of 

ltSA^good^brick  should  not  absorb  to  exceed  .067  of  its  weight  of  water. 

The  courses  of  brick  walls  should  be  of  same  height  in  front  and  rear,  whether 
front  is  laid  with  stretchers  and  thin  joints  or  not. 

In  ashlar-facing  the  stones  should  have  a width  or  depth  of  bed  at  least  equal  to 

Hard  bricks  set  in  cement  and  3 months  set  will  sustain  a pressure  of  40  tons 

P€The  compression  to  which  a stone  should  be  subjected  should  not  exceed  .1  of  its 

The  extreme  stress  upon  any  part  of  the  masonry  of  St.  Peter’s  at  Rome  is  com  - 
puted at  15.5  tons  per  sq.  foot ; of  St.  Paul’s,  London,  14  tons  ; and  of  piers  of  bew 
York  and  Brooklyn  Bridge,  5.5  tons.  ao.  fioo  ~ _ 

The  absorption  of  water  in  24  hours  by  granites,  sandstones,  and  limestones  ot  a 
durable  description  is  1,  8,  and  12  per  cent,  of  volume  of  the  stone. 

Color  of  Bricks  depends  upon  composition  of  the  clay,  the  molding  sand,  tem- 
perature of  burning,  and  volume  of  air  admitted  to  kiln. 

Pure  clay  free  of  iron  will  burn  white , and  mixing  of  chalk  with  the  clay  will 

Pl Presence  of  iron  produces  a tint  ranging  from  red  and  orange  to  light  yellow , 

“^large  proportion  of  oxide  of  iron,  mixed  with  a pure  clay,  will  produce  a bripld 
red,  and  when  there  is  from  8 to  ,o  per  cent.,  and  the  brick  is  exposed  to  an  intense 
heat,  the  oxide  fuses  and  produces  a dark  blue  or  purple,  and  w'lth  .a  small „ 
of  manganese  and  an  increased  proportion  of  the  oxide  the  color  is  darkened,  e\en 

t0 Small6  volume  of  lime  and  iron  produces  a cream  color , an  increase  of  iron  pro- 
duces red , and  an  increase  of  lime  brown. 

c\ayU(m  n ta\n  ing  alkal  i es  anTbumedat  a high  temperature  produces  a bluish  green. 
For  other  notes  on  materials  of  masonry,  their  manipulation,  etc.,  see  “Limes, 
Cements,  Mortars,  and  Concretes,”  pp.  588-597- 

Bointiiag.— Before  pointing,  the  joints  should  be  reamed,  and  in  close  ma- 
sonry they  must  be  open  to  .2  of  an  inch,  then  thoroughly  saturated  with  water, 
and  maintained  in  a condition  that  they  will  neither  absorb  Water  ^ 
or  impart  any  to  it.  Masonry  should  not  be  allowed  to  dry  rapidly  after  pointi  g, 
but  it  should  be  well  driven  in  by  the  aid  of  a calking  iron  and 
I11  pointing  of  rubble  masonry  the  same  general  directions  are  to  be  obser\  ed. 


MASONRY.  £gg 

Sand  is  Argillaceous , Siliceous , or  Calcareous , according  to  its  composition 
Its  use  is  to  prevent  excessive  shrinking,  and  to  save  cost  of  lime  or  cement.  Or- 
d manly  it  is  not  acted  upon  by  lime,  its  presence  in  mortar  being  mechanical  and 
To  thl1  «nr?Ull°  1fmGS cen?euts  jt  weakens  the  mortar.  Rich  lime  adheres  better 
modarSUrfaCe  SaUd  ^an  t0  ltS  °wn  Part^c^esj  hence  the  sand  strengthens  the 

It  is  imperative  that  sand  should  be  perfectly  clean,  freed  from  all  impurities 
and  of  a sharp  or  angular  structure.  Within  moderate  limits  size  of  grain  does 
not  affect  the  strength  of  mortar;  preference,  however,  should  be  given  to  coarse 
Calcareous  sand  is  preferable  to  siliceous  given  10  coarse. 

required  for^mortTr^  from  in  Sharpn8SS 

saft'ean  Whe“  rUbbed  UP°U  th8m’  aDd  ,he  pres8nce  of 

aud  Cinder’  When  pr0perIy  crushed  and  used>  make  good 

V18  I?ixiD?  of  concrete,  slake  lime  first,  mix  with  cement  and  then 
with  the  chips,  etc.,  deposit  in  layers  of  6 ins.,  and  hammer  down.  ’ 

Bricks. 

Of  Wenl;t°"of  lhdi-mins!°?s  by  'TioUS  manufacturers,  and  different  degrees 
of  intensity  of  their  burning,  render  a table  of  exact  dimensions  of  different 
manufactures  and  classes  of  bricks  altogether  impracticable. 

averages  arePgiven  :h°WeVer’  °f  the  ran^es  of  their  dimensions,  following 


Description.  | Ins, 

Description. 

Ins. 

Baltimore  front 
Philadelphia  “ i 
Wilmington  “ 
Croton  “ 

Colabaugh 

Eng.  ordinary. . . 

[ “ Lond.  stock 
Butch  Clinker. . . 

j 8.25  x 4-125  x 2.375 

8-5  X4  X 2.25 

8.25  X 3.625  X 2.375 
9 X 4.5  X2.5 

8.75X4.25  X2.5 

6.25X3  X1.5 

Maine 

Milwaukee 

North  River 

Ordinary 

Stourbridge  1 

fire-brick j 

Amer.  do.,  N.  Y. . 

7- 5  X 3-375X2.375 

8.5  X 4.125X2.375 
8 X3.5  X 2.25 

{ 7-75  X 3.625X2.25 
(8  X 4.125X2.5 

9- 125  X 4.625  X 2.375 

8- 875  X 4.5  X 2.625 

tha  l L 7 variations  in  aimensxons  of  bricks,  and  thickness  of 

the  layer  of  mortar  or  cement  m which  they  may  be  laid,  it  is  also  impracti- 

R hpi°  glVe  any  rU  6+uf  8lneral  aPPH.cati°n  for  volume  of  laid  brickwork, 
t becomes  necessary,  therefore,  when  it  is  required  to  ascertain  the  volume 
of  bricks  m masonry,  to  proceed  as  follows : 

To  Compute  Volume  of  Bricks,  and  Number  in  a Cube 
Foot  of  NX asonry. 

f^Ce  dimensions  of  particular  bricks  used,  add  one  half  thick- 
ness of  the  mortar  or  cement  in  which  thev  are  laid,  and  compute  the  area  • 
divide  width  of  wall  by  number  of  bricks  of  which  it  is  composed ; multiply 
this  area  by  quotient  thus  obtained,  and  product  will  give  volume  of  the 
mass  of  a brick  and  its  mortar  in  ins. 

cube  foot.1728  by  thiS  v0lume’  and  <luotient  will  give  number  of  bricks  in  a 

Examcle . — W i d th  of  a wall  is  to  be  12.75  ins.,  and  front  of  it  laid  with  PhiladeT 
face  anTbacking  .“nTcuS  fc?  iDCh  ta  d8Ptli;  b°W  many  bricks  " iU  «><™  in 
Philadelphia  front  brick,  8.25  X 2.375  ins.  face. 

825  +^5X24-2=8.25  + .25  = 8.5.  _ length  of  trick  and  joint  ; 

2„375  + '25  X 2 2 = 2-375  + .25  = 2.625  = width  of  brick  and  joint. 

rvidt™/waU)=liji%'3125  im=area  °f  Mo;  12.75  -1-  3 ( number  of  bricks  in 
Hence  22.3125  x 4. 25  = 94.83  cube  ins. ; and  1728=94,83  = 18.22  bricks. 


6oo 


MASONRY. 


One  rod  of  brick  masonry  (Eng.)  = 11.33  cube  yards  and  weighs  15  tons,  or  272 
superficial  feet  by  13.5  thick,  averaging  4300  bricks,  requiring  3 cube  yards  mortar 
and  120  gallons  water. 

Bricklayers’  hod  will  contain  16  bricks  or  .7  cube  feet  mortar. 

I^ire— "briclis. 

Fire-clay  contains  Silica,  Alumina,  Oxide  of  Iron,  and  a small  proportion 
'of  Lime,  Magnesia,  Potash,  and  Soda.  Its  fire-resisting  properties  depend- 
ing  upon  the  relative  proportions  of  these  constituents  and  character  of  its 
grain. 

A good  clay  should  be  of  a uniform  structure,  a coarse  open  grain,  greasy 
to  the  hand,  and  free  from  any  alkaline  earths. 

The  Stourbridge  clay  is  black  and  is  composed  as  follows : 

Silica 63.3  | Alumina 23.3  I Lime 73  I Protoxide  of  iron. ...  1.8 

Water  and  organic  matter 10.3 

Newcastle  clay  is  very  similar. 

Stone  Masonry. 

Masonry  is  classed  as  Ashlar  or  Rubble. 

Ashlar  is  composed  of  blocks  of  stone  dressed  square  and  laid  with 
close  joints. 

Coursed  Ashlar  consists  of  blocks  of  same  height  throughout  each  course. 


Fig.  x. 


•i 

MM 

Fig.  2. 


Fig.  1.— Coursed,  with  chamfered  and 
rusticated  quoins  and  plinth. 


Fig.  2. — Regular  Coursed. 


Fig.  3- 


pig.  3. — Irregular  Coursed. 


Fig.  5- 


Fig.  6. 


ri 

pig,  5._ Ranged  Random , level,  and 
broken  courses. 


Fig.  6.— Random,  level,  and  broken. 


MASONRY. 


601 


Rn'b'ble  -A^slilar 
Is  ashlar  faced  stone  with  rubble  backing. 

R.izb'ble  Masonry 

Is  composed  of  small  stones  irregular  in  form,  and  rough. 


[V‘% 

tt:  .S, 

/'in.''1;. 

k:y, 

lg£s 

i\,vv 

f,  V, 
l \ Vj 

KY:  '"1 

k* 

Fig.  7.  Block  Coursed.—  Large  blocks 
in  courses  (regular  or  irregular),  Beds 
and  Joints  roughly  dressed. 


Fig.  8. 


Fig.  8. — Coursed  and  Ranged  Random. 


Fig.  9.  Ranged  Random. — Squared 
rubble  laid  in  level  and  broken 
courses. 


Fig.  10.  Coursed  Random. — Stones  laid 
in  courses  at  intervals  of  from  12  to  18 
ins.  in  height. 


Fig.  11. 


Fig.  11.  Uncoursed  or  Random.  — 
Beds  and  Joints  undressed,  projections 
knocked  off,  and  laid  at  random.  In- 
terstices filled  with  spalls  and  mortar. 


Fig.  12.  Dry  Rubble.— Without  mortar 
or  cement. 


Dry  Rri'b'ble 

Is  a wall  laid  without  cement  or 
mortar. 


Fig.  13- 


Fig.  13.  Laced  Coursed. — Horizontal 
bands  of  stone  or  bricks,  interposed  to 
give  stability. 


Fig.  14.  Rustic  or  Rag. — Stones  of 
irregular  form,  and  dressed  to  make 
close  joints. 


Note.— Rustic  or  Rag  work  is  frequently  laid  in  mortar. 

3E 


602 


MASONRY. 


Terra  Cotta. 

Terra  Cotta  in  blocks  should  not  exceed  4 cube  feet  in  volume.  When 
properly  burned,  it  is  unaffected  by  the  atmosphere  or  by  fumes  of  any  acid. 


^Arclies  and  AValls. 

Springing. — Point  si  Fig.  1 5,  on  each  side,  Fig.  1 5. 

from  which  arch  springs. 

Crown. — Highest  point  of  arch. 

Haunches.— Sides  of  arch,  from  springing 
half-way  up  to  crown. 

Spandrel. — Space  between  extrados, a hor- 
izontal line  drawn  through  crown  and  a ver- 
tical line  through  upper  end  of  skewback. 

Skewbaclc  is  upper  surface  of  an  abut-  Pier  Abutment 

ment  or  pier  from  which  an  arch  springs, 
and  its  face  is  on  a line  radiating  from  centre  of  arch. 

Abutment  is  outer  body  that  supports  arch  and  from  which  it  springs. 

Pier  is  the  intermediate  support  for  two  or  more  arches. 

Jambs  are  sides  of  abutments  or  piers. 

Voussoirs  are  the  blocks  forming  an  arch. 

Key-stone  is  centre  voussoir  at  crown. 

Span  is  horizontal  distance  from  springing  to  springing  of  arch. 

pise. — Height  from  springing  line  to  under  side  of  arch  at  key-stone. 

Length  is  that  of  springing  line  or  span.  . 

Ring-course  of  a wall  or  arch  is  parallel  to  face  of  it,  and  in  direction  of 

its  span.  . 

String  and  Collar  courses  are  projecting  ashlar  dressed  broad  stones  at 
right  angles  to  face  of  a wall  or  arch,  and  in  direction  of  its  length. 

c Camber  is  a slight  rise  of  an  arch  as  .125  to  .25  of  an  inch  per  foot  of 


span. 

Quoin  is  the  external  angle  or  course  of  a wall. 

Plinth  is  a projecting  base  to  a wall.  e . 

Footing  is  projecting  course  at  bottom  of  a wall,  in  order  to  distribute  it3 
weight  over  an  increased  area.  Its  width  should  be  double  that  of  base  of 
wall,  diminishing  in  regular  offsets  .5  width  of  their  height. 

Blocking  Course.— A course  placed  on  top  of  a cornice. 

Parapet  is  a low  wall,  over  edge  of  a roof  or  terrace. 

Extrados—  Back  or  upper  and  outer  surface  of  an  arch. 

Intrados  or  Soffit  is  underside  of  lower  surface  of  arch  or  an  opening. 

Groined  is  when  arches  intersect  one  another. 

Invert. — An  inverted  arch,  an  arch  with  its  intrados  belowT  axis  or  spring- 
ing line. 

Ashlar  masonry  requires  .125  of  its  volume  of  mortar.  Rubble , 1.2  cube 
yards  stone  and  .25  cube  yard  mortar  for  each  cube  yard. 

Rubble  masonry  in  cement,  160  feet  in  height,  will  stand  and  bear  20  000  { 
lbs.  per  sq.  inch. 

Stones  should  be  laid  with  their  strata  horizontal. 

When  “ through”  or  “ thorough  bonds ” are  not  introduced,  headers  should  , 
overlap  one  another  from  opposite  sides,  known  as  dogs'  tooth  bond. 

Aggregate  surface  of  ends  of  bond  stones  should  be  from  .125  to  .25  of 
area  of  each  face  of  wall.  . , 

Weak  stones,  as  sandstone  and  granular  limestone,  should  not -have  a 
length  over  3 times  their  depth.  Strong  or  hard  stones  may  ha\  e a length 
from  4 to  5 times  their  depth. 


MASONRY. 


603 


Gallets  are  small  and  sharp  pieces  of  stone  stuck  into  mortar  joints,  in 
which  case  the  work  is  termed  galleted. 

Snapped  work  is  when  stones  are  split  and  roughly  squared. 

Quarry  or  Rock-/ heed.— Quarried  stones  with  their  faces  undressed. 

Pitch-faced. — Stones  on  which  the  arris  or  angles  of  their  face,  with  their 
sides  and  ends,  is  defined  by  a chisel,  in  order  to  show  a right-lined  edge. 

Drafted  or  Drafted  Margin  is  a narrow  border  chiselled  around  edges  of 
faces  of  a block  of  rough  stone. 

Diamond-faced  is  when  planes  are  either  sunk  or  raised  from  each  edge 
and  meet  in  the  centre. 

Squared  Stones. — Stones  roughly  squared  and  dressed. 

Rubble. — Unsquared  stones,  as  taken  from  a quarry  or  elsewhere,  in  their 
natural  form,  or  their  extreme  projections  removed. 

Cut  Stones. — Stones  squared  and  with  dressed  sides  and  ends. 

Dressed.  Stones. 

The  following  are  the  modes  of  dressing  the  faces  of  ashlar  in  engineering : 

Rough  Pointed. — Rough  dressing  with  a pick  or  heavy  point. 

Fine  Pointed. — Rough  dressing,  followed  by  dressing  with  a fine  point. 

Crandalled. — Fine  pointing  in  right  lines  with  a hammer,  the  face  of 
which  is  close  serried  with  sharp  edges. 

Cross  Crandalled. — When  the  operation  of  crandalling  is  right  angled. 

Hammered. — The  surface  of  stone  may  be  finished  or  smooth  dressed  by 
being  Axed  or  Bushed ; the  former  is  a finish  by  a heavy  hammer  alike  to  a 
crandall,  the  latter  is  a final  finish  by  a heavy  hammer  with  a face  serried 
'with  sharp  points  at  right  angles. 


Tliicliiiess  of  Briclc  "Walls  for  Warehouses.  ( Molesworth .) 


Length, 

Height. 

Thickness. 

Length. 

Height. 

Thickness. 

Length. 

Height. 

Thickness. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Unlimited. 

25 

13 

Unlimit’d. 

100 

34 

45 

30 

13 

do. 

30 

17-5 

60 

40 

17-5 

30 

40 

13 

do. 

40 

21.5 

70 

50 

21.5 

40 

50 

17-5 

do. 

50 

26 

50 

60 

21.5 

35 

60 

17.5 

do. 

60 

26 

45 

70 

21.5 

30 

70 

17-5 

do. 

70 

26 

60 

80 

26 

45 

80 

21.5 

do. 

80 

30 

70 

90 

30 

60 

90 

26 

do. 

90 

34 

70 

100 

30 

55 

100 

26 

For  drawings  and  a description  of  stone-dressing  tools,  see  a paper  by  J.  R.  Cross, 
"W.  E.  Merrill,  and  E.  B.  Van  Winkle,  “A.  S.  Civil  Engineer  Transactions*”  Nov.  1877. 

Walls  not  exceeding  30  feet  in  height,  upper  story  walls  may  be  8.5  ins.  thick. 
From  16  feet  below  top  of  wall  to  base  of  it,  it  should  not  be  less  than  the  space 
defined  by  two  right  lines  drawn  from  each  side  of  wall  at  its  base  to  16  feet  from 
top. 

Thickness  not  to  be  less  in  any  case  than  one  fourteenth  of  height  of  story. 


Datlis. 

Laths  are  1.25  to  1.5  ins.  by  4 feet  in  length,  are  usually  set  .25  of  an  inch 
apart,  and  a bundle  contains  100. 


6 04 


MAS0NKY. 


Plastering. 

Volumes  required  for  Various  Thickness. 


Material. 

Sq’ 

•5 

uare  Yar 
•75 

ds. 

1 

Material. 

Square  Yards. 

•5  I -75  I 1 

Cube  Feet. 

Cement  1 

Ins. 

2.25 

4-5 

6-75 

Ins. 

i-5 

3 

4-5 

Ins. 

I-I5 

2.25 

3-33 

Cube  Feet. 

Lime  1,  sand  2, ) 
hair  3.75 j ** 

Ins.  1 Ins.  | Ins. 
75  yards,  sup’l  ren- 
dered and  set  on 
j brick  or  70  on  lath. 

Cement  1,  sand  1. . . 
Cement  1,  sand  2. . . 

Estimate  of  Materials  and.  Labor  for  lOO  Sq.  Yards  of 
Latli  and  Plaster. 


Materials 
and  Labor. 

Three  Coats 
Hard  Finish. 

Two  Coats 
Slipped. 

Materials 
and  Labor. 

Three  Coats 
Hard  Finish. 

Two  Coats 
Slipped. 

Lime 

4 casks. 

3. 5 casks. 

White  sand 

2.5  bushels. 

Lump  lime 

.66  “ 

Kails 

13  lbs. 
4 days. 

13  lbs. 

3. 5 days. 

Plaster  of  Paris. . 

•5  “ 

Masons 

Laths 

2000. 

4 bushels. 

2000. 

Laborer 

Hair 

3 11 

2 “ 

3 ousneis. 

Sand 

7 loads. 

6 loads. 

Cartage 

1 “ 

•75  “ 

Rough  Cast  is  washed  gravel  mixed  with  hot  hydraulic  lime  and 
water  and  applied  in  a semi-fluid  condition. 


.A^rclies  and  vtIixl exit s . 


To  Compute  Depth  of  Keystone  of  Circnlar  or  Elliptic 
A.rcli. 


\/R  + S-r-  2 


_|_  .25  —d.  R representing  radius , s span , and  d depth , all  in  feet 


This  is  for  a rise  of  about  .25  of  span;  when  it  is  reduced,  as  to  .125,  add  =5  instead 
of  .25. 


Illustration.  — Arch  ofWasliington  aqueduct  at ‘‘Cabin  John”  has  a span  of  220 
feet,  a rise  of  57.25,  and  a radius  of  134.25 ; what  should  be  depth  of  its  keystone  . 


Vi34. 25 + 220-^2  + 25  _ 15^3  _|_  25  4 l6  feet.  Depth  is  4. 16  feet. 

4 4 


Viaducts  of  several  arches  increase  results  as  determined  above  by  add- 
ing .125  to  .15  to  depth. 

For  arches  of  2d  class  materials  and  work,  and  for  spans  exceeding  10 
feet,  add  .125  to  depth  of  keystone,  and  for  good  rubble  or  brick- work 
add  .25. 

Note  — It  is  customary  to  make  the  keystones  of  elliptic  arches  of  greater  depth 
than  that  obtained  by. above  formula.  Trautwine,  however,  who  is  high  authority 
in  this  case,  declares  it  is  unnecessary. 


To  Compute  Radius  of  an  Arcla,  Circular  or  Ellipse. 


2 _|_  r 2 -f-  2 r = R.  r representing  rise. 


Railway  Arches. 

For  Spans  between  25  and  70  feet.  Rise  .2  of  span.  Depth  of  arch  .055  of  span. 
Thickness  of  abutments  .2  to  .25  of  span,  and  of  pier  .14  to  .16  ol  span. 


Abutments. 

When  height  does  not  exceed  1. 5 times  base.  R-^5  + -1  r + 2 — thickness  at  spring 
of  arch  in  feet.  ( Trautwine. ) 

Batter.—  From  .5  to  1.5  ins.  per  foot  of  height  of  wall. 


MASONRY. — MECHANICAL  CENTRES. GRAVITY.  605 

To  Compute  Depth,  of  Arch.  (Hurst.) 
c \/R  = D.  c = Stone  (block)  .3.  Brick  = .4.  Rubblerir.45. 

When  there  are  a series  of  arches,  put  .3  = .35,  .4  = .45,  and  .45 ".5. 

NT  i 11  i in  vim  Thickness  of  Abutments  for  Bridge  and 
similar  Arches  of  120°.  (Hurst.) 

When  depth  of  crown  does  not  exceed  3 feet.  Computed  from  formula. 
/ 6 R+  (j-g)  — = T.  H representing  height  of  abutment  to  springing  in  feet. 


Radius 

Height  of  Abutment  to  Springing. 

Radius 

Heig 

ht  of  Abutment  to  Springing. 

of  Arch. 

5 

7-5 

10 

20 

30 

of  Arch. 

5 

7-5 

10 

20 

30 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

4 

3-7 

4.2 

4-3 

4.6 

4-7 

12 

5-6 

6.4 

6.9 

7.6 

7-9 

4-5 

3-9 

4.4 

4.6 

4.9 

5 

15 

6 

7 

7-5 

8.4 

8.8 

5 

4.2 

4.6 

4.8 

5-1 

5-2 

20 

6-5 

7-7 

8.4 

9.6 

10 

6 

4-5 

4-7 

5-2 

5-6 

5-7 

25 

6.9 

8.2 

9.1 

10.5 

11. 1 

7 

4-7 

5-2 

5-5 

6 

6.1 

30 

7.2 

8.7 

9-7 

11.4 

12 

8 

4.9 

5-5 

5-8 

6.4 

6.5 

35 

7-4 

9.1 

10.2 

11. 8 

12.9 

9 

5-i 

5-8 

6. 1 

6.7 

6.9 

40 

7.6 

9.4 

10.6 

12.8 

13.6 

10 

5-3 

6 

6.4 

7- 1 

7-3 

45 

7.8 

9-7 

11 

i3-4 

i4-3 

11 

5-5 

6.2 

6.6 

7-3 

7-6 

50 

7-9 

10 

11.4 

14 

15 

Note. — Abutments  in  Table  are  assumed  to  be  without  counterforts  or  wing- 
walls.  A sufficient  margin  of  safety  must  be  allowed  beyond  dimensions  here 
given. 


Culverts  for  a road  having  double  tracks  are  not  necessarily  twice  the 
length  for  a single  track. 

For  other  and  full  notes,  tables,  etc.,  see  Trautwine’s  Pocket  Book,  pp.  341-356. 


MECHANICAL  CENTRES. 

There  are  four  Mechanical  centres  of  force  in  bodies,  namely,  Centre 
of  Gravity,  Centre  of  Gyration,  Centre  of  Oscillation,  and  Centre  of 
Percussion. 

Centre  of  Grravity. 

Centre  of  Gravity  of  a body,  or  any  system  of  bodies  rigidly  con- 
1 nected  together,  is  point  about  which,  if  suspended,  all  parts  will  be  in 
equilibrium. 

A body  or  system  of  bodies,  suspended  at  a point  out  of  centre  of  gravity, 
will  rest  with  its  centre  of  gravity  vertical  under  point  of  suspension.  ’ 

A body  or  system  of  bodies,  suspended  at  a point  out  of  centre  of  gravity, 
and  successively  suspended  at  two  or  more  such  points,  the  vertical  lines 
through  these  points  of  suspension  will  intersect  each  other  at  centre  of 
gravity  of  body  or  bodies. 

Centre  of  gravity  of  a body  is  not  always  within  the  body  itself. 

If  centres  of  gravity  of  two  bodies,  as  B C,  be  connected  by  a line,  dis- 
tances of  B and  C from  their  common  centre  of  gravity,  c,  is  as  the  weiahts 
of  the  bodies.  Thus,  B : C ::  C c : c B.  y 

To  Ascertain  Centre  of  Gravity  of  any  Plane  Figure  Mechanically . 

Suspend  the  figure  by  any  point  near  its  edge,  and  mark  on  it  direction 
of  a plumb-line  hung  from  that  point;  then  suspend  it  from  some  other 
point,  and  again  mark  direction  of  plumb-line.  Then  centre  of  gravity  of 
surface  will  be  at  point  of  intersection  of  the  two  marks  of  plumb-line 
3E* 


6o6 


MECHANICAL  CENTRES. GRAVITY. 


Centre  of  gravity  of  parallel-sided  objects  may  readily  be  found  in  this 
way.  For  instance,  to  ascertain  centre  of  gravity  of  an  arch  of  a bridge, 
draw  elevation  upon  paper  to  a scale,  cut  out  figure,  and  proceed  with  it  as 
above  directed,  in  order  to  find  position  of  centre  of  gravity  in  elevation  of 
the  model.  In  actual  arch,  centre  of  gravity  will  have  same  relative  position 
as  in  paper  model.  . 

In  regular  figures  or  solids,  centre  of  gravity  is  same  as  their  geometrical 

centres. 

Line. 

Circular  Arc.  ™ = distance  from  centre , r representing  radius , c chord , and  l 
length  of  arc. 

Surfaces. 

Square , Rectangle , Rhombus , Rhomboid,  Gnomon,  Cube , Regular  Polygon, 
Circle,. Sphere,  Spheroid  or  Ellipsoid , Spheroidal  Zone,  Cylinder,  Circular 
Ring,  Cylindrical  Ring,  Link , Helix , Plain  Spiral,  Spindle , all  Regular  Fig- 
ures, and  Middle  Frusta  of  all  Spheroids,  Spindles , etc. 

The  centre  of  gravity  of  the  surfaces  of  these  figures  is  in  their  geometri- 
cal centre. 

Triangle. — On  a line  drawn  from  any  angle  to  the  middle  of  opposite  side , 
at  two  thirds  of  the  distance  from  angle. 

Trapezium.-- Draw  two  diagonals,  and  ascertain  centres  of  gravity  of  each 
of  four  triangles  thus  formed ; join  each  opposite  pair  of  these  centres,  and  it 
is  at  intersection  of  the  lines. 


Trapezoid.  X — = distance  from  B on  a line  joining  middle  of  two 

parallel  sides  B b,  m representing  middle  line. 


Circular  Arc.  ~ = distance  from  centre  of  circle. 


Sector  of  a Circle.  . 4244  r = distance  from  centre  of  circle. 

Semicircle.  . 4244  r = distance  from  centre. 

Semi- semicircle.  .4244  r ==  distance  from  both  base  and  height  and  at  their  inter- 


section. 


Segment  of  a Circle.  = distance  from  centre , a representing  area  of  segment.  , 


Sector  of  a Circular  Ring.  ± X — ^4  X = distance  from  centre  of 


3 arc  Z. 
arcs , r and  r'  representing  the  radii. 

Illustration.— Radii  of  surfaces  of  a dome  are  5 and  3.5  feet,  and  angle  «)  at 
centre  = 1300. 

4 sin.  65°  125  — 42-875  _4  y -9o63  x 82.12$ __  feet. 

7 X arc  1300  X 25  — 12.25  3 X 2.2689  6.8067X12.75 


Hemisphere , Spherical  Segment , and  Spherical  Zone , At  centre  of  their 
heights. 

Circular  Zone.— Ascertain  centres  of  gravity  of  trapezoid  and  segments 
comprising  zone;  draw  a line  (equally  dividing  zone)  perpendicular  to 
chords;  connect  centres  of  segments  by  a line  cutting  perpendicular  to  | 
chords. 


Then  centre  of  gravity  of  figure  will  be  on  perpendicular , toward  lesser 
chord , at  such  proportionate  distance  of  difference  between  centres  of  gravity 
of  trapezoid  and  line  connecting  centres  of  segments , as  area  of  segments 
bears  to  area  of  trapezoid. 


MECHANICAL  CENTRES. — GRAVITY. 


607 


Prism  and  Wedge. — When  end  is  a Parallelogram,  in  their  geometrical 
centres ; when  the  end  is  a Triangle,  Trapezium,  etc.,  it  is  in  middle  of  its 
length,  at  same  distance  from  base,  as  that  of  triangle  or  trapezoid  of  which 
it  is  a section. 

Parabola  in  its  axis  = .6  distance  from  vertex. 

Prismoid. — At  same  distance  from  its  base  as  that  of  the  trapezoid  or 
trapezium,  which  is  a section  of  it. 

Lune. — On  a line  connecting  centres  of  gravity  of  arcs  at  a proportionate 
< point  to  respective  areas  of  arcs. 


Solids. 


Cube , Parallelopipedon , Hexahedron,  Octahedron,  Dodecahedron,  Icosahe- 
dron, Cylinder , Sphere,  Right  Spherical  Zone,  Spheroid  or  Ellipsoid,  Cylin- 
drical Ring,  Link,  Spindle , all  Regular  Bodies , and  Middle  Frusta  of  all 
Spheroids  and  Spindles,  etc.  Centre  of  gravity  of  these  figures  is  in  their 
geometrical  centre. 

Tetrahedron. — In  common  centre  of  centres  of  gravity  of  the  triangles  made  by  a 
section  through  centre  of  each  side  of  the  figures. 

Cone  and  Pyramid.  .25  of  line  joining  vertex  and  centre  of  gravity  of  base  = dis- 
tance from  base. 


(v  “4—  ^ —l—  2 1 

Frustum  of  a Cone  or  Pyramid.  - — — - --- , X - h = distance  from  centre 

(i'-\-r’)z  — r r'  4 J 

of  lesser  end , r and  r’,  in  a cone  representing  radii,  and  in  a pyramid  sides,  and  h 
height. 


Cone,  Frustum  of  a Cone , Pyramid , Frustum  of  a Pyramid,  and  Ungula. — 
At  same  distance  from  base  as  in  that  of  triangle,  parallelogram,  or  semicir- 
cle, which  is  a right  section  of  them. 

Hemisphere.  . 375  r :=  distance  from  centre. 


Spherical  Segment. 


3.1416  vs2  (r — -T-  v = distance  from  centre , vs  repre- 


senting versed  sine,  and  v volume  of  segment, 
vertex. 

Spherical  Sector.  .75  (r  — .5  h)  = distance  ji'om  centre.  2 r 3 ^ __  distance 

8 

from  vertex. 

Spirals. — Plane,  in  its  geometrical  centre.  Conical,  at  a distance  from  the 
base,  .25  of  line  joining  vertex  and  centre  of  gravity  of  base. 

Frustum  of  a Circular  Spindle.  v = distance  from  centre  of  spindle . 

2 (h  — u.z)  ’ 

h representing  distance  Vetween  two  bases , D distance  of  centre  of  spindle  from  centre 
of  circle,  and  z generating  arc , expressed  in  units  of  radius. 

7.2 

Segment  of  a Circular  Spindle.  — — jfz)  ~ ^s^ance  Jrom  centre  of  spindle. 

Semi-spheroids. — Prolate.  .375  a. — Oblate.  .375  a — distance  from  centre. 

Semi-spheroid  or  Ellipsoid  and  its  Segment. — See  HaswelVs  Mensuration,  pages 
281  and  282.  ’ ^ 6 


( 8 V 3 X h = distance  from 
\i2  r — 4/7  J 


Frusta  of  Spheroids  or  Ellipsoids.  Prolate.  .75  gj2**2  ^ = distance  from 

3 CL^  ““ 

centre  of  spheroid,  a representing  semi-transverse  diameter  in  a prolate  frustum,  and 
semi-conjugate  in  an  oblate  frustum. 


608  MECHANICAL  CENTRES. GRAVITY. 


Segments  of  Spheroids.— Prolate.  .75  ^ ^ • — Oblate.  .75  ^ — distance 

from  centre  of  spheroid,  d and  d'  representing  distances  of  base  of  segments  from 
centre  of  spheroid. 


Any  Frustum.  .75  + d ) X (2  a2  d 4 -d  ) __  distance  from  cmire  0j  sphe- 

' 3 a2  — dz-\-d  d-\-d 2 

void,  d and  d'  representing  distances  of  base  and  end  of  segments  from  centre  of  the 
spheroid. 

Segment  of  an  Elliptic  Spindle  at  two  thirds  of  height  from  vertex. 

Paraboloid  of  Revolution , at  two  thirds  of  height  from  vertex. 

Segment  of  a Hyperbolic  Spindle,  at  75  of  height  from  vertex. 

Frustum  of  Paraboloid  of  Revolution.  X — = distance  from  base,  r and 

r'  representing  radii  of  base  and  vertex. 

Segment  of  Paraboloid  of  Revolution,  at  two  thirds  of  height  from  vertex. 

Segments  of  a Circular  and  a Parabolic  Spindle. — -See  Haswell  s Mensuration, 
pages  192  and  199. 

Parabola.  .4  of  height  = distance  from  base. 

Hyperboloid  of  Revolution.  ^ ^ ^ Xh  = distance  from  vertex,  b representing 
diameter  of  base. 

(d  -4-  d')  (2  a2  — d'2  -f-  d2)  . 

Frustum  of  Hyperboloid  of  Revolution.  .75  3 ” distance 

from  centre  of  base,  a representing  semi-transverse  axis,  or  distance  from  centre  of 
curve  to  vertex  of  figure ; d and  d’  distances  from  centre  of  curve  to  centre  of  lesser 
and  greater  diameter  of  frustum. 

Segment  of  Hyperboloid  of  Revolution.  ^ ^ 4!  X h = distance  from  vertex. 


Of  Two  Bodies.  dV-  — distance  from  V or  volume  or  area  of  larger  body,  d rep- 
J V -|~  v 

resenting  distance  between  centres  of  gravity  of  bodies,  and  v volume  or  area  of  less 
body. 


Cycloid.  — . 833  of  radius  of  generating  circle  = distance  from  centre  of 
chord  of  curve. 


A ny  Plane  Figure.— Divide  it  into  triangles,  and  ascertain  centre  of  grav- 
ity of  each ; connect  two  centres  together,  and  ascertain  their  common  cen- 
tre ; then  connect  this  common  centre  and  centre  of  a third,  and  ascertain 
the  common  centre,  and  so  on,  connecting  the  last-ascertained  common  centre 
to  another  centre  till  whole  are  included,  and  last  common  centre  will  give 
centre  required. 


Of  an  Irregular  Body  of  Rotatioil. 

Divide  figure  into  four  or  six  equidistant  divisions ; ascertain  volume  of 
each,  their  moments  with  reference  to  first  horizontal  plane  or  base,  and 
then  connect  them  thus : 

(A  + 4 Ar  + e A2  + 4A3  + A4)  ~ = v,  a A„  etc.,  representing  volume  of  dims- 
ions,  and  h height  of  body  from  base  ; 

a (0A+1X4.  Ax  4-2X2  A2 + 3X4  A3 + 4 A4)  h _ djstance  0j  centre  of 
and  A + 4 Ax-j-2  A2  + 4 A3  + A4  4 

gravity  from  base. 


MECHANICAL  CENTRES. — GYRATION. 


609 


Centre  of  Gry-ration. 

Centre  of  Gyration  is  that  point  in  any  revolving  body  or  system 
of  bodies  in  which,  if  the  whole  quantity  of  matter  were  collected,  the 
Angular  velocity  would  be  the  same  ; that  is,  the  Momentum  of  the  body 
or  system  of  bodies  is  centred  at  this  point,  and  the  position  of  it  is  a 
mean  proportional  between  the  centres  of  Oscillation  and  Gravity. 


If  a straight  bar  of  uniform  dimensions  was  struck  at  this  point,  the 
stroke  would  communicate  the  same  angular  velocity  to  the  bar  as  if  the 
whole  bar  was  collected  at  that  point. 

The  A ngular  velocity  of  a body  or  system  of  bodies  is  the  motion  of  a line 
connecting  any  point  and  the  centre  or  axis  of  motion : it  is  the  same  in  all 
parts  of  the  same  revolving  body. 

In  different  unconnected  bodies,  each  oscillating  about  a common  centre, 
their  angular  velocity  is  as  the  velocity  directly,  and  as  the  distance  from 
the  centre  inversely.  Hence,  if  their  velocities  are  as  their  radii,  or  distances 
from  the  axis  of  motion,  their  angular  velocities  will  be  equal. 

When  a body  revolves  on  an  axis,  and  a force  is  impressed  upon  it  suffi- 
cient to  cause  it  to  revolve  on  another,  it  will  revolve  on  neither,  but  on  a 
line  in  the  plane  of  the  axes,  dividing  the  angle  which  they  contain ; so  that 
the  sine  of  each  part  will  be  in  the  inverse  ratio  of  the  angular  velocities 
with  which  the  bodies  would  have  revolved  about  these  axes  separately. 

Weight  of  revolving  body,  multiplied  into  height  due  to  the  velocity  with 
which  centre  of  gyration  moves  in  its  circle,  is  energy  of  body,  or  mechani- 
cal power,  which  must  be  communicated  to  it  to  give  it  that  motion. 

Distance  of  centre  of  gyration  from  axis  of  motion  is  termed  the  Radius 
of  gyration ; and  the  moment  of  inertia  is  equal  to  product  of  square  of 
radius  of  gyration  and  mass  or  weight  of  body. 

The  moment  of  inertia  of  a revolving  body  is  ascertained  exactly  by  as- 
certaining the  moments  of  inertia  of  every  particle  separately,  and  adding 
them  together ; or,  approximately,  by  adding  together  the  moments  of  the 
small  parts  arrived  at  by  a subdivision  of  the  body. 


To  Compute  Moment  of  Inertia  of  a Revolving  Body. 

Rule.  Divide  body  into  small  parts  of  regular  figure.  Multiply  mass 
or  weight  of  each  part  by  square  of  distance  of  its  centre  of  gravity  from 
axis  ot  revolution.  The  sum  of  products  is  moment  of  inertia  of  body. 

NoTE  -The  value  of  moment  of  inertia  obtained  by  this  process  will  be  more 
exact,  the  smaller  and  more  numerous  the  parts  into  which  body  is  divided. 


To  Compute  Radius  of  Gyration  of  a Revolvin 
al;> out  its  -A-xis  of  Revolution. 


Body 


Rule.  Divide  moment  of  inertia  of  body  by  its  mass,  or  its  weight,  anc 
square  root  of  quotient  is  length  of  radius  of  gyration. 

Note.— When  the  parts  into  which  body  is  divided  are  equal,  radius  of  gyratior 
™f! determined  by  taking  mean  of  all  squares  of  distances  of  parts  from  axis 
ot  revolution,  and  taking  square  root  of  their  sum. 

Or,  VR2  -f-  r2  -4-  2 = G.  R and  r representing  radii. 
example.— A straight  rod  of  uniform  diameter  and  4 feet  in  length,  weighs  4 lbs. 
vhat  is  its  inertia,  and  where  is  its  radius  or  centre  of  gyration  ? 

f°0t  ofjen,gth  weiShs  1 lb-,  and  if  divided  into  4 parts,  centre  of  gyration  of 
each  is  respectively  .5,  1.5,  2.5,  and  3.5  feet.  Hence, 

*X  .5^=  .25I 

i5i'52=  l 21  ==  inertia,  which  -4-4  = 5. 25,  and  ^5. 25  = 2. 291 

1 X 2. 5 = 6. 25  f feet  radius. 

1 X 3.5s  = 12.25  J 


6io 


MECHANICAL  CENTRES. GYRATION. 


Following  are  distances  of  centres  of  gyration  from  centre  of  motion  in 
various  revolving  bodies : 


Straight , uniform  Rod  or  Cylinder  or  thin  Rectangular  Plate  revolving  about  one 
end;  length  X -5773,  and  revolving  about  their  centre;  length  x .2886. 

The  general  expression  is,  when  revolving  at  any  point  of  its  length, 

(l3  + l'3\ 

\3  i*'+n/‘ 


76 


l and  V representing  length  of  the  two  points. 


Circular  Plane , revolving  on  its  centre;  radius  of  circle  X .7071 ; Circle  Plane , as 
a Wheel  or  Disc  of  uniform  Thickness , revolving  about  one  of  its  diameters  as  an 
axis;  radius  X .5. 


Solid  Cylinder,  revolving  abotit  its  axis;  radius  X •7°7I- 

Solid  Sphere,  revolving  about  its  diameter  as  an  axis;  radius  X .6325. 

Thin,  hollow  Sphere,  revolving  about  one  of  its  diameters  as  an  axis;  radius 
X .8164.  Surface  of  sphere  .8615  r. 

Sphere  and  Solid  Cylinder  (vertical),  at  a distance  from  axis  of  revolution  = 
y/t 2 -f . 4 r2  for  sphere , and  Vl2-\~- 5 r2  for  cylinder , l representing  length  of  connec- 
tion to  centre  of  sphere  and  cylinder. 


Cone,  revolving  about  its  axis  ; radius  of  base  X -5447  5 revolving  about  its  ver- 
tex = Vi2  /i2-f  3 r'2 -r-  20,  li  representing  height , and  r radius  of  base;  revolving 
about  its  base  = V2  h2  -j-  3 r2  -4-  20. 

Circular  Ring , as  Rim  of  a Fly-wheel  or  Hollow  Cylinder,  revolving  about  its 
diameter  = Vlt2  + r2'f2,  R representing  radius  of  periphery,  and  r of  inner  circle 
of  ring. 

76  W (R2  + r2)  4-  w (4  r»  _ , ..  ...  , 

Flv-wheel  = / — — — ! — : W and  io  representing  weights  of 

J V 12  (W-f  w) 

nm  and  of  arms  and  hub , and  l length  of  arms  from  axis  of  wheel. 

Section  of  Rim.  + r2  -}-  r d.  d representing  depth  and  c periphery 

of  rim. 

j.  (2-i- b2 

Parallelopiped,  revolving  about  one  end,  distance  from  end  = W — — — , b rep- 


resenting breadth. 

Illustration. — In  a solid  sphere  revolving  about  its  diameter,  diameter  being 
2 feet,  distance  of  centre  of  gyration  is  12  X -6325  = 7.59  ins. 

To  Compute  ]Elemen.ts  of  Gyration. 

GWv_.  ?rtg_  GWv T>rtg_  G ¥j>=;(, 

rtg  ’ ’ W«  “ ’ P tg  ’ Gv  ’ 

^ ? ^ ^ = u G representing  distance  of  centre  of  gyration  from  axis  of  rotation , 
GW  . 

W weight  of  body,  t time  power  acts  in  seconds,  v velocity  in  feet  per  second  acquired 
by  revolving  body  in  that  time,  and  r distance  of  point  of  application  of  power  fi  om 
axis  of  body,  as  length  of  crank,  etc. 

Illustration  i.  — What  is  distance  of  centre  of  gyration  in  a fly-wheel,  pover  , 

224  lbs.,  length  of  crank  7 feet,  time  of  rotation  10  seconds,  weight  of  wheel  5000 
lbs.,  and  velocity  of  it  8 feet  per  second? 

224  X 7 X 10  X 32- 166  _ 504  373  a feet 

5600  X 8 42  800 


2.— What  should  be  weight  of  a fly-wheel  making  12  revolutions  per  minute,  its 
diameter  8 feet,  power  applied  at  2 feet  from  its  axis  84  lbs.,  time  of  rotation  6 sec- 
onds, and  distance  of  centre  of  gyration  of  wheel  3.5  feet? 

84  X 2 X 6 X 32.166 


8 X 3-1416  X 12 
60 


= 5.0265  feet  — velocity.  Then  - 


3.5  X 5.0265 


; 1843.2  IbS. 


MECHANICAL  CENTRES. — GYRATION. 


6ll 


When  the  Body  is  a Compound  one.  Rule.— Multiply  weight  of  several 
particles  or  bodies  by  squares  of  their  distances  in  feet  from  centre  of  mo- 
tion or  rotation,  and  divide  sum  of  their  products  by  weight  of  entire  mass  • 
the  square  root  of  quotient  will  give  distance  of  centre  of  gyration  from 
centre  of  motion  or  rotation.  J 

Example.  -If  two  weights,  of  3 and  4 lbs.  respectively,  be  laid  upon  a lever  (which 
is  here  assumed  to  be  without  weight)  at  the  respective  distances  of  1 and  2 feet 
what  is  distance  of  centre  of  gyration  from  centre  of  motion  (the  fulcrum)  ? ’ 

3 + 16 


3Xi2  = s;  4X22  = i6; 


3 + 4 


_ ^9  _ 
" 7 


2.71,  and  ^2.71  =+.64  feet. 


That  is,  a single  weight  of  7 lbs.,  placed  at  1.64  feet  from  centre  of  motion  and  re 
respective  places. time’  WOUld  have  same  mor*entm  as  the  two  weights  ’in  their 

When  Centre  of  Gravity  is. given.  Rule. -Multiply  distance  of  centre  of 
oscillation  from  centre  or  point  of  suspension,  by  distance  of  centre  of  grav- 
ity from  same  point,  and  square  root  of  product  will  give  distance  of  centre 
01  gyration.  0 

Example.— Centre  of  oscillation  of  a body  is  9 feet,  and  that  of  its  gravitv  d foot 
centre  olr^  011  °F  P°int  of  ^pension;  ^ It  what  distance  fronfthis JotnUa 

9 X 4 — 36,  and  +36  = 6 feet 

To  Compute  Centre  of  Gyration  of  a Water-wheel. 

Rule.— Multiply  severally  twice  weight  of  rim,  as  composed  of  buckets 
shrouding,  etc.,  and  twice  that  of  arms  and  that  of  water  in  the  buckets 
(when  wheel  is  m operation)  by  square  of  radius  of  wheel  in  feet-  divide 
by  twice  sum  of  these  several  weights,  and  square  root  of  quotient  will 
give  distance  m ieet. 

Example.  — In  a wheel  20  feet  in  diameter,  weight  of  rim  is  2 tons  weight  of 
' in  buckets  1 ton;  what  is  aistanc°  °f 

Buckets  = 2 tons  x 102  X 2=  4™  3 + 2 + 1 X 2 = 12  sum  of  weights. 

Water  =iton  X 102  — IOO  TT  /IIOO 

noo  Hence  y.  i f f^9*:6T=9-57  feet. 

^S^1RA7L  Formulas.— p ^Presenting  power , H horses’  power,  F force  amlied  to 
rotate  m lbs.,  M mass  of  revolving  body  in  lbs.,  r radius  upon  which  F acts  in 
feet,  d distance  from  axis  of  motion  to  centre  of  gyration  in  feet,  t time  force  is  an 
plied  in  seconds , n number  of  revolutions  in  time  t,  x angular  velocity,  or  number  of 

revolutions  per  minute  at  end  of  time  t,  and  G = 32-166  F r2 


v 


- = t: 


i 4 pm 
G 

153-5  t Fr 
'.lld2  / : 


2 pr2  x 
60  G 


244  t P 

+2  d 2 


Mxd2 
*53-5  l r 
= M; 


= F; 


lid2  ' 
Hind2 
2+6  *2  F = 
M d2  _ 


2.56  t2  F r 
Hid2  = 
2 Hid2 

i = H. 


244  t ’ 134 100 1 " 

reet^^fa°».rRil?  °f  ? fly'wheel  weighing  7ooo  lbs.  has  radii  of  6.5  and  5 7S 
from  of  mil-  ,tre  of  ?yration-  and  What  force  must  be  applied  to  it  2 feet 
ao Teconds?  h™  “ t0  g,TC  an  anSu,ar  Telocity  of  130  revolutions  per  minute  in 
power?  * °W  many  rcv°lutions  will  it  make  in  40  seconds?  and  what  is  its 


1302  X 7000  X 6. 142 
134 100  X 40 


4459  862680 
5364000 


: 829. 7 horses. 


34306636 
12  280 


of  gyration  = -^Jr1 


Centre  of  gyration  = + 5‘752  __  ^.fee^  Then  F __  130  X 7000  X 6.  i42 


= 2793.7  lbs.,  and  X 40  X 2793.7X2 


153-5  X 40  X 2 


7000X6.142 = 86-  e7  revolutions. 


612  MECHANICAL  CENTRES. OSCILLATION,  ETC. 


Centres  of  Oscillation  and  Percussion. 

Centre  of  Oscillation  of  a body,  or  a system  of  bodies,  is  that  point 
in  axis  of  vibration  of  a vibrating  body  in  which,  if,  as  an  equivalent 
condition,  the  whole  matter  of  vibrating  body  was  concentrated  it  would 
continue  to  vibrate  in  same  time.  It  is  resultant  point  of  whole  vibrat- 
ing energy,  or  of  action  of  gravity  in  producing  oscillation. 

As  narticles  of  a bodv  further  from  centre  of  its  suspension  have  greater 
velocity  of  vibration  than  those  nearer  to  it,  it  is  apparent^  that  centre  of 
oscillation  is  further  from  its  centre  than  centre  of  gravity  is  from  axisof 
susnension  but  it  is  situated  in  centre  of  a line  drawn  from  axis  of  a body 
Sih  centre  of  gravity.  It  further  differs  from  centre  of  gyration 
in  this  that  while  motion  of  oscillation  is  produced  by  gravity  of  a body, 
that  of  gyration  is  caused  by  some  other  force  acting  at  one  place  only. 

Radius  of  oscillation,  or  distance  of  centre  of  oscillation  from  axis  of  sus- 
pendoris  a thlrd  pro’portional,  to  distance  of  centre  of  gravity  from  axis 
of  suspension  and  radius  of  gyration. 

Centre  of  Percussion  of  a body,  or  a system  of  bodies,  revolving 
about  a point  or  axis,  is  that  point  at  which,  if  resisted  by  an  imm  - 
able  obstacle  all  the  motion  of  the  body,  or  system  of  bodies,  would  be 
£ S without  impulse  on  the  point  of  suspension.  It  is  also 
that  point  which  would  strike  any  obstacle  with  greatest  effect,  and 
from  this  property  it  has  been  termed  percussion. 

Centres  of  Oscillation  and  Percussion  are  in  same  point.—  If  a blow  is 

That  is  centre  of  percussion  is  identical  with  centre  of  osciUation  ai  d 

centre  of  percussion,  its  motion  will  be  absolutely  destroyed,  so  that  the  body 
will  not  incline  either  way.  „ , . 

As  in  bodies  at  rest,  the  entire  weight  may  be  considered  as  conected  m 
centre  of  gravity;  so  in  bodies  in  vibration,  the  entire  force  may  hecOTSid- 
ered  as  concentrated  in  centre  of  oscillation;  and  in  bodies  in  motion,  the 
whole  force  may  be  considered  as  concentrated  m centre  of  percussion. 

If  centre  of  oscillation  is  made  point  of  suspension,  point  of  suspension 
will  become  centre  of  oscillation. 


rill  UtJbUlUC  LCI1WO  V/X.  , , 

Angle  of  Oscillation  or  Percussion  is  determined  by  angle  delineated  by 
vertical  plane  of  body  in  vibration,  in  plane  of  motion  of  body. 


Cl  LlCctl  piCtllVv  V/X  - ' • 1 1 

equal  in  height  to  versed  sine  of  the  arc. 


To  Compute  Centre  of  Oscillation  »rper™^io,i  of  a 
Body  of  Uniform  Density  and  F » 


Pin  f —Multiply  weight  of  body  by  distance  of  its  centre  of  gravity JErom 
point  |of'  suspension  ;W multiply  a J weight  of  body  by  square  of  its  length, 

31  DkidfurS  quotient  by  product  of 

its  centre  of  gravity,  and  quotient  is  distance  of  centres  front  po 
pension. 


MECHANICAL  CENTRES. OSCILLATION,  ETC.  613 

W l~ 

Or,  — -4-  W X <7  = distance  from  axis.  Or,  square  radius  of  gyration  of  body 

and  divide  by  distance  of  centre  of  gravity  from  axis  of  suspension. 

Example.— Where  is  centre  of  oscillation  in  a rod  0 feet  in  length  from  its  Doint 
of  suspension,  and  weighing  9 lbs.  ? F 


= 4°*  5 = product  of  weight  and  its  centre  of  gravity  ; 92 


= 243  = quo- 


tient of  product  of  weight  of  body  and  square  of  its  length--  3 ; ---  — 6 feet 

40-5 

When  Point  of  Suspension  is  not  at  End  of  Rod.  Rule. — To  cube  of 
distance  of  point  of  suspension  from  top  of  rod  or  bar,  add  cube  of  its  dis- 
tance from  lower  end,  and  multiply  sum  by  2. 

Divide  product  by  three  times  difference  of  squares  of  these  distances  and 
quotient  is  distance  of  point  of  oscillation  from  point  of  suspension.  ’ 

Example.— A homogeneous  rod  of  uniform  dimensions,  6 feet  in  length  is  sns 
E 0ef  suspensln°?m  US  UPP®r  ^ What  iS  distance  of  Point  of  oscillation  from 

. 2 (4-53+  i-53)  i8q 

o 1. 5 — 4-5.  — ■ ■ — - — — - — 3. 5 feet. 

■ 3 (4-5J  I-5  ) 54 

Centres  of  Oscillation  and.  Percussion  in  Bodies  of 
Various  Figures. 

When  Axis  of  Motion  is  in  Vertex  of  Figure,  and  when  Oscillation  or  Motion 
is  Facewise. 

Right  Line , or  any  figure  of  uniform  shape  and  density  — 66  l 
Isosceles  Triangle  = .75  h.  Circle  =1  2 c r 

Parabola  = .714  h.  Cone  = .8h. 

When  Axis  of  Motion  is  in  Centre  of  Body.  Wheel  ==  .75  radius. 

When  Oscillation  or  Motion  is  Sidewise.  Right  Line,  or  any  figure  of  uni- 
agonai  aPe  ^ densit'J  — 66  l-  Rectangle,  suspended  at  one  angle  — .66  of  di- 

o’fTts  + 33  parameter;  IT  suspended 

Sector  of  a Circle  — c representing  chord  of  arc,  and  r radius  of  base. 


Circle  = .75  d. 


Sphere  ~ - 


Cone  = - axis-1 

5 s axis 


7^+7}  + r + c’  c representing  length  of  cord  by  which  it  is  suspended. 

To  Ascertain  Centres  of  Oscillation  and  IPercussion 
experimentally. 

Suspend  body  very  freely  from  a fixed  point,  and  make  it  vibrate  in  small  arcs 
noting  number  of  vibrations  it  makes  in  a minute,  and  let  number  made  in  a min’ 
ute  be  represented  by  n ; then  will  distance  of  centre  of  oSatlon  from  poin?  of 

suspension  be  = -4-  — ins. 

n2 

For  length  of  a pendulum  vibrating  seconds,  or  60  times  in  a minute  beimr 
39-i25  ins.,  and  lengths  of  pendulums  being  reciprocally  as  the  squares  of  number 
of  vibrations  made  in  same  time,  therefore  n 2 : 602  : : 39. 125 : 6°2  * 39-I25  _ 140850 

*«"*“*•  ordistance  of  Zntr’e 


3 F 


6 1 4 mechanical  centres— mechanics. 

To  Compute  Centres  of  Oscillation  or  Percussion  of  a 
System  of  particles  or  Bodies. 

Rui  e.— Multiply  weight  of  each  particle  or  body  by  square  of  its  distance 
from  point  of  suspension,  and  divide  sum  of  their  products  by  sum  of  weights, 
multiplied  by  distance  of  centre  of  gravity  from  point  of  suspension,  and 
quotient  will  give  centre  required,  measured  from  point  of  suspension. 

Or  y ^ _ distance  of  centre. 

’ W0  + W'0' 

Fkamplf  i —Length  of  a suspended  rod  being  20  feet,  and  weight  of  a foot  in  length 
of  it  eqiial  xoo  oz  has  a ball  attached  at  under  end  weighing  100  oz. ; at  what  point 
of  rod  from  point  of  suspension  is  centre  of  percussion  . 

IOo  X 30  = 2000  = weight  of  rod  ; 2000  X ??  = 20000  = momentum  of  rod,  or  prod- 

„ #1  2000  X 202  ,,  ,,r  __ 

uct  of  its  weight,  and  distance  of  its  centre  of  gravity  ; - — 266  66  . 

force  of  rod  ; 1000  X 202  = 400000  = force  of  hall. 

Then  266  666.66  400000  _ l6'66feet 

20  000-1-20  000  , 

, loncrth  and  weighing  2 lbs.  for  each  foot  of  its  length, 

th^^alV^of^bs'^ch-onl  fixed  6 feet  from  the  point  of  suspension,  and  the 
other  aUhe  end  of  the  rod;  what  is  the  distance  between  the  points  of  suspension 
and  percussion?  a 

I2  x 2 x = 144  = momentum  of  rod,  = ^-=1152  =force  of  rod. 

3Xi2  =36=  “ °o/^dhall.  3X  62  = 3 X 36  = i°8  — liofistball, 

ig8  sum  of  moments.  3Xi22=3Xi44  — of  vd  hall, 

n o a**  1602  sum  of  forces. 

Then  1692  198  = 8. 545  feet. 

MECHANICS. 

Mechanics  is  the  science  which  treats  of  and  investigates  effects  of 
forces,  motion  and  resistance  of  material  bodies,  and  of  equilibrium, 
it  is  divided  into  two  parts— Statics  and  Dynamics. 

Statics  treats  of  equilibrium  of  forces  or  bodies  at  rest.  Dynamics 
of  forces  that  prodmee  motion,  or  bodies  m motion. 

These  bodies  are  further  divided  into  Mechanics  of  Solid,  Fluid , and  Aert- 
form  bodies ; hence  the  following  combinations : 

1 Statics  of  Solid  Bodies,  or  Geostatics. 

.2!  Dynamics  of  Solid  Bodies , or  Geodynamics. 

-I  Statics  of  Fluids , or  Hydrostatics. 

T Dynamics  of  Fluids , or  Hydrodynamics.  _ 
ce  Statics  of  Aeriform  BoSes,  or  Aerostatics, 
t.  iLamics  of  Aeriform  Mies,  Pneumatics  or  Aerodynamics, 
mines  me  various,  and  are  divided  into  moving  forces  or  resistances;  as 
■<Gram1y,  Heat  or  Caloric,  TM 

MmcdUr,  Magnetism,  Coh 

(Gourde  —Two  forces  of  equal  magnitude  applied  to  or  operating  upon  - 

Moment. -=Q usmtity  of  motion  in  a moving  body,  which  is  alwajs  eq 

tersely  as  their  quantities  of 

i matter,  their  momenta  ate  equal. 


STATICS. 

Composition  and.  rtesolu-tion  of*  Forces. 

When,  two  forces  act  upon  a body  in  same  or  in  an  opposite  direc- 
tion, effect  is  same  as  if  only  one  force  acted  upon  it,  being  sum  or 
difference  of  the  forces.  Hence,  when  a body  is  drawn  or  projected  in 
directions  immediately  opposite,  by  two  or  more  unequal  forces,  it  is  affected 
as  if  it  were  drawn  or  projected  by  a single  force  equal  to  difference  between 
the  two  or  more  forces,  and  acting  in  direction  of  greater  force. 

This  single  force,  derived  from  the  combined  action  of  two  or  more  forces, 
is  their  Resultant. 

The  process  by  which  the  resultant  of  two  or  more  forces,  or  a single 
force  equidistant  in  its  effect  to  two  or  more  forces,  is  determined,  is  termed 
the  Composition  of  Forces , and  the  inverse  operation ; or,  when  combined 
effects  of  two  or  more  forces  are  equivalent  to  that  of  a single  given  force, 
the  process  by  which  they  are  determined  is  termed  the  Decomposition  or 
Resolution  of  Forces.  Two  or  more  forces  which  are  equivalent  to  a single 
force  are  termed  Components. 

When  two  forces  act  on  same  point  their  intensities  are  represented  by  sides 
of  a parallelogram , and  their  combined  effect  will  be  equivalent  to  that  of  a 
single  force  acting  on  point,  in  direction  of  diagonal  of  parallelogram,  the 
intensity  of  which  is  proportional  to  diagonal. 

Illustration.— Attach  three  cords  to  a fixed  point,  c,  Fig.  i ; let  c a and  c b pass 
over  fixed  rollers,  and  suspend  weights  A and  B therefrom. 
i Point  c will  be  drawn  by  the  forces  A and  B in  directions  a c 
and  b c.  Now,  in  order  to  ascertain  which  single  force,  P,  would 
produce  the  same  effect  upon  it,  set  off  the  distances  c m and 
cn  on  the  cords  in  the  same  proportion  of  length  as  weights 
of  A and  B ; that  is,  so  that  cm  : cn::  A : B ; then  draw  par- 
allelogram cm  on  and  diagonal  o c,  and  it  will  represent  a sin- 
gle force,  P,  acting  in  its  direction,  and  having  same  ratio  to 
weights  A or  B as  it  has  to  sides  c m or  c n of  parallelogram. 
Consequently,  it  will  produce  same  effect  on  point  c as  com- 
bined actions  of  A and  B. 

A parallelogram,  constructed  from  lateral  forces,  and  diagonal  of  which  is 
mean  force,  is  termed  a Parallelogram  of  Forces. 

Illustration.  — Assume  a weight,  W,  Fig.  2,  to  be 
suspended  from  a ; then,  if  any  distance,  a o,’is  set 
off  in  numerical  value  upon  the  vertical  line,  a W, 
and  the  parallelogram,  o r a s,  is  completed,  a s and 
a r,  measured  upon  the  scale,  a o,  will  represent 
strain  upon  ac  and  ae  in  same  proportion  that  a 0 
bears  to  weight  W. 

If  several  forces  act  upon  same  point , and  their  intensities  taken  in  order 
are  represented  by  sides  of  a polygon,  except  one , a single  force  proportioned 
to  and  acting  in  direction  of  that  one  side  will  be  their  resultant. 

To  Resolve  a Single  Force  into  a Pair  of  Forces. — Figs.  3 and  4. 

The  ends  of  a cord,  Fig.  3,  are  led  over  two  points,  a and  6,  and  in  centre  of 
cord  at  c a weight  of  4 lbs.  is  suspended.  If  distances  a c,  b c,  are  each  1 foot  dis- 


Fig.  2 


Fig.  4. 


tance  a b should  be  18  ins. 

When  cord  is  in  this  posi- 
tion, weight  at  c draws  upon 
c a and  c b with  a force  of 
3 lbs. ; hence  c of  4 lbs.  is 
equal  to  two  forces  of  3 lbs. 
each  in  direction  of  a c and  b c. 

Apply  ends  of  cord  to  ef,  Fig.  4,  distance  being  22  ins.,  then  the  strain  once  cf 
are  each  5 lbs. ; hence  one  force  of  4 lbs.  is  4qual  to  two  of  5 lbs.  each. 


6i6 


MECHANICS. STATICS. DYNAMICS. 


Equilibrium  of  Forces. 

Two  bodies  which  act  directly  against  each  other  in  same  line  are  in  equi- 
librium when  their  quantities  of  motion  are  equal ; that  is,  when  product  of 
mass  of  one,  into  velocity  with  which  it  moves  or  tends  to  move,  is  equal  to 
product  of  mass  of  other,  into  its  actual  or  virtual  * velocity.  # 

When  the  velocities  with  which  bodies  are  moved  are  same,  their  forces 
are  proportional  to  their  masses  or  quantities  of  matter.  Hence,  when  equal 
masses  are  in  motion,  their  forces  are  proportional  to  their  velocities. 

Relative  magnitudes  and  directions  of  any  two  forces  may  be  represented 
bv  two  right  lines,  which  shall  bear  to  each  other  the  relations  of  the  forces, 
- ® insiUnoH  tn  othpr  in  an  aneie 


Fig. 


snail  uttti  IV  

and  which  shall  be  inclined  to  each  other  in  an  angle 
equal  to  that  made  by  direction  of  the  forces. 

Illustration.  — Assume  a body,  W,  to  weigh  150  lbs.,  and 
resting  upon  a smooth  surface,  to  be  drawn  by  two  forces,  a 
and  6 Fig.  5.  = 24  and  30  lbs.,  which  make  with  each  other 
an  angle 5 cl  W 6 = 105°,  in  which  direction  and  with  what 
acceleration  will  motion  occur? 


Cos.  a W b = 105°,  and  cos.  180°  - 
force.. 


: cos.  75°,  mean 


V = V302  -f  242  — 2 X 30  X 24  cos.  75°'=  V goo  + 576  — H4°  cos-  75° 
= V1476  — (1440  X 258  82)  = V 1103. 3 = 33  21  lbs. 

33.21  X 32.166 


P g 

The  acceleration  is 


150 

Angle  of  Repose  is  greatest  inclination  of  a plane  to  horizon  at  which  a 
body  will  remain  in  equilibrium  upon  it.  . 

Hence  greatest  angle  of  obliquity  of  pressure  between  two  planes,  consist- 
ent with  stability,  is  the  angle  tangent  of  which  is  equal  to  coefficient  ox 
friction  of  the  two  planes. 

Inertia,  is  resistance  which  a body  at  rest  offers  to  an  external  powei  to 
be  put  in  motion  or  to  change  its  velocity  or  direction  when  in  motion. 

To  Compute  Inertia  of  a Revolving  Body. 

Divide  it  into  small  parts  of  a regular  figure,  multiply  weight  of  each  part 
by  square  of  its  distance  of  its  centre  of  gravity  from  axis  of  revolution, 
and  sum  of  products  will  give  moment  of  inertia  of  body. 

DYNAMICS. 

Dynamics  is  the  investigation  of  the  laws  of  Motion  of  Solid  Bodies , 
or  of  Matter , Force,  Velocity , Space,  and  Time. 

Mass  of  a body  is  the  quantity  of  matter  of  which  it  is  composed. 

Force  is  divided  into  Motive,  Accelerative,  or  Retardative. 

Motive  Force,  or  Momentum , of  a body,  is  the  product  of  its  mass  and 
its  velocity,  and  is  its  quantity  of  motion.  This  force  can,  therefoie,  be 
ascertained  and  compared  in  any  number  of  bodies  when  these  two 
quantities  are  known. \ 

Accelerative  or  Retardative  Force  is  that  which  respects  velocity  of 
motion  only,  accelerating  or  retarding  it;  and  it  is  denoted  by  quotient 
of  motive  force,  divided  by  mass  or  weight  of  body.  Thus,  ll  a body 

* Virtual  velocity  is  the  velocity  which  a body  in  equilibrium  would  acquire  were  the  equilibrium 
t0+bHdUUcrompkred,  because  it  is  not  referable  to  any  standard,  as  a 

a cannon-ball  weighing  15  lbs.,  projected  with  a velocity  of  1500  feet f0[  weight 
body,  its  momentum,  according  to  the  above  rule,  wou.d  be  X 1^00  2-  3 > l > b 

is  a pressure  with  which  it  cannot  be  compared. 


> -r  ± 

- = 7.1215  feet. 


MECHANICS. DYNAMICS. 


617 


of  5 lbs.  is  impelled  by  a force  of  40  lbs.,  accelerating  force  is  8 lbs.  • 
but  if  a force  of  40  lbs.  act  upon  a body  of  10  lbs.,  accelerating  force 
is  only  4 lbs.,  or  half  former,  and  will  produce  only  half  velocity. 

With  equal  masses,  velocities  are  proportional  to  their  forces. 

With  equal  forces,  velocities  are  inversely  as  the  masses. 

With  equal  velocities,  forces  are  proportional  to  the  masses. 

Work  is  product  of  force,  velocity,  and  time. 

Motion.— The  succession  of  positions  which  a body  in  its  motion  pro- 
gressively  occupies  forms  a line  which  is  termed  the  trajectory  or  path 
of  the  moving  body.  ’ 

A motion  is  Uniform  when  equal  spaces  are  described  by  it  in  equal 
times,  and  Variable  when  this  equality  does  not  occur.  When  spaces 
described  in  equal  times  increase  continuously  with  the  time,  a variable 
motion  is  termed  accelerated , when  spaces  decrease,  retarded , and  when 
equal  spaces  are  described  within  certain  intervals  only,  the  motion  is 
termed  periodic , and  intervals  periods.  Uniform  motion  is  illustrated 
m progressive  motion  of  hands  of  a watch ; variable  in  progressive  ve- 
ocity  of  falling  and  upwardly  projected  bodies ; and  periodic  by  oscil- 
lation of  a pendulum  or  strokes  of  a piston  of  a steam-engine. 


Formulas,  fv,  H 550,  and  — -P; 

* t 


Uniform  Motion. 

P W 


=- , — =£_  and 


/ 

and 

W 

550  t 


H55° 

/ 

fs 

H 550 


W 


. P t 
vt'  /’ 


V ■ 


and 


and 


H 550  t 


f t'  /’  /’  " / 

= t]  fs,  H550  l,  P t,  and  fv  t = W; 


P 

55o ! 


H55°_  , 
v ~7’ 
sf  s 
p’  7’ 
fv  fs 
550  ’ 550 1 


t ’ 
W 
Jv' 

, and 


^ „ = H-  p representing  power  in  effect , body , or  momentum,  f force  in  lbs. , v and 

s velocity  and  space  in  feet  per  second , t time  in  seconds,  H horse-power , and  W work 
\Yl  jOOt-CuS, 

If  two  or  more  bodies , etc.,  are  compared,  two  or  more  corresponding  letters 
as  l , p,  p , \ , v,  v1,  etc.,  are  employed.  ^ 

Illustration  i -Two  bodies,  one  of  20,  the  other  of  10  lbs.,  are  impelled  by  same 
momentum,  say  60.  They  move  uniformly,  first  for  8 seconds,  second  for  6-  what 
are  the  spaces  described  by  both?  5 o,  waai, 

60  -f-  20  = 3 = Y,  and  6o  -f-  io  = 6 = v. 

Then  T \ =3  X 8 = 24  = S,  and  tv  — 6 x 6 = 36  = s,  spaces  respectively. 

2.  — If  a power  of  12  800  effects  has  a velocity  of  10  feet  per  second,  what  is  its 
lorce'  12  800  -4-  10  = 1280  lbs. 


Uniform  'V'aria'ble  Motion. 

Space  described  by  a body  having  uniform  variable  motion  is  represented 
by  sum  or  difference  of  velocity,  and  product  of  acceleration  and  time  ac- 
cording as  the  motion  is  accelerated  or  retarded.  ’ 


Illustration  1.— A sphere  rolling  down  an  inclined  plane  with  an  initial  velocitv 
of  25  feet  acquires  in  its  course  an  additional  velocity  at  each  second  of  time  of  < 
feet;  what  will  be  its  velocity  after  3 seconds?  5 

25  + 5 X 3 = 40  feet. 

,ht,~nA2°i0m0tll  h.Vine  a"  initla'  velocity  of  30  feet  per  second  is  so  retarded 
that  in  each  second  it  loses  4 feetj  what  is  its  velocity  after  6 seconds? 


30  — 4 x 6 = 6 feet. 


6i8 


MECHANICS. DYNAMICS. 


Uniform  HVIotion.  Accelerated. 

In  this  motion,  velocity  acquired  at  end  of  any  time  whatever  is  equal  to  prod- 
uct of  accelerating  force  into  time,  and  space  described  is  equal  to  product  of  half 
accelerating  force  into  square  of  time,  or  half  product  of  velocity  and  time  of  ac- 
quiring the  velocity. 

Spaces  described  in  successive  seconds  of  time  are  as  the  odd  numbers,  i,  3,  5,  7, 
9,  etc. 

Gravity  is  a constant  force,  and  its  effect  upon  a body  falling  freely  in  a vertical 
line  is  represented  by  9,  and  the  motion  of  such  body  is  uniformly  accelerated. 

The  following  theorems  are  applicable  to  all  cases  of  motion  uniformly  acceler- 
ated by  any  constant  force,  F: 


v2  2 s v I s 

.5 1 .,=,5  a F =.I7T.  =.«•  - = jg  = yj  .777-  - *• 

2 S V 2 S V2 

T=SF(=^ffF,  = , - = — = — = F. 


When  gravity  acts  alone,  as  when  a body  falls  in  a vertical  line,  F is  omit- 
ted. Thus, 


V 2 . , v /2S 

•5  *t,  = r,  = t g = ^T=t 

t representing  time  in  seconds , and  s velocity  in  feet  per  second. 


v 2 s 1>2 


If,  instead  of  a heavy  body  falling  freely,  it  be  projected  vertically  upward 
or  downward  with  a given  velocity,  v,  then  s = tv  .5/7 1~ ; an  expression 
in  which  — must  be  taken  when  the  projection  is  upward,  and  + when  it  is 
downward. 

Illustration  i.  — Tf  a body  in  10  seconds  has  acquired  a velocity  by  uniformly 
accelerated  motion  of  26  feet,  what  is  accelerating  force,  and  what  space  described, 
in  that  time? 

2 6 

26=10  = 2.6  = accelerating  force  ; -A-  x io2  = 1 30  feet  = space  described. 

2. — A body  moving  with  an  acceleration  of  15.625  feet  describes  in  1.5  seconds  a 

15.625  X (1- 5)2  0 j.  , 

space  = = 17.578  feet, 

3.  — A body  propelled  with  an  initial  velocity  of  3 feet,  and  with  an  acceleration 

7 2 

of  5 feet,  describes  in  7 seconds  a space  — 3X7  + 5X  — = 143- 5 feet. 

2 

4.  — A body  which  in  180  seconds  changes  its  velocity^  from  2.5  to  7.5  feet,  trav- 
erses in  that  time  a distance  of  2-5  7 — x 180  = 900  feet. 

5. — A body  which  rolls  up  an  inclined  plane  with  an  initial  velocity  of  40  feet  per 
second,  by  which  it  suffers  a retardation  of  8 feet,  ascends  only  ^ = 5 seconds , and 

O 

402  2 x 8 = 100  feet  in  height,  then  rolls  back,  and  returns,  after  10  seconds,  with 

a velocity  of  40  feet,  to  its  initial  point;  and  after  12  seconds  arrives  at  a distance 
of  40  X 12  — 4 X i22  = 9 6 feet  below  point,  assuming  plane  to  be  extended  backward. 

Circular  ^Motion. 

2 p r n _ 2 p r nf  _ ^ 5500  FP_  W frn_f  2pm  __  jp. 

60  — t ~~  ’ rn  ~2  prn'~J'  5500  — 550  X 60  r 

f 2 p r n'  — = W.  r representing  radius  in  feet,  n number  of  revolutions 

60 

of  circle  per  minute,  n ' total  revolutions,  f force  in  lbs.,  i time  in  seconds,  and  H* 
horse-power. 


Fig.  6. 


nVXotion  on  an  Inclined.  Plane. 
To  Ascertain  Conditions  of  Motion  by  Gravity. 


0 


Assume  A B,  Fig.  6,  an  inclined  plane,  13  C its  base 
AC  its  height,  and  b a body  descending  the  plane;  from 
dot,  centre  ol  gravity  of  body,  draw  b a perpendicular 
to  BC,  representing  pressure  of  b by  gravity;  draw  bo 
parallel  and  b r perpendicular  to  A 13,  and  complete 
j j K parallelogram;  then  force  ba  is  equal  to  both  b o.  b r, 

. , i,  ~ nf  which  b r is  sustained  by  reaction  of  plane,  and 

iorce  b o is  wholly  effective  in  accelerating  motion  of  body. 

Let  tUis  force  be  represented  byf  and  b a,  by  g or  force  of  gravity,  then  by  similar 

A i \s  s* 


J ^ '.will/  V v y y ut  jut  Go  OJ  f 

tr. angle,  / : gy.bo  : ba  : AC:  AB.  Hence,  A-L—  9 — f 

’ A B ~J‘ 


Put  A B _ AC  — A and  Z.ABC  = fl,  then  force  which  produces  motion  on  the 
plane  on /becomes  g - , and  g sin  .a. 

Therefore,  accelerating  force  on  an  inclined  plane  is  constant  and  equations  of 
motion  will  be  obtained  by  substituting  its  value  of  / for  g in  equations  i 2 and 
3,  page  6 1 8.  5 ’ 


ght*  lv 2 

2 l ’ 2 g iC 

2/  g h t j 2 g h s 

t ’ i ’ 
l v J2I  s 


•5  tv,  ■5fft2sin.a,  and 


2 S lv  /: 

V ’ g a’  V 9 /* 


<7  sin.  a 


, and 


g t sin.  a, 


V /?  sin.  a 


2 g sin.  a 
and  V2  g s sin.  a = v. 

= t.  a representing  /_  ABC. 


When  a Body  is  projected  doom  or  up  an  Inclined  Plane,  with  a (riven  Ve- 
limfwm  be'6  diStance  wh“h(  H wil*  be  from  point  of  projection  in  a given 


t v ±- 


2 t 


■ , and  — (2  Z v ± g li  t)  — s. 


Illustration  1. —Length  of  an  inclined  plane  is  100  feet,  and  its  an^le  of  inclina- 
t,on  60  ; what  is  time  of  a body  rolling  down  it,  and  velocity  acquired? 

s:n..6o°—  866. 

2 X 100 

3^6  x7866  = V 7- 18  = 268  seconds,  and  32. 16  X 2.68  X 866  = 74.64  feet 

of2'7flLab^ ’s  Projected  up  an  inclined  plane,  which  rises  i in  6,  with  a velocity 
50  feet  per  second,  what  will  be  its  place  and  velocity  at  end  of  6 seconds  ? 7 

32.16X1X6^  / xX 

2 ~x  6 / — 2°3-  52  feet  from,  bottom , and  50  — (32. 16  X 6 x - J = 


V: 


6X50 

50  — 32. 16  = 17. 84  feet. 
tnSESgK53”d  r’,ane  in  l0ast  ,ime-  » lcl‘StlP  to  its  height, 

Work  Accumulated  in  Moving  Bodies. 

Quantity  of  work  stored  in  a body  in  motion  is  same  as  that  which  would 

AccunXte  6d  "V  by  ffrav!,v  Sfjt  from  the  height  due  to  the  velocity. 
Accumulated  work  expressed  in  foot-lbs.  is  equal  to  product  of  hem-lit  so 
found  in  feet,  and  weight  of  body  in  lbs.  Height  due  to  velocity  is”equal 

dutSSdtwrt  W;,0e,ty  dividi!d  l’y  64'4,  a.nd  "°'k  and  velocity  may  be  de- 
duced oi.ectly  from  each  other  hv  following  rules: 

To  Compute  A dcumulated  Work. 

arid  dh'hiT^r,diiply  wiigllt  V1  lb?‘  bv  S(luare  of  velocity  in  feet  per  second, 
an.l  divide  by  64.4,  and  quotient  is  accumulated  work  in  foot-lbs. 
r2  x m 

~b4.4  • °L  = w X h.  W representing  worlc,  w weight  in  lbs.,  and 


Or.  W = - 


h height  due  to  velocity  in  feet  per  second. 


6 20 


MECHANICS. DYNAMICS. 


60 


Work  ky  Percussive  Force. 

If  a wedge  is  driven  by  strokes  of  a hammer  or  other  heavy  mass,  effect 
of  percussive  force  is  measured  by  quantity  of  work  accumulated  in  stricken 
body  This  work  is  computed  by  preceding  rules,  from  weight  of  body 
and  velocity  with  which  a stroke  is  delivered,  or  directly  from  height  of 
fall,  if  gravity  be  percussive  power. 

Useful  work  done  through  a wedge  is  equal  to  work  expended  upon  it, 
assuming  that  there  is  no  elastic  or  vibrating  reaction  from  the  stroke,  as  if 
the  work  had  been  exerted  by  a constant  pressure  equal  to  weight  of  strik- 
ing body,  exerted  through  a space  equal  to  height  of  fall,  or  height  due  to 
its  final  velocity. 

If  elastic  action  intervenes,  a portion  of  work  exerted  is  absorbed  in  an 
elastic  stress  to  resisting  body ; and  the  elastic  action  may  be,  in  some  cases, 
so  great  as  to  absorb  the  work  expended. 

T he  principle  of  action  of  a blow  on  a wedge  is  alike  applicable  to  action 
of  the  stroke  of  a monkey  of  a pile-driver  upon  a pile.  . 

If  there  be  no  elastic  action,  the  work  expended  being  product  of  weight 
of  monkey  by  height  of  its  fall,  is  equal  to  work  performed  in  driving  the 
pile:  that  is,  to  product  of  resistance  to  its  descent  by  depth  through  which 
it  is  driven  by  each  blow  of  monkey. 

Illustration. — If  a horse  draws  200  lbs.  out  of  a mine,  at  a speed  of  2 miles  per 
hour,  how  many  units  of  work  does  he  perform  in  a minute,  coefficient  of  friction  .05  r 

2 X 5280  __  x^feet  per  minute.  Hence,  176  X 200  .05  X 200  = 35  210  units. 

Decomposition  of  Force. 

By  parallelogram  of  force  it  is  il- 
lustrated how  a vessel  is  enabled  to 
be  sailed  with  a free  wind  and  against 
one. 

Assume  wind  to  be  free  or  in  direction 
of  arrows,  Fig.  7,  and  perpendicular  to 
line  A B,  the  course  of  vessel. 

Let  line  m o represent  direction  and 
force  of  wind,  and  rs  plane  of  sail;  from 
0 draw  0 u perpendicular  to  r s , and 
from  m perpendicular,  m v on  r s,  and 
mu  on  ou. 

By  principle  of  parallelogram  of  forces, 
force  m 0 may  be  decomposed  into  ov 
and  ou.  since  they  are  the  sides  of  parallelogram  of  which  m o,  representing force 
of  wind,  is  diagonal.  Force  of  wind,  therefore,  is  measured  by  ou,  both  in  magm- 
tude  and  direction,  and  represents  actual  pressure  on  sail.  Q 

Draw  u n and  u x parallel  to  0 A and  o w,  thus  fornung  parallclog  am  w n o x 
1 Hence  force  0 u is  equal  to  the  two,  0 n 

and  o x.  Force  o n acts  in  a direction 
perpendicular  to  vessel’s  course  and  that 
of  0 x is  to  drive  vessel  onward. 

It  can  thus  be  shown  that  when  di- 
rection of  sail  bisects  angle  m o B,  the 
effect  of  o x is  greater  than  wffien  sail  is 
in  any  other  position. 

Assume  wind  to  be  ahead  as  in  direc- 
tion of  arrow’s,  Fig.  8.  Let  0 m repre- 
sent direction  and  force  of  w'ind,  and  r s 
direction  of  sail;  from  0 draw  on,  and 
proceed  as  before,  and  o u represents  the 
effective  force  that  acts  upon  the  sail, 
0 n that  which  drives  her  to  leeward,  and 
0 x that  which  drives  her  on  her  course. 

For  full  treatises  on  this  subject,  see  John  C.  Trautwine’s  Engineer's  Pocket-book  1872  ; Bull’s  Ex- 
perimental Mechanics,  London,  1871 ; and  Dynamics,  Construction  of  Machinery,  etc.,  by  G.  Finden 

Warr,  London,  1851. 


MECHANICS. MOMENTS  OF  STRESS  ON  GIRDERS,  ETC.  621 


MOMENTS  OF  STRESS. 

To  Describe  and  Coixipate  Moments  of  Stress  in  GJ-irders 
or  Beams. 


Fig.  i. 


Beam  Supported  at  Both  Ends. 


Loaded  in  Middle , Fig.  i. — Assume 
A B beam.  At  middle-  erect  W c — 
Wl  „ ~ 

. Connect  A c and  c B,  and  ver- 

4 

tical  distances  between  them  and  A B 
“JB  will  give  moment  required. 

ib'  w* 

I bus,  — — - = M at  any  point.  W rep- 

. , , „ resenting  weight  or  load , l length  of 

span,  x horizontal  distance  from  nearest  support  at  which  M is  required , and  M mo- 
ment of  stress. 

Illustration. — Assume  l^z  io  feet,  W = io  lbs.,  and  x — % feet. 

Then,  W c = -°--X  10  = 25  lbs.  at  centre  of  span  ; ■—  l5  at  x. 

2 

Loaded  at  Any  Point , Fig  2. 

Proceed  as  for  previous  figure. 
W a b 

— r—  or  vv  c — maximum  load. 

B l 

W xb  „ 

— Y~  — M between  A and  W. 

a representing  least  distance  of  W to  support , Wxa  _ vr  w „ 

and  b greatest  distance.  1 — ^ between  W and  B. 

Illustration. — Take  elements  as  before  with  a = 3 feet,  and  x = 1.5  and  3.5  feet. 

Then,  We  = ^3X7  = 2i  ^ ^ ^ xoX±s>Lz  = ^ ^ ’ 

IO  IO 

between  A and  W,  and  — * 3 5 * 3 — IO  ^ af  x iefween  an ^ g 
10 

Note.  — x must  be  taken  from  the  pier  which  is  on  the  same  side  of  W as  * is. 

Loaded  with  Two  Equal  Weights  at  Equal  Distances  from  Ends , alike  to  a Trans- 
verse Girder  as  for  a Single  Line  of  Railway.—  Fig.  3. 

At  point  of  stress  of  weights 
erect  W c and  W d , each  ~ W a. 
Connect  A c d and  B,  and  vertical 
-t-j2  distances  between  A B,  as  defined 
^ by  cd,  will  give  moments. 

W (l  — 0)  TTr 

= W a = w b = M at 

r, 1 2 

any  point  between  weights. 

Loaded  with  Four  Equal  Weights , symmetrically  bearing  from  Centre , alike  to  a 
Transverse  Girder  as  for  a Double  Line  of  Railway.—  Fig.  4. 


Fig.  3- 


t 


w 


IP 


4-  d e At  W and  to"  erect  W c,  and 

w"  i = 2 W a,  and  at  and  re' 
erect  w cZ,  «/e,  each  ==  W (2  <2  + a'). 

Connect  A c d ei  and  B,  and  or- 
dinates to  A B will  give  mo- 
ments. 

W (2  a-\-  a')  =1  M at  w and  w'\ 
2 W a — M at  W and  w". 

Illustration. — Assume  W each 
10  lbs.  2 feet  apart,  and  1 10  feet. 
Then,  10  (2  x 2 -f  2)  = 60  at  w or  w\  and  2 x 10  x 2 = 40  at  W or  w". 


3-  T 

c 

c 

1 

i 

\ 

622  MECHANICS. MOMENTS  OE  STRESS  ON  GIRDERS,  ETC. 

Fj  m Loaded  at  Different  Points.—  Fig.  5. 

JS-— Locate  three  weights,  W,  w , and 
/|  ! \ w',  as  at  a b , az  bx,  a2  b2. 

Draw  A c B,  A d B,  and  A e B,  for 
three  separate  cases,  as  by  formula, 

W ab  „ 

— — , Fig.  2. 

Produce  W c until  Wo=Wr,W s, 
IB  and  Wc;  W d until  wu  — wn , w v 
w and  w d,  and  w'  e to  w' m in  like 
^ manner. 

Connect  A oum  and  B,  and  an  or- 


A/m 


0 

s 1 

s 

f y 

<2^ 

v 

\ \ 

r 

n. 

% Www'.  f manner. 

0--- 


f a1--—-y- bx-- :| 

i a2 — x- b2 — 


% dinate  therefrom,  to  A B will  give 
l moment  or  stress  at  the  point  taken. 

Illustration. — Take  a = 2 feet,  az  = 4,  a2  ==  6,  b = 8,  Z>i  = 6,  = 4>  x 2> 


ty,  and  w'  each  10  lbs.,  and  l = 10  feet. 

Then  y(Wax-)-waIa:-l-w,&2x)  = Mat  as. 

__  280  Q 7, 

Take  x = 2.  Then  — (10  X 2 X 2 + 10  X 4 X 2 + 10  X 8 X 2)  = — — 28  s. 

10 

440  „ 

x = 4.  Then  — (10X2X2  + 10X4X2  + 10X8X4)  = — = 44 

Take  a;  = 5.  Then  ^ (10  X 2 X 5 + 10  X 4 X 5 + 10  X 4 X 5)  = ~ = 5°  lbs. 

Loaded  with  a Rolling  Weight.— 

Fig.  6.  Fig.  6. 

Define  parabola  A c B as  deter- 
W l 

mined  by  — = the  ordinate  at  c, 

and  vertical  distances  between  A B 
will  S>ve  moments. 

& W x(l  — x)  . . . 

==  M at  any  point. 

Loaded  Uniformly  its  Entire  Length.—  Define  parabola  as  at  Fig.  6,  ordinate  of 
which  at  c = — — • L representing  stationary  or  dead  load  per  unit  of  length. 


L x 


(l  — x)  — M at  any 


and  = M at  centre. 


Loaded  with  Two  Connected  .Weights,  moving  in  either  Direction,  alike  to  a Locomo- 
tive or  Car  on  a Railway. — Fig.  7. 

pig  7 e Define  parabola  A c B as  deter- 


( W -4-  w)  l 
mined  by  = c, 


At  A and  B erect  Ae,  B i = rv  d 
rJl  JL  \ i connect  A i and  B e,  and  vertical 

A,  v—'—+r  d • di 

% 1 „„  — — ij? w 


<|pr  ~ Y 


distances  between  A 0 B and  A c B 
will  give  moments. 


w d 


at  any  point. 

Position  of  W at  greatest  moment , when  x=-±  2 (W-f«q 


| [(W  + w)  {l  — x)  — wd]  = M 
Or  if  W and  w are 


l . d 

equal,  when  x = - ± — 
“ 4 


Illustration.— Assume  x = 3,  d = 4,  and  W w each  10  lbs.,  and  1 10  feet. 
Then  A (w+75  X io— ^ - 1^X4)  = M any  point,  as  at  W r,  to  r. 


MECHANICS.  MOMENTS  OF  STRESS  ON  GIRDERS,  ETC.  623 


Shearing,.  Stress. 

To  Determine  Shearing  Stress  at  any  Dart  of  a Girder 
or  and  under  any  Distribution  of  Load. 

Fl£-  8*  c Required  to  determine  stress  of  a 

AT  | i -R  beam  at  any  point  as  c,  Fig.  8. 

/yA J f|F  Assume  W = load  between  A and 

S c,  and  w that  between  B and  c. 

Then  Sx  at  c = P — W,  or  P'  — iv. 


The  greater  of  the  two  values  to  be  taken. 

S x representing  shearing  stress  at  any  point  x,  P and  P'  the  reaction  on  «/„««.•/« 
any  poM  to‘“*  m bmm  between  suPPorts,  W dnd  » loads  or  stress  concentrated  at 

'i’o  Describe  and  Ascertain  Shearing  Stress  in  a 
Girder  or  Beam. 


Loaded  Uniformly.  Fig.  9. 

At  A and  B,  erect  A c,  B e,  each 
equal  to  — . Connect  c and  e at 

middle  of  span  as  at  n,  and  vertical 
distances  between  A B and  cne  will 
give  shearing  stresses  as  determined 
by  the  ordinates  tocne. 

l r 


be  disregarded.  L representing  distributed  load  per  unit  of  length. 
Illustration. — Assume  W = 10  lbs.  per  foot,  l — 10,  and  x = 2.5  feet. 

Then  10  ^ — 2.5^  = 25  lbs. 


S.  Sign  of  result  to 


Note.— The  moment  of  rupture  at  any  point,  produced  by  several  loads  aetino- 

“=teSieiy.eqUa‘  t0  th0  — 

AD^EfLondonTs^  Diagrams  see  Straius  in  GirdCTS>  by  William  Humber, 


Operation  deduced  by  Graphic  Delineation  of  Greatest  Stress,  with  a 
Uniformly  Distributed  Load  of  4000  Lbs.— Fig.  10. 


Determine  moment  of  weights  by 

formulas.^?,  211  and  lill 
l l ’ l 


IB 

Assume  W = 7ooo  lbs. , w = 4000, 
and  w'  = 3000,  m — j feet,  n = 13 
r—  T3;  s—7i  °—3 , v=i7,  and  1= 20. 


7000  X 13  X 7 

— = 31  850, 


Then  W = 

w — 4000  X 13  X 7 , 3000X3X17 

2o  — 18200,  and  w __ — — 7650,  and  let  fall  perpendic 

ulars  thereto,  as  3 d,  2 c,  and  1 b. 


upo^perpendimilars  frl?’  “n,fUm  °f distances  of  intersections  of  these  lines 
these  points  ’ 3’  ’ 1 resPectlveIy.  will  give  stress  upon  A B at 

To  determine  Greatest  Stress  at  Greatest  Load. 

Stress  at  3d  =31850  I Stress  at  1 b = i7  ; 7650  ; 3 - x „0 

“20=13:18200:7=9800  I / 3 — 1 350 

7XX3X4000X.5  „ 43000 

52  100  lbs.,  concentrated  load  at  W,  and  proportion 


of  uniformly  distributed  load  of  4000  lbs 


624 


MECHANICAL  POWERS. LEVER. 


MECHANICAL  POWERS. 

Mechanical  Power  is  a compound  of  Weight , or  Force  and  Velocity: 
it  cannot  be  increased  by  mechanical  means. 

The  Powers  are  three  in  number — viz.,  Leyer,  Inclined  Plane,  and 
Pulley. 

>^OTEi a Wheel  and  Axle  is  a continuous  or  revolving  lever , a Wedge  a double  in- 

clined plane,  and  a Screw  a revolving  inclined  plane. 

LEVER. 

Levers  are  straight,  bent,  curved,  single,  or  compound. 

To  Compute  Lengtli  of  a Lever. 

When  Weight  and  Power  are  given.  Rule.— Divide  weight  by  power, 
and  quotient' is  leverage,  or  distance  from  fulcrum  at  which  power  supports 
weight. 

Or,  ^ —p.  w representing  weight , P power , and  p distance  of  power  from  fulcrum. 

Example.— A weight  of  1600  lbs.  is  to  be  raised  by  a power  or  force  of  80;  re- 
quired length  of  longest  arm  of  lever,  shortest  being  1 foot. 

1600  -T-  80  = 20  feet. 

To  Compute  Weight  tliat  can  He  raised,  "by-  a Lever. 

When  its  Length , Power , and  Position  of  its  Fulcrum . are  given.  Rule.-— 
Multiply  power  by  its  distance  from  fulcrum,  and  divide  product  by  dis- 
tance of  weight  from  fulcrum. 

Or  — = W.  w representing  distance  of  weight  from  fulcrum. 

’ w 

Example.— What  weight  can  be  raised  by  375  lbs.  suspended  from  end  of  a lever 
8 feet  from  fulcrum,  distance  of  weight  from  fulcrum  being  2 feet? 

375  X^2=  1500  lbs. 


To  Compute  ^Position.  of  Fulcrum. 

When  Weight  and  Poioer  and  Length  of  Lever  are  given , and  when  Ful- 
crum is  between  Weight  and  Power.  Rule— Divide  weight  by  power,  add 
1 to  quotient,  and  divide  length  by  sum  thus  obtained. 

j,  -4-  _j-  — w.  L representing  entire  length  of  lever. 

Example.— A weight  of  2460  lbs.  is  to  be  raised  with  a lever  7 feet  long  and  a 
power  of  300;  at  what  x>art  of  lever  must  fulcrum  be  placed  ? 

2460  -4-  300  = 8. 2,  and  8. 2 + 1 = 9. 2.  Then  (7  X 12)  84  9. 2 = 9. 13  ins. 

When  Weight  is  between  Fulcrum  and  Power.  Rule.— Divide  length 
by  quotient  of  weight,  divided  by  power. 


To  Compute  Length.  of  Arm  of  Lever  to  which 
Weight  is  attached. 

When  Weight , Power , and  Length  of  Arm  of  Lever  to  which  Power  is  ap- 
plied are  given.  Rule.  — Multiply  power  by  length  of  arm  to  which  it  is 
applied,  and  divide  product  by  weight. 


MECHANICAL  POWERS. LEVER. 


Example. — A weight  of  1600  lbs.,  suspended  from  a lever,  is  supported  by  a power 
of  80,  applied  at  other  end  of  arm,  20  feet  in  length ; what  is  length  of  arm  ? 

80  x 20-i-  1600  = 1 foot. 

Note.— These  rules  apply  equally  When  fulcrum  {or  support)  of  lever  is  between 
weight  and  power  ;*  when  fulcrum  is  at  one  extremity  of  lever,  and  power , or  weight , 
at  the  other  ; t and  when  arms  of  lever  are  equally  or  unequally  bent  or  curved. 

To  Compute  Power  Required.  to  Raise  a given  Weight. 

When  Length  of  Lever  and  Position  of  Fulcrum  are  given.  Rule. — Mul- 
tiply weight  to  be  raised  by  its  distance  from  fulcrum,  and  divide  product 
by  distance  of  power  from  fulcrum. 

. W w 
Or,  = P. 

P 

Example — Length  of  a lever  is  10  feet,  weight  to  be  raised  is  3000  lbs.,  and  its 
distance  from  fulcrum  is  2 feet;  what  is  power  required? 


3000  x 2 


6000 

Z~S~Z 


: 750  lbs. 


To  Compute  Length  of  Arm  of  Lever  to  -which.  Lower 
is  applied. 

When  Weight , Power,  and  Distance  of  Fulcrum  are  given.  Rule. — Mul- 
tiply weight  by  its  distance  from  fulcrum,  and  divide  product  by  power. 

„ W w 
Or,  -p-  =p. 

Example.— A weight  of  400  lbs.,  suspended  15  ins.  from  fulcrum,  is  supported  by 
a power  of  50,  applied  at  other;  what  is  length  of  the  arm  ? 

400  X 15  -r-  50=  120  ins. 


Fig.  1. 


'yf  are  computed  directly  as  ah  to  b c. 


When  Arms  of  a Lever  are  bent  or  curved , 
Distances  taken  from  perpendiculars,  drawn 
from  lines  of  direction  of  weight  and  power, 
must  be  measured  on  a line  running  horizon- 
c tally  through  fulcrum,  as  a b c,  Figs.  1 and  2. 

When  A rms  of  a Lever  are  at  Right  A ngles , 
and  Power  and  Weight  are  applied  at  a Right 
Angle  to  each  other , 

Fig.  3,  The  moments 


Fig.  2. 


Thrust,  or  press- 
ure on  fulcrum, 
is  in  this  case  less 
than  sum  of  pow- 
er and  weight ; 
and  it  may  be 
i determined  by 
drawing  a paral- 
lelogram upon 
the  two  arms  of 

O^S  lever, arms  repre-  | 
w P senting  inverse- 

ly their  respec- 
tive forces.  That  is,  a b represents  magnitude  and  direction  of  weight  W, 
and  b e of  power  P.  Diagonal  ob  of  parallelogram  represents  magnitude 
and  direction  of  third  force,  or  thrust  upon  fulcrum. 

* Pressure  upon  fulcrum  is  equal  to  sum  of  weight  and  power, 
t Pressure  upon  fulcrum  is  equal  to  difference  of  weight  and  power. 

3 G 


626  MECHANICAL  POWERS. LEVER. WHEEL. 

Fig.  4.  When  same  Lever  is  bovne  into  an  Oblique 

Position , Power  continuing  to  act  Horizontally, 
Fig.  4,  Draw  vertical  a v through  end  0 of 
lever,  and  produce  the  power  line  p c to  meet 
it  at  a.  Complete  parallelogram  avbr ; then 
sides  r b and  b v are  perpendiculars  to  direc- 
tions to  power  and  weight,  on  which  moments 
are  computed. 

P Consequently,  moment  P x r b . = moment 
W X a v,  and  a diag(  nal,  b a,  is  resultant  thrust 
at  fulcrum. 


Fig-  5- 


When  Power  does  not  act  Horizon- 
tally, Fig.  5,  but  in  some  other  direc- 
tion, a p , produce  the  power  - line  p a 
and  draw  b c perpendicular  to  it ; draw 
b 0 , then  moments  are  computed  on 
perpendiculars  b c,b  0,  and  Pxci  = 
\V  x b o. 

If  several  weights  or  powers  act 
upon  one  or  both  ends  of  a lever,  con- 
dition of  equilibrium  is 

V p -J-  P ' p'  -j-  P " p",  etc.,  = W w -f- 
W'  w\  etc. 

In  a system  of  levers,  either  of  similar,  compound,  or  mixed 
kinds,  condition  is  P p p' p"  ^ 


Illustration. — Let  P = 1 lb.,  p and  p ' each  10  feet,  p"  1 foot;  and  if  w and  10' 
be  each  1 foot,  and  w"  1 inch,  then 


==  1200;  that  is,  1 lb.  will  support  1200,  with  levers 


1 X 120  X 120  X 12 172  800 

12  X 12  X 1 144 

of  the  lengths  above  given. 

Note.  — Weights  of  levers  in  above  formulas  are  not  considered,  centre  of  gravity 
being  assumed  to  be  over  fulcrums. 

* • • bli'PJ  ’ J i 1 f ‘ ' t 

General  Rule,  therefore,  for  ascertaining  relation  of  Power  to 
Weight  in  a lever,  whether  straight  or  curved,  is.  Power  multiplied  by  its 
distance  from  fulcrum  is  equal  to  weight  multiplied  ly  its  distance  from 
fulcrum.  0r>  p . w . . w . or  P p = W w ; and 


W w 


p. 


p p 


— w. 


W w 


=zp. 


P p 

Wz 


WHEEL  AND  AXLE. 

A Wheel  and  Axle  is  a revolving  lever. 

Power,  multiplied  bv  radius  of  wheel,  is  equal  to  weight,  multiplied  by 
radius  of  axle. 

As  radius  of  wheel  is  to  radius  of  axle,  so  is  effect  to  power. 

R R P 

Or,  P R = W r.  Or,PV  = Wv.  Or,R:r::W;P.  Or,P---W;  -^-  = r 

— R.  R and  r representing  radii , and  V and  v velocities  of  wheel  and  axle . 


MECHANICAL  POWERS. WHEEL  AND  AXLE. 

When  a series  of  wheels  and  axles  act  upon  each  other,  either  by  belts  or 
teeth,  weight  or  velocity  will  be  to  power  or  unity  as  product  of  radii,  or 
circumferences  of  wheels,  to  product  of  radii,  or  circumferences  of  axles. 

Illustration.— If  radii  of  a series  of  wheels  are  9,  6,  9,  10,  and  12,  and  their  pin- 
ions have  each  a radius  of  6 ins.,  and  power  applied  is  10  lbs.,  what  weight  will 
they  raise? 

10  X 9 X 6 X 9 X 10  X 12  __  583  200  _ 

6X6X6X6X6  7776  ~ 75  S' 

Or,  if  1st  wheel  make  10  revolutions,  last  will  make  75  in  same  time. 

To  Compute  Tower  of  a Combination  of  Wheels  and  an 
Axle  or  Axles,  as  in  Cranes,  etc. 

# Rule.— Divide  product  of  driven  teeth  by  product  of  drivers,  and  quo- 
tient is  their  relative  velocity ; which,  multiplied  by  length  of  lever  or  arm 
and  power  applied  to  it  in  pounds,  and  divided  by  radius  of  barrel,  will  give 
weight  that  can  be  raised. 

v IV  w r 

^r,  ~ = W ; Or,  W r = vl  P ; Or,  ~j  — P.  I representing  length  of  lever  or 

arm , r radius  of  barrel,  P power,  v velocity,  and  W weight. 

Example  i. — A power  of  18  lbs.  is  applied  to  lever  or  winch  of  a crane,  length  of 
it  being  8 ins.,  pinion  having  6 teeth,  driving-wheel  72,  and  barrel  6 ins.  diameter. 

= 12,  and  12X8X18  = 1728,  which,  -1-  3,  radius  of  barrel,  = 576  lbs. 

2. — A weight  of  94  tons  is  to  be  raised  360  feet  in  15  minutes,  by  a power,  velocity 
of  which  is  220  feet  per  minute;  what  is  power  required  ? 

360  -r- 15  = 24  feet  per  minute.  Hence  — X 94  ==  10. 2545  tons. 


Compound  -Axle,  ox*  Oliixiese  "Windlass. 

Axle  or  drum  of  windlass  consists  of  two  parts,  diameter  of  one 
being  less  than  that  of  the  other. 

The  operation  is  thus : At  a revolution  of  axle  or  drum,  a portion  of  sus- 
taining rope  or  chain  equal  to  circumference  of  larger  axle  is  wound  up,  and 
at  same  time  a portion  equal  to  circumference  of  lesser  axle  is  unwound. 
Effect,  therefore,  is  to  wind  up  or  shorten  rope  or  chain,  by  which  a weight 
or  stress  is  borne,  by  a length  equal  to  difference  between  circumferences^of 
the  two  axles.  Consequently,  half  that  portion  of  the  rope  or  chain  will  be 
shortened  by  half  difference  between  circumferences. 


To  Compute  Elements  of  a Wlieel  and  Compound 
Axle,  or  Cliinese  Windlass.— Eig.  6. 


Rule.— Multiply  power  by  radius  of  wheel,  arm,  or  f,v  6 

bar  to  which  it  is  applied,  and  divide  product  by  half  ° 
difference  of  radii  of  axle,  and  quotient  is  weight  that  ai- 
can  be  sustained.  • P 

_ PR  TTT  ^ 

0r>  ~ r7)  ~W‘  R representing  radius  of  wheel,  etc.,  and  r and  r' 
radii  of  axle  at  its  greatest  and  least  diameters. 

Example.— What  weight  can  be  raised  by  a capstan,  radius  of  its  bar  a 
5 feet,  power  applied  50  lbs.,  and  radii,  r r',  of  axle  or  drum  6 and  5 ins.  ? ’ 


50  X 5 X 12 
•5(6-5) 


3°°°  * 7, 

— — = 0000  lbs. 
■5 


628  MECHANICAL  POWERS. INCLINED  PLANE. 


"Wheel  and  Pinion  Combinations,  or  Complex 
Wheel-work. 

Power,  multiplied  by  product  of  radii  or  circumferences,  or  number  of 
teeth  of  wheels,  is  equal  to  weight,  multiplied  by  product  of  radii  or  circum- 
ferences, or  number  of  teeth  or  leaves  of  pinions. 

Or,  PRR'  R",  etc.,  ==  W r r'  r",  etc. 

Note. — Cogs  on  face  of  wheel  are  termed  teeth , and  those  on  surface  of  axle  are 
termed  leaves  ; the  axle  itself  in  this  case  is  termed  a pinion. 

Liacli  and  Pinion. 


Xo  Compute  Power  of  a Ftachc  and  iPinion. 

Rule. — Multiply  weight  to  be  sustained  by  quotient  of  radius  of  pinion, 
divided  by  radius  of  crank,  and  product  is  power  required. 

Or,  W^-  = P. 
li 

When  Pinion  on  Crank  Axle  communicates  with  a Wheel  and  Pinion. 
Rule. — Multiply  weight  to  be  sustained  by  quotient  of  product  of  radii  of 
pinions,  divided  by  radii  of  crank  and  wheel,  and  product  is  power  required. 

r r' 

°r,Win?  = P. 

Example. — If  radii  of  pinions  of  a jack-screw  are  each  one  inch;  of  crank  and 
wheel  io  and  5 ins. ; what  power  will  sustain  a weight  of  750  lbs.  ? 


1 X 1 750 

750  X — — - = = 15  Ms. 

10X5  50 


INCLINED  PLANE. 

Xo  Compute  Length,  of  Base,  Height,  or  Length. 
When  any  Two  of  them  are  given , and  ivhen  Line  o f Direction  of  Power 
or  Traction  is  Parallel  to  Face  of  Plane. — Proceed  as  in  Mensuration  or 
Trigonometry  to  determine  side  of  a right-angled  triangle,  any  two  of  three 
being  given. 

Xo  Compute  Power  necessary-  to  Support  a "Weight  on 
an  Inclined.  Plane. 

When  Height  and  Length  are  given.  Rule. — Multiply  weight  by  height 
of  plane,  and  divide  product  by  length. 

W h 

Or,  — — ==  P.  h and  l representing  height  and  length  of  plane. 

Example. — What  is  power  necessary  to  support  1000  lbs.  on  an  inclined  plane 
4 feet  in  height  and  6 feet  in  length? 

IOOO  x 4 -r-  6 = 666.67  lbs. 


Xo  Compute  "Weight  that  may  he  Sustained  by  a given 
Lower  on  an  Inclined  Plane. 

When  Height  and  Length  of  Plane  are  given.  Rule. — Multiply  power 
by  length  of  plane,  and  divide  product  by  height. 


Example.— What  is  weight  that  can  be  sustained  on  an  inclined  plane  5 feet  in 
height  and  7 feet  in  length  by  a power  of  700  lbs.  ? 

700  X 7-^5  = 980  lbs. 

Note. — In  estimating  power  required  to  overcome  resistance  of  a body  being 
drawn  up  or  supported  upon  an  inclined  plane,  and  contrariwise,  if  body  is  de- 
scending; weight  of  body,  in  proportion  of  power  of  plane  (i.  e.,  as  its  length  to  its 
height),  must  be  added  to  resistance , if  being  drawn  up  or  supported,  or  to  the  mo- 
ment if  descending. 


MECHANICS. INCLINED  PLANE. 


To  Compute  Heiglit  or  Length  of  an  Inclined.  Plane. 

When  Weight,  and  Power  ancl  one  of  required  Elements  are  given , and 
when  Height  is  required.  Rule.— Multiply  power  by  length,  and  divide 
product  by  weight. 

When  Length  is  required.  Rule.— Multiply  weight  by  height,  and  divide 
product  by  power. 

^ PI  . . Wh  , 

Or,  — = h,  and  — p-  — /. 

To  Compute  Pressure  on  an  Inclined  Plane. 

Rule. — Multiply  weight  by  length  of  base  of  plane,  and  divide  product 
bv  length  of  face. 


A W6 

Or,  — — —pressure. 


b representing  length  of  base  of  plane. 


Example.— Weight  on  an  inclined  plane  is  ioo  lbs.,  base  of  plane  is  4 feet,  and 
length  of  it  5;  required  pressure  on  plane. 

100  X 4 -y*  5 = 80  lbs. 

When  Two  Bodies  on  Two  Inclined  Planes  sustain  each  other , as  by  Connection 
of  a Cord  over  a Pulley , their  Weights  are  directly  as  Lengths  of  Planes. 
Illustration. — If  a weight  of  50  lbs.  upon  an  inclined  plane,  of  10  feet  rise  in  100 
of  an  inclination,  is  sustained  by  a weight  on  another  plane  of  10  feet  rise  in  90, 
what  is  the  weight  of  the  latter  ? 

100  : 90  : : 50  : 45  = weight  that  on  shortest  plane  would  sustain  that  on  largest 

When  a Body  is  Supported  by  Two  Planes , as  Fig.  7,  pressure  upon  them 
7 will  be  reciprocally  as  sines  of  inclinations  of  planes. 

D Thus,  Aveight  is  as  sin.  A B D. 

A Pressure  on  A B as  sin.  1)  B i. 

Pressure  on  B D as  sin.  A B h. 

Assume  angle  A B D to  be  900,  and  D B i,  6o°;  then  angle 
A B h will  be  300;  and  as  sines  of  900,  6o°,  and  300  are  respec- 
tively .1,  .866,  and  .5,  if  weight-- 100  lbs.,  then  pressures  on 
A B and  B D will  be  86.6  and  50  lbs.,  centre  of  gravity  of  weight  assumed  to  be  in  its 
centre. 

When  Line  of  Direction  of  Power  is  parallel  to  Base  of  Plane , power  is 
to  weight  as  height  of  plane  to  length  of  its  base. 

Or,  P : W ::  h : b. 


Hence,  P = 


W h 


P b 
~h 


W = ; h = 


P h 


7 W h 

6=-r- 


When  Line  of  Direction  of  Power  is  neither  parallel  to  Face  of  Plane  nor 
to  its  Base , but  in  some  other  Direction , as  P',  Fig.  8,  power  is  to  weight  as 
sine  of  angle  of  plane’s  elevation  to  cosine  of  angle  which  line  of  power  or 
traction  describes  with  face  of  plane. 

Thus,  P' : W : : sin.  A : cos.  P'  e c. 

Sin.  A : cos.  P'  e c : : P'  : W. 

Cos.  P ' e c : sin.  A : : W : P'. 
Illustration. — A weight  of  500  lbs.  is  required  to  be 
sustained  on  a plane,  angle  of  elevation  of  which, 
c AB,  is  io°;  line  of  direction  of  power  or  traction, 
P' e c,  is  50;  what  is  sustaining  power  required? 

Cos.  P'ec  (50)  = . 996  19  : sin.  A (io°)  = . 173  65  : : 500  : 87. 16  lbs. 

Or,  draw  a line,  B s,  perpendicular  to  direction  of  power’s  action  from  end 
of  base  line  (at  back  of  plane),  and  intersection  of  this  line  on  length,  A c. 
will  determine  length  and  height  (n  r)  of  the  plane. 

3 G* 


n B 


630 


MECHANICS. WEDGE. — SCREW. 


Illustration. — By  Trigonometry  (page  385),  A B,  assumed  to  be  1,  A r and  nr  are 
= .985  and  .171. 


Note.— When  line  of  direction  of  power  is  parallel  to  plane,  power  is  least. 


1.  When  One  Body  is  to  be  Forced  or  Sustained.  Rule.— Multiply  weight 
or  resistance  to  be  sustained  by  depth  of  back  of  wedge,  and  divide  product 
I by  length  of  its  base. 

Example.  — What  power,  applied  to  the  back  of  a wedge  6 ins.  deep,  will  raise  a 
weight  of  15000  lbs.,  the  wedge  being  100  ins.  long  on  its  base? 


2.  When  Two  Bodies  or  Two  Parts  of  a Body  are  Forced  or  Sustained  in  a 
Direction  Parallel  to  Bach  of  Wedge.  Rule.— Multiply  weight  or  resist- 
ance to  be  sustained  by  half  depth  of  back  of  wedge,  and  divide  product  by 
length  of  wedge. 


Note. — The  length  of  a single  wedge  is  measured  on  its  base,  and  of  a double 
wedge,  from  centre  of  its  head  to  its  point. 

Example. — The  depth  of  the  back  of  a double-faced  wedge  is  6 ins.,  and  the 
length  of  it  through  the  middle  10;  what  power  applied  to  it  is  necessary  to  sus- 
tain or  overcome  a resistance  of  150  lbs.  ? 


Note. — As  power  of  wedge  in  practice  depends  upon  split  or  rift  in  wood  to  be 
cleft,  or  in  rise  of  body  to  be  raised,  the  above  rules  as  regards  length  of  wedge  are 
only  theoretical  when  a rift  or  rise  exists. 


A Screw  is  a revolving  inclined  plane. 

To  Compute  Xj&ngtli  and.  Tleiglit  of  Plane  of  a Screw. 

As  a screw  is  an  inclined  plane  wound  around  a cylinder,  length  of  plane 
is  ascertained  by  adding  square  of  circumference  of  screw  to  square  of  dis- 
tance between  threads,  and  taking  square  root  of  sum. 

The  Pitch  or  height  of  a screw  is  distance  between  its  consecutive  threads. 

To  Compute  Power. 

Rule. — Multiply  weight  or  resistance,  to  be  sustained  by  pitch  of  threads, 
and  divide  product  by  circumference  described  by  power. 


Example.— What  is  power  requisite  to  raise  a weight  of  8000  lbs.  by  a screw  of  12 


Hence 


500  X .171 
•985 


= 86.8  lbs.  = product  of  weight  X height  of  plane  -r-  length  of  it. 


"W^edge. 

A Wedge  is  a double  inclined  plane. 

To  Compute  Power. 


15  000X  6 QOOQO 

— = = 900  lbs. 

100  100 


Or,  ^ d ' --  = P.  d representing  depth  of  back , and  l length. 


,50x6^450^.,^ 


10 


To  Compute  Elements  of  a Wedge. 


SCREW. 


ins.  circumference  and  1 inch  pitch? 


8000  X 1 12  = 666.66  lbs. 


To  Compute  'Wei glut. 

Rule.— Multiply  power  by  circumference  described  by  it,  and  divide 
product  by  pitch  of  threads. 

Or,—  = W. 

P 

To  Compute  IPitcli. 

Rule.— Multiply  power  by  circumference  described  by  it,  and  divide 
product  by  weight. 

To  Compute  Circumference. 

Rule.— Multiply  weight  by  pitch,  and  divide  product  by  power. 

„ W p Wp 

urj  — c-  Or , - p .=  r.  r representing  radius. 

When  Power  is  applied  by  a Lever  or  Wheel , substitute  radius  of  power 
for  circumference. 

Illustration.— If  a lever  30  ins.  in  length  was  added  to  circumference  of  screw 
in  preceding  example, 

Then,  12-7-3.416  = 3.819,  and  ^-^  + 30  = 31.9095  — radius  of  power. 

TT  8000  X 1 
Hence 

Compound  Screw. 


W : P : : R n : r n. 

Or,  r n : R n : : P : W.  n representing  continued  product  of  number  of  wheels  or 
axles. 

Illustration.— If  a power  of  150  lbs.  is  applied  to  a crank  of  20  ins.  radius  turn- 
ing an  endless  screw  with  a pitch  of  half  an  inch,  geared  to  a wheel  pinion  of 
which  is  geared  to  another  wheel,  and  pinion  of  second  wheel  is  geared  to  a third 
wheel,  to  axle  or  barrel  of  which  is  suspended  a weight;  it  is  required  to  know 
what  weight  can  be  sustained  in  that  position,  diameter  of  wheels  beinsr  and 
pinions  and  axle  2 ins.  8 f 

150X20X2X3-1416  ... 

"7 — = 37  699-2  lbs.  = power  applied  to  face  of  first  wheel. 

Diameters  of  wheels  and  pinions  being  18  and  2,  their  radii  are  9 and  1 . 

Hence,  1 X 1 X 1 : 9 X 9 X 9 :*•  37 699.2  : 27 482 716.8  lbs. 


632 


MECHANICS. SCREW. PULLEY. 


Differential  Screw. 

When  a hollow  screw  revolves  upon  one  of  less  diameter  and  pitch  (as 
designed  by  Mr.  Hunter),  effect  is  same  as  that  of  a single  screw,  in  which 
the  distance  between  threads  is  equal  to  difference  of  distances  between 
threads  of  the  two  screws. 

Therefore  power,  to  effect  or  weight  sustained,  is  as  difference  between 
distances  of  threads  of  the  two  screws  to  circumference  described  by  power. 

Illustration.  — If  external  screw  has  20  threads,  and  internal  one  21  threads  in 
pitch  of  1 inch,  and  power  applied  describes  a circumference  of  35  ins.,  the  result  or 

power  is  as  — co  — = , or  .002  38.  Hence  — = 14  706. 

1 21  20  420  .00230 


PULLEY. 

Pulleys  are  designated  as  Fixed  and  Movable , according  as  cord  is  passed 
over  a fixed  or  a movable  pulley.  A movable  pulley  is  when  cord  passes 
through  a second  pulley  or  block  in  suspension ; a single  movable  pulley  is 
termed  a runner ; and  a combination  of  pulleys  is  termed  a system  of  pulleys. 

A Whip  is  a single  cord  over  a fixed  pulley. 

To  Compute  Dower  Required  to  Raise  a given  Weiglit. 

When  Number  of  Parts  of  Cord  supporting  Lower  Bloch  are  given , and 
when  only  one  Cord  or  Rope  is  used.  Rule. — Divide  weight  to  be  raised  by 
number  of  parts  of  cord  supporting  lower  or  movable  block. 

Or,  w -4-  n — P.  Or,  n P = W.  n representing  number  of  parts  of  cord  sustain- 
ing lower  block. 

Example. — What  power  is  required  to  raise  600  lbs.  when  lower  block  contains 
six  sheaves? 

When  Cord  is  attached  to  Upper  or  Fixed  Bloch. 

— — go  lbs.  = weight  -4-  number  of  parts  of  rope  sustaining  lower  block. 

6X2 

When  Cord  is  attached  to  Lower  or  Movable  Bloch. 

- 46.15  lbs.  =a  weight  -7-  number  of  parts  of  rope  sustaining  lower  block. 


600 


6X2  + 1 

To  Compute  Weiglit  a given  Dower  will  Raise. 

When  Number  of  Parts  of  Cord  supporting  Lower  Bloch  are  given.  Rule. 
— Multiply  power  by  number  of  parts  of  cord  supporting  lower  block. 

Or,  P n = W. 

To  Compute  1ST  umber  of  Cords  necessary  to  Sustain 
Lower  Block. 

When  Weight  and  Power  are  given. 

Or,  W -. 


Rule. — Divide  weight  by  power. 
-P  = «. 


Fig.  10. 


When  more  than  one  Coifl  is  used. 

In  a Spanish  Burton , Fig.  10,  where  ends  of 
one  cord,  a P,  are  fastened  to  support  and  power, 
and  ends  of  the  other,  c 0,  to  lower  and  upper 
blocks,  weight  is  to  power  as  4 to  1. 

In  another,  Fig.  11,  where  there  are  two  cords, 
a and  0,  two  movable  pulleys,  and  one  fixed 
pulley,  with  ends  of  one  rope  fastened  to  sup- 
port and  upper  movable  pulley,  and  ends  of 
other  fastened  to  lower  block  and  power,  weight 
is  to  power  as  5 to  1. 


Fig.  12. 


Compound  or  Fast  and  Loose  Fallens. 

When  Cord  is  attached  to  Fixed  Block,  Fig.  12.  Rule. — 
Multiply  power  by  the  power  of  2,  of  which  the  index  is 
number  of  movable  pulleys. 

Or,  P 2*  = W. 

Or,  Multiply  power  successively  by  2 for  each  pulley. 

Example  i. — What  weight  will  one  pound  support  in  a system 
of  three  movable  pulleys,  the  cords  being  connected  to  a fixed 
block  on  Fig.  12.  ..  , 0 7, 

1 X 23  — 8 lbs. 

Example  2.— What  would  a like  power  support,  fixed  block  be- 
ing made  movable  and  cord  attached  thereto? 

1 X 2*  — 1 = 15  lbs. 

If  fixed  pulleys  were  substituted  for  hooks  a b c,  Fig.  12,  power 
would  be  increased  threefold;  hence  1 x 33  = 27. 

In  a System  of  Pulleys,  Figs.  13  and  14,  with  any  Number  of  Cords , 0 0,  e ey 
Ends  being  fastened  to  Support. 


Fig.  13. 


W-f-2n=  P; 


W 

“XP  = W;  ~=z2n. 


n rep- 


Fig.  14. 


resenting  number  of  distinct  cords. 

Illustration.  — What  weight  will  a power 
of  1 lb.  sustain  in  a system  of  two  movable  pul- 
leys and  two  cords  ? 

1 X 2 X 2 = 4 lbs. 

When  fixed  Pulleys,  e e,  are  used  in  Place  p 
of  Hooks,  to  Attach  Ends  of  Rope  to  Sup- 
port.— Fig.  14. 

W-h3”  = P;  3n  X P = W;  W=P  = 3«  g 

Illustration. -What  weight  will  a power  of  5 lbs.  sustain  with  two  movable  and 
three  fixed  pulle5rs,  and  two  cords?  * v v 

5 A 3 A 3 — 45 

When  Ends  of  Cord  or  Fixed  Pulleys  are  fastened  to  Weight , as  by  an  Inver- 
sion of  the  last  Figures,  putting  Supports  for  Weights,  and  contrariwise. — 
Jb  igs.  13  and  14. 

_.  W w 

7T»_-.»=p;  (2*-x)P  = W;  - = (2^-1). 


Fig.  14. 


(2* -I) 

w 


= P: 


(3«-i)P  = W; 


P 
W 

W=(3w-i). 


(3n~  1) 

Illustration  —What  weight  will  a power  of  1 lb.  sustain  in  a system  of  two  mov- 
able pulleys  and  two  cords,  and  one  of  two  movable  and  two  fixed  pulleys  and  two 


1X2X2- 


1 = 0 lbs. 


1 X 3 X 3 — 1 = 8 lbs. 


When  Cords  sustaining  Pulleys  are  not  in  a Vertical  Direction.— Fig.  15. 

Fig-  x5-  Fig.  15,  is  vertical  line  through  which  weight  bears  and 

from  0 draw  or,  os  parallel  to  D e and  A e. 

; Forces  acting  at  e are  represented  by  lines  e s,  e r,  and  e o • 
and  as  tension  of  every  part  of  cord  is  same,  and  equal  to 
power  P.  sides  o s and  0 r of  parallelogram  must  be  equal  and 
■ therefore  diagonal  e 0 divides  the  angle  ros  into  two  equal 
dk  portions  Hence  the  weight  will  always  fall  into  the  position 
V m which  the  two  parts  of  cord  A e and  e D will  be  equally 
P inclined  to  vertical  line,  and  it  will  bear  to  power  same  ratio 
aseotoei 

Therefore  W : P : : 2 cos.  .50:1.  e representing  angle  A e D. 

P*„C0S7  5p  = W'  ThatJs’ twice  P°wer-  multiplied  by  cosine  of  half  angle 
of  cord,  at  point  of  suspension  of  weight,  is  equal  to  weight.  ° 


634 


METALS. — ALLOYS  AND  COMPOSITIONS. 


Illustration  —What  weight  will  be  sustained  by  a power  of  5 lbs.,  with  an  ob- 
lique movable  pulley,  Fig.  15,  having  an  angle,  A e D,  of  300  ? 

5 X 2 X -965  93  = 9-6593  lbs.  = twice  power  X cos.  150. 

When  Direction  of  Cord  is  Irregular , Weight  not  resting  in  Centre  of  it. 


p sm.  a 
W ““  sin.  (a  4-  b)  ’ 


P sin.  w. 


sin.  a 

greater  and  lesser  angles  of  cord  at  e. 


W sin.  a 
sin.  (a-f-  b) 


— P.  a and  b representing 


METALS. 

ALLOYS  AND  COMPOSITIONS. 

A-lloy-  is  the  proportion  of  a baser  metal  mixed  with  a finer  or  purer, 
as  copper  is  mixed  with  gold,  etc. 

Amalgam  is  a compound  of  Mercury  and  a metal— a soft  alloy. 

Compositions  of  copper  contract  in  admixture,  and  all  Amalgams  ex- 
pand. 

In  manufacture  of  Alloys  and  Compositions,  the  less  fusible  metals 
should  be  melted  first. 

In  Compositions  of  Brass,  as  proportion  of  Zinc  is  increased,  so  is 
malleability  decreased. 

Tenacity  of  Brass  is  impaired  by  addition  of  Lead  or  Tin. 

Steel  alloyed  with  one  five-hundredth  part  of  Platinum,  or  Silver,  is 
rendered  harder,  more  malleable,  and  better  adapted  for  cutting  instru- 
ments. _ 

Specific  gravity  of  alloys*  does  not  follow  the  ratios  of  those  of  their 
components ; it  is  sometimes  greater  and  sometimes  less  than  the  mean. 

Composition  Tor  "Welding  Cast  Steel. 

Borax  qi  parts;  Sal-ammoniac,  9 parts.  Grind  or  pound  them  roughly  together; 
fufe  them  in  a metal-pot  over  a clear  fire,  continuing  heat  until  a 1 spume  has  dis- 
appeared from  surface.  When  liquid  is  clear,  pour  composition  out  to  cool  and  con- 
crete, and  grind  to  a fine  powder;  then  it  is  ready  for  use. 

To  use  this  composition,  the  steel  to  be  welded  should  be  raised  to  a bright  yellow 
heat  then  dip  it  in  the  welding  powder,  and  again  raise  it  to  a like  heat  as  before, 
it  is  then  ready  to  be  submitted  to  the  hammer. 


ITxisible  Compounds. 


Compounds. 


Rose’s,  fusing  at  2000 

Fusing  at  less  than  2000 

Newton’s,  fusing  at  less  than  2120. 
Fusing  at  1500  to  1600 


33-3 


Tin. 


25 

19 


Lead. 


23 

33-3 

31 

25 


Bismuth. 


50 

33-4 

50 

50 


Cadmium. 


Solders. 


Solder  is  an  alloy  used  to  make  joints  between  metals,  and  it  must  be 
more  fusible  than  the  metals  it  is  designed  to  unite,  and  it  is  distinguished 
as  hard  and  soft,  according  to  the  temperature  of  its  fusing. 

The  addition  of  a small  portion  of  Bismuth  increases  its  fusibility. 

. For  a table  of  Alloys,  having  densities  different  from  a mean  of  their  components,  see  D.  K.  Clark’s 
Manual,  London,  1877,  page  201. 


METALS. — ALLOYS  AND  COMPOSITIONS.  635 


.Alloys  and.  Compositions. 


Copper. 

Zinc. 

Argentan 

55 

24 

Aluminum,  brown 

95 

— 

Babbitt’s  metal  * 

3-7 

— 

Brass,  common 

84-3 

5-2 

u 44  

75 

25 

“ u hard 

79-3 

6.4 

“ instruments 

92.2 

— 

“ locomot.  bearings. 

90 

I 

44  Pinchbeck 

80 

20 

“ red  Tombac 

88.8 

II. 2 

“ rolled 

74-3 

22.3 

“ Tutenag 

50 

31 

11  very  tenacious... 

88.9 

2.8 

“ wheels,  valves. .. . 

9° 

— 

“ white 

10 

80 

“ “ 

3 

90 

il  wire 

7 

— 

67 

33 

11  yellow,  fine 

66 

34 

Britannia  metal 

— 

— 

When  fused  add 

— 

— 

Bronze,  red 

n rr 

87 

13 

86 

11. 1 

“ yellow 

67.2 

31.2 

“ Gun  metal,  large 

9° 

“ 44  small 

93 

— 

“ 44  soft. 

95 

— 

“ Cvmbals 

80 

— 

“ Medals 

93 

— 

44  Statuary  

9J,4 

5-5 

Chinese  silver 

58.1 

17.2 

4i  white  copper. . . 

40.4 

25-4 

Church  bells 

80 

5-6 

44  44  

69 

Clocks,  Musical  bells 

87-5 

— 

Clock  bells 

72 

— 

German  silver 

33-3 

33-4 

“ 44  

40.4 

25-4 

44  “ fine 

49-5 

24 

Gongs 

81.6 

House  bells 

77 

— 

Lathe  bushes 

80 

' 

Machinery  bearings 

“ 44  hard. 

Metal  that  expands  in) 

87-5 

— 

77-4 

7 

cooling ) 

Muntz  metal,  10  oz.  lead. 

60 

40 

Pewter,  best 

— 

44  





Sheathing  metal 

56 

45 

Speculum  “ 

66 

50 

21 

Telescopic  mirrors 

66.6 

— 

Temper  t 

33-4 

— 

Type  metal  and  stereo- ) 

— 

type  plates ] 

— 

— 

White  metal 

7-4 

7-4 

“ “ hard..'.,...,. 

69.8 

25-8 

Oreide 

• 73, 

12.3 

Tin. 

Nickel. 

Lead. 

mony. 

muth. 

minum. 

— 

21 

_ 

— 

— 

— 

— 

— 

— 

— 

— 

5 

89 

— 

7-3 

— 

10.5 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

14- 3 

— 

— 

■ — 

— 

— 

7.8 

— 

— 

— 

— 

— 

9 

— 

— 

— 

’ — 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

— 

3-4 

— 

— 

— 

— 

— 

— 

19 

— 

— 

— 

— 

8-3 

— 

— 

— 

— 

— 

10 

— 

— 

— 

— 

— 

10 

— 

— 

— 

— 

— 

— 

— 

— 

7 

— 

— 

— 

— 

46 

47 

— 

— 

— 

— ' 

— 

— 

— 

— 

. — 

— 

— 

— 

— 

— 

25 

— 

— 

25 

— 

— 

— 

— 

— 

25 

25 

— 

— 

— 

■ — 

— •' 

— 

— 

2.9 

— 

— 

— ' 

— 

— 

1.6 

— 

— 

— : 

— 

c 

10 

— 

— 

— ' 

— 

0 

7 

— 

— 

0 

5 

— 

— 

— 

' — ' 

£ 

20 

— 

— 

— 

1 

7 

— 

— 

— 

> 

0 

1.4 

• — 

i-7 

— 

02 

— 

— 

11. 6 

— 

— 

2 

II. I 

2.6 

31.6 

— 

— 

— 

— 

10. 1 

— 

4-3 

— 

— 

c 

3i 

— 

— 

— 

— 

£ 

12.5 

— 

— 

— 

— 

— 

26.5 

— 

— 

•~T- 

— 

1-5 

— 

33-3 

— 

— 

— 

— 

— 

31.6 

— 

— 

, 

2.6 

— 

24 

— 

— 

— 

2-5: 

18.4 

— 

— 

— 

23 

— 

— 

_ • 

rO 

— . ■ 

20 

— 

— 

— 

3 

— 

12.5 

— 

— 

— 

a 

— 

15-6 

— 

— 

— 

S 

— 

— 

— 

75 

16.7 

8.3 

— 





— 

,g 

86 

— 

— 

14 

— 

B 

80 

— 

20 

— 

— 

< 

— 

— 

— 

— 

— 

— 

22 

— 

— 

' 

— 

12 

29 

— 

— 

— 

— 

— 

33-4 

— 

— 

— 

— 

— 

66.6 

— 

— 

— 

— 

— , 

— 

75 

25 

— 

— 

— 

— 

87-5 

12-5 

— 

— 

28.4 

— 

56.8 

— 

— 

4.4 

— 

— 

— 

— 

Magnesia 4.4  Cream  of  tartar  .6.5 

Sal-ammoniac  . 2.5  Quicklime .1.3 


See  page  636  for  directions. 


t For  adding  small  quantities  of  copper. 


METALS. ALLOYS  AND  COMPOSITIONS. 


Solders. 


Copper. 

Tin. 

Lead. 

Zinc. 

Silver. 

Bis- 

muth. 

Gold. 

Cad- 

mium. 

Anti- 

mony. 

Tin 



25 

75 

— 

— 

16 

— 

— 

— 

“ 

— 

58 

16 

— 

— 

— 

— 

10 

“ coarse,  melts ) 
at  5000 . . . j 

- 

33 

67 

- 

- 

- 

- 

- 

— 

ordi’y,  melts) 
at  360° . . . j 

- 

67 

33 

- 

- 

— ■ 

— 

Spelter,  soft 

“ hard 

So 

— 

— 

50 

— 

— 

— 

— 

— 

65 

— 

— 

35 

— ■ 

— 

— 

— 

— 

Lead 

33 

67 

— 

82 

— 

— 

— 

— 

Steel 

13 

— 

5 

— 

— 

— 

— 

Brass  or  Copper . . . 

50 

— 

— 

50 

— 

— 

— 

— 

— 

Fine  brass 

47 

— 

— 

47 

6 

— 

— 

— 

— 

Pewterers’  or  Soft . 

33 

45 

— 

— ' 

22 

— 

— 

— 

’ “ “ . 

— 

50 

25 

— 

— 

25 

— 

— 

— 

Plumbers’  pot- 1 
metal. . j 

- 

33 

67 

- 

- 

- 

- 

— 

- 

“ coarse 

— 

25 

75 

— 

— 

— 

— 

— 

— 

“ fine 

— 

67 

33. 

— 

— 

— 

— 

— 

— 

“ fusible... 

— • 

50 

50 

— 

— 

— 

— 

— 

— ' 

“ very  “ ... 

— 

25 

25 

— 

— 

50 

89 

— 

— 

Gold 

.4 

— 

— 

— 

7 

— 

— 

— 

“ hard 

66 

— 

— 

34 

— 

— 

— 

— 

— 

“ soft 

— 

66 

34 

— 

— 

— 

— 

— 

— 

Silver,  hard 

20 

— 

— 

80 

— 

— 

— 

— 

“ soft 

12 

— 

— 

— 

67 

— 

— 

21 

— 

Pewter 

— 

40 

20 

— 

40 

— 

— 

— 

Iron 

66 

— 

— 

j 33 

— 

— 

— 

— 

1 

Copper  

53 

47 

— 

— 

— 

— 

— 

— 

A Plastic  Metallic  Alloy.— See  Journal  of  Franklin  Institute,  vol.  xxxix.,  page  55, 


for  its  composition  and  manufacture. 


Soldering  Fluid  for  use  with  Soft  Solder. 

To  2 fluid  oz.  of  Muriatic  acid  add  small  pieces  of  Zinc  until  bubbles  cease  to  rise. 
Add  .5  a teaspoonful  of  Sal-ammoniac  and  two  fluid  oz.  of  Water. 

By  the  application  of  this  to  Iron  or  Steel,  they  may  be  soldered  without  their  sur- 
faces being  previously  tinned. 


Fluxes  for  Soldering  or  Welding. 


Iron Borax. 

Tinned  iron Resin. 

Copper  and  Brass Sal-ammoniac. 


Zinc Chloride  of  zinc. 

Lead Tallow  or  resin. 

Lead  and  tin Resin  and  sweet  oil. 


Baiybitt’s  .Anti-attrition  Metal. 

Melt  4 lbs.  Copper;  add  by  degrees  12  lbs.  best  Banca  tin,  8 lbs.  Regulus  of  anti- 
mony, and  12  lbs.  more  of  Tin.  After  4 or  5 lbs.  Tin  have  been  added,  reduce  heat 
to  a dull  red,  then  add  remainder  of  metal  as  above. 

This  composition  is  termed  hardening  ; for  lining,  take  1 lb.  of  this  hardening , 
melt  with  it  2 lbs.  Banca  tin,  which  produces  the  lining  metal  for  use.  Hence,  the 
proportions  for  lining  metal  are  4 lbs.  of  copper,  8 of  regulus  of  antimony,  and  96 
of  tin. 

Brass. 


Brass  is  an  alloy  of  copper  and  zinc,  in  proportions  varying  with  purpose 
of  metal  required,  its  color  depending  upon  the  proportions. 

It  is  rendered  brittle  by  continued  impacts,  more  malleable  than  copper 
when  cold,  but  is  impracticable  of  being  forged,  as  its  zinc  melts  at  a low 


temperature.  . _ . „ ...  . 

Its  fusibility  is  governed  by  its  proportion  of  zinc ; a small  quantity  ol 


phosphorus  gives  it  fluidity. 


METALS. — ALLOYS  AND  COMPOSITIONS. — IRON.  637 


Bronze. 

Bronze  is  an  alloy  of  copper  and  tin ; it  is  harder,  more  fusible,  and 
stronger  than  copper.  It  is  usually  known  as  Gun-metal . 

Aluminum  Bronze  contains  90  to  95  per  cent,  of  copper,  and  5 to  10  per 
cent,  aluminum. 

Phosphor  Bronze  contains  copper  and  tin  and  a small  proportion  of  phos- 
phorus. It  wears  better  than  bronze. 


IRON. 

Foreign  substances  which  iron  contains  modify  its  essential  proper- 
ties. Carbon  adds  to  its  hardness,  but  destroys  some  of  its  qualities, 

| and  produces  Cast  Iron  or  Steel,  according  to  proportion  it  contains. 
Thus,  .25  per  cent,  renders  it  malleable,  .5  steel,  1.75  is  limit  of  weld- 
ing steel,  and  2 is  lowest  limit  of  cast  iron.  Sulphur  renders  it  fusible, 
difficult  to  weld,  and  brittle  when  heated,  or  “ hot  short.”  Phosphorus 
renders  it  “cold  short”  but  may  be  present  in  proportion  of  .002  to 
.003,  without  affecting  injuriously  its  tenacity.  Antimony,  Arsenic,  and 
Copper  have. same  effect  as  sulphur,  the  last  in  a greater  degree.  Sili- 
con  renders  it  hard  and  brittle.  Manganese,  in  proportion  of  .02,  ren- 
ders it  “ cold  short”  and  Vanadium  adds  to  its  ductility. 

Cast  Iron. 


Process  of  making  Cast  Iron  depends  much  upon  description  of  fuel  used ; 
whether  charcoal,  coke,  bituminous,  or  anthracite  coals.  A larger  yield  from 
same  furnace,  and  a great  economy  in  fuel,  are  effected  by  use  of  a hot  blast. 
The  greater  heat  thus  produced  causes  the  iron  to  combine  with  a larger 
percentage  of  foreign  substances. 

Cast  Iron  for. purposes  requiring  great  strength  should  be  smelted  with 
a cold  blast  Pig-iron,  according  to  proportion  of  carbon  which  it  contains, 
is  di\  ided  into  Foundry  Iron  and  Forge  Iron , latter  adapted  only  to  conver- 
sion into  malleable  iron ; while  former,  containing  largest  proportion  of  car- 
bon, can  be  used  either  for  castings  or  bars. 

High  temperature  in  melting  injures  gun-metal. 


There  are  many  varieties,  of  . Cast  Iron,  differing  by  almost  insensible 
shades ; the  two  principal  divisions  are  gray  and  white,  so  termed  from 
color  of  their  fracture.  Their  properties  are  very  different. 

Gray  Iron  is > softer  and  less  brittle  than  white;  it  is  in  a slight  degree 
malleable  and  flexible,  and  is  insonorous ; it  can  easily  be  drilled  or  turned 
and  does  not  resist  the  file.  It  has  a brilliant  fracture,  of  a gray,  or  some- 
times a bluish-gray,  color;  color  is  lighter  as  grain  becomes  closer,  and  its 
hardness  increases.  It  melts  at  a lower  heat  than  white,  and  preserves  its 
flmdity  longer.  Color  of  the  fluid  metal  is  red,  and  deeper  in  proportion  as 
the  heat  is  lower;  it  does  not  adhere  to  the  ladle;  it  fills  molds  well,  con- 
tracts less,  and  contains  fewer  cavities  than  white;  edges  of  its  castings 
are  sharp,  and  surfaces  smooth  and  convex.  It  is  used  for  machinery  and 
ordnance  where  the  pieces  are  to  be  bored  or  fitted.  Its  tenacity  and  specific 
gravity  are  diminished  by  annealing. 

,,,2/''™  Is.  very  brittle  and  sonorons;  it  resists  file  and  chisel,  and  is 
susceptible  of  high  polish ; surface  of  its  castings  is  concave ; fracture  pre- 
s ,ry  appearance,  generally  fine  grained  and  compact,  sometimes 
radiating  or  lamellar.  When  melted  it  is  white,  throws  off  a great  number 
of  sparks,  and  its  qualities  are  the  reverse  of  those  of  grav  iron  ; it  is  there- 
in for  machinery  purposes.  Its  tenacity  is'  increased,  and  its 

specific  gravity  diminished,  by  annealing. 

3 H 


METALS. — IRON. 


638 

Mottled  Iron  is  a mixture  of  white  and  gray ; it  has  a spotted  appear- 
ance; flows  well,  and  with  few  sparks;  its  castings  have  a plane  surface, 
with  edges  slightly  rounded.  It  is  suitable  for  shot,  shells,  etc.  A fine  mot- 
tled is  only  kind  suitable  for  castings  which  require  great  strength.  The 
kind  of  mottle  will  depend  much  upon  volume  of  the  casting.  A medium- 
sized  grain,  bright  gray  color,  fracture  sharp  to  touch,  and  a close,  compact 
texture,  indicate  a good  quality  of  iron.  A grain  either  very  large  or  very 
small,  a dull,  earthy  aspect,  loose  texture,  dissimilar  crystals  mixed  together, 
indicate  an  inferior  quality. 

Besides  these  general  divisions,  the  different. varieties  of  pig-iron  are  more 
particularly  distinguished  by  numbers,  according  to  their  relative  hardness. 

No.  j —Fracture  dark  gray,  crystals  large  and  highly  lustrous,  alike  to 
new  surface  of  lead.  It  is  the  softest  iron,  possessing  in  highest  degree  the 
qualities  belonging  to  gray  iron ; it  has  not  much  strength,  but  on  account 
of  its  fluidity  when  melted,  and  of  its  mixing  advantageously  with  scrap 
iron  and  with  the  harder  kinds  of  cast  iron,  it  is  of  great  use  to  a foundry. 

No.  2 is  harder,  closer  grained,  and  stronger  than  No.  1 ; it  has  a gray 
color  and  considerable  lustre.  It  is  most  suitable  for  shot  and  shells. 

No.  3 is  harder  than  No.  2.  Fracture  white,  crystals  larger  and  brighter 
at  centre  than  at  the  sides;  color  gray,  but  inclining. to  white;  has  consid- 
erable strength,  but  is  principally  used  for  mixing  with  other  kinds  of  iron 
and  for  large  castings. 

No.  4 or  Bright. — Fracture  light  gray,  with  small  crystals  and  little  lustre, 
and  not  being  sufficiently  fusible  for  castings  it  is  used  for  conversion  to 
wrought  iron. 

No.  5.  Mottled.' — Fracture  dull  white,  with  gray  specks,  and  a line  of 
white  around  edge  or  sides  of  fracture. 

No.  6.  White. — Fracture  white,  with  little  lustre,  granulated  with  radiat- 
ing crystalline  surface.  It  is  hardest  and  most  brittle  of  all  descriptions, 
and  is*unfit  for  use  unless  mixed  with  other  grades,  cr  for  being  converted 
to  an  inferior  wrought  iron. 

Qualities  of  these  descriptions  depend  upon  proportion  of  carbon,  and  upon 
state  in  which  it  exists  in  the  metal ; in  darker  kinds  of  iron,  where  propor- 
tion is  sometimes  7 per  cent.,  it  exists  partly  in  state  of  graphite  or  plumbago, 
which  makes  the  iron  soft.  In  white  iron  the  carbon  is  thoroughly  com- 
bined with  the  metal,  as  in  steel. 

Cast  iron  frequently  retains  a portion  of  foreign  ingredients  from  the  ore, 
such  as  earths  or  oxides  of  other  metals,  and  sometimes  sulphur  and  phos- 
phorus, which  are  all  injurious  to  its  quality. 

Foreign  substances,  and  also  a portion  of  the  carbon,  are  separated  by 
melting  iron  in  contact  with  air,  and  soft  iron  is  thus  rendered  harder  and 
stronger.  Effect  of  remelting  varies  with  nature  of  the  iron  and  character 
of  ore  from  which  it  has  been  extracted ; that  from  hard  ores,  such  as  mag- 
netic oxides,  undergoes  less  alteration  than  that  from  hematites,  the  latter 
being  sometimes  changed  from  No.  1 to  white/  by  a single  remelting  in  an  , 
air  furnace.  ' 

Color  and  texture  of  cast  iron  depend  greatly  upon  volume  of  casting  and  . 
rapidity  of  its  cooling ; a small  casting,  which  cools  quickly,  is  almost  alwav  s 
white , and  surface  of  large  castings  partakes  more  of  the  qualities  of  white  j 
metal  than  the  interior. 

All  cast  iron  expands  at  moment  of  becoming  liquid,  and  contracts  in  cool- 
ing; gray  iron  expands  more  and  contracts  less  than  other  iron. 

Remelting  iron  improves  its  tenacity;  thus,  a mean  of  14  cases  for  two 
fusions  gave,  for  1st  fusion,  a tenacity  of  29284  lbs.;  for  2d  fusion,  33  79° 
lbs.  For  two  cases — for  first  fusion,  15  129  lbs. ; for  2d  fusion,  35  7S6  lbs. 


METALS. — IEOX. 


639 


IVT a.llea'ble  Castings. 

. Malleable  cast  iron  is  made  by  subjecting  a casting  to  a process  of  anneal- 
in&  by  enclosing  it  m a box  with  hematite  iron  ore  or  black  oxide  of  iron 
volume  m a°  e<Pia^e  *ieat  ^or  a period  depending  upon  form  and 

Wrought  Iron. 

Wrought  iron  is  made  from  pig-iron  in  a Bloomeiy  Fire  or  in  a Puddllno 
Furnace— generally  in  latter.  Process  consists  in  melting  and  keeping  it 
exposed  to  a great  heat,  constantly  stirring  the  mass,  bringing  every  part  of 
it  under  action  of  the  flame  until  it  loses  its  remaining  carbon!  wh^it  be- 
comes malleable  iron-  W hen,  however,  it  is  desired  to  obtain  iron  of  best 
quality,  pig-iron  should  be  refined.  ue&u 

Refining*  —This  operation  deprives  iron  of  a considerable  portion  of  its 
carbon ; it  is  effected  in  a Blast  Furnace,  where  iron  is  melted  by  means  of 
charcoal  or  coke,  and  exposed  for  some  time  to  action  of  a great  heat  • the 
metal  is  then  run  into  a cast-iron  mold,  by  which  it  is  formed  into  a far>e 

Sftbirn  " SUrfaCG  °f  Pkte  " Chmed’  C°Id  Water  is  Po-edone 

A Bloomerl/  resembles  a large  forge  fire,  where  charcoal  and  a strong  blast 
are  used ; and  the  refined  metal  or  pig-iron,  after  being  broken  into  pifces  of 
1S  PJaced  before  the  blast,  directly  in  contact  with  charcoal  • as 
the  metal  fuses,  it  falls  into  a cavity  left  for  that  purpose  below  the  blast 
u here  the  bloomer  works  it  into  the  shape  of  a ball , which  he  places  ao  ain 
before  the  blast,  with  fresh  charcoal ; this  operation  is  generally  agaiif  re- 
peated, when  ball  is  ready  for  the  “shingler.”  b y ‘gam  re 

Shingling  is  performed  in  a strong  squeezer  or  under  a trip-hammer.  Its 
ho!fCt  *°  Pr>SS  ,out/s  Perfectly  as  practicable  the  liquid  cinder  which  a 
ball  contains ; it  also  forms  a ball  into  shape  for  the  puddle  rolls  A heavy 
hammer, we^hingfrem  6 to  7 tons,  effects  this  object  most  Ihor^ughly^ 
t so  cheaply  as  the  squeezer.  A ball  receives  from  i s to  20  blows  of  a 
hammer,  being  turned  from  time  to  time  as  required:  ft  is  now  terned  a 
Bloom,  and  is  ready  to  be  rolled  or  hammered ; or  a ball  is  passed  once 
ro[igU®  1 1G  S(iueezer’  and  is  still  hot  enough  to  be  passed  through  the  puddle 

A Puddling  Furnace  is  a reverberatory  furnace,  where  flame  of  bituminous 
coal  is  brought  to  act  directly  upon  the  melted  metal.  The  “ puddler  ” then 
stirs  it,  exposing  each  portion  in  turn  to  action  of  flame,  and  continues this 
as  ong  as  he  is  able  to  work  it.  When  it  has  lost  its  fluidity,  he  forms  i!  into 
bt’-18hlDg  fr°m  80  t0  IO°  lbs*’  which  are  then  Passed  ^ the  “shingle/” 
Puddle  Rolls.-  By  passing  through  different  grooves  in  these  rolls  n 
bloom  is  reduced  to  a rough  bar  from%  to  4 feet  in  length  hi  tlZ  ™ ’ 
mg  an  idea  of  its  condition,  which  is  rough  and  imperfect.’  ° e"  " 

nf  ^W.-To  prepare  rough  bars  for  this  operation,  they  are  cut  by  a pair 
of  shears,  into  such  lengths  as  are  best  adapted  to  the  volume  « , 
required ; the  sheared  bars  are  then  piled'  one  over the  Xr 
■volume  required,  when  pile  is  ready  for  balling.  ° 1 

to  nnfe;r7h‘S  (Peration  is  performed  in  balling  furnace,  which  is  similar 
to  tinm  v F h rT’  vX“Pt  *h,at  ltS  •b°,t0m  or  1,earth  is  made  up,  from  tithe 

tearing.11  f " UStd  ‘°  S‘Ve  " W6lding  h6at  to  pile"  P-P-e 

Finishing  Rolls  — The  balls  are  passed  successively  between  rollers  of  va- 
ncus  forms  and  dimensions,  according  to  shape  of  finished  bar  required. 

u.  in  ',r0i.1  depends  upon  description  of  pig-iron  used  skill  of  the 
‘puddler,”  and  absence  of  deleterious  substances  in  the  furnace. 


640 


METALS.— IKOX. LEAD. STEEL. 


Strongest  cast  irons  do  not  produce  strongest  malleable  iron. 

For  many  purposes,  such  as  sheets  for  tinning,  best  boiler-plates,  and  bars 
for  converting  into  steel,  charcoal  iron  is  used  exclusively ; and,  generally, 
this  kind  of  iron  is  to  be  relied  upon,  for  strength  and  toughness,  with  greater 
confidence  than  any  other,  though  iron  of  a superior  quality  is  made  from 
pigs  made  with  other  fuel,  and  with  a hot  blast.  Iron  for  gun-barrels  has 
been  lately  made  from  anthracite  hot-blast  pigs. 

Iron  is  improved  in  quality  by  judicious  working,  reheating  hammering, 
or  rolling : other  things  being  equal,  best  iron  is  that  which  has  been  wrought 
the  most. 

Best  quality  of  iron  has  greatest  elasticity. 

Tests. — It  will  not  blacken  if  exposed  to  nitric  acid.  Long  silky  fibres  in 
a fracture  denote  a soft  and  strong  metal ; short  black  fibres  denote  a badly 
refined  metal,  and  a fine  grain  denotes  hardness  and  condition  known  as 
« coltl  short.”  Coarse  grain  with  bright  and  crystallized  fracture,  with  dis- 
colored spots,  also  denotes  “ cold  short”  and  brittle  metal,  working  easily  and 
welding  well.  Cracks  upon  edges  of  a bar,  etc.,  indicate  hot  short.  Good 
iron  heats  readily,  is  worked  easily,  and  throws  off  but  few  sparks.. 

A high  breaking  strain  may  not  be  conclusive  as  to  quality,  as  it  may  be 
due  to  a hard,  elastic  metal,  or  a low  one  may  be  due  to  great  softness. 

When  iron  is  fractured  suddenly,  a crystalline  surface,  is  produced,  and 
when  gradually,  a fibrous  one.  Breaking  strain  of  iron  is  increased  by  heat- 
ing it  and  suddenly  cooling  it  in  water.  Iron  exposed  to  a welding  or  white 
heat  and  not  reduced  by  hammering  or  rolling  is  weakened. 

Specific  gravity  of  iron  is  a good  indication  of  its  quality,  as  it  indicates 
very  correctly  its  relative  degree  of  strength. 

LEAD. 

Sheet  Lead  is  either  Cast  or  Milled, , the  former  in  sheets  16  to  18  feet  in 
length  and  6 feet  in  width ; the  latter  is  rolled,  is  thinner  than  the  former, 
is  more  uniform  in  its  thickness,  and  is  made  into  sheets  25  to  35  feet  in 
length,  and  from  6 to  7.5  feet  in  width. 

Soft  or  Rain  Water,  when  aerated,  Silt  of  rivers,  Vegetable  matter,  Acids 
Mortar  and  Vitiated  Air  will  oxidize  lead.  The  waters  which  act  with 
greatest  effect  on  it  are  the  purest  and  most  highly  oxygenated,  also  nitrites, 
nitrates,  and  chlorides,  and  those  which  act  with  least  effect  are  such  as  con- 
tain carbonate  and  phosphate  of  lime. 

Coating  of  Pipes , except  with  substances  insoluble  in  water,  as  Bitumen 
and  Sulphide  of  lead,  is  objectionable. 

Lead-encased  Pipes.— An  inner  pipe  of  tin  is  encased  in  one  of  lead. 

STEEL. 

Steel  is  a compound  of  Iron  and  Carbon,  in  which  proportion  of  latter 
is  from  1 to  5 per  cent,  and  even  less  in  some  descriptions  It  is  dis- 
tinguished  from  iron  by  its  fine  grain,  and  by  action  of  diluted  nitric 
acid,  which  leaves  a black  spot  upon  it. 

There  are  many  varieties  of  steel,  principal  of  which  are : 

Natural  Steel , obtained  by  reducing  rich  and  pure  descriptions  of  iron 

ore  with  charcoal,  and  refining  cast  iron  so  as  t0  dePny®  ll  f ^ fiks  and 
portion  of  carbon  to  bring  it  to  a malleable  state.  It  is  used  tor  hies  an 

other  tools.  , . . « 

Indian  Steel , termed  Wootz,  is  said  to  be  a natural  steel,  containing  a small 
portion  of  other  metals. 


METALS. STEEL. 


641 

. Blistered  Steel,  ov  Steel  of  Cementat  ion,  is  prepared  by  direct  combination  of 
iron  and  carbon.  For  this  purpose,  iron  in  bars  is  put  in  layers,  alternating 
with  powdered  charcoal,  m a close  furnace,  and  exposed  for  7 or  8 days  to 
a high  temperature,  and  then  put  to  cool  for  a like  period.  The  bars,  on 
being  taken  out,  are  covered  with  blisters,  have  acquired  a brittle  quality, 
and  exhibit  in  fracture  a uniform  crystalline  appearance.  The  degree  of 
carbonization  is  varied  according  to  purposes  for  which  the  steel  is  intended 
and  the  very  best  qualities  of  iron  are  used  for  the  finest  kinds  of  steel. 

Tilted  Steel  is  made  from  blistered  steel  moderately  heated,  and  subjected 
creased 11  °f  * ^ hatnmer’  b-v  which  means  its  tenacity  and  density  are  in- 

Shear  Steel  is  made  from  blistered  or  natural  steel,  refined  by  piling  thin 
bars  into  fagots,  which  are  brought  to  a welding  heat  in  a reverberatorv 
furnace, and  hammered  or  rolled  again  into  bars;  this  operation  is  repeated 
several  times  to  produce  finest  kinds  of  shear  steel,  which  are  distinguished 
by  the  terms  of  Half  shear,  Single  shear,  and  Double  shear,  or  steel  of  i 2 or 
3 mans , etc.,  according  to  number  of  times  it  has  been  piled. 

hammered^'  *S  blister  steel  beated  to  al1  orange  red  color  and  rolled  or 

Cr™Ve  S'eel  is  made  hy  breaking  blistered  steel  into  small  pieces 
and  melting  it  m close  crucibles,  from  which  it  is  poured  into  iron  molds; 

Lmrt  rf  i'ri  redl'c®d  ,t0  a bar,b>'  hammering  or  rolling.  Cast  steel  is  best 
kind  of  steel,  and  best  adapted  for  most  pu. posts;  it  is  known  by  a very 
fine,  er en,  and  close  grain,  and  a silvery,  homogeneous  fracture;  it  is  very 
of  affi.v  ff'v  ftr|T  bardness’  bl,t  is  fMfflcult  to  weld  without  use 
0t  ki"lds  °l  stee  have  a Slmilar  appearance  to  cast  steel,  but 
gTam  is  coarser  and  less  homogeneous : they  arc  softer  and  less  brittle,  and 
weld  more  read.lv.  A fibrous  or  lamellar  appearance  in  fracture  indicates 
hnrdne^?1  St T j ^ uatmal  of  K«‘at  toughness  and  elasticity,  as  well  as 
s',adc  by.  f°rP"8  together  steel  and  iron,  forming  the  celebrated 
Damasked  Steel,  which  is  used  for  sword-blades,  springs,  etc. ; damask  an- 
themed while  th* produce?  by  adUuted  acid,  which  gives  a black  tint  to 
tne  steel,  while  the  iron  remains  white. 

with  diem St  Stee1’  kreakm£  8trength  is  greater  across  fibres  of  rolling  than 

P‘?C€SS  jf,an  improvement  on  this  method,  and  consists  in  addins  to 
molten  metal  a small  quantity  of  carburet  of  manganese  ® 

remove>carlTOn<)and sulca!*8  addi”S  Ditrate  °f  soda  t0  mo,te“  i«*  order  ,0 

anf£coaV°CSM:_“alleabIe  ir°n  iS  meUed  in  crucib,es  with  oxide  of  manganese 

Puddled  Steel  is  produced  by  arresting  the  puddling  in  the  manufacture 
of  the  wrought  iron  before  all  the  carbon  has  been  removed,  the  small 
State, toL1  remammg’  -3  t0  1 per  cent-’  be“g  sufficient  to  make  an 

enUtiZ iTll'Z'sZ:2  t0  '5  Per  Cent°f  Carb°n;  Whe“  “ Fres- 

in  ordctT  Sue-  iS  made  direbt  from  P'>iron-  The  carbon  is  first  removed, 
in  order  to  obtain  pure  wrought  iron,  and  to  this  is  added  the  exact  quantity 
of  carbon  required  for  the  steel.  The  pig  should  be  free  from  sulphur  and 
phosphorus  It  is  melted  in  a blast  or  cSpola,  and  run  into  a Zl  a 
pear-shaped  iron  vessel  suspended  on  hollow  trunnions  and  lined  with  fire- 
bnck  or  clay)  where  it  is  subjected  to  an  air  blast  for  a period  of  lo  mtol 

gelds^  is  adde°d  SP  ® “ °n’ after  which  fro;"  3 t0  P«  cent,  of  spie- 
3 H* 


METALS. — STEEL. 

The  blast  is  then  resumed  for  a short  period,  to  incorporate  the  two  metals, 
when  the  steel  is  run  off  into  molds.  The  moment  at  which  all  the  carbon 
has  been  removed  is  indicated  by  color  of  the  flame  at  mouth  of  conv  erter. 
The  ingots,  when  thus  produced,  contain  air  holes,  and  it  becomes  necessary 
to  heat  them  and  render  them  solid  under  a hammer. 

Siemens  Process.— Pig-iron  is  fused  upon  open  hearth  of  a regenerative 
furnace,  and  when  raised  to  a steel-melting  temperature,  rich  and  pure  ore 
and  limestone  are  added  gradually,  whereby  a reaction  is  established  between 
the  oxvgen  of  the  ferrous  oxide  and  the  carbon  and  silicon  m the  metal.  Ihe 
silicon" is  thus  converted  into  silicic  aild,  which  with  the  lime  forms  a fusible 
slag,  and  the  carbon,  combining  with  oxygen,  escapes  as  carbonic  acid,  and 
induces  a powerful  ebullition. 

MnrNfi ration  of  this  process. — The  ore  is  treated  in  a separate  rotatory  furnace 
with  carbonaceous  material,  and  converted  into  balls  of  malleable  iron,  which  are 
transferred  from  the  rotatory  to  the  bath  of  the  steel-melting  furnace. 

process  is  adapted  to  the  production  of  steel  of  a very  high  quality,  because 
th^SSKd  ptoptoS  of  the  ore  are  separated  from  the  metal  in  the  rotatory 
furnace. 

Siemens -Martin  Process.- Scrap-iron  or  steel  is 
highly  heated  condition  to  a bath  of  about  .25  its  weight,  of  highly  heated 
pig  and  melted.  Samples  are  occasionally  taken  from  the  bath,  in  order  to 
ascertain  the  percentage  of  carbon  remaining  in  *he  }S  a W d 

in  small  quantities,  in  order  to  reduce  the  carbon  to  about  .1  per  cent. 

At  this  stage  of  the  process,  siliceous  iron,  spiegeleisen,  or  ferro-manganese 
is  added  in  sSch  proportions  as  are  necessary  to  produce  steel  of  the  required 
degree  of  hardness.  The  metal  is  then  tapped  into  a ladle. 

Landore-Siemen’s  Steel  is  a variety  of  steel  made  by  the  A^cation  of 
Siemen’s  Process.  Its  great  value  is  due  to  its  extreme  ductility,  and  its 
having  nearly  like  strength  in  both  directions  ot  its  plates. 

Whitworth'*  Compressed  Steel  is  molten  steel  subjected  to  a pressure  of 
about  6 tons  per  square  inch,  by  which  all  its  cavities  are  dispelled,  and  it  is 
compressed  to  about  .875  of  its  original  volume,  its  density  and  strength  be- 
ing  proportionately  increased. 

Chrome  and  Tungsten  Steel  are  made  by  adding  a small  percentage  of 
Chromium  or  Tungsten  to  crucible  steel,  the  result  producing  a steel  of 
great  hardness  and  tenacity,  suitable  for  tools,  such  as  drills,  etc. 

Homogeneous  Steel  is  a variety  of  cast  steel  containing  .25  per  cent,  of 
carbon. 

Remarks  on  Manufacture  of  Steel , and  Mode  of  Working  it. 

(D.  Chernoff ’ 1868). 

Steel  when  cast  and  allowed  to  cool  quietly,  assumes  a . crj-stalline  structure 
Higher  temperature  to  which  it  is  heated,  softer  it  becomes,  and  gieater  is  libeity 
its  particles  possess  to  group  themselves  into  crystals.  . 

Steel  however  hard  it  may  be,  will  not  harden  if  heated  to  a temperature | lower 
than  what  may  be  distinguished  as  dark  cherry-red,  a,  however  quickly  it  is  cooled, 
on  contrary,  it  will  become  sensibly  softer,  and  more  easily  woi  ked  w ith  a tile. 

Steel  heated  to  a temperature  lower  than  red,  but  not  sparkling,  6,  does  not 
chamre  its  structure  whether  cooled  quickly  or  slowly.  When  temperature  has 
reached  b substance  of  steel  quickly  passes  from  granular  or  crystalline  condition 
to  amorphous,  or  wax-like  structure,  which  it  retains  up  to  its  melting-point,  c.  ^ 
Points  a b and  c have  no  permanent  place  in  scale  of  temperature,  but  their  posi 
tions  vary’ with  quality  of  steel;  in  pure  steel,  they  depend  directly  on  quantity  of 
constituent^ carbon  Harder  the  steel,  lower  the  temperatures  Tints  above  speci- 
fied have  reference  only  to  hard  and  medium  qualities  of  steel;  in  very  soft  kinds 
of  steel,  nearly  approaching  to  wrought  iron,  points  a and  b range  very  high,  and  m 
wrought  iron  point  b rises  to  a white  heat. 


METALS. — STEEL. 


643 

Assumption  of  the  crystalline  structure  takes  place  entirely  in  cooling,  between 
temperatures  c and  6;  when  temperature  sinks  below  b there  is  no  change  of  struc- 
ture. For  successful  forging,  therefore,  heated  ingot,  after  it  is  taken  out  of  furnace 
must  be  forged  as  quickly  as  practicable,  so  as  not  to  leave  any  spot  untouched  by 
hammer,  where  the  steel  might  crystallize  quietly,  as  formation  of  crystals  should 
be  hindered,  and  the  steel  should  be  kept  in  an  amorphous  condition  until  tem- 
perature sinks  below  point  b. 

Below  this  temperature,  if  piece  is  cooled  in  quiet,  mass  will  no  longer  be  disposed 
to  crystallize,  but  will  possess  great  tenacity  and  homogeneousness  of  structure 
When  steel  is  forged  at  temperatures  lower  than  6,  its  crystals  or  grains  bein°- 
driven  against  each  other,  change  their  shapes,  becoming  elongated  in  one  direction0 
and  contracted  in  another;  while  density  and  tensile  strength  are  considerably  in- 
creased. But  available  hammer-power  is  only  sufficient  for  treatment  of  small  steel 
forgings;  and  object  of  preventing  coarse  crystalline  structure  in  large  forgings 
is  more  easily  and  more  certainly  effected,  if,  after  having  given  forging  desired 
shape,  its  structure  be  altered  to  an  homogeneous  amorphous  condition  by  heating 
it  to  a temperature  somewhat  higher  than  6,  and  the  condition  be  fixed  by  rapid 
coo  mg  to  a temperature  lower  than  6,  the  piece  should  then  be  allowed  to  finish 
cooling  gradually,  so  as  to  prevent,  as  far  as  practicable,  internal  strains  due  to 
sudden  and  unequal  contraction. 

Alloys  of  steel  with  Silver , Platinum , Rhodium , and  Aluminum  have  been 
made  with  a view  to  imitating  Damascus  steel,  Wootz,  etc.,  and  improving 
fabrication  of  some  finer  kinds  of  surgical  and  other  instruments. 

Properties  of  Steel.— After  being  tempered  it  is  not  easily  broken;  it  welds 
readily;  does  not  crack  or  split;  bears  a very  high  heat/and  preserves  the 
capability  of  hardening  after  repeated  working. 

Hardening  and  Tempering.— Upon  these  operations  the  quality  of  manu- 
factured steel  in  a great  measure  depends. 

Hardening  is  effected  by  heating  steel  to  a cherry-red,  or  until  scales  of 
oxide  are  loosened  on  surface,  and  plunging  it  into  a cooling  liquid;  decree 
of  hardness  depends  upon  heat  and  rapidity  of  cooling.  Steel  is  thus  ren- 
dered so  hard  as  to  resist  files,  and  it  becomes  at  same  time  extremely 
brittle.  Degree  of  heat,  and  temperature  and  nature  of  cooling  medium 
must  be  chosen  with  reference  to  quality  of  steel  and  purpose  for  which  it 
is  intended.  Cold,  water  gives  a greater  hardness  than  oils  or  like  sub- 
stances, sand,  wet-iron  scales,  or  cinders,  but  an  inferior  degree  of  hardness 
to  that  given  by  acids.  Oil,  tallow,  etc.,  prevent  cracks  caused  by  too  rapid 
cooling.  Lower  the  heat  at  which  steel  becomes  hard,  the  better. 

Tempering.  Steel  in  its  hardest  state  being  too  brittle  for  most  purposes 
the  requisite  strength  and  elasticity  are  obtained  by  tempering— or  “lettinq 
down  the  temper  ” which  is  performed  by  heating  hardened  steel  to  a certain 
degree  and  cooling  it  quickly.  Requisite  heat  is  usually  ascertained  by  color 
winch  surface  of  the  steel  assumes  from  film  of  oxide  thus  formed.  Degrees 
or  heat  to  which  these  several  colors  correspond  are  as  follows : 

At43oO  very  faint  yellow..  (Suitable  for  hard  instruments;  as  hammer -faces 
At  4500,  pale  straw  color. . . . ( drills,  lancets,  razors,  etc  ’ 

At  47oO,  full  yellow ; For  instruments  requiring  hard  edges  without  elastic!- 

At  54?o°;  brown,  with 'purple  (A*’  “ ShearS’  SCiSS°rS’  turai“gtools,  penknives,  etc. 

spots ) for  tools  for  cutting  wood  and  soft  metals;  such  as 

At  538°,  purple * l plane-irons,  saws,  knives,  etc. 

it  cSo’  minuniUe fFo,r  t?oIs  re(*uirinS  strong  edges  without  extreme 

Tf  I6  o’  f 1 hardness;  as  cold-chisels,  axes,  cutlery  etc 

A n6^;f,aylSh  b ue’  verg‘  { For  sPr*n£-temper,  which  will  bend  before  breaking- 
ing  on  black \ as  saws,  sword-blades,  etc.  ’ 

If  steel  is  heated  to  a higher  temperature  than  this,  effect  of  the  hardening 
process  is  destroyed.  ° 

d t fhlgl!  br^aking  strain  may  not  be  conclusive  as  to  quality,  as  it  may  be 
due  to  a hard,  elastic  metal,  or  a low  one  may  be  due  to  great  softness. 


644 


METALS. TIN. ZINC. MODELS. 


Case-hardening. 

This  operation  consists  in  converting  surface  of  wrought  iron  into  steel, 
by  cementation,  for  purpose  of  adapting  it  to  receive  a polish  or  to  bear  fric- 
tion, etc. ; it  is  effected  by  heating  iron  to  a cherry-red,  in  a close  vessel,  in 
contact  with  carbonaceous  materials,  and  then  plunging  it  into  cold  water. 
Bones,  leather,  hoofs,  and  horns  cf  animals  are  generally  used  for  this  pur- 
pose, after  having  been  burned  or  roasted  so  that  they  can  be  pulverized. 
Soot  is  also  frequently  used. 

The  operation  reduces  strength  of  the  iron. 

TIN. 

Tin  is  more  readily  fused  than  any  other  metal,  and  oxidizes  very  slowly. 

Its  purity  is  tested  by  its  extreme  brittleness  at  high  temperature. 

Tinplate  is  iron  plate  coated  with  tin. 

Block  Tin  is  tin  plate  with  an  additional  eoating  of  tin. 

ZINC. 

Zinc,  if  pure,  is  malleable  at  220°  ; at  higher  temperatures,  such  as  400°, 
it  becomes  brittle.  It  is  readily  acted  upon  by  moist  air,  and  when  a film 
of  oxide  is  formed,  it  protects  the  surface  from  further  action.  When,  how- 
ever, the  air  is  acid,  as  from  the  sea  or  large  towns,  it  is  readily  oxidized  to 
destruction. 

Iron,  Copper,  Lead,  and  Soot  are  very  destructive  of  it,  in  consequence  of 
the  voltaic  action  generated,  and  it  should  not  be  in  contact  with  calcareous 
water  or  acid  woods. 

The  best  quality,  as  that  known  as  u Vielle  Montague,”  is  composed  of  zinc 
.995,  iron  .004,  and  lead  .001.  Its  expansion  and  contraction  by  differences 
of  temperature  is  in  excess  of  that  of  any  other  metal. 


STRENGTH  OF  MODELS. 

The  forces  to  which  Models  are  subjected  are,  \ 

1.  To  draw  them  asunder  by  tensile  stress.  2.  To  break  them  by  trans- 
verse stress.  3.  To  crush  them  by  compression. 

The  stress  upon  side  of  a model  is  to  corresponding  side  of  a structure  as 
cube  of  its  corresponding  magnitude.  1 bus,  if  a structure  is  six  times  greater 
than  its  model,  the  stress  upon  it  is  as  63  to  1 = 216  to  1 : but  resistance  of 
rupture  increases  only  as  squares  of  the  corresponding  magnitudes,  or  as 
62it0  j =36  to  1.  A structure,  therefore,  will  bear  as  much  less  resistance 
than  its  model  as  its  side  is  greater. 


To  Compute 


Dimensions  of*  a Beam,  etc.,  which,  a 
Structure  can  hear. 

Rule  —Divide  greatest  weight  which  the  beam,  etc.  (including  its  weight), 
in  the  model  can  bear,bv  the  greatest  weight  which,  the  structure  is  required 
to  bear  (including  its  weight),  and  quotient,  multiplied  by  length  ot  beam, 
etc.,  in  model,  will  give  length  of  beam,  etc.,  in  structure. 

Example. — A beam  in  a model  7 inches  in  length  is  capable  of  bearing  a weight 
of  26  lbs.  but  it  is  required  to  sustain  only  a weight  or  stress  of  4 lbs. ; what  is  the 
greatest  length  that  a corresponding  beam  can  be  made  in  the  structuie  / 

26^4  = 6.5,  and  6.5  X 7 = 45-5  ins • 


MODELS. — MOTION  OF  BODIES  IN  FLUIDS. 


645 


Resistance  in  a model  to  crushing  increases  directly  as  its  dimensions ; 
but  as  stress  increases  as  cubes  of  dimensions,  a model  is  stronger  than  the 
structure,  inversely  as  the  squares  of  their  comparative  magnitudes. 

Hence,  greatest  magnitude  of  a structure  is  ascertained  by  taking  square 
root  of  quotient,  as  obtained  by  preceding  rule,  instead  of  quotient  itself. 

Example. — If  greatest  weight  which  a column  in  a model  can  sustain  is  26  lbs., 
and  it  is  required  tt>  bear  only  4 lbs. ; height  of  column  being  18  ins.,  what  should 
be  height  of  it  in  structure? 


.5  = 2.55,  and  2.55  X 18  = 45.9  ins.,  height  of  column  in  structure. 


If,  when  length  or  height  and  breadth  are  retained,  and  it  is  required  to 
give  to  the  beam,  etc.,  such  a thickness  or  depth  that  it  will  not  break  in  con- 
sequence of  its  increased  dimensions, 


Then\/(T)  = V6 

ness  required. 


•5  = 2-  55,  which,  x square  of  relative  size  of  model  = thick - 


To  Compute  Resistance  of*  a Bridge  from  a ]VLodel. 
n2  W — (n  — 1)  ioJ  = load  bridge  will  bear  in  its  centre. 

Example. — If  length  of  the  platform  of  a model  between  centres  of  its  repose 
upon  the  piers  is  12  feet,  its  weight  30  lbs.,  and  the  weight  it  will  just  sustain  at  its 
centre  350  lbs.,  the  comparative  magnitudes  of  model  and  bridge  as  20,  and  actual 
length  of  bridge  240  feet ; what  weight  will  bridge  sustain  ? 

[400  j 

— X (20  — 1)  x 30  = 140  000  — 3800  X 30  — 26  000  lbs. 


MOTION  OF  BODIES  IN  FLUIDS. 

If  a body  move  through  a fluid  at  rest,  or  fluid  move  against  body  at 
rest,  resistance  of  fluid  against  body  is  as  square  of  velocity  and  density 
of  fluid  ; that  is,  R = d v2.  For  resistance  is  as  quantity  of  matter  or 
particles  struck,  and  velocity  with  which  they  are  struck.  But  quan- 
tity or  number  of  particles  struck  in  any  time  are  as  velocity  and  density 
of  fluid ; therefore,  resistance  of  a fluid  is  as  density  and  square  of 
velocity. 


v2  a d v2  _ 

— = h,  and  = R.  h representing  height  due  to  velocity , d density  of  fluid, 

2 g 2 9 

and  R resistance  or  motive  force. 


Resistance  to  a plane  is  as  plane  is  greater  or  less,  and  therefore  resistance 
to  a plane  is  as  its  area,  density  of  medium,  and  square  of  velocity;  that  is, 
'R=.adv‘2. 


Motion  is  not  perpendicular,  but  oblique,  to  plane  or  to  face  of  body  in  any 
angle,  sine  of  which  is  .9  to  radius  1 ; then  resistance  to  plane,  or  force  of 
fluid  against  plane,  in  direction  of  motion,  will  be  diminished  in  triplicate 
ratio  of  radius  to  sine  of  angle  of  inclination,  or  in  ratio  of  1 to  s3. 


tt  adv2s3  adv2s3 

nence, = K,  and  = F.  w representing  weight  of  body , and  F 


retarding  force. 


2 gw 


m Progression  of  a solid  floating  body,  as  a boat  in  a channel  of  still  water, 
gives  rise  to  a displacement  of  water  surface,  which  advances  with  an  un- 
dulation in  direction  of  body,  and  this  undulation  is  termed  Wave  of  Dis- 
placement. 


MOTION  OF  BODIES  IN  FLUIDS. 


646 

Resistance  of  a fluid  to  progression  of  a floating  body  increases  as  velocity 
of  body  attains  velocity  of  wave  of  displacement,  and  it  is  greatest  when  the 
two  velocities  are  equal. 

In  the  motion  of  elastic  fluids,  it  appears  from  experiments  that  oblique 
action  produces  nearly  same  effect  as  in  motion  of  water,  m the  passage  01 
curvatures,  apertures,  etc. 

Resistance  to  an  Area  of  One  Sq.  Foot  moving  through 
Water,  or  Contrariwise. 

Angle  of  | . _ 

Surface  Pressure  per  Sq.  Foot  for  following  V e- 
wjth  j locities per  Foot  per  Minute. 

Plane  of 

480 


Angle  of 
Surface 
with 
Plane  of 
Current. 

Pressu 

lo 

120 

O 

Lbs. 

6 

.09 

8 

•133 

9 

.156 

10 

.179 

15 

•355 

20 

.608 

25 

•94 

30 

i-353 

35 

1.798 

40 

2.258 

240 


Lbs. 

•359 

•53 

.624 

.718 

1.42 

2- 434 

3- 76 
5-4I3 
7-I92 
9.032 


480 
Lbs. 
1-435 
2.122 
2.496 
2.87 
5.678 
9-734 
1 5- 038 
21.653 
28.766 
36- 1 3 


900 


Current. 


Lbs. 

5.046 

7- 459 

8- 775 
10.091 
19.963 
34.222 
52.869 
76.123 

101.132 

127.018 


45 

50 

55 

60 

65 

70 

75 

80 

85 

90 


Lbs. 

2.66 

2- 995 

3- 249 
3-455 
3.607 
3.728 
3.8! 
3-857 
3.892 
3-9 


240 


Lbs. 

10.639 

11.981 

12.995 

13.822 

14-43 

14- 9I4 
15.241 
15.428 

15- 569 

15.6 


Lbs. 

42-557 

47-923 

51-979 

55.286 

57-72 

59-654 

60.965 

61.714 

62.275 

62.4 


Resistance  to  a plane,  irom  a niua  acung  m * ------ 

its  face,  is  equal  to  weight  of  a column  of  fluid,  base  of  which  is  plane  and 
altitude  equal  to  that  which  is  due  to  velocity  of  the  motion,  01  through 
which  a heavy  body  must  fall  to  acquire  that  velocity . 

Resistance  to  a plane  running  through  a fluid  is  same  as  force  of  fluid  m 
motion  with  same  velocity  on  plane  at  rest.  But  force  of  fluid  m motion  is 
equal  to  weight  or  pressure  which  generates  that  motion,  and  this  is  equal  to 
weight  or  pressure  of  a column  of  fluid,  base  of  which  is  area  of  the  plane, 
an  (Fits  altitude  that  which  is  due  to  velocity. 

Illustration.— If  a plane  i foot  square  be  moved  through  water  at  rate  of  32.166 
feet  per  second,  then  ■32,i6>-  = 16.083,  space  a body  would  require  to  fall  to  acquire 
a velocity  of  32.166  feet  per  second;  therefore  1 X 62.5  (weight  of  a cube  foot  of 

water)  x 32' l66--  = 1005  lbs.  = resistance  of  plane. 

64-333 

Resistance  of  different  Fig-ures  at  different  Velocities  in 
Air. 


Veloci- 
ty per 
Second. 

C01 

Vertex. 

ae. 

Base. 

Sphere. 

Cylin- 

der. 

Hemi- 

sphere. 

Round. 

V eloci- 
ty  per 
Second. 

Co: 
Vertex . 

ne. 

Base. 

Sphere. 

Cylin- 

der. 

Hemi- 

sphere. 

Round. 

Feet. 

3 

4 

5 
8 

9 

10 

Oz. 
.028 
.048 
.071 
. 168 
.2x1 
.26 

Oz. 
.064 
. 109 
. 162 
• 382 
.478 

•587 

Oz. 

.027 

.047 

.068 

.162 

.205 

•255 

Oz. 

•05 

.09 

•143 

•36 

•456 

•565 

Oz. 

.02 

•039 

.063 

.16 

.199 

.242 

Feet. 

12 

14 

15 
x6 
18 
20 

Oz. 

•376 

.512 

•589 

•673 

.858 

1.069 

Oz. 

.85 

1. 166 

1- 346 
1.546 
2.002 

2- 54 

Oz. 

•37 

.505 

.581 

.663 

.848 

i-o57 

Oz. 
.826 
1145 
1.327 
1.526 
j I-986 
! 2.528 

Oz. 

•347 

.478 

•552 

! .634 
| .818 
' 1033 

Diameter  of  all  the  figures  was  6.375  ins.,  and  altitude  of  the  cone  6.625  ins. 

Angle  of  side  of  cone  and  its  axis  is,  consequently,  250  42'  nearly. 

From  the  above,  several  practical  inferences  may  be  drawn. 

1.  That  resistance  is  nearly  as  surface,  increasing  but  a very  little  above 
that  proportion  in  greater  surfaces. 


MOTION  OF  BODIES  IN  FLUIDS. 


647 

2.  Resistance  to  same  surface  is  nearly  as  square  of  velocity,  but  gradu- 
ally increasing  more  and  more  above  that  proportion  as  velocity  increases. 

3.  TV  hen  after  parts  of  bodies  are  of  different  forms,  resistances  are  differ- 
ent, though  fore  parts  be  alike. 

4.  The  resistance  on  base*  of  a cone  is  to  that  on  vertex  nearly  as  2 a to 
I.  And  m same  ratio  is  radius  to  sine  of  angle  of  inclination  of  side  of  cone 
to  its  path  or  axis.  So  that,  111  this  instance,  resistance  is  directly  os  sine 
of  angle  of  incidence,  transverse  section  being  same,  instead  of  square  of  sine. 

Resistance  on  base  of  a hemisphere  is  to  that  on  convex  side  nearly  as 
2.4  to  1,  instead  or  2 to  1,  as  theory  assigns  the  proportion. 

<d  f,l’!*7®;_ReSiS^ntie  t0  a.  sphere  moving  through  a fluid  is  but  half  re- 
sistance to  its  great  circle,  or  to  end  of  a cylinder  of  same  diameter,  moving 
with  an  equal  velocity,  being  halt  of  that  of  a cylinder  of  same  diameter.  S 


fzgx^dx- 


X — n 


n 


- V.  d representing  diameter  of  sphere , and  N and  n spe- 


cific gravities  of  sphere  and  resisting  fluid. 

isr  4 

nX  3 d==  S'  S resenting  space  through  which  a sphere  passes  while  acquir- 
ing its  maximum  velocity , in  falling  through  a resisting  fluid. 

bal1  -0f  lead  1 inch  in  diameter,  specific  gravity  u.qo  be  set 
free  in  water,  specific  gravity  1,  what  is  greatest  velocity  it  will  attain  in3desrpnu. 
ing,  and  what  space  will  it  describe  in  attaining  this  velocity  ? 

<7  = 32.166,  d = ~ foot,  N = 11.33,  and  n — 1. 

~ V 7-  *48  X 10. 33  = 8. 593  feet  per  sec. 


Then  . /2  x 32. 166  X — of  — x - 1-33 1 

v 3 12  1 

Hence,  — — X — of  — — 1.250  feet. 

1 3 12  ^ 


3 nr  V2 

Q-n  \!  —J  — retardive  force  = . 

o g a 2 g s 


_ n a v2  n a v2 

y mder.  — = R,  and  a representing  area  or  p r2,  and 


2 9 


w weight  of  body. 

Illustration.— Assume  a = 32  sq.  feet,  v = 10  feet  per  second,  and  n = .0012 
.0012  X 32  X 10" 


Then 


64-33 


- = - ° 6 of  a cube  foot  of  water  — .06  of  62.5  = 3.75  lbs. 


Conical  Surface.  nav2s*  _ R ^ n p d2  v2  s 2 p ^ n p d2  v2  s2 

r ...  2 9 8 g ’ 8 Q w 

J.  s 1 epi  esenting  sine  of  inclination , and  a convex  surface  of  cone. 

Curved  End  as  a Splrere  or  Hemispherical  End.  ? 71  V*  ^ 
= R,  and  Circle  .5  of  spherical  end.  9 

In  general,  when  n is  to  water  as  a standard,  result  is  in  cube  feet  of  wafpr  if 
a .s  in  sq.  feet;  and  in  cube  ins.  of  water,  if  a is ’in  sq.  ins.,  v in ^s^gT^l 

If  n is  given  in  lbs.  in  a cube  foot,  a is  in  sq.  feet,  v and  g are  in  feet,  result  is  in  lbs. 

To  Compute  Altitude  of  a Column  of  Air,  Pressure  of 
which  shall  be  equal  to  Resistance  of  a BodyaoviiiB 
tlirouglx  it,  witli  any  Velocity. 

5 r 

-X  —=x  — altitude  in  feet,  ax  = volume  of  column  in  feet,  and  — ax  — weight 

tiiiZrl  dLr,ein?ZtiVrea  °f  ofbody'  similar  10  any  in  table,  perpen - 

^ 

thl  £?U  taSSif0*  °r  the  P°pU''U'  aS3erti0"  thal  a taPer  6Par  ca”  “>wed  In  wnter  easiest  when 


648 


MOTION  OF  BODIES  IN  FLUIDS. 


When  a — — of  a foot,  as  in  all  figures  in  table,  * becomes  ^ r when  r _ re - 

9 

oietancc  tw  table  to  similar  body.  , 

Illustration.- Assume  convex  face  of  hemisphere  resistance  = .634  oz.  at  a ve- 
locity  of  16  feet  per  second. 

Then  r = .634,  and  * = ^ r = *.ynsM  = <***>*  of  column  of  air,  pressure  of 

which  = resistance  to  a sphtrical  surface  at  a velocity  of  16  feet 

. -1  „ prPc!sure  of  Air  in.  rear  of  a Projectile 

To  Cot]^Pj^|e^or  to  Pressure  '!"<■  to  its  Velocity. 

Assume  height  of  barometers.  5 feet,  and  weight  of  atmospheres  14.7  lbs. 

Weight  of  cube  inch  of  mercury  = ^ = -49  «*.,  “d  wei&ht  of  cube  mch  °f  “r 
= .00004357  lbs.;  hence,  .49-000043  57  = » ' wbl«*  X *-5  feet-zBns  feet. 

Then  16.08  : V*  «5  ::  = *>  a 

awstt 

x"  n*.‘«  H»«,V . VW* •••■  • — ** 

To  Compute  Velocity  lost  by  a projectile. 

If  a bod^  is  projected  with  any  velocity  in  a medium  of  same  density  with  itsel , 
and  it  describes  a space  = 3 of  lts  diameters,  ^ ^ 

Then  * = 3 d,  and  b = g^  = gd* 


9 nnd  c-__r  _21o8  = velocity  lost  nearly . 66  of  projectile  velocity. 
Hence,  b x = -x- , ana  bx  — a o8 


cbx~ 1 _ 

c=base  of  Nap.  system  of  log. ; hence  c^  = number  corresponding  to  Nap.  log. 
I x.  Hence,  if  b x X -4343i  result  — com.  log.  ol  c . 

6 z =5  = 1. 125,  which  x -4343  = -488  587  5,  and  number  to  this  com.  log.=3.°8o3. 
3.0803  I 2.08 

Hence,  velocity  lost  g— - 

Illustration. -If  an  iron  ba"“'c^  5<T feet  of  °pac£? 


feet  per^econd^  ^a^would  be  vdo'city  lo  ™afternmving  through  500  feet  of  space  ? 
d = -=g,  * = 5oo,  N = 7A,  andn  = .ooi2. 

mx*  T1XI2X  500X  3X6_  81  and  v = l?^  = 99z  feet  per 
Hence,  b x — g ^ — 8 X 21X10000  44°  ’ CTF5 


second,  having  lost  coo  feet,  or  nearly  i of  its  initial  velocity. 

12  _ oQi2  A and  - = - and  - inverted,  because  N and  n are  in  denominator. 

10  000  ’22  • 3 6 

To  Compute  Time  and  Velocity. 

i-f-—  = g^d  = 5’  and  ^ = U 

its  velocity  at  end  of  that  time?  x l 

3X12X  3 X 6 ___x_  and  bx  = ^;  hence  - = — 5 ^ =^»  ““ 
6“  8X22X10^  2716’  2716  1 . 

i.  - - ,x:Z3Zi  = J-  nearly.  v = 690  and  <=2716  X ( - —)  - *-67  sec* 

5 a 1200  090 


NAVAL  ARCHITECTURE. 


649 


NAVAL  ARCHITECTURE. 

Results  of  Experiments  upon  Form  of  Vessels. 
( Wm . Bland.) 

Cubical  Models.  Head  Resistance. — Increases  directly  with  area 
of  its  surface.  Weight  Resistance. — Increases  directly  as  weight. 

"Vessels’  Models.  Lateral  Resistance.  — About  one  twelfth  of 
length  of  body  immersed,  varying  with  speed. 

Order  of  Superiority  of  Amidship  Section. — Rectangle,  Semicircular, 
Ellipse,  and  Triangle. 

Centre  of  lateral  resistance  moves  forward  as  model  progresses. 
Centre  of  gravity  has  no  influence  upon  centre  of  lateral  resistance. 


Relative  Speeds. 

Length. — Increased  length  gives  increased  speed  or  less  resistance. 

Depth  of  Flotation. — Less  depth  of  immersion  of  a vessel,  less  the  resistance. 
Amidship  Section. — Curved  sections  give  higher  speed  than  angled. 

Sides.  -Slight  horizontal  curves  present  less  resistance  than  right  lines. 
Curved  sides  with  one  fourth  more  beam  give  equal  speeds  with#  straight 
sides  of  less  beam.  Keel.— Length  of  keel  has  greater  effect  than  depth. 
Stern. — P arallel-sided  after  bodies  give  greater  speed  than  taper-sided. 


Form  of  Bow. 


Order  of  Speed. 


Isosceles  triangle , sides  slightly  convex 

“ “ “ right  lines 

“ “ slightly  concave  at  entrance  and  running) 

out  convex ( 


Spherical  equilateral  triangle  compared  to  Equilateral  triangle , speed  is 
as  11  to  12.  Equilateral  triangle , with  its  isosceles  sides  bevelled  off  at  an 
angle  of  45 °,  compared  to  bow  with  vertical  sides,  is  as  5 to  4. 

When  bow  has  an  angle  of  14°  with  plane  of  keel,  compared  with  one  of 
70,  its  speed  is  greater. 


Bodies  Inclined  Upwards  from  Amidship  Section. 

1.  Model  with  bow  inclined  from  IS,  has  less  resistance  than  model  with- 
out any  inclination. 

2.  Model  with  stern  inclined  from  IS,  has  less  resistance  than  model  with- 
out any  inclination. 

Model  1 had  less  resistance  than  model  2.  Model  with  both  bow  and 
stern  inclined  from  IS,  has  less  resistance  than  either  1 or  2. 


Stability. 

Results  of  Experiments  upon  Stability  of  Rectangular 
Blocks  of  Wood  of  Uniform  Length  and  Depth,  "but 
of  Different  Breadths.  (Wm.  Bland.) 

Length  15,  Depth  2,  and  Depression  1 inch. 


Width. 

Weight. 

As  Observed. 

Ratio  0 
With  like 
Weights. 

f Stability. 

By  Squares  of 
Breadth. 

By  Cubes  of 
Breadth. 

Ins. 

Oz. 

3 

24 

1 

1 

I 

1 

4-5 

£ 

35 

2-5 

2.4 

2.25 

3-375 

0 

45 

7 

3-7 

4 

8 

7 

55 

11 

4.8 

6.25 

15-625 

NAVAL  ARCHITECTURE. 


650 


Hence  it  appears  that  rectangular  and  homogeneous  bodies  of  a uniform 
length,  depth,  weight,  and  immersion  in  a fluid,  but  of  different  breadths,  have 
stability  for  uniform  depressions  at  their  sides  (heeling)  nearly  as  squares 
of  their  breadth ; and  that,  when  weights  are  directly  as  their  breadths, 
their  stability  under  like  circumstances  is  nearly  as  cubes  of  their  breadth. 

With  equal  lengths,  ratio  of  stability  is  at  its  limit  of  rapid  increase  when 
width  is  one  third  of  length,  being  nearly  in  cube  ratio;  afterwards  it  ap- 
proaches to  arithmetic  ratio. 


Results  of*  Experiments  upon  Stability  and  Speed,  of 
IMIodels  having  ,A_miclsliip  Sections  of  different  Forms, 
"but  Uniform  Length,  Breadth,  and.  Weights.  ( W.  Bland.) 
Immersion  different , depending  upon  Form  of  Section. 


Form  of  Immersed  Section. 

Stability.  ] 

Speed. 

Half-depth  triangle,  other  half  rectangle 

12 

14 

A 

Rectangle  . 

T 

3 

Right-angled  triangle :i! 

7 

3 

Semicircle 

9 

2 

* Draught  of  water  or  immersion  double  that  of  rectangle. 


Statical  Stability  is  moment  of  force  which  a body  in  flotation  exerts  to 
attain  ite  normal  position  or  that  of  equilibrium,  it  having  been  deflected 
from  it,  and  it  is  equal  to  product  of  weight  of  fluid  displaced  and  horizontal 
distances  between  the  two  centres  of  gravity  of  body  and  of  displacement,  or 
it  is  product  of  weight  of  displacement,  height  of  Meta-centre , and  Sine  of 
angle  of  inclination. 

Dynamical  Stability  is  amount  of  mechanical  work  necessary  to  deflect  a 
body  in  flotation  from  its  normal  position  or  that  of  equilibrium,  and  it  is 
equal  to  product  of  sum  of  vertical  distances  through  which  centre  of  grav- 
ity of  body  aseends  and  centre  of  buoyancy  descends,  in  moving  from  ver- 
tical to  inclined  position  by  weight  of  body  or  displacement. 

To  Determine  Measure  of  Stability  of  Hull  of  a Vessel 
01*  Bloating  Body.— Fig.  1. 


Measure  of  stability  of  a floating  body  depends  essentially  upon  horizontal  dis- 
tance, G s,  of  meta-centre  of  body  from  centre 
of  gravity  of  body;  and  it  is  product  of  force 
of  the  water,  or  resistance  to  displacement  of 
it,  acting  upward,  and  distance  of  G s , or  P x 
G 5.  If  distance'  c M,  represented  by  r,  and 
/ angle  of  rolling,  cM  r,  by  M°,  measure  of  sta- 
bility, or  S is  determined  by  P r,  sin.  M°  = S ; 
and  this  is  therefore  greater,  the  greater  the 
weight  of  body,  the  greater  distance  of  meta- 
centre  from  centre  of  gravity  of  body,  and  the 
greater  the  angle  of  inclination  of  this  or  of 
c M r. 


Assume  figure  to  represent  transverse  section  of  hull  of  a vessel,  G centre  of 
gravity  of  hull,  w l water-line,  and  c centre  of  buoyancy  or  of  displacement  of  im- 
mersed hull  in  position  of  equilibrium.  Conceive  vessel  to  be  heeled  01  inclined 
over,  so  that  ef  becomes  water-line,  and  s centre  of  buoyancy;  produce  .<?  M,  and 
point  M is  meta-centre  of  hull  of  vessel. 

Transverse  meta-  centre  depends  upon  position  of  centre  of  bll«>’anc>;  f«rf 
vertical  line  drawn  from  centre  intersects  a line  passing  through  centre  of  grawtj  ot  hull  ot  vessel 
perpendicular  to  plane  of  keel. 

Point  of  meta-centre  may  be  the  same,  or  it  may  differ  slightly  for  different  angles  of  heeling.  Angle 
of  direction  adopted  to  ascertain  position  of  n*eta- centre  shouiji  be  greatest  wjhich,  iinder  oroinary  cir- 
cumstances, is  of  probable  occurrence  ; in  different  vessels  tbit  angle  ranges  from  20  to  60  ..  ^ 

If  meta-centre  is  above  centre  of  gravity,  equilibrium  is  Stable;  if  it  coincides  with  it,  equilibrium  is 
Indifferent ; and  if  it  is  below  it.,  equilibrium  i$  Unstable. 


Comparative  Stability  of  different  hulls  of  vessels  is  proportionate  to  the  distance 
of  G M for  same  angles  of  heeling,  or  of  distance  G s.  Oscillations  of  hull  of  a ves- 
sel may  be  resolved  into  a rolling  about  its  longitudinal  axis,  pitching  about  its 
transverse  axis,  and  vertical  pitching,  consisting  in  rising  and  sinking  below  and 
above. position  of  equilibrium. 

If  transverse  section  of  hull  of  a vessel  is  such  that,  when  vessel  heels,  level  of 
centre  of  gravity  is  not  altered,  then  its  rolling  will  be  about  a permanent  longi- 
tudinal axis  traversing  its  centre  of  gravity,  and  it  will  not  be  accompanied  by  any 
vertical  oscillations  or  pitchings,  and  moment  of  its  inertia  will  be  constant  while 
it  rolls.  But  if,  when  hull  heels,  level  of  its  centre  of  gravity  is  altered,  then  axis 
about  which  it  rolls  becomes  an  instantaneous  one,  and  moment  of  its  inertia  will 
vary  as  it  rolls;  and  rolling  must  then  necessarily  be  accompanied  by  vertical  os- 
cillations. 

Such  oscillations  tend  to  strain  a vessel  and  her  spars,  and  it  is  desirable,  therefore, 
that  transverse  section  of  hull  should  be  such  that  centre  of  its  gravity  should  not 
alter  as  it  rolls,  a condition  which  is  always  secured  if  all  water-lines,  as  ivl  and  ef 
are  tangents  to  a common  sphere  described  about  G;  or,  in  other  words,  if  point  of 
their  intersections,  o,  with  vertical  plane  of  keel,  is  always  equidistant  from  centre 
of  gravity  of  hull. 

To  Compute  Statical  Stability. 
stability  ^ M = S’  D rePresenting  displacement , M angle  of  inclination , and  S 

Illustration  r.  Assume  a ship  weighing  6000  tons  is  heeled  to  an  angle  of  q° 
distance  c M = 3 feet,  ® y ’ 

Sin.  9°  = . 1564.  Then  6000  X 3 X -1564  — 2815.2  foot-tons. 

Q«r>~Teigll^0f  a floatin„g  bo(Jy  is  5515  l^s.,  distance  between  its  Centre  of  gravity 
and  meta-centre  is  11.32  feet,  and  angle  M — 200.  J 

Sin.  M = .342  02.  Hence  5515  x 11.32  x .34202  = 21  352.24  foot-lbs. 

Statical  Surface  Stability-. 

Moment  of  Statical  surface  stability  at  any  angle  is  c z D.  Assuming 

at  anv^iSt'  o^h^  COmcld,ed  \v'ith  o ; coefficient  of  a vessel’s  stability 
ferMrnl  r«  13.e^resscd  'Ihe"  the  displacement  is  multiplied  by 

gravity,  or meta"cenU'e  for  angle  of  heel  above  centre  of 

Approximately . Rule.— Divide  moment  of  inertia  of  plane  of  flotation 
for  upright  position  relatively  to  middle  line  bv  volume  of  displacement- 
and  quotient  multiplied  by  sine  of  angle  of  heel  will  give  result. 

Per  F°0t  °f  LenVth  °f  Vessel<  § (B3  Sin.  M).  B representing  half  breadth. 
Dynamical  Surface  Stability. 
0fvSr!ifemiCalSl,r^  stability  is  expressed  bv  product  of  weight 

M^n:^s^^epre8sio,“ of  centre  °f  b"°"anc^  durhl^he 

To  Compute  Dynamical  Stability  of  a Vessel. 

ab^^t^fi’ra^tULE'lMuIaPl7.d“^cment  b-v  he'ght  of  meta-cHitre 
abov  e centre  of  gravitv , and  product  by  versed  sine  of  angle  of  heel. 

Or  multiply  statical  stability  for  given  angle  by  tangent  of  .5  angle  of  heel. 

To  Compute  Elements  of  Stability  of  a Floating  Body. 

A a — s>  sin.  M ~ r’  sin.  M ~ aD(i  sin<  Mr==c-  A representing  area  of 

Gc  m;  \zt 

ity  and  of  line  of  disnlaremprtt  nf’-t  ° h°7  lzonia^  ^stance,  G s,  between  centre  of  grav- 
Of  gravity  and  buoyancy,  all  “"<r" 


652 


NAVAL  ARCHITECTURE. 


Note.— When  centre  of  gravity,  G,  is  below  that  of  displacement,  c,  then  e is 
when  it  is  above  c it  is  — ; and  when  it  coincides  with  c it  is  o;  or  e is  — when 

and  a body  will  roll  over  when  e sin.  M = or  >s. 

Assumed  elements  of  figure  illustrated  are  A = 86,  A'  = 21. 5,  b = 21. 5,  and  e = . 5. 

The  deduced  arc  $ = 3.7,  c = 3.87,  g=  10.82,  a = 14.9,  and,  r = 11.32.  b repre- 
senting breadth  at  water-tine  or  beam  in  feet,  and  P weight  or  displacement  in  Lbs. 
or  tons. 

TheQS  = ?^Xl4-9  = 3-7 /«*, 

= 10. 82  /eef,  c = . 342  02  x 1 1. 32  — 3-  87  feet 


Of  Hull  of  a Vessel. 

63 


( . b*  d:  p sin.  M = S ; d cos.  ,5  M = d 

\10.7  to  13*  A ) ’ 

= p(fc“+Iinni)=Si  and 

P (s  dz  e sin.  M)  = S.  <2  representing  depth  of  centre  of  gravity  of  displacement  un- 
der water  in  equilibrium,  and  d'  depth  when  out  of  equilibrium,  both  in  feet. 


10.7  to  13  (11.93)  A 


= 9, 


Illustration  i. — Displacement  of  a vessel  is  10000000  lbs. ; breadth  of  beam,  50 
feet;  area  of  immersed  section,  800  sq.  feet;  vertical  distance  from  centre  of  grav- 
ity of  hull  up  to  centre  of  buoyancy  or  displacement,  1.0  feet,  and  horizontal  dis- 
tance a between  centres  of  gravity  of  areas  immersed  and  emerged,  when  careened 
to  an  angle  of  90  10'  = 33.4  feet,  immersed  area  being  50  sq.  feet. 


Sin.  90  10'  = . 1593.  Then  s = ~ X 33-4  = 2.0875  feet,  800X2.0875  = 50X33-4) 
r = .2' 39.  _ jh  feet.  g — — — — — = 13. 1 feet,  S = ( - — \ -J-  1.9  X 

.1593  5 J 11.93X800  \ii.93X8oo/ 

10 000 000  X • 1 593  — 23 9°5 39^  lbs->  and  e = (10 1! ocx! “ 2' 087 5)  =1^feet 


2.— Assume  a ship  having  a displacement  of  5000  tons,  and  a height  of  meta-centre 
of  3.25  feet,  to  be  careened  to  6°  12'.  What  is  her  statical  stability? 

Sin.  6 6 12' = .1079.  Then  5000  X 3-25  X . 1079  = 1753.37  foot-tons. 


3. — Assume  a weight,  W,  of  50  tons  to  be  placed  upon  her  spar  deck,  having  a 
common  centre  of  gravity  of  15  feet  above  her  load-line, 

Then  5000  X 3-  25  — 50  -f-  1 5 X • 1079  = 1745. 29  foot-tons. 

4.  — Assume  100  tons  of  water  ballast  to  be  admitted  to  her  tanks  at  a common 
centre  of  gravity  of  15  feet  below  her  load-line, 

Then  5000  X 3-25  + 100  X 15  X .1079  = 1915. 22  foot- tons. 

5.  — Assume  her  masts,  weighing  6 tons,  to  be  cut  down  20  feet, 

Then  10  X — = — foot  = fall  of  centre  of  gravity,  and  5000  X ( 3-  25  + ) X • io79 

5000  50  \ 5°/ 

= 1774.95  tons. 

To  Compute  Elements  of  Bower,  etc.,  reqnireci  to 
Careen,  a Body-  or  "Vessel. 

Sin.  M (h  - n sin.  M)  + » sec.  M - * = l - - fj = m- 

W lrz=zFc,  and  W l = S.  W representing  weight  or  power  exerted  and  l distance  ; 
at  which  weight  or  power  acts  to  careen  body , taken  from  centre  of  gravity  of  displace- 
ment perpendicular  to  careening  force,  h vertical  height  from  centre  of  gravity  of  dis- 
placement to  centre  of  weight  or  power  to  careen  body  when  it  is  in  equilibrium , 
n horizontal  distance  from  centre  of  vessel  to  centre  of  weight  or  power,  L length  of 
vessel,  m meta-centre,  and  S as  in  preceding  case , all  in  feet. 


* Unit  for  section  of  a parallelogram  is  10.7  ; of  a semicircle  12,  and  of  a triangle  12.8. 


Illustration. — A weight  is  placed  upon  deck  of  a vessel  at  a mean  height  of  o 87 
feet  from  centre  line  of  hull;  height  at  which  it  is  placed  is  n.32,  and  other  ele- 
ments as  in  first  case  given.  J 

Sec.  20  .342.  Then  h = n.32,  n = 3.87,  and  l = .342  (1 1.3  —TfyX~w)  + 

3.87  x 1.0642 -3.7  = .342  x 10  + 4.12  — 3.7  = 3.84/^. 

Assume  W = 5515.  Then  5515  X 3-84  = 21 187.6  foot-lbs. 


Or  P (iv  cos.  M + A sin.  M)  =.  S.  iv  representing  distance  of  weight  from  centre  of 
vessel , and  h height  of  w above  water-line , both  in  feet.  U J n re  of 

H0!fST?Ay°f'If  a weight  of  30  tons  placed  at  20  feet  from  centre  of  hull  or 
deck,  10  feet  above  water-line,  careens  it  to  an  angle  of  2°  9',  what  is  its  stability? 
cos.  2°  9'  = .9993 ; Sin.  2°  9'  = .0375. 

30  (20  X. 9993  + 10  X .0375)  = 30  x 20.361  =610. 83 /oo^ons. 


Bottom,  and.  Immersed  Surface  of  Hnll  of ‘Vessels. 

To  Compute  Bottom  a nd  Side  Surface  of  Bull 
Bottom  and  Side  Rule.  Multiply  length  of  curve  of  amidship  sec'tion, 
taken  from  top  of  tonnage  or  mam  deck  beams  upon  one  side  to  same  point 
upon  other  (omitting  width  of  keel),  by  mean  of  lengths  of  keel  and  be- 
tween perpendiculars  m feet,  multiply  product  by  .85  or  .9  (according  to  the 
capacity  of  vessel),  and  product  will  give  surface  required  in-  sq.  feet. 

Example.— Lengths  of  a steamer  are  as  follows:  keel  201  feet,  and  between  ner- 
pendiculars  210  feet,  curved  surface  of  amidship  section  76  feet;  what  is  surface? 

Coefficient  .87.  210  + 201 2 = 205.5,  and  76  X 205.5  X .87  = i3  587  sq.  feet. 


Note.— Exact  surface  as  measured  was  13650  sq.  feet. 

Bottom  Surface.  Rule.— Multiply  length  of  hull  at  load-line  by  its 
breadth,  and  this  product  by  depth  of  immersion  (omitting  the  depth  ol 
keel)  m feet ; and.  this  product  multiplied  by  from  .07  to  .08  (according  tc 
capacity  of  vessel)  will  give  surface  required  in  sq.  feet. 

Example.— Length  upon  load-line  of  a vessel  is  310  feet,  beam  40  feet,  depth  of 
keel  1 foot,  and  draught  of  water  20  feet;  what  is  bottom  or  wet  surface? 


Coefficient  assumed  .073.  310  X 40  X 20  — 1 X .073  = 17  199  sq.  feet. 

To  Compute  Resistance  to  Wet  Surface  of*  Bull. 

C a v2  — R C representing  a coefficient  of  resistance,  a area  of  wet  surface  in  sa. 
feet,  and  v velocity  of  hull  in  feet  per  second. 

Values  of  C I-00/*  clean  copper.  I .014,  iron  plate. 

’ 1-oij  smooth  paint.  | .019,  iron  plate,  moderately  foul. 


1 75 f ! “‘VWV.UVWJ 

Power  required  to  propel  one  sq.  foot  of  immersed  amidship  section  at  53  is  071 
that  of  smooth  wet  surface.  ^ /j 


To  Compute  Elements  of  a Vessel. 

Displacement  and  its  Centre  of*  Gravity. 

Displacement  of  a vessel  is  volume  of  her  body  below  water-line. 

Centre  of  Gravity,  or  Centre  of  Buoyancy  of  Displacement , is  centre  of 
gravity  of  w^ater  displaced  by  hull  of  vessel. 

For  Displacement.  Rule.— Divide  vessel,  on  half  breadth  plan,  into  a 
nUK5<r  eQuidistant  sections,  as  one,  twro,  or  more  frames,  commencing 
at  & and  running  each  side  of  it.  Add  together  lengths  of  these  lines  in 
both  fore  and  aft  bodies,  except  first  and  last,  by  Simpson’s  rule  for  areas 
(see  page  344) ; multiply  sum  of  products  by  one  third  distance  between 
sections,  and  product  will  give  area  of  water-line  between  fore  and  aft  sections. 

Then  compute  areas  contained  in  sections  forward  and  aft  of  sections  taken  in- 
eluding  stern  and  rudder-post,  rudder  and  stem,  and  add  sum  to  area  of  body-sec- 
tions already  ascertained.*  J 

* To  Compute  Area  of  a Water-line , see  Mensuration  of  Surfaces,  page  344. 

3 I* 


NAVAL  ARCHITECTURE. 


Compute  area  of  remaining  water-lines  in  like  manner.  Tabulate  results,  and 
multiply  them  by  Simpson’s  rule  in  like  manner  as  for  a water-line,  and  again  by 
consecutive  number  of  water-lines,  and  sum  of  products  between  water-line  and 
product  will  give  volume  between  load  and  lower  water-line. 

Add  area  of  lower  water-line  to  area  of  upper  surface  of  keel;  multiply  half  sum 
by  distance  between  them,  and  product  will  give  volume;  then  compute  areas  con- 
tained in  sections  forward  and  aft  of  sections  taken  as  before  directed. 

If  keel  is  not  parallel  to  lower  water-line,  take  average  of  distance  between  them. 

Compute  volume  of  keel,  rudder-post  and  rudder  below  water-line;  add  to  volume 
already  ascertained;  multiply  product  by  two,  for  full  breadth,  and  product  will 
give  volume  required  in  cube  feet,  all  dimensions  being  taken  in  feet. 

Example. -Assume 

Fig.  2. 


a vessel  ioo  feet  in 
length  by  20  feet  in 
extreme  breadth,  on 
load-line  of  8 feet  9 
inches  immersion. 
Figs.  2 and  3. 

Distance  between 
sections,  for  purpose 
of  simplifying  this 
example,  is  taken 
at  10  feet;  usually 
frames  are  18  to  30 


ins.  apart,  and  two  or  more  included  in  a section.  Water-lines  2 feet  apart. 


1 st  Water-line. 

2 d Water-line. 

3 d Water-line. 

4 

5 

= 

5 

4 

2.7 

= 

2.7 

4 

i-5 

= 

i-5 

3 

7-7  X 

4 

= 

30.8 

3 

6.9 

X 

4 f=i 

27.6 

3 

5 X 

4 

= 

20 

2 

9-5  X 

2 

== 

*9 

2 

8.7 

X 

2 = 

17.4 

2 

6.6  X 

2 

= 

13.2 

1 

9.9  X 

4 

= 

39-6 

1 

9-5 

X 

4 ~ 

38 

1 

8.7  X 

4 

— 

34-8 

0 

10  X 

2 

20 

0 

9.6 

X 

2 = 

19.2 

0 

8.9  X 

2 

= 

17.8 

A 

9.6  x 

4 

= 

38- 4 

A 

9 

X 

4 = 

36 

A 

7.6  X 

4 

= 

30-4 

B 

7.8  X 

2 

= 

15.6 

B 

7 

X 

2 = 

14 

B 

7 X 

2 

= 

14 

C 

6.8  X 

4 

= 

27.2 

C 

5 

X 

4 — 

20 

C 

3 X 

4 

= 

12 

D 

4 

== 

4 

D 

2 

= 

2 

D 

1.2 

= 

1.2 

199.6 

176.9 

144.9 

IO  -r 

'3 

= 

3j 

10  —7 

"3  . = 

10  -r 

"3 

3^ 

Abaft  section  4,  rud- 
der and  post 

Forward  section  D 
and  stem 


665.3 


20.7 

711 


589-7 

Abaft  section  4,  rud- 
der and  post 13.2 

Forward  section  D 

and  stem 9.1 

612 


483 

Abaft  section  4,  rud- 
der and  post 7 

Forward  section  D 
and  stern 5.4 


495-4 


4 th  Water-line. 


4 

5 

•7 

2 

X 

4 

•7 

8 

2 

4-3 

X 

2 

. = 

8.6 

1 

6-5 

X 

4 

= 

26 

0 

6.8 

X 

2 

= 

13.6 

A 

5 

X 

4 

— 

20 

B 

3.6 

X 

2 

= 

7.2 

C 

•9 

X 

4 

== 

3-6 

D 

•3 

= 

•3 

104-3  ^ 3& 

293.3 


Abaft  section  4,  rud 

der  and  post 3.2 

Forward  section  D 
and  stem 8 


297-3 


Keel. 

Half  breadth  = .25  X length  of  98  feet : 
Rudder-post  and  rudder 


24- 5 
•3 
24.8 


Results. 


1st  water  line  71 1 


2d 

3<* 

4th 

Keel 


612  X 4 = 2448  X 1 = 2448 
495.4  X 2 = 990.8  X 2 = 1981.6 

297  3 X 4 = 1189.2  X 3 = 3567-6 
24.8  _24-8  X 4 — 99-2 

5363.8  8096.4 


3)10727.6 

Displacement , 3575.9  X 2 = 7i5i.8cu&e;r. 


NAVAL  ARCHITECTURE. 


To  Compute  Centre  of  Gravity  of  Displacement. 

Rule. — Divide  sum  of  products  obtained  as  above,  by  consecutive  water- 
lines,  by  sum  of  products  obtained  in  column  of  products  by  Simpson’s  mul- 
tipliers, and  quotient,  multiplied  by  distance  between  water-lines,  will  give, 
depth  of  centre  below  load  water-line. 


Illustration  ] 
Or, 


V an) 


8096.4,  from  above,  -4-  5363.8  = 1.5,  which  X 2 = 3 feet. 

— d.  n representing  draught  of  water  exclusive  of  any  drag  of 


keel,  a area  of  immersed  surface  of  hull  in  sq.  feet,  and  D displacement  in  cube  feet. 

2.— Assume  draught  of  water  8 feet,  displacement  7152  cube  feet,  and  area  of  im- 
mersed surface  of  hull  uoo  sq.  feet. 


Then 


8 


(»T  — 

\ IIOI 


2 X I.187 


= 3-3 1 fat. 


100  X 8/ 


To  Compute  Displacement  -Approximately. 

Coefficient  of  Displacement  of  a vessel  is  ratio  that  volume  of  displacement 
bears  to  parallelopipedon  circumscribing  immersed  body. 

V 

D — C.  V representing  volume  of  displacement  in  cube  feet , L length  at  im- 

nursed  water-line , B extreme  breadth , and  D draught  in  depth  of  immersion  boih 
in  feet. 

Coefficient  of  Area  of  A midship  Section  in  Plane  of  a Water-line  is  ratio 
which  their  areas  bear  to  that  of  circumscribing  rectangle. 

L representing  length  of  water-line,  and  D distance  between  water-lines , both  in  feet. 

Coefficients.  [By  S.  M.  Poole , Constructor  tT.  S.  Navy. ) 

Rule.— Multiply  length  of  vessel  at  load-line  bv  breadth,  and  product  by 
depth  (from  load-line  to  under  side  of  garboard-strake)  in  feet,  and  this 
product  by  coefficient  for  vessel  as  follows  : divide  by  35  for  salt  water  36 
for  fresh  water,  and  quotient  will  give  displacement  in  tons. 

Amidship  sections  range  from  .7  to  .9  of  their  circumscribing  square  and  tneau 
of  horizontal  lines  from  . 55  to  .75  of  their  respective  parallelograms.  Hence  ranges 
for  vessels  of  least  capacity  to  greatest  are  .7y.ee  — anH 


Merchant  ship,  very  full 6 to  .7 

“ “ medium 58  to. 62 

River  steamer,  stern-wheel. . . .6  to  .65 

Ship  of  the  line 5 to  6 

Naval  steamer,  first  class 5 to  .6 

“ “ . • 52  to  .58 

Merchant  steamer,  sharp 54  to  .58 

Half  clipper 52  to  .56 

Brigs,  barksj  etc 52  to  .56 

River  steamer,  tugboat,  med’m  .52  to. 56 


Merchant  steamer,  medium. . . . c2  to 

Clipper to  .54 

Schooner,  medium 48  to  .52 

River  steamer,  tug  boat,  sharp  .45  to  .5 

“ medium 45  to  .5 

“ “ sharp 42  to  .45 

Schooner,  sharp 46  to  .5 

Yachts,  sharp 4 to '45 

“ very  sharp .3  toil 

River  steamers,  very  sharp. . . .36  to  .42 
In  steam  launch  Miranda , when  making  16.2  knots  per  hour,  with  a displace- 
ment of  58  tons,  her  coefficient  was  3.  1 

To  Compute  Change  of  Trim 

W d L 

IT  Xm~d'  D rePresentin9  displacement  at  line  of  draught  in  tons,  L length 
at  same  line  in  feet,  and  m longitudinal  meta-centre. 

at  daughter  25.5  feet,  lias  11=380  feet,  >11  = 475  feet 
and  D _ 8625  tons.  If,  then,  a weight  of  20  tons  was  shifted  fore  and  aft  100  feet,  ’ 


20  X IOO  380 

- x — = . 1856  feet  = 2.22  ins. 
475 


8625 


Illustration. — Vertical  Plane  at  53  and  Horizontal  at  Load-line. 


ARCHITECTURE, 


NAVAL 


To  Compute  Centre  of  GJ-ravity  or  Buoyancy  Approxi- 
mately. 

2 Q 

j t°  — °f  mean  draught  of  hull,  using  larger  coefficient  for  full-bodied  vessels. 

To  Delineate  Curve  of  Displacement. 

# This  curve  is  for  purpose  of  ascertaining  volume  of  water  or  tons  weight, 
displaced  by  immersed  hull  of  a vessel  at  any  given  or  required  draught;  or 
weight  required  to  depress  a hull  to  any  given  or  required  draught.  From, 
results  of  computation  for  displacement  of  vessel,  proceed  as  follows,  Fig.  4 : 
FiS-  4-  On  a vertical  scale  of  feet  and  ins., 

as  A B,  set  off  depths  of  keel  and  water- 
lines,  draw  ordinates  thereto  represent- 
ing displacement  of  keel,  and  at  each 
water-line,  in  tons. 

< Through  points  1,  2,  3,  4,  and  5 de- 
lineate curve  A 5,  which  will  represent 
displacement  at  any  given  or  required 
draught. 

Draw  a horizontal  scale  correspond- 
ing to  weight  due  to  displacement  at 
load-line,  as  A C.  and  subdivide  it  into  tons  and  decimals  thereof,  and  a ver- 
tical line  let  fall  from  any  point,  as  x , at  a given  draught,  will  indicate 
weight  of  displacement  at  depth,  on  scale  AC,  and,  contrariwise,  a line  raised 
from  any  point,  as  2,  on  A C will  give  draught  at  that  weight. 

Illustration.— Displacement  of  hull  (page  654)  at  load-line  = 7151.8  cube  feet, 
which  -4-  35  for  salt  water  = 204.3  tons,  hence  A C represents  tons,  and  is  to  be  sub- 
divided accordingly. 

Assume  launching  draught  to  have  been  4 feet,  then  a vertical  let  fall  from  4 will 
indicate  weight  of  hull  in  tons  on  A C. 


Coefficients . {By  C.  MacJcrow , M.  I.  N.  A.) 


Description  of  Vessel. 


Iron-Clads. 


Hail  Steamers. , 


Merchant,  small 

Gunboats 

Troop  Ships 

Swift  Naval  Steamers. . 
Fast  Steamers,  R.  N 


Length. 


225 

325 

^50 

385 

368.27 

220 

90 

125 

160 

350 

340-5 

337-3 

270 

300 


45 

59 

35 

42 

42.5 

27 

i5 

23 

3i-3 

49.12 

46.13 
50.28 
42 

40.27 


Mean 

Draught. 


15 

24-75 


18.71 

8 

4 

8 


23-5 

15-75 

22.75 

*9 

14 


Coefficient. 

Displace-  Amidship  Water- 
lines. 


•715 

.64 

.687 

.516 

.702 

•637 

•536 

.466 

•47 

•4 

•483 

•497 

.414 


•932 

.81 

.85 

.88 

.812 

.912 

.914 

.87 

•745 

.674 

.68 

•787 

.792 

.711 


755 

7i 

84 

8 

635 

742 

704 

616 

603 

7 

582 

614 

62» 

711 


Curve  of  Weight. 

To  Compute  Number  of  Tons  required  to  Depress  a 
vessel  One  Inch  at  any  Draught  of  Water  Parallel 
to  a Water-line. 

ro^V“'7Uivide  area  of  plane  by  12,  and  again  by  35  or  36,  as  may  be 
required  for  salt  or  fresh  water.  J 

,.a^rE^hi^Vwafer7ter  liDe  °fa'Vff  * 1423  ^ feet;  what  is  its 

1422-7-12  = 118.5,  which  -4-35  = 3.38  tons. 


NAVAL  ARCHITECTURE. 


To  Compute  Common.  Centre  of  Grravity  of*  Hull,  Ar- 
mament, Engine,  Boilers,  etc.,  of  a "Vessel. 

Rule. — Compute  moments  of  the  several  weights,  relatively  to  assigned 
horizontal  and  vertical  planes,  by  multiplying  weight  of  each  part  by  its 
horizontal  and  vertical  distance  from  these  planes. 

Add  together  these  moments,  according  to  their  position  forward  or  aft,  or 
above  or  below  these  planes,  and  difference  between  these  sums  will  give  po- 
sition forward  or  aft,  above  or  below,  according  to  which  are  greatest. 

Divide  results  thus  ascertained  by  total  weight  of  vessel,  and  product  will 
give  horizontal  and  vertical  distances  of  centre  of  gravity  from  these  planes. 

It  is  customary  to  assume  vertical  plane  at  0,  and  horizontal  plane  at 
load-line. 

Note. — In  following  illustration,  in  order  to  simplify  computation  in  table,  com- 
mon centre  of  gravity  of  hull,  machinery,  etc.,  is  taken,  instead  of  centres  of  indi- 
vidual parts,  as  engine,  boiler,  propeller,  etc. 


Illustration. — Assume  half-girths  as  in  following  table,  and  distance  between 
sections  io  feet. 


Sec- 

tion. 

Half- 

Girths. 

FOR^ 

Multi- 

pliers. 

fARD. 

Prod- 

uct. 

Multi- 

pliers. 

Mo- 

ments. 

Sec- 

tion. 

Half- 

Girths. 

ABJ 

Multi- 

pliers. 

lFT. 

Prod- 

uct. 

Multi- 

pliers. 

Mo- 

ments. 

No. 

&... 

Feet. 

25 

1 

25 





No. 

Feet. 

23 

4 

92 

1 

92 

A 

23 

4 

92 

1 

92 

2 . . . 

20 

2 

40 

2 

80 

B.... 

21 

2 

42 

2 

84 

3 ••• 

18 

4 

72 

3 

216 

C.... 

x9 

4 

76 

3 

228 

4 ... 

16 

2 

32 

4 

128 

D. ... 

E. ... 

17 

2 

34 

4 

136 

5 ••• 

H 

,1 

14 

5 

70 

i5 

1 

15 

5 

75 

615 

534 

586 

Moments  forward,  615  — moments  abaft,  586  = 29 -4- sum  of  product  534  = .054, 
which  X 10  feet  ==  .54  feet  forward  of 


Centre  of*  Lateral  Resistance. 

Centre  of  Lateral  Resistance  is  centre  of  resistance  of  water,  and  as  its  po- 
sition is  changed  with  velocity  of  vessel,  it  is  variable.  It  is  generally  taken 
at  centre  of  immersed  vertical  and  longitudinal  plane  of  vessel  when  upon 
an  even  keel. 

If  vessel  is  constructed  with  a drag  to  her  keel,  the  centre  will  be  moved  r 
proportionately  abaft  of  longitudinal  centre. 

Yacht  America  had  a drag  to  her  keel  of  2 feet,  and  centre  of  lateral  re- 
sistance of  her  hull  was  8.08  feet  abaft  of  centre  of  her  length  on  load-line. 

Centre  of*  Effort. 

Centre  of  Effort  is  centre  of  pressure  of  wind  upon  sails  of  a vessel  in  a 
vertical  and  longitudinal  plane.  Its  position  varies  with  area  and  location  i 
of  sails  that  may  be  spread,  and  it  is  usually  taken  and  determined  by  the  ( 
ordinary  standing  sails,  such  as  can  be  carried  with  propriety  in  a moderateh 
fresh  breeze. 

Iii  computing  this  position,  the  yards  are  assumed  to  be  braced  directly  fore  5 
and  aft  and  the  sails  flat. 

Note.— Centre  of  effort  of  sails,  to  produce  greatest  propelling  effect,  must  accord 
with  capacity  of  vessel  at  her  load-line,  compared  with  fullness  of  her  immersed 
body  at  its  extremities.  Thus,  a vessel  with  a full  load-line  and  sharp  extremities 
below,  will  sustain  a higher  centre  of  effort  than  one  of  dissimilar  capacity  and  con- 
struction. 


NAVAL  ARCHITECTURE. 


659 


To  Compute  Location  of  Centre  of  Effort. 

Rule— Multiply  area  of  each  sail  in  square  feet  by  height  of  its  centre  of 
gra\  lty  abo\  e centre  of  lateral  resistance  in  feet,  divide  sum  of  these  prod- 
ucts  (moments)  by  total  area  of  sails  in  square  feet,  and  quotient  will  give 
height  of  centre  in  feet.  b 

2.  Multiply  area  of  each  sail  in  square  feet,  centre  of  which  is  forward  of 
a vertical  plane  passing  through  centre  of  lateral  resistance,  by  direct  dis- 
tance of  its  centre  from  that  plane  in  feet,  and  add  products  together. 

3.  Proceed  in  like  manner  for  sails  that  are  abaft  of  this  plane,  add  their 

products  together,  and  centre  of  effort  will  be  on  that  side  which  has  greatest 
moment  of  sail.  b 

Example.— Assume  elements  of  yacht  America  as  rigged  when  in  U.  S.  Service. 

Distance  of  Centre 
of  Gravity  of  Sails. 

• Foreward.  I Abaft. 


Area. 


Flying  Jib. 

Jib 

Foresail. .. 
Mainsail. . . 


Sq.  Feet. 
656 
1087 
*455 
2185 


I 5383 

Vertical  moments  172575 

Area  of  sails 5383 

sistance. 


Height  of 
Cent,  of  Grav- 
ity of  Sails. 


Feet. 

28 

26 

34 

35 


Vertical 

Moments 


18368 
28  262 
49470 
76475 


*72  575 


52 

32 


Moments. 


Foreward. 


34  ns 
34  784 


Abaft. 


4 365 
87  400 


68  896  j 91  765 


32.06  — height  of  centre  above  centre  of  lateral  re- 

Xf  j f 91  765  -v.  68  896 

sistance  1 53*5 = 4' 25  “ distance  °f  c™tre  abaft  centre  of  lateral  re- 

Relative  Positions  of  Centre  of  Effort  and  of  Lateral 
Resistance. 


Square  Riff.  Ft?.?  f'+f'l 


~=  E. 


and 


4 A 

5 d 


Fore  and  Aft  Rig. 


L 


E, 


I^F+O  

= E-.  L representing  length  of  load-line,  d distance  of  centre  of  buoyancy 


fanTeof  ^mhfofbnon^  ?f?entre  °f  latelal  resistance  abaft  centre  of  it,  d"  dis - 
Meta-Centre. 

Meta-centre  of  a vessel’s  hull  is  determined  by  location  of  centre  of  erav 
21^7  °f lmmersed  bottom  of  hull,  for  it  is  that  pltTn  traSlfe 
section  of  hull,  where  a vertical  line  raised  from  its  centre  of  gravitv  o? 

F^page^o  a ' paSSi"g  through  centre  of  g™vity  of  hullj  as 

To  Compute  Height  of  Meta-Centre. 

By  Moment  of  Inertia.  ~ — M.  I representing  moment  of  inertia  of  area 
of  water-line  or  plane  of  flotation,  and  D volume  of  displacement  in  cube  feet. 

'area0by  square  re  r“  area  '•*  sa,m  of  Products  of  each  element  of  that 

computed  Ce  fr0m  ax,s>  about  vvhich  moment  of  area  is  to  be 

T°  Ascert£^  Moment  of  Inertia  approximately. 


Rectangle  = C L B 3 • c = 


when  L = 4 B ; C = ~ when  L = 5 B ; and  C : 


200  wlieu  L 6B-  With  ^ery  fine  lines  and  great  proportionate  length  C = — . 
L and  B measured  at  load-line.  25 


66  o 


NAVAL  ARCHITECTURE. 


Illustration.— Assume  length  of  vessel  233  feet,  breadth  43,  draught  16,  and 
displacement  2700  tons.  Length  = 5. 65  beams;  hence  C is  taken  at  — . Volume 
of  displacement  = 2700  X 35  = 92  500  cube  feet. 

Exact  height  of  moment  was  10.44  feet. 


Then  21  * *33  X 43^  ,., 


400  X 92  500 

By  Ordinates.  Rule.— Divide  a half  longitudinal  section  of  load  water- 


line by  ordinates  perpendicular  to  its  length,  of  such  a number  that  area 
between  any  two  may  be  taken  as  a parallelogram.  Multiply  sum  of  cubes 
of  ordinates  by  respective  distances  between  them,  and  divide  two  thirds 
of  product  by  volume  of  immersion,  in  cube  feet. 

Illustration.— Take  dimensions  from  Figs.  2 and  3,  page  654. 

Cube. 

51  460 


Length. 

Cube. 

Length. 

Cube. 

5 

. 125 

A . . 

7-7 

• 456 

B .. 

....7.8.... 

••  475 

9-5 

• 857 

C .. 

9-9 

1 10  

. 970 
.1000 

D . . 

....4  .... 

..  64 

5146  X 10 

3)102  920 

7I5I-8)  34306.6  = 4.77  ft. 


If  there  are  more  ordinates,  their  coefficients  must  be  taken  in  like  manner,  as 
1 — 4 — 2 — 4 — 2 — 4 — 1. 

For  operation  of  this  method,  see  Simpson's  rule  for  areas,  page  342. 

Or,  — I*  X — M.  y representing  ordinates  of  half -breadth  sections  at  load- 

line,  d*x  increment  of  length  of  load-line  section  or  differential  of  x,  and  D displace- 
ment of  immersed  section  in  cube  feet. 

— (a3  4 d>3  -J-  2 C3  -p  4 d3  e3)  — + F -f-  A 

A 3 3 _ ft  5 c g 

By-  Areas.  — jy 


and  e representing  ordinates  of  1st  or  load  water-line,  F area  of  irregular  section 
between  1st  frame  and  stem , and  A area  of  like  section  between  ^ 
stern-post , both  in  sq.  feet , D displacement , in  cube  feet,  and  l distance  between  frame* 
or  sections  of  water-line,  as  may  be  taken,  in  feet. 


To  Ascertain  Areas  of  F and  A. 

—Big.  £5. 


— a&X&c3A-4  = F,  and— deXc5'3-^4  = A. 

3 3 


Elements  of  Capacity  and  Speed  of  Several  Types  of 
Steamers  of  Ft.  dST.  (W.  H.  White.) 

T HP  tO 

Length 

Displacement. 


Iron-clads. 
Recent  types. 


do.  twin  sc. 
Unarmored. 
Swift  cruisers 

Corvettes 

Ships 

Gun-vessels. . 
Gun-boats  . . . 

Merchant. 
Mail,  large. . . 

“ smaller. 
Cargo,  large. . 
“ smaller. 


Feet. 

300  to  330 
280  to  320 


270  to  340 
200  to  220 
160 

125  to  170 
80  to  90 


400  to  500 
300  to  400 
250  to  350 
200  to  300 


Breadth. 


5-25*05-75 
4-5*0  5 


6.5  to6-75 

6 

5 

5. 5 to  6. 25 
3 to  3-25 


9 to  1 1 
8 to  10 
7.5  to  10 
7 to  9 


Tons. 

7500  to  9000 
6000  to  9 000 


3000  to  5 500 
1 800  tO  2 OOO 
850  to  950 
420  to  800 
200  tO  250 


7000  to  IOOOO 
5000  to  7000 
3000  to  6 000 
1 500  to  4 000 


Speed. 


Knots. 

14  to  15 


to  15 


15  to  16 
12.75  to  13.25 


to  I 
to 


to  15 
to  14 
to  15 
to  II 


.9  to  I 
. 7 to 


1.3*01.5 
[ to  I.  2 
I tO  I.  2 
.8  to  1.4 
.8  to  1. 1 


.5  to  .6 

.4*0  -5 

-3*o  .5 
• 2 to  .4 


Displace-  [Displace- 
ment. ment  ■%. 


16  to  20  \ 
151019  j 


20  to  24  • 
13  to  14 
10  to  1 1 
7 to  11  | 
5 *°  7 


10  to  11 
7 to  10 


5 to  9 
3 to  6 


NAVAL  ARCHITECTURE. 


To  Compute  Power  Required  in  a Steam  Vessel,  capac- 
ltV  °T  another  Vessel  "being  given. 

In  vessels  of  similar  models.  — = V ; — = y ' ; V __  q . an(1  _ R . 

VTOdT \°fvolumei  of  given  and  required  cylinders  and  reuo- 
lutions  in  cube  feet,  a and  A areas  o/  immersed  section  of  given  and  required 
vessel  in  sq.  feet  at  like  revolutions  and  speed  of  given  vessel , s and  S speeds  ofqiven 
and  required  vessel  at  revolutions  of  given  vessel, , both  in  feet  per  minute  rand  r' 
) e?°lutwns  of  given  and  required  vessel  per  minute , and  C product  of  volume  of  com 
lined  cylinder  and  revolutions  for  required  vessel.  J V0LUme  °J  com * 

i.lL^fRA,TIT--A/team  Yessel  having  an  area  of  amidship  section  of  67s  sq  feet 
has  two  cylinders  of  a combined  capacity  of  53?  aa  cube  feet  and  a cnJod  nf  l 
knots  per  hour  with  ,5  revolutions  of  her  e^in\\  Requhed  volunfe  of  L\m5 

wVth  14‘^^luUons  °f  IO  feet’  f°r  a SeCli°"  °f  700  feet  and  a sl’eed  of  ’3  knots 

tt  = 533-33  X 15  = 8000  mbefeel,  —fl-  = S296.3  cube  feet,  ;33  X 8296.3^ 
i X ^*75  * 10.53 

1574S -2  cube  feet,  — _*5745  2 = 16388.!  and  = 56:.66  cu&e 


I4-5  * 2-X  14.5 

feet  which +10  stroke  of  piston , 12  for  ins.,  and  X 1728  ins.  in  a cube  foot  — 
561.66  X 1728  n n J 

- 8087.9  *«•  area  of  each  cylinder  = diameter  of  101.5 


= v’  V yy = V;  and  ^ = IIP-  0r> = v; and  Lc-=IfP- 


10  X 12 

Approximate  Rules  to  Compute  Speed  and  IIP  of  Steam 
Vessels. 

V3 ®?  = c-  IIP  - V3 Di 

V c.°^ienl  ofvasd.  A area  of  immersed  amidship  section  in  sq  feet 

\ velocity  of  vessel  in  knots  per  hour , and  D displacement  of  vessel  in  tons.  ^ J ’ 

Note.— When  there  exists  rig,  an  unusual  surface  in  free  board, deck- houses  etc 
or  any  element  that  effects  coefficient  for  class  of  vessel  given,  a corresponding  ad’ 
dition  to,  or  decrease  of,  following  units  is  to  be  made:  6 P d g ad* 

Range  of  Coefficients  as  deduced  from  observation  is  as  follows  : 


SIDE-WHEEL. 


Steamboat. 
Medium  lines 


Fine  lines. . . . 

Steamer. 
Medium  full  lines* 


Fine  linest. 


Sq.F 

43 

150 

136 

675 


3600 
5233 
* Full  rigged. 


V3  A 


v3d§ 


PPOPELLER. 


Steamboat. 
Medium  lines. 


Fine  lines  . 

Steamer. 
Medium  full. . . 


Torpedo  bout.. 


A 

D 

y 

( 

V3  A 

v3dI 

IH? 

IH? 

45 



12 

500 

150 

~ 

15 

530 

55o 

2532 

9 

194 

570 

390 

1475 

IO 

180 

470 

3600 

13 

210 

— 

27 

20 

170 

500 

t Bark  rigged. 


Coefficients  as  Determined  by  Several  Steamers  of  H.  B.  M.  Service. 

(C.  Mackrow , M.  I.  N.  A.) 


Length. 

Length 

Beam. 

A rea  of 
Section  at 

Displace- 

ment. 

IH? 

Speed. 

V3A 
I FP  ~ 

Feet. 

185 

212 

360 

270 

380 

400 

362 

400 

6- 53 

5- 89 

7- 33 
6*43 
6.52 

6- 73 

7- 33 
6-73 

Sq.  Feet. 
236 
377 

814 

632 

1308 

1198 

778 

1185 

Tons. 

775 

I554 

5898 

3057 

9487 

9*52 

5600 

9071 

3K 

782 

1070 

2084 

2046 

3205 

5971 

3945 

6867 

Knots. 

IO*34 

10.89 

”•5 

12.3 

12.05 
13.88 

14.06 
*5-43 

333 

456 

598 

574 

7i4 

536 

548 

634 

552  NAVAL  ARCHITECTURE. 

Approximate  Rule  for  Speed  of  Screw  Propellers. 
i Moles  worth.) 


PN 


S3  v 

= v-  and  -*r-aP. 


88 


(Moles  worth.) 

;*2Lr=  N;  ^?  = V;  “i-  = P;  ^ = 

V and  » representing  velocities  in  knots  and  miles  per  hour,  r pitch  of  propeller  in 
feet  and  N number  of  revolutions  per  minute. 

This  does  not  include  slip,  which  ranges  from  io  to  30  per  cent. 

TPitcli  of  Screw  IPropeller. 

Pitch  ranges  with  area  of  circle  described  by  diameter  of  screw  to  that  of 
a midship  section. 


Area  of  screw  circle  to  amidsliip  j I 6 
section  = 1 to ) 1 


4-5 


3-5 


2-5 


Two  Blades. 


| 1.02  | i.ii  | 1.2  | 1.27  I I-3I  I i-4  I x*47 


Pitch  to  diameter  of  screw  = 1 to  1 1 * -7  ; ' j ~ j n 1 _ q8 

Four  Blades.  I x.08  | 1.38  | 1 5 1 *.62  I x.7x  I **77  1 *-89  I J‘98 


Length  = .166  diameter. 

Slip  of  Side-wlieels. 

Radial  Blades.  ?Aip^  = S.  Feathering. 
length  of  arc  of  immersed  circumference  of  blades,  c length  of  chord  of  immersed  arc. 
and  S slip,  alt  in  feet.  Area  cf  Blades. 


2(A-c)_s  Wenthemna.  1,5  (A  ’ - = S.  A representing 


75  HT  . A Sea  Service.  S = A.  D representing  diameter 


River  Service. 

of  wheel  in  fed,  and  A area  of  each  blade  in  square  feet. 

Length  of  Blades.  .7  in  Ri™r  service  and  6 in  Sea  service. 

Distances  between  Radial  Blades.  2.25  in  River  serv.ee  and  3 feet  m service, 
between  Feathering  blades , 4 to  6 feet. 

Proportion  of  Bower  Utilized  iir  a Steam  Vesse  . 

P — z __  c p representing  gross  IIP,  2 loss  of 


diameter  of  wheels  at  centre  of  effect, 
■O  uV  A noari  r.  mpMcient  for  vessel 


effect  by  slip  ana  ooixgue  

r revolutions  per  minute,  and  C coefficient  for  vessel 

Illustration.  IIP  of  engines  of  of  effect 'of  wheels  is 

aad  what 

power  applied  to  propel  vessel ? c in  this  case  is  is  37  per  cent., 


by  wheels.  „ . . Q _ _ 

1120  — (1120X  28.74  — 100)  _ 798^11  _ 6^.63  coefficient. 

Speed  of 

to  propel  vessel  at  this  speed  = 65.63  X i7-°5  - 19 ‘V0-1* 
iq 076. 13  X i7-°5  X 60  2ft 


33  000 

Friction  of  engines  1.5  lbs.  upon  3848  sq.  ins.  X x3-5  revolu- 

FrV^Uon^onoad  6percent^  upon  pressuri  of  s^an^ess 

for  friction  of  engine,  as  above ■V 

Oblique  action  01  wheels 

Slip  of  wheels 

Absorbed  by  propulsion  of  vessel 


BP. 

Per  cent, 
of  Power. 

l*|  94-45] 

x8.83 

5’}  60.45  j 

1 

18 

X5-37 

52.8 

Screw  IPropeller.  Friction  of  engines. . 


“ of  screw  surface  and  resistance  of  edges  of  blades. 

Slip  of  propeller 

Absorbed  by  propulsion  of  vessel 


663 

IP. 

Per  cent, 
of  Power. 

96. 06 ) 
81.48} 

18.83 

53-44 

6.83 

205.55 

26.27 

375-92 

48.04 

782.45 

100 

Note. — From  experiments  of  Mr.  Froude,  he  deduced  that,  as  a rule,  only  37  to 
40  per  cent,  of  whole  power  exerted  was  usefully  employed. 

With  an  auxiliary  propeller,  essential  differences  are  in  friction  of  surfaces  and 
edges  of  blades  of  propeller  and  slip  of  propeller,  being  as  12  to  6 83  in  excess  in  first 
case,  and  as  13.7  to  26.27  in  second  case,  or  50  per  cent  less. 

Resistance  of  Bottoms  of  Hulls  at  a Speed  of  one  Knot  per  Hour. 

Smooth  wood  or  painted 01  lb.  I Copper. 007  lb. 

Smooth  plank 016  “ Moderately  foul 019  k‘ 

Iron  bottom,  painted 014  “ | Grass  and  small  barnacles 06  “ 

Sailing. 

Ratio  of*  Effective  Area  of*  Sails  and  of*  'Vessel’s  Speed, 
under  Sail  to  Velocity  of  Wind. 


Course. 

Ratio  of 
Effective 
Area 
of  Sails. 

Ratio  of 
Speed  of 
Vessel 
to  Wind. 

Course. 

Ratio  of 
Effective 
Area 
of  Sails. 

Ratio  of 
Speed  of 
Vessel 
to  Wind. 

5 points  of  wind 

2 abtftbeam.. 

•59 

•33 

Wind  abeam 

.82 

.6 

•91 

•5 

“ astern 

I 

.5 

6 “ of  wind 

.68 

•5 

“ on  quarter 

•96 

.66 

Then  = ^/i-52=  1 139,  hence  area  of  sails  a'  = — - 


Rropnlsioix  and  Area  of  Sails. 

Plain  sails  of  a vessel  are  standing  sails,  excluding  royals  and  gaff  topsails. 

Resistance  of  vessels  of  similar  models  but  of  different  dimensions  for  equal 
speeds  =-Dt 

Hence  ^ . a and  a'  representing  areas  of  sails  of  known  and  given  ves- 

sels, and  D and  D'  their  displacements  in  tons. 

Illustration. — Assume  D and  D'==24<x>  and  1600. 

878  per  centum. 

1-139 

In  Vessels  of  Dissimilar  Models . — Plain  sail  area  should  be  a multiple 
of  D*. 

Multiples  for  Different  Classes  of  Vessels,  R.  N. 

Sailing.  | Steamers. 

Ships  of  Line 100  to  120  j Ships,  iron-clad 60  to  80 

Frigates.. i I Frigates ) 

Sloops > 120  to  160  | Sloops ( 80  to  120 

Brigs ) I Brigs ) 

English  Yachts , designed  for  high  speed,  have  multiples  from  180  to  200, 

and  when  designed  for  ordinary  speed  from  130  to  180. 

When  Area  of  Sail  to  Wet  Surface  of  Hull  is  taken. — American  yacht  Sappho  had  a 
ratio  of  2.7  to  1,  and  several  English  yachts  nearly  the  same,  while  in  some  others 
it  was  but  2 to  1. 


664 


NAVAL  ARCHITECTURE. 


Location  of  Masts,  etc.  Load-line  = ioo. 


Vessel. 

Di 

Fore. 

[stance  from  Ster 
Main. 

n. 

Mizzen. 

Foot  of  Sail.* 

Height  of  Centre 
of  Effect  above 
Water-line  = 
Breadth.* 

Ship 

Bark 

Brig 

Schooner  

Sloop 

10  to  20 
12  to  20 
17  to  20 
16  to  22 

53  to-  58 

54  to  60 
64  to  65 

55  to  61 
36  to  42 

80  to  90 

81  to  91 

125  to  160 
130  to  160 
160  to  165 
160  t(M70 
170  to  190 

1.5  tO  2 
1.5  to  1.95 
1.5  to  1.75 
1.5  to  1.75 
1.25  to  1.75 

* Measured  from  Tack  of  Jib  to  Clew  of  Spanker  or  Mainsail. 


Rake  of  Masts. 

Ships  - Foremast  o to  .28  of  length  from  heel,  Main  and  Mizzen  o to  .25. 
Schooners.— Foremast  .1  to  .25,  Mainmast  .63  to  .77.  Sloops.— .08  to  .11. 


Area  of  Sails. 


Sails. 

3 Yards  upon 
each  Mast. 

4 Yards  upon  1 
each  Mast. 

Sails.  J 

3 Yards  upon  1 
each  Mast. 

4 Yards  upon 
each  Mast. 

J4b 

.08 

.295 

.417 

.08 

.295 

•417  1 

1 Mizzenmast 1 

Spanker  or  ) ! 

j Driver. . . J 

.127 

.081 

.14 

.068 

Jfnyfi  mast, 

Mainmast 

Proportional  Area  of  Sails  upon  each  Mast  under  above  Divisions. 


Sail.  | Fore. 

Main. 

Mizzen. 

Proportion  to  1. 

Course 

Topsail 

Topgallant  sail 

Royal 

Spanker  or  Driver 

Jib 

115 
105 
• 075 

.08 

.097 

.063 

• 045 

.08 

. 162 
.149 
. 106 

.138 

.127 

.089 

.063 

•075 

.052 

.081 

.063 

•045 

.032 

.068 

•389 

•358 

•253 

•33 

•303 

.215 

.152 

375 

•375 

.417 

•4*7 

.208 

.208 

1 

1 

Balance  of  Sails.— lUiect  01  jiu  is  equal  ^ ^ — 

mast,  and  sails  upon  mizzenmast  balance  those  of  foremast. 

Areas  of  sails  upon  masts  of  a ship  should  be  in  following  proportion : 

Fore 1.4x4  I Main 2 | Mizzen. ....... . x 

When,  therefore,  main  yard  has  a breadth  of  sail  of  ioo  feet,  fore  yard 
should  have  70.71  feet,  and  mizzen  50  feet  -,  topgallant  and  royal  yards  and 
sails  being  in  same  proportion. 

Angles  of  Heel  for  Different  Vessels. 
Approximately.  D representing  displacement  of  vessel  in  lbs., 

M height  of  meta- centre  above  centre  of  gravity  in  feet,  a angle  of  heel  of  Vf^tiUntor- 
culaf  measure  * and  H height  of  centre  of  effect  above  centre  of  lateral  resistance, 

in  feet. 

Moment  of  sail  should  he  equal  to  moment  of  stability  at  a defined  angle 
of  heel. 


Frigates,  etc. 4° 

Corvettes • ••  5° 


. , Circular 

Angie-  Measure. 


.07 

.087 


. . Circular 

Angle.  Measure. 

Schooners,  etc 6°  .105 

Yachts 6°  to  90  .105  to  .107 


Illustration.  -Assume  displacement 170 . tons ' 6’75  feCt’ 

H = 36  feet,  and  angle  of  heel  90 ; what  should  be  area  of  sails . 

170  X 2240  = 380  800  lbs.  90  = • i°7- 
380800X6-75  x.  107  _ 7g39.8  sq.feet. 

36  


* See  rule,  page  113. 


NAVAL  ARCHITECTURE. 


665 


Trimming  of  Sails. 

That  a vessel’s  sail  may  have  greatest  effect  to  propel  her  forward,  it  should 
be  so  set  between  plane  of  wind  and  that  of  her  course,  that  tangent  of  angle 
it  makes  with  wind  may  be  twice  tangent  of  angle  it  makes  with  her  course. 

Or,  tan.  a — 2 tan.  b.  a representing  angle  of  sail  ivith  wind,  and  b angle  of  sail 
and  course  of  vessel. 


-Angles  of  Course  and  Sails  with.  Wind. 


Wind 

Ahead. 

Angle 

of 

Course. 

Tan- 

gent. 

Half 

Tan- 

gent. 

Angles 

with 

Wind. 

of  Sail 
with 
Course. 

Wind 

Abaft. 

Angle 

Course. 

Tan- 

gent. 

Half 

Tan- 

gent. 

Angles 

with 

Wind. 

of  Sail 
with 
Course. 

Points. 

4 

45° 

.562 

.281 

290  18' 

150  42' 

Points. 

2 

1120  30' 

2.166 

1.082 

65°  13' 

47°  17' 

5 

56°  i5' 

•732 

•365 

36°  i2' 

20°  3' 

3 

!23°  45' 

2-737 

1.368 

69°  56' 

53°  49' 

6 

67°  3°' 

•923 

.461 

42°  43' 

240  45' 

4 

135° 

3-562 

1.781 

74°  17' 

6o°  43' 

Abeam 

900 

I-4I5 

.707 

54°  45' 

35°  16' 

6 

157°  3o' 

7-511 

3-754 

82°  25" 

75°  S' 

Kffective  Impulse  of  Wind. 

-AX  Fig.  6.  Let  P 0,  Fig.  6,  represent  direction  by  com- 

x pass  and  force  of  wind  on  sail,  AB;  from  P 

draw  P C parallel  to  A B,  from  o draw  0 C per- 
pendicular to  A B ; o C is  effective  pressure 
of  wind  on  sail  A B,  and  r C,  perpendicular  to 
plane  of  vessel,  is  component  of  o C,  which  pro- 
duces lateral  motion,  as  heel  and  leeway,  and 
r 0 is  component  of  0 C,  which  propels  vessel. 

I sin.  a=P;  Pcos.  * = L;  and  Psin.  a = E. 
I representing  direct  impact  and  P effective 
pressure  of  wind  on  sail , L effective  impact 
producing  leeway , and  E effective  impact  which 
propels  vessel. 

Note. — The  law  as  usually  given  is  sin.2.  This  is  manifestly  incorrect,  as  it  gives 
results  less  than  normal  pressure  for  angles  of  small  incidence.  At  an  angle  of  in- 
cidence of  wind  of  250,  the  law  of  sin.  is  exact.  Hence,  although  it  may  not  be 
exact  at  all  angles,  it  is  sufficiently  so  for  practical  purposes. 

Illustration  1.— Assume  wind  5 points  ahead,  and  I = 100  lbs. 


By  preceding  table  angle  of  course  with  wind  56°  15';  hence  angle  of  sail  a , with 
wind  36°  12',  as  tan.  36°  12'  = 2 tan.  200  3',  and  angle  x 56°  15'  — 36°  12' = 20°  3'. 

Then,  100  X sin.  36°  12'  = 100  X .5906  = 59.06;  59.06  X cos.  200  f — 59.06  X 

•9394  = 5S-48,  and  59.06  X sin.  200  3'  = 59.06  X .3426  = 20.23  lbs. 

2.— Assume  wind  4 points  abaft,  and  I — - 100  lbs. 

Then,  iooXsin.274°  17'=  100 X .9626s  ==  92.66 ; 92.66  X cos.  1800  — 74O I7'_j_45o. 
= 6o°  43r  = 92.66 X -49  = 45.41,  and  92.66xsin.  6o°  43' = 92.66  X .8722  = 80.82  lbs. 

To  Compute  Sailing  Power  of  a "Vessel. 

F /sin.  w,  sin.  5 = P. 

To  Compute  Careening  Power  of  a Sailing  Vessel. 

F / sin.  w,  cos.  s — P.  F representing  area  of  sails  in  sq.  feet,  f force  of  wind  in 
lbs.  per  sq.  foot,  w angle  of  vjind  to  sails , and  s angle  of  sails  to  course  of  vessel. 

To  Compute  Angle  of  Steady  Heel. 

Within  a Range  of  8°. 

a PE  . TT  . J 

- j)  = sin.  H.  a representing  area  of  plain  sail  in  sq.feet,  P pressure  of  wind 

in  lbs.  per  sq.  foot , E height  of  centre  of  effect  above  mid-draught , in  feet,  D displace- 
ment of  hull , in  lbs. , and  M height  of  meta-centre  in  feet. 

P assumed  at  1 lb.  per  sq.  foot , or  that  due  to  a brisk  wind. 


Illustration.— Assume  a = 15600,  draught  =±20,  and  M = 62 


72,  D = 6 800  000,  and  M = 3. 

Then  x5  600  X x X 72 1x23200 

6 800  000  X 3 20  400  000 


= -0555  = 3°  10'. 
3K* 


hence  62  4-  — = 
10 


666 


NAVAL  ARCHITECTURE. 


Course  and.  Apparent  Course  of  Wind. 

Apparent  course  of  a wind  against  sails  of  a vessel  is  resultant  of  normal 
course  of  wind  and  a course  equal  and  directly  opposite  to  that  of  vessel. 

Illustration. — If  P,  Fig.  7,  repre- 
ss* 7- — sent  direction  by  compass  and  force  of 

wind,  and  a b direction  and  velocity  of 
vessel,  from  P draw  P c parallel  and 
equal  to  a b , join  c a and  it  will  repre- 
sent direction  and  force  of  apparent 
wind. 

Or  — — ratio  of  velocity  of  apparent 
’ c P 

wind  to  that  of  vessel,  ^ = ratio  of  velocity  oj  wind  to  that  of  vessel. 

Resistance  of  Air.  {Mr.  Fronde.) 

Resistance  of  wind  to  a vessel  is  estimated  as  equivalent  to  square  of  its 

In  a calm  resistance  of  air  to  a steamer  = one  thirty-fourth  part  of  resist- 
ance of  water,  and  when  a steamer’s  course  is  head-to, and  combined  veloc- 
ity of  vessel  and  wind  = 15  knots,  resistance  is  one  ninth  of  that  of  the  water. 

Resistance  of  air  to  a sq.  foot  of  surface  at  right  angles  to  course  of  a ves- 
sel is  about  33  lb.,  and  when  surface  is  inclined  to  direction  of  wind,  press- 
ure  varies  as  sine  of  angle  of  incidence. 

Mean  of  angles  of  surface  of  a steamer  exposed  to  wind  may  be  taken  at 
450  ; hence  their  resistance  is  about  .25  lb.  per  sq.  foot  when  wind  has  a 
locity  of  10  knots  per  hour. 

If  sectional  area  of  a steamer’s  hull  above  water .is  750  s^feet,  fibf^nd 
to  air  at  a speed  of  10  knots  in  a calm  would  be  750  X .25  — 187.5  lbs.,  and 
resistance  to  smoke-pipe,  spars,  and  rigging  (brig  rigged)  would  be  201  lbs. 

Leeway. 

Annie  of  Leeway  in  good  sailing  vessels,  close  hauled,  varies  from  8°  to 
12°,  and  in  inferior  vessels  it  is  much  greater. 

Ardency  is  tendency  of  vessel  to  fly  to  the  wind,  a consequence  of  the 

centre  of  effort  being  abaft  centre  of  lateral  resistance. 

Slackness  is  tendency  of  vessel  to  fall  off  irom  the  wmd,  a consequence  of 
the  centre  of  effort  being  forward  centre  of  lateral  resistance. 

Results  of  Experiments  upon  Resistance  of  Screw  ■propdlers,  at  High  Velocities 
and  Immersed  at  Varying  Depths  of  \\  ate) . 


Immersion  of 
Screw. 

Resistance. 

Immersion  of 
Screw. 

Resistance. 

Surface. 

1 

2 feet. 

7 

1 foot. 

5 

3 “ 

7-5  1 

Immersion  of  1 

Resistance. 

Screw. 

4 feet. 

7.8 

5 “ 

1 8 

Slip  of  Propeller,  15  per  cem. ; 01  w 

ing  axes  of  blades  as  the  centre  of  pressure,  23  per  cent. 

Freeboard. 

Measured  from  Spar-de.de  stringer  to  surf  ace  of  water  D, epth  of  Hold  from  under • 

J side  of  spar  deck  to  top  of  ceiling. 


Hold. 

II  Hold. 

| Hold.  | 

Feet. 

Ins.  Feet. 

Ins. 

Feet.  | 

8 

1-5  12 

2.25 

16  | 

*o 

2 ||  14 

2-5 

18  | 

Ins. 

2.75 

3 


Ins. 

3-125 

3-25 


Hold. 


Feet. 

24 

26 


Ins. 

3-375 

3-5 


Feet. 
28 
I 3° 


3.625 

3-75 


NAVAL  ARCHITECTURE. 


667 


_ „ VI a, ting  Iron.  Hulls. 

D L 

Soob  d = T‘  D rePresentin9  displacement  in  tons , L length  of  hull,  b breadth , and 

d depth.  Or,  .o$fy/d  = T.  f representing  distance  between  centres  of  frames,  and 
d depth  of  plate  below  load-tine,  all  in  feet,  and  T thickness  of  plate  in  ins. 


NLasts  and.  Spars. 

Lower  masts at  spar  deck.  I 

Bowsprit “ stem. 

Topmasts “ lower  cap. 

Topgallant  masts “ topmast  cap.  | 

Fore  and  main  masts,  when  of  pieces,  D „„ 

length.  Mizzenmast  .66  diameter  of  mainmast.  Masts  of  one  piece“i  inch  for  each 
3-5  t0  3-75  feet  of  whole  length. 

Bowsprit,  depth,  equal  diameter  of  mainmast;  width,  diameter  equal  to  foremast. 
Main  and  fore  topmasts 1 inch  for  each  3 to  3.25  ) 


Diameter  for  Dimensions. 

Jib-boom at  bowsprit  cap. 

Yards in  middle. 

Gaft’s at  inner  end. 

Main  and  Spanker  booms  at  taffrail. 
inch  for  each  3 to  3.25  feet  of  whole 


Mizzen  topmast 

Topgallant  masts 1 

Royal  masts 1 

Topgallant  poles 1 

Jib-boom 1 

Fore  and  main  yards 

Topsail  yards 

Cross -jack,  Topgallant,  and) 

Royal  yards j 

Main  and  Spanker  booms 

Gaffs 

Studding-sail  yards  and  booms. 


3-33  I 

3.33  }-feet  of  whole  length. 


875  ‘ 


3-25  “ 3-33  I 
3-25  “ 3-33  Y 
3.66  | 

2.87  J 

2 ft.  of  length  beyond  bowsprit  cap. 
4 


Pd  = T;  .196  C D3 


4 

5 

3-5 

3-  5 to  4 

4- 5  t0  4-75 
Rudder  Head.  ( Mackrow .) 

T 


feet  of  whole  length. 


= M;  3 / — T — — D; 
’ V *196  C ’ 


, A v2 

and  — P.  P representing  press- 

ure on  rudder  when  hard  over,  in  tons,  d distance  of  geometrical  centre  of  rudder  from 
axis  of  motion,  in  ins.,  T stress  on  head,  and  M moment  of  resistance  of  head,  both  in 
inch-tons,  A immersed  area  of  rudder  in  sq.  feet , v velocity  of  water  passing  rudder 
in  knots  per  hour,  and  C coefficient  = 3. 5 per  sq.  inch  for  Iron,  and  .125  for  Oak. 

Illustration. — Assume  area  of  wooden  rudder  24  sq.  feet,  distance  of  its  geomet- 
rical centre  from  centre  of  pintles  2 feet,  and  velocity  of  water  10  knots. 


1X2X12  = 24  inch-tons. 


- ==  9.93  ins. 


24°°  ' V - *96  X .125" 

Memoranda. 

Weights.  — A man  requires  in  a vessel  a displacement  or  488  lbs.  per  month  for 
baggage,  stores,  water,  fuel,  etc.,  in  addition  to  his  own  weight,  which  is  estimated 
at  175  lbs.  A man  and  his  baggage  alone  averages  225  lbs. 

A ship,  150  feet  in  length,  32  beam,  and  22.83  in  depth,  or  664  tons,  C.  H.  (0  M ) 
has  stowed  2540  square  and  484  round  bales  of  cotton.  Total  weight  of  cargo 

I 254448  lbs.,  equal  to  4.57  bales,  weighing  1889  lbs.,  per  ton  of  vessel. 

A full  built  ship  of  1625  tons,  N.  M.,  can  carry  1800  tons’  weight  of  cargo  or  stow 
4500  bales  of  pressed  cotton. 

Hull  of  iron  steamboat  John  Stevens  — length'  245  feet,  beam  31  feet,  and  hold 

II  feet;  weight  of  iron  239440  lbs.  And  of  one  other— length  175  feet  beam  24 
feet,  and  8 feet  deep;  weight  of  iron  159  190  lbs. 

Weight  of  hull  of  a vessel  with  an  iron  frame  and  oak  planking  (composite)  com- 
pared with  a hull  entirely  of  wood,  is  as  8 to  15. 

An  iron  hull  weighs  about  45  per  cent,  less  than  a wooden  hull. 

Iron  ship,  254  feet  in  length,  42  beam,  and  23.5  hold,  1800  tons  register,  has  a stow- 
age of  3200  tons  cargo  at  a draught  of  22  feet.  Weight  of  hull  in  service  1450  tons. 

Loss  by  Weight  per  Sq.  Foot  per  Month  of  Metalling  of  a Vessel's  Bottom  in  Service. 
Copper  .0061  lb. ; Muntz  metal  .0045  lb. ; Zinc  .007  lb. ; and  Iron  .0204  lb. 
Comparison  between  Iron  and  Steel  plated  Steamers.—  In  a vessel  of  5000  tons 
displacement,  hull  of  steel-plated  will  weigh  320  tons  less  = 6.^6  pep  centum  less. 


668 


OPTICS. 


OPTICS. 

Mirrors,  in  Optics,  are  either  Plane  or  Spherical.  A plane  mirror  is  a 
plane  reflecting  surface,  and  a spherical  mirror  is  one  the  reflecting  surface 
of  which  is  a portion  of  surface  of  a sphere.  It  is  concave  or  convex,  ac- 
cording as  inside  or  outside  of  surface  is  reflected  from.  Centre  of  the 
sphere  is  termed  Centre  of  curvature. 

Focus— Point  in  which  a number  of  rays  meet,  or  would  meet  if  produced. 
Fig.  i.  Principal  Focal  Distance  is  half  radius 

of  curvature,  and  is  generally  termed  the 
j focal  distance.  Line  ac  is  termed  the 

principal  axis , and  any  other  right  line 


itz  through  c which  meets  the  mirror  is  termed 
a Secondary  axis.  When  the  incident 
rays  are  parallel  to  the  principal  axis , the 
reflected  rays  converge  to  a point,  F. 
Conjugate  Foci  are  the  foci  of  the  rays  proceeding  from  any  given  point 
in  a spherical  concave  mirror,  and  which  are  reflected  so  as  to  meet  in  an- 
other point,  on  a line  passing  through  centre 
of  sphere.  Hence,  their  relation  being  mu- 
tual, they  are  termed  conjugate. 

Let  P be  a luminous  point  on  principal  axis. 
Fig.  2,  and  P i a ray;  draw  the  normal  line  c ?, 
which  is  a radius  of  the  sphere;  then  c i P is  an- 
gle of  incidence,  and  ci  0 the  angle  of  reflection, 
equal  to  it;  hence  c i bisects  an  angle  of  triangle 
i P c P 

P i 0,  and  therefore, 

’ ’ i 0 c O 

When  conjugate  focus  is  behind  a mirror,  and  reflected  rays  diverge,  as 
if  emanating  from  that  point,  such  focus  is  termed  Virtual,  and  a focus  in 
which  they  actually  meet  is  termed  Real. 

As  a luminous  point,  as  P,  Fig.  3,  is 
moved  to  the  mirror,  the  conjugate  focus 
moves  up  from  an  indefinite  distance  at 
back,  and  meets  it  at  surface  of  mirror. 

If  an  incident  ray  converges  to  a point 
s,  at  back  of  mirror,  it  will  be  reflected 
to  a point  P in  front.  The  conjugate 
foci  P s having  changed  places, 

Pencil. — Rays  which  meet  in  a focus  and  are  taken  collectively. 

Objects.— As  regards  comparative  dimensions  or  volumes,  it  follows,  from 
similar  triangles,  that  their  linear  dimensions  are  directly  as  their  distances 
from  centre  of  curvature. 


Xo  Compute  Dimension  or  "Volnme  of  ail  Image. 

When  Dimensions  and  Position  of  Object  are  Given , and  for  either  Convex 
or  Concave  Mirrors . 

L 5.  or  _L  — 0L  L and  l representing  lengths  of  image  and  object , F focal 

l F ’ L F . . , 

length , and  D and  d respectively , distances  of  image  and  object  from  principal  Jocus.  , 

Refraction  . 

Deviation. — Angle  at  which  a ray  is  diverted  from  its  original  or  normal 
course  when  subjected  to  refraction  is  thus  termed. 

Indices  of  Refraction. — Ratio  of  sine  of  angle  of  incidence  to  sine  of  angle 
of  refraction,  when  a ray  is  diverted  from  one  medium  into  another,  is  termed 
relative  index  of  refraction  from  former  to  latter. 


OPTICS. 


669 


When  a ray  is  diverted  from  vacuum  into  any  medium,  the  ratio  is  greater 
than  unity,  and  is  termed  absolute  index  or  index  of  refraction . 

Mean  Indices  of  Refraction. 


Glass,  lead,  3 flint 2.03 

“ lead  2,  sand  1 1.99 

“ “ 1,  flint  1 1.78 

Ice 1. 31 

Quartz........ 1.54 


Eye,  vitreous  humor 1.339 

crystalline  lens,  under 1.379 

“ “ “ central 1.4 

Diamond  2.6 

Glass,  flint 1.57 

For  indices  of  other  substances,  see  page  584. 

Heat  increases  refractive  power  of  fluids  and  glass. 

Critical  Angle. — Its  sine  is  reciprocal  of  index  of  refraction,  the  incident 
ray  being  in  the  less  refractive  medium. 


Visual  Angle  is  measure  of  length  of  image  of  a straight  line  on  the  retina. 

Total  Reflection  is  when  rays  are  incident  in  the  more  refractive  medium, 
at  an  angle  greater  than  the  critical  angle. 

Mirage. — An  appearance  as  of  water,  over  a sandy  soil  when  highly  heated 
by  the  sun. 

Caustic  Curves  or  Lines  are  the  luminous  intersections  from  curve  lines,  as 
shown  on  any  reflective  surface  in  a circular  vessel. 


To  Compute  Index  of  Refraction. 

^ = Index.  I representing  angle  of  incidence,  and  R that  of  refraction. 

To  Compute  Refraction. 

Concave-Convex  and  Meniscus. — Effect  of  a concave-convex  in  refracting 
light  is  same  as  that  of  a convex  lens  of  same  focal  distance,  and  that  of  a 
meniscus  is  same  as  a concave  lens  of  same  focal  distance. 

2 R 7* 

Meniscus,  with  parallel  rays  — — - = F. 

Magnifying  Power. — In  Telescopes  the  comparison  is  the  ratio  in  which  it 
apparently  increases  length.  In  Microscopes  the  comparison  is  between  the 
object  as  seen  in  the  instrument  and  by  the  eye,  at  the  least  distance  of 
vision,  which  is  assumed  at  10  ins.,  and  the  magnifying  power  of  a micro- 
scope is  equal  to  the  distance  at  which  an  object  can  be  most  distinctly  ex- 
amined, divided  by  the  focal  length  of  the  lens  or  sphere. 

Linear  po-wer  is  number  of  times  it  is  magnified  in  length,  and  Super- 
ficial, number  of  times  it  is  magnified  in  surface. 

Magnifying  power  of  microscopes  varies,  according  to  object  and  eye- 
glass, from  40  to  350  times  the  linear  dimensions  of  object,  or  from  1600  to 
122500  times  its  superficial  dimensions. 

Apparent  Area. — As  areas  of  like  figures  are  as  the  squares  of  their  linear 
dimensions,  the  apparent  area  of  an  object  varies  as  square  of  visual  angle 
subtended  by  its  diameter. 

The  number  expressing  Magnification  of  Apparent  Area  is  therefore 
square  of  magnifying  power  as  above  described. 

Illustration. — If  diameter  of  a sphere  subtends  i°  as  seep  by  the  eye,  and  io° 
as  seen  through  a telescope,  the  telescope  is  said  to  have  a power  of  10  diameters. 


OPTICS, 


67O 


To  Compute  Elements  or  Mirrors  and.  Lenses. 

Or  l r 

Mirrors.  Spherical  Concave.*  • — — D ; =r  = L. 

r — il  r — zl 

Or  Lr  c22 

Spherical  Convex. t -y_^-  = D ; .-T-  Parabolic  Concave. 


Unequally.  Convex,  t 


2 B r 


2L-fr 


=r  F. 


Plano-  Convex.  § 2 B - 
Sphere. 


16  A 
.66  £ = F. 

I 


= F. 


! 1 I 


i=  F. 


K + r 

Hyperbolic  Concave.W  Elliptic  Concave .If 

0 representing  object  — 1,  r radius  of  convexity,  l and  L length  or  distance  of  object 
from  vertex  of  curve,  and  from  external  vertex,  D dimension  of  object,  d.  diameter  of 
base,  ¥ focal  distance,  and  h depth  of  mirror  in  lilce  dimensions,  I index  of  refraction, 
and  t thickness  of  lens. 

Illustration  i. — Before  a concave  mirror  of  5 feet  radius  is  set  an  object  at  1.5 
feet  from  vertex  of  curve;  what  is  ratio  of  apparent  dimension  of  image,  and  what 
is  length  of  and  distance  of  object  from  external  vertex  ? Object  = 1. 

— 1X5 — = 2.5  feet,  and  - X 5 = 3. 75  feet. 

5 — 2X15  5 — 2X1.5 

2.— If  object  is  set  at  4.5  feet  from  vertex  of  a like  mirror,  what  is  length  of  and 
distance  of  inverted  object  from  internal  vertex? 

*X5  4-5X5 


= 1.25  feet,  and 


; = 5.625  feet. 


2X4-5  — 5 ' 2 X 4f- 5 _ 

3. — Before  a convex  mirror  of  3.5  feet  radius  is  set  an  object  at  3 feet  from  ver- 
tex of  curve;  what  is  length  of  and  distance  of  object  from  external  curve? 

. . **  .3‘.5_^  — .368  foot,  and  3Xj5-~—  * 

2 X 3 + 3-  5 6 2 X 3 4"  3-5 


: 1. 105  feet. 


4. — A parabolic  reflector  has  a depth  of  1.25  feet  and  a diameter  of  2 feet;  what 
is  its  focal  distance  from  vertex  of  internal  curve? 


16  X 1 25 

Lenses.  Double  Convex. 


= .2  feet  or  2.4  ins. 
R r 


0 F 


= D; 


l ¥ 


¥ — l 7 ¥ — l 

Double  Concave. 


= L; 


Rr 


to  — i X R + r 
S-fF  OF 

F ~ ’ F — 0 " 

- , 


— F.  When  R — r 


= F; 


= V ; and 
Fo-D 


S F 
S + F 


= 0. 


T A hF  7 

= L;  and  - — r-=  = l. 

-JXR+?  u L + F 

Optical  centres  are  in  centres  of  lens.  Plano  - Convex  and  Plano  - Concave. 

— F.  Optical  centres  are  respectively  centres  of  convex  and  concave  sur- 

Rr 


TO  — I 

faces.  Convex  Concave  ( Meniscus ) and  Concavo-Convex. 


to  — 1 X R — t 


z = ¥. 


Optical  Centres.  Convex  Concave.  Delineate  lens  in  half  section,  draw  R from 
its  centre  to  circumference  of  lens  (intersection  of  radii),  draw  r parallel  thereto 
and  extending  to  its  circumference,  connect  R and  r at  these  external  points  of 
contact  with  circumference  and  external  curve,  extend  line  to  axis  of  lens,  and  point 
of  contact  is  centre  required.  Concavo-Convex.  Proceed  in  like  manner,  but  in 
this  case  r extends  to,  or  delineates,  the  inner  surface  of  the  lens,  and  point  of  con- 
tact with  axis  is  centre  required. 


* D or  image  disappears  when  l — .5  r. 
Lr 


t When  0 is  beyond  F,  it  will  be  inverted,  as 


2 I — r 


§ When  convex  side  is  exposed  to  parallel  rays 


and  — - - = l.  t When  equally  convex  F = R. 

and  when  parallel  rays  fall  upon  plane  side,  F = 2 R.  (1  Rays  of  light,  heat,  or  sound,  reflected  from 
focus  of  a liyperbola,  will  diverge  from  its  concave  surface,  ‘ft  and  when  from  the  focus  of  an  ellipse, 
will  be  refracted  by  surface  of  the  other. 


OPTICS. — PILE-DRIVING. 


67I 


When  object  is  beyond  focal  distance  (F),  its  image  (D)  will  be  inverted,  as  ^ — — = D,  and  j — = l. 


P representing  magnifying  power  of  lens,  S limit  of  normal  sight , 10  to  12  ins.  for 
far-sighted  eyes  and  6 to  8 for  near-sighted , ordinarily  10  ins. , V limit  of  distinct 
vision , 0 extreme  distance  of  object  from  optical  centre  at  distinct  vision,  and  m index 
of  refraction. 

Illustration  i. — If  a double  convex  lens  of  flint  glass  lias  radii  of  6 and  6.25  ins  , 


2. — If  a double  concave  lens  has  a focal  distance  of  2 ins.,  and  object  is  6 ins.  from 
vertex  of  curve,  what  is  its  dimension  and  what  is  its  distance  from  vertex  of  inner 
curve  ? 


3.— If  focal  distance  of  a single  microscope  is  4 ins.,  what  is  its  limit  of  distinct 


ing  length  of  focal  distance  from  object  lens. 

Illustration. — Principal  focal  distance  of  ocular  lens  of  a telescope  is  .9  in.,  of 
objective  lens  90  ins. ; what  is  its  magnifying  poiver? 


Effect  of  blow*  of  a ram,  or  monkey,  of  a pile-driver,  is  as  square  of 
its  velocity ; but  the  impact  is  not  to  be  estimated  directly  by  this  rule, 
as  the  degree  and  extent  of  the  yielding  of  the  pile  materially  affects  it. 
The  rule,  therefore,  in  application,  is  of  value  only  as  a means  of  com- 
parison. 

By  my  experiments  in  1852,  to  determine  the  dynamical  effect  of  a fall- 
ing body,  it  appeared  that  while  the  effect  was  directly  as  the  velocity,  it 
was  far  greater  than  that  estimated  bv  the  usual  formula  Vs  2 y,  which,  for 
a weight  of  1 lb.  falling  .2  feet,  would  be  11.34  lbs.,  giving  a momentum  of 
11.34  foot-lbs. ; vdiereas,  by  the  effect  shown  by  the  record  of  actual  obser- 
vations, it  would  be  W v 4.426  = 50  lbs. 

Piles  are  distinguished  according  to  their  position  and  purpose:  thus, 
Gauge  Piles  are  driven  to  define  limit  of  area  to  be  enclosed,  or  as  guides  to 
the  permanent  piling. 

Sheet  or  Close  Piles  are  driven  between  gauge  piles  to  form  a compact  and 
continuous  enclosure  of  the  work. 

Weight  which  each  pile  is  required  to  sustain  should  be  computed  as  if  the 
pile  stood  unsupported  by  any  surrounding  earth. 

A heavy  ram  and  a lowr  fall  is  most  effective  condition  of  operation  of  a 
pile-driver,  provided  height  is  such  that  force  of  blow  will  not  be  expended 
in  merely  overcoming  friction  of  leader  and  inertia  of  pile,  and  at  same  time 
not  from  such  a height  as  to  generate  a velocity  which  will  be  essentially 
expended  in  crushing  fibres  of  head  of  pile. 


what  is  its  focal  distance  ? 


Index  of  refraction  = 1.57,  see  page  584. 


1.57  — 1 X 6 + 6.25 


vision,  and  what  its  magnifying  pow*er? 


0 = 2. 857  ins. 


2.857  X 4 • A IO+  4 ,. 

— — r=  10  ins.,  and  — — = 3-5  times. 
4 — 2.857  5 4 


Telescopes,  Opera-glasses,  etc. 


90-4-  .9  = 100  times  the  object. 


PILE-DRIVING. 


*+  for  telescopes  and  — for  opera-glasse9,  etc. 


PILE-DRIVING. 


672 

Refusal  of  a pile  intended  to  support  a weight  of  13.5  tons  can  be  : 
taken  at  10  blows  of  a ram  of  1350  lbs.,  falling  12  feet,  and  depressm5  ^ 
pile  .8  of  an  inch  at  each  stroke.* 

Pneumatic  Piles.- A hollow  pile  of  cast  iron,  2.5  feet  in  diameter,  was  depressed 
into  the  Goodwin  Sands  33  feet  7 ins.  in  5.5  hours. 

Nasmyth's  Steam  Pile-hammer  has  driven  a pile  14  ins.  square,  and  18  feet  m 
length  15  feet  into  a coarse  ground,  imbedded  in  a strong  clay,  in  17  seconds,  with 
20  blows  of  monkey,  making  70  strokes  per  minute. 

Morin  computed  work  of  a ram  in  foot-lbs.,  in  raising  a monkey  for  8 hours  per 
day  as  follows:  Tread-wlieel  3900,  Winch  2600. 

French  engineers  estimate  the  safe  load  for  a pile,  when  driven  to  refusal  of  .4 
inch  under  30  blows,  to  be  25  tons. 

Shaw's  Gunpowder  Pile-driver  is  operated  by  cartridges  of  powder  on  head 
of  pile,  which  are  ignited  by  fall  of  the  ram.  30  to  40  blows  per  minute 
have  been  made  under  a fall  of  5 aud  10  feet.  27  piles  have  been  duven  m 
rough  gravel  and  clay  7.2  feet  in  one  day. 

To  Compute  Safe  Load  tliat  may  toe  Borne  toy  a Bile. 
(Maj.  John  Sanders , U.  S.  E.) 

Approximately.  = W.  E representing  weight  of  ram  in  lbs. , h height  of 

fall  and  d distance  pile  is  depressed  by  blow , both  in  feet. 

Illustration.— A ram  weighing  3500  lbs.,  falling  3.5  feet,  depressed  a pile  4.2  ins. 

Then  35oo  X (42  4- a)  _ 35og>  _ 43?5  ibs ^ weight  which  pile  would  bear  with 

8 8 

safety. 

Molesworth  gives  this,  but  with  a variation  in  symbols  and  their  expression. 

To  Compute  Coefficient  of  Resistance  of  the  Earth. 

— C.  R representing  resistance  of  the  earth , and  d as  preceding. 

Weisbach  gives  following  formula : Resistance  of  bed  of  earth  being  con- 
stant, mechanical  effect  expended  in  penetration  of  pile  will  be  p 
P representing  weight  of  pile  in  lbs. 

Illustration. — Assuming  elements  of  preceding  case,  with  addition  of  weight  ot 
pile  at  ,500 lbs.,  ■ 3500^x  3-5.  _ 43875 ooo_01^„,,,_ 

1500  -f-  35°°  X (4* 2 I2)  *75° 

To  Compute  "Weight  of*  Ram.  (Molesworth.) 

P (JlK. = R.  P representing  weight  of  pile  in  lbs .,  h height  of  fall  and  L 

\ 5 A L / ; 

length  of  pile,  both  in  feet,  and  A area  of  section  of  pile  m sq.  ins. 


Fall. 

1000 

Wei 

1200 

ght. 

150a 

2000 

Fall. 

IOOO 

Wei 

1200 

gbt. 

1500 

Feet. 

1 

5 

10 

Lbs. 
8000 
17  920 
25  360 

Lbs. 

9 600 
21  504 
3°432 

Lbs. 
12  000 
26  880 
33°40 

Lbs. 
16000 
35  840 
50  720 

Feet. 

15 

20 

25 

Lbs. 
31  060 
35860 
40  100 

Lbs. 
37  272 
43032 
48  720 

Lbs. 
46  590 
53  79° 
60150 

Lbs. 

62  120 
71720 
80200 

Sheet  Riling. 

Bevelling 1200  | Shoeing 25° 

Ringing  Engine 

Requires  1 man  to  each  40  lbs.  weight  of  ram,  which  varies  from  450  to 
900  lbs. 


PILE-DRIVING. PNEUMATICS. AEROMETRY.  673 


Dile-sinliing. 

Mitchell's  Screw  Piles  are  constructed  of  a wrought-iron  shaft  of  suitable 
diameter,  usually  from  3 to  8 ins.,  with  1.5  turns  of  a cast-iron  thread  of 
from  1.5  to  3 feet  diameter. 

Hydraulic  Process  is  effected  by  the  direction  of  a stream  of  water  under 
pressure,  within  a tube  or  around  the  base  of  a pile,  by  which  the  sand  or 
earth  is  removed. 

Pneumatic  and  Plenum  Process. — For  illustration  and  details,  see  Traut- 
wine’s  Engineer’s  Pocket-book,  page  326. 

Dr.  Wkewell  deduced  the  following  results : 

1.  A slight  increase  in  hardness  of  a pile  or  in  weight  of  a ram  will  con- 
siderably increase  distance  a pile  may  be  driven. 

2.  Resistance  being  great,  the  lighter  a pile  the  faster  it  may  bQ  driven. 

3.  Distance  driven  varies  as  cube  of  the  weight  of  ram. 

Relative  Resistance  of  Formations  to  Driving  a Pile. 

Coral  100  I Hard  clay 60  I Light  clay  and  sand. . . 35 

Clay  and  gravel 83  | Clay  and  sand 45  | River  silt 25 


PNEUMATICS.— AEROMETRY. 

Motion  of  gases  by  operation  of  gravity  is  same  as  that  for  liquids. 
Force  or  effect  of  wind  increases  as  square  of  its  velocity. 

If  a volume  of  air  represented  by  1,  and  of  320,  is  heated  t degrees  without 
assuming  a different  tension,  the  volume  becomes  (1  + .002088  t)=Y;  and 
if  it  requires  a temperature  in  excess  of  t'  320,  it  will  then  assume  volume 
(r  -f  .002088  t'  — 32°).  All  aeriform  fluids  follow  this  law  of  dilatation  as 
well  as  that  of  compression  proportional  to  weight. 

When  air  passes  into  a medium  of  less  density,  its  velocity  is  determined 
by  difference  of  its  densities.  Under  like  conditions,  a conduit  will  discharge 
30.55  times  more  air  than  water. 

To  Compute  the  Degree  of  Rarefaction  that  may  "be  ef- 
fected. in.  a 'Vessel. 

Let  quantity  of  air  in  vessel,  tpbe,  and  pump  be  represented  by  1,  and 
proportion  of  capacity  of  pump  to  vessel  and  tube  by  .33 ; consequently,  it 
contains  .25  of  the  air  in  united  apparatus. 

Upon  the  first  stroke  of  piston  this  .25  will  be  expelled,  and  .75  of  original 
quantity  will  remain ; .25  of  this  will  be  expelled  upon  second  stroke,  which 
is  equal  to  .1875  of  original  quantity;  and  consequently  there  remains  in 
apparatus  .5625  of  original  quantity.  Proceeding  in  this  manner,  following 
Table  is  deduced : 


No.  of  Strokes. 

Air  Expelled  at  each  Stroke. 

Air  Remaining  in  Vessel. 

1 

to 

U\ 

II 

cln 

•75  — -75 

3 _ 3 

9 _3  X 3 

2 

16  4X4 

16  4X4 

9 _ 3X3 

27  3X3X3 

3 

64  ~ 4 X 4 X 4 

64  4X4X4 

And  so  on,  multiplying  air  expelled  at  preceding  stroke  by  3,  and  dividing 
it  by  4;  and  air  remaining  after  each  stroke  is  ascertained  by  multiplying 
air  remaining  after  preceding  stroke  by  3,  and  dividing  it  by  4. 
2 L 


PNEUMATICS.—  AEROMETRY. 


674 


Feet. 

Miles. 

460 

.087 

15  840 

3 

16000 

3.02 

10  560 

2 

15  840 

3 

575  000 

90 

Distances  at  which.  Different  Sounds  are  -A/udible. 

A full  human  voice  speaking  in  open  air,  calm 

In  an  observable  breeze,  a powerful  human  voice  with  the! 

wind  can  be  heard • ... j 

Report  of  a musket. . . . . . 

Drum 10560 

Music,  strong  brass  band 15  840 

Cannonading,  very  heavy 

In  Arctic  Ocean,  conversation  has  been  maintained  over  water  a distance 
of  6696  feet. 

In  a conduit  in  Paris,  the  human  voice  has  been  heard  3300  feet. 

For  an  echo  to  be  distinctly  produced,  there  must  be  a distance  of  55  feet. 

Coefficients  of  Efflux  of  Discharge  of  Air.  (D1  Aubuisson.) 

Orifice  in  a thin  plate. . 65  .751 

Cylindrical  ajutage .93  -958 

Slight  conical  ajutage 94  1.09 

To  Compute  Volume  of  Air  Discharged  through  an  Open- 
ing into  a Vacuum,  per  Second. 

a C V2  g h = V in  cube  feet  a representing  area  of  opening  in  square  feet,  C co- 
efficient of  efflux,  and  V 2 g h — 1347.4,  as  shown  at  page  428. 

Illustration. — Area  of  opening  1 foot  square,  and  C = . 707. 

Then  1 X .707  X 1347.4  = 952.61  cube  feet. 

Inversely,  V -4-  a = velocity  in  feet  per  second. 

■Velocity  and  Pressure  of  Wind. 

Pressure  varies  as  square  of  velocity,  or  P oc  V2. 

V2X-oo5  = P;  V200  P — V;  d2x.oo23  = P;  and  .0023  v2  sin.  x = P. 

V representing  velocity  in  miles  per  hour , v in  feet  per  second , P pressure  in  lbs. 
per  sq.foot,  and  x angle  of  incidence  of  wind  with  plane  of  surface. 

Table  deduced  from  above  Formulas. 


Velocity 


Feet. 

88 

176 

264 

352 

440 

528 

7°4 

880 

1320 

1760 


Pressure 
on  a 

Sq.  Foot. 


Lbs. 
.005 
.02  ) 
•045) 
.08 
.125) 
.x8 

.32  ; 
• 5 

1.125 


Character  of  the  Wind. 


Barely  observable. 
Just  perceptible. 
Light  breeze. 

Gentle,  pleasant 
wind. 

Fresh  breeze. 
Brisk  blow. 

Stiff  breeze. 


Velocity 


Miles. 

25 

30 

35 

40 

45 

50 

60 

80 

9° 


Feet. 

2200 

2640 

3080 

3520 

3960 

4400 

5280 

7040 

7920 

8800 


Pressure 
on  a 

Sq.  Foot. 


Lbs. 

3<I25 
.4-5  1 

6.125  j 
8 

10.125 

12.5 

18 

32  •> 
4o-5l 

50  j * 


Character  of  the 
Wind. 


Very  brisk. 
High  wind. 

Very  high  wind. 
Gale. 

Storm. 

Great  storm. 
Hurricane. 

Tornado. 


Illustration. — What  is  pressure  per  sq.  foot,  when  wind  has  a velocity  of  18 
miles  per  hour?  l82  x od5  _ l 6z  ^ 

To  Compute  Force  of  Wind  upon  a Surface. 

- sini — — p.  v representing  velocity  of  wind  in  feet  per  second,  a area  of 

440 

surface  in  sq.feet , and  A angle  of  incidence  of  wind. 

At  Mount  Washington  wind  has  been  observed  to  have  had  a velocity  of  150  miles 
per  hour. 

Extreme  pressure  of  wind  at  Greenwich  Observatory  for  a period  of  20  years  was 
41  lbs.  per  sq.  foot. 


PNEUMATICS. — AEEOMETEY. 


675 


Force  of  wind  upon  a surface,  perpendicular  to  its  direction,  has  been  ob- 
served as  high  as  57.75  lbs.  per  sq.  foot;  velocity  = 159  feet  per  second. 

Dr.  Hutton  deduced  that  resistance  of  air  varied  as  square  of  velocity 
nearly,  and  to  an  inclined  surface  as  1.84  power  of  sine  X cosine. 

Figure  of  a plane  makes  no  appreciable  difference  in  resistance,  but  con- 
vex surface  of  a hemisphere,  with  a surface  double  the  base,  has  only  half 
the  resistance. 

At  high  velocities,  experiments  upon  railways  show  that  the  resistance 
becomes  nearly  a constant  quantity. 


Direction  in 
Northern  Hemisphere. 


Course  of  AWixid.. 

Cyclones. 

Wind  has  its  direction  nearly  at 
right  angles  to  line  between  points  of 
highest  and  lowest  pressure  of  air,  or 
barometer  readings,  and  its  course  is 
with  the  point  of  lowest  pressure  at 
its  left,  and  its  velocity  is  directly  as 


Direction  in 
Southern  Hemisphere. 


difference  of  the  pressures. 

In  Northern  Temperate  zone,  winds  course  around  an  area  of  low  pressure 
in  reverse  direction  to  course  of  hands  of  a watch,  and  they  flow  away  from 
a location  of  high  pressure,  and  cause  an  apparent  course  of  the  winds  in  di- 
rection of  course  of  the  hands. 


To  Compute  Resistance  of*  a Diane  Surface  to  Air. 

.0022  av2  = P in  lbs.  a representing  area  of  plane  in  sq.feet , v velocity  in  direc- 
tion of  wind  in  feet  per  second , -f-  when  it  moves  opposite , and  — when  with  the  wind. 


To  Compute  Resistance  of  a Plane  Surface  when  moving 
at  an  Angle  to  Air. 

v a sin.  x _ p ^ x re^resenting  angle  of  incidence. 

45o 

To  Compute  Height  of  a Column  of  Mercury  to  induce 
an  iCfflnx  of  Air  through  a given  Nozzle. 

Barometer  assumed  at  2. 46  feet  = 29. 52  ins. , and  Temperature  520. 


* , — — H,  and  48.073  d2  -^/H  = P.  d representing  diameter  of  nozzle  and  H 

48.0732  d4 

height  of  column  of  mercury,  both  in  feet , and  P volume  of  air  in  lbs.  per  one  second. 
Illustration. — Assume  d — .19,  and  P = .7  lbs. 

- = .1511  foot.  48.073  X .iff  .1511  = -7- 


8.0732  X • 19 


To  Compute  Pressure  or  'Weigh. t of  Air  under  a given 
Height  of  Barometer  and  Temperature,  Discharged  in 
One  Second. 


30.787  d2  B —pressure  in  lbs.  Or,  48.073  d2  fB  — lbs.  b representing 

height  of  barometer  in  external  air , B manometer  or  pressure  of  air  in  reservoir  in 
mercury , both  in  feet,  and  t temperature  of  air  or  gas  in  degrees. 

Illustration. — Assume  b = 2. 5 feet ; d = . 25  foot ; B = . 1 foot ; and  t = 1.055  feek 


Then  30.787  X .0625 


f-i  X 


2.5-f.I 


= 1.924  x V-2465  = -9543 


i-o55 


676 


PNEUMATICS. AEKOMETKY. 


To  Compute  Temperature  for  a.  given.  Latitude  and  Ele- 
vation. 

82. 8 cos.  I — .001  981  E — . 4 — t.  E representing  elevation  in  feet. 
Illustration.—  Assume  1 = 450;  cos.  =.707;  and  E = 656  feet. 


Then  82.8  X -707  — .001  981  X 656  — .4  = 58-54  — 1-299  — -4  = 58-54  — -899  = 
57.641. 

To  Compute  Volume  of  Air  or  Gras  Discharged  through, 
an  Opening  and  under  a Pressure  above  that  of*  Ex- 
ternal  -A-ir. 

A ir.  1347.4  C ^ VB  (6'  + B)  T = V in  cube  feet  per  second. 

T = 1 -f-  .002  22  ( t — 320),  and  6'  = 2.5  — .00009  elevation. 

Or,  621.28  d2  \/B  = V. 

Illustration. — What  would  be  volume  of  air  that  would  flow  through  a nozzle 
.246  foot  in  diam.  from  a reservoir  under  a pressure  of  .098  foot  of  mercury,  into 
air  under  a barometric  pressure  of  2.477  feeb  temperature  of  air  55. 40,  location  450 
of  latitude,  and  at  an  elevation  of  650  feet  above  level  of  sea? 

C = . 75 ; b' = 2.5  — .00009  X 650  = 2.4415  (2.44);  and  T = i.c>502. 

Then  1347.4  X .75  V'098  (2.44  + .098)  X 1.0502  = 24.689  X V-^7  - 12.63 

2-477 

cube  feet. 

When  Densities  of  External  Air  and  that  in  Reservoir  are  Equal. 

1 347<  4 C ^ V B(6  + B)T  = V.  b'  representing  height  of  mercury  in  reservoir. 

Qas  — — V.  p representing  specif  c gravity  of  gas  compared 

Vp  V H42X^ 

with  air , and  L length  of  pipe  or  conduit  in  feet. 

Illustration.— If  a pipe  .05  feet  in  diameter  and  420  feet  in  length,  communi- 
cates with  a gasometer  charged  with  carburetted  hydrogen  (illuminating  gas),  under 
a water  pressure  as  indicated  by  a manometer  of  .1088  foot,  what  would  be  the  dis- 
charge per  second  ? 

d = .05  foot ; L = 420  feet ; and  B = = .008  foot.  Specific  gravity  of  gas 

.5625. 

4231  / .008X  .055  = 4231  /.poo 006 002  50^0  = .013  71  cube  foot. 

V.5625  V42o+i2XT^  -75  V 420H-2.X 

Resistance  of  Curves  and  Angles—  Curves  and  angles  increase  resistance 
to  discharge  of  air  or  gas  very  materially.  By  experiment  of  D’Aubuisson 
7 angles  of  45 0 reduced  discharge  of  gas" one  fourth. 


To  Compute  Diameter  of*  Discharge-pipe  or  Nozzle. 

When  Length  and  Diameter  of  Pipe , Volume,  and  Pressure  are  given. 

. / 42  V2  . 

4 / a— . - — d in  feet. 

\ 42302  Bd5  — L V2 

Illustration. — If  a pipe  1000  feet  in  length,  and  .4  foot  in  diameter,  leads  to  a 
reservoir  of  air,  under  a mercurial  manometric  pressure  of  .18  foot,  what  diameter 
must  be  given  to  a nozzle  to  discharge  4 cube  feet  per  second? 

Then  «/■-•  42  X 42  X -4s  " 4/  6-88»*8  = ^ = 

V 4*30*  x 18  X -45—  1000  X 4a  V 32  980. 19- >6ooo 
.1418/00^  = 1.703  ins. 

Volumes  of  two  gases  flowing  through  equal  orifices,  and  under  equal  pressures, 
are  in  inverse  ratio  of  square  roots  of  their  respective  densities. 


Specific  gravity  of  mercury  compared  with  water. 


RAILWAYS. 


677 


Fig.  1. 


RAILWAYS. 

To  Define  a Curve.— Dig.  1.  ( Molesworth .) 

^ &?*?-$  or  Z tan.  a? = R ; R (cotan.  a?)  ==  * ; 


1719  c 


= a;  R (cosec.  x — i)  = d-, 


R (cosin.  *)  = s ; R (coversin.  a)  = V ; 


— w,  and  (5400  — x)  .000582  R — Z. 


c representing  any  chord , « length  of  tangent,  d distance  of  centre  of  curve  from,  in- 
tersection of  tangents,  s half  chord  of  curve,  and  l length  of  curve,  all  inlike  dimensions 
a tangential  angle  ofc  in  minutes,  n number  of  chords  in  curve,  and  x half  angle  of 
intersection,  but  in  form'ulas  for  number  of  chords  and  length  of  curve  to  be  expressed 
in  minutes . 

Illustration. — Assume  radius  900  and  chord  400  feet;  angle  of  intersection .= 
120  44'  = 764  minutes,  and  x = 56°  15'  5". 

Tangent  of  56°  15'  5"  = x. 496.33.  Cotangent  ==  .668  14. 

1719  X 4°°  __  r — gQO  feet ; I7I9  X 4°^  _ ^4  minutes  ; 900  X • 668  14  = t = 

764  [ 9 00 


601.33  feet;  900  X 1.20269  — 1 = d==  182,42 jM;  900  X • 555  55  — $ — $oofeet; 
900  X .16833  = V = 161. 5 feet; 


— 2.645  times,  and  .000.582  X 900  X 

764 


5400  — 3379  = 1058 .6  feet 

Tangential  Angles  for  Chords  of  One  Chain . 


Radius  of 
Curve. 

Tangential 

Angle. 

Radius  of 
Curve. 

Tangential 

Angle. 

Radius  of 
Curve. 

Tangential 

Angle. 

Radius  of 
Curve. 

Tangential 

Angle. 

Chains. 

5 

8 

9 

10 

12 

5°  43-8', 
3°  34-87 
3°  , 

2°  51.9  / 
2°  23.25 

Chains. 

15 

20 

25 

30 

35 

i°  54-6' 
i°  25-95 
1°  8.76' 

57-3', 
49s11 

Chains. 

40 

45 

50 

60 

70 

42.9/ 

38.2 

34.38' 

28.65 

24-55' 

1 mile 
1.25  mil’s 
1.5  miles 
1. 75  “ 

2 

21.48' 

I4-33 

12.28 

10.74' 

INvJl £1.  a ^ ^ . . 0 

chain  chords  is  .5  the  angle  for  1 chain  chords. 

Curves  of  less  than  20  chains  radius  should  be  set  out  in  . 5 chain  chords.  Curves 
of  more  than  1 mile  radius  may  be  set  out  in  2 chain  chords. 

Angles  in  above  Table  are  in  degrees,  minutes,  and  decimals  of  minutes. 

Fig.  2. 

I 


Sidings. 

2 y/d  R — (.5  df  = l.  R representing  radius  of 
curve,  l length  of  curve  over  points,  and  d distance 
between  tracks, 


all  in  feet. 


Fig.  3- 


Turn-out  of*  Unequal  Dadii. 

-y\  x—y  = Z)  o + b = Z;  r — y = A; 


r x 

R ~+r~ 


y/ y {r -\- A)  — a\  R1-  z = B;  y/z  (R  + B)  _ 6. 


R and  r representing  radii  of  the  curves  re- 
spectively as  to  length , x distance  between  outer 
rails  of  tracks  and  other  symbols  as  shown,  all 
in  feet. 

3 L* 


6y  8 


RAILWAYS. 


Fig.  4. 


Points  and.  Grossings. 

. l G 

V (R-j-s)  G z=  l]  — =sin.  a;  

’ R ver.  sin.  a 


— R.  R repre- 
senting radius  of  curves,  G gauge  of  road,  a angle  of  crossing, 
and  x — R — G,  all  in  feet. 

In  horizontal  curves,  width  required  for  clearance  of 
flange  of  wheel,  and  for  width  of  rail  at  heel  of  switch, 
render  it  necessary  to  make  an  allowance  in  length  of  l, 
as  ascertained  by  formula. 

For  other  diagrams  and  formulas,  see  Molesworth’s  Pocket- 
book,  pp.  208-18,  21st  edition. 

To  Compute  Tangential  Angle  For  Curves.  — a.  c 

representing  chord  in  feet , and  a angle  in  minutes. 

Illustration. — What  is  angle  for  a curve  with  a radius  of  900  feet,  and  a chord 
of  400  feet? 

i7T9  X 4°°  — 764  minutes. 

90Q 

Curving  oF  Hails. 

= v.  I representing  length  of  rail  in  feet , v versed  sine  at  centre,  when 


1-56  l 2 
R 


curved,  in  ins. 

Illustration. — What  is  curve  for  a rail  20  leet  jn  length,  with  a radius  of  900  feet? 
1.5  X 20^ 


900 


- =.666  ins. 


Curves  'by'  Offsets  in  Equal  Cliords. 


Fig.  5- 


Chord  2 
2R 


- — 0 offset. 


Chord2 
R = 


: 2,  0 offset. 


Illustration.— Assume  chords  150,  and  ra- 
f dius  900  feet. 

6" 


22  500 


. 22500  ,. 

112.5  feet;  — - — = -2$  feet. 
— o 

o 

(•scp 


2 X 900  ~ ' 900 

To  Compute  "Versed  Sines  and  Ordinates  oF  Curves. 
Fig.  Vw  R — VR2  — ( .5  C)2 


-f  v = D ; and 

\/R2  — ce2 — (R  — v)  = o.  D representing  diameter  of 
\ circle , and  v versed  sine  of  curve. 

I R 'x  1 Illustration. — Assume  radius  900,  and  chord  400  feet. 

L — j 

D 900  — v 810  000  — 40  000  = 900  — 877. 5 = 12. 5 feet. 

Relation  oF  Base  oF  Driving  or*  Rigid  ‘Wheels  to  Curve. 
R 

— — . B.  R representing  minimum  radius  of  curve,  G gauge  of  road,  and  B base, 
iri  feet. 

To  Compute  Elevation  oF  Outer  Rail. 

For  any  Radius  or  Combination  o f Curve  with  Straight  Line. 

•5  F y/G  — c.  V representing  velocity  of  train  in  feet  per  second , G gauge  of  road, 
and  c length  of  a chord , both  in  feet,  the  versed  sine  of  which  — elevation  in  ins. 

On  Curves, 

— G = E.  E representing  elevation  of  outer  rail  in  ins. 


alf’ 


RAILWAYS. 


679 


Radii  of  Carves  set  oat  in  Tangential  Angles. 


Angle  for 
Chord  of 
1 00  Feet. 

Radius 

of 

Curve. 

Angle  for 
Chord  of 
100  Feet. 

Radius 

of 

Curve. 

Angle  for 
Chord  of 
100  Feet. 

Radius 

of 

Curve. 

Angle  for 
Chord  of 
100  Feet. 

Radius 

of 

Curve. 

0 ' 

Feet. 

0 ' 

Feet. 

0 ' 

Feet. 

O r 

Feet. 

30 

5729.6 

2 30 

II45-9 

4 3° 

636.6 

6 30 

440.7 

1 

2864.8 

3 

954-9 

5 

573 

7 

409'3 

1 30 

1909.9 

3 30 

818.5 

5 3° 

520.9 

7 3°  s 

382 

2 

1432.4 

4 

716.2 

6 

447-5 

8 

358.1 

Note.— If  chords  of  less  length  are  used,  radius  will  be  proportional  thereto. 

To  Ascertain  Radius  of  Curve  in  Inches  for  Scale , in  Feet  per  Inch . 
Divide  radius  of  curve  in  feet  by  scale  of  feet  per  inch. 

To  Compate  Repaired  Weiglit  of  Rail. 

Rule.— Multiply  extreme  load  upon  one  driving-wheel  in  lbs.  by  .005, 
and  product  will  give  weight  of  rail  in  lbs.  per  yard. 


To  Compate  Radias  of  Carve  and.  Wheel  Base. 

q B G = R.  = B.  B representing  maximum  rigid  wheel  base  of  cars , and  G 
9 G 

gauge  of  way , both  in  feet. 

To  Determine  Elevation  of  Oater  Rail. 

For  any  Radius  or  Construction  of  Curve  with  Straight. — Fig.  7. 

Fig.  7.  Y .5  t/G  = c.  V representing  speed  of  train  in  feet  per  sec- 

ond. G gauge  of  rails  in  feet,  and  c length  of  chord,  versed  sine 
v of  which  will  give  at  its  centre  the  elevation  required. 

Thus,  determine  chord  c,  align  it  on  inner 

c . side  of  rail,  and  distance  of  rail  from  it  at 

--cY~~  , centre  of  its  length  will  give  elevation  re- 

^ quired,  whatever  the  radius  of  rail. 


For  Cun 


[.782  V2  (N  D W)]— 4 P R 


= E ; 


V2 

Or,  W = E.  D representing 

1.25  R 


N D R 

diameter  of  wheels,  W width  of  gauge , P lateral  play  between  flange  and  rail , and 
R radius  of  curve , all  in  feet , i-^N  ratio  of  inclination  of  tire , V velocity  of  train  in 
miles  per  hour , and  E elevation  of  outer  rail  in  ins.  (Molesworth. ) 

WC(d+ 1) 


2 R 


- = resistance  due  to  curve,  and  W representing  weight  of  body , both  in 


lbs.,  C coefficient  of  friction  of  wheels  upon  rails  = . 1 to  .27,  according  to  condition  of 
weather , d distance  of  rails  apart , l length  of  rigid  wheel  base,  and  R radius  of  curve, 
all  in  feet.  (Morrison.) 

Illustration. — Assume  weight  of  locomotive  30  tons,  radius  of  curve  1000  feet, 
distance  of  rails  apart  4 feet  8.75  ins.,  length  of  base  10  feet,  and  rails,  dry,  C ==  1. 

30  X 2240  X .1  X (4.73  + IO)_ 

— — — 494-93  wS. 

2 X 1000 

To  Compute  Resistance  doe  to  Gravity'  upon  an  In- 
clination. 


r — = lbs.  per  ton  of  train. 

gradient 


Rise  per  Male,  and  Resistance  to  Gravity-,  in  EDs.  per 
Ton. 


Gradient  of  1 inch. . 

20  I 25 

30 

35 

40 

45 

50 

60 

70 

80 

90 

100 

Rise  in  feet 

264  21 1 

176 

151 

132 

117 

106 

88 

75 

66 

59 

53 

Resistance 

112  1 89.6 

74-7 

64 

56 

50 

00 

't- 

37-3 

32 

28 

24.8 

22.4 

68o 


RAILWAYS. 


To  Coxnpuite  Load  which,  a Locomotive  will  Draw  up 
an  Inclination. 

T-r-  r-f- r'  — W = L.  T representing  tractive  power  of  locomotive  in  lbs.,  r re- 
sistance due  to  gravity , and  r'  resistance  due  to  assumed  velocity  of  train  in  lbs.  per 
ton , W weight  of  locomotive  and  tender , and  L load  locomotive  can  draw,  in  tons , ex- 
clusive of  its  own  weight  and  tender. 

Coefficients  of  Traction  of  Locomotives.—  Railroads  in  good  order,  etc.,  4 to  6 lbs. ; 
in  ordinary  condition,  8 lbs. 

In  coupled  engines  adhesion  is  due  to  load  upon  wheels  coupled  to  drivers. 

To  Compute  Traction,  Retraction,  and  Adhesive  Power 
of  a Locomotive  or  Train. 

When  upon  a Level.  asP-rD  aT:  a representing  area  of  one  cylinder  in 

sq.  ins.,  s stroke  of  piston  and  D diameter  of  driving-wheels,  both  in  feet,  P mean 
pressure  of  steam  in  lbs.  per  sq.  inch , and  T traction , in  lbs. 

When  upon  an  Inclination,  a s P -4-  D — r w h = T.  r representing  resistance 
per  ton , w weight  of  locomotive  upon  driving-wheels , in  tons , h height  of  rise  in  feet 
per  100  of  road , and  R z=.r  w h — retraction,'  in  lbs. 

C w b -f-  iqo  — A.  b representing  base  of  inclination  in  feet  per  100  of  road. 

C io  = A.  C = coefficient  in  lbs.  per  ton,  and  A adhesion , in  lbs. 

When  Velocity  of  a Train  is  considered. 

When  upon  a Level,  W (C  + W)  = R.  When  upon  an  Inclination, 
W(rh  + C + VY)  = R.  V representing  velocity  of  train  in  miles  per  hour. 

Illustration.— A train  weighing  200  tons  is  to  be  driven  up  a grade  of  52.8  feet 
per  mile,  with  a velocity  of  16  miles  per  hour;  required  the  retractive  power  ? 

52.8  per  mile  = 1 in  100  feet  r'=  22.4  lbs.  0 = 5. 

200  (22.4  X 1 + 5 V1^)  = 200  X 22.4 -f-  9 = 6280  lbs. 


Velocity  of  Trains. 


Miles  per  hour 

IC> 

15 

20 

30 

40 

50 

60 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Resistance  upon  straight ) 
line  per  ton ) 

8-5 

9-25 

10.25 

i3-25 

17-25 

22.5 

29 

Do.,  with  sharp  curves) 
and  strong  wind* j 

13 

14 

i5-5 

20 

26 

34 

43-5 

7° 
Lbs. 

36-5 

. 55 

* Equal  to  50  per  cent,  added  to  resistance  upon  a straight  line. 

Friction  of  locomotive  engines  is  about  9 per  cent.,  or  2 lbs.  per  ton  of  weight. 
Case-hardening  of  wheel-tires  reduces  their  friction  from  .14  to  .08  part  of  load. 

To  Compute  Maximum  Load  that  can  be  drawn  by  an 
Engine,  up  the  Maximum  Grade  that  it  can  .Attain, 
Weight  and  Grade  "being  given.  (Maj.  McClellan , U.S.A.) 

.2  A — 8 L 


.2  A 


= L,  and 


- = G.  A representing  adhesive  weight  of  engine, 


.4242  G + 8 ’ .4242  L 

in  lbs.,  G grade  in  feet  per  mile,  and  L load,  in  tons. 

Note  1.— When  rails  are  out  of  order,  and  slippery,  etc.,  for  .2  A,  put  .143  A. 

2. —With  an  engine  of  4 drivers,  put  .6  as  weight  resting  upon  drivers;  with  6 
drivers  the  entire  weight  rests  upon  them. 

Illustration. — An  engine  weighing  30  tons  has  6 drivers;  what  are  the  maximum 
loads  it  can  draw  upon  a level,  and  upon  a grade  of  250  feet,  and  what  is  its  maxi- 
mum grade  for  that  load  ? 

. 2 X 2240  X 10  i q 440  , ,7  • 2 X 2240  X 3°  *3  44° 

^ 4 J — ...  P.  4t_  — 1505.4  tons  upon  a level.  — -7-7;  = — 

8.4242  ^ 4 • .4252X250  + 8 114.05 


* X 2240  X 30  — 8 X 117  8 _ 12497  _ 


.4242  + 8 

117.  8 tons  Up  a grade  of  250  feet.  “ 49.97 

Adhesion  of  a 4-wheeled  locomotive,  compared  with  one  of  6 wheels,  is  as  5 to  8. 


250.1  feet. 


railways. 


68  i 


OPERATION  OF  LOCOMOTIVES.  (O.  Chanute,  Am.  Soc.  C.  E.) 

zaires  ion. 

Adhesion  of  a locomotive  is  friction  of  its  driving-wheels  upon  the  rails, 
varying  with  condition  of  the  surface,  and  must  exceed  traction  of  the  engine 
upon  them,  otherwise  the  wheels  will  slip. 

T ’ Wpfnfnrp  made  in  the  construction  of  locomotives  and 

tracks^ave1  gradually  increased  the  proportion  which  the  adhesion  bears  to 
the  insistent  weight  upon  the  driving-wheels. 

The  first  accurate  experiments  were  those  of  Mr  Wood  upon  the  early  English 
coal  radwaya  He  deduced  the  adhesion  to  be  as  follows : 

y . . .,  ta  of  weight  on  drivers. 

Upon  perfectly  dry  rails. . £ u u a “ 

“ damp  or  muddy  rails u u u u 

“ very  greasy  rails 04 

t o a n h tatrobe  indicated  .13  as  a safe  working  adhesion,  while  modern 
In  1838,  B.  H.  Latrooe  inQlc»>~u  f iwht  as  maximum,  and  .11  as  a minimum, 
European  practice  assumes q jo°ns  subject  to  mists.  Thus,  on  the  Seem- 

SS-SS 

Materially  better  results  are  obtained  m United Snh^American 

=^£23H:ssf“ 

From  these  data  the  following  tables  have  been  computed. 


Coefficients  of  Adhesion  upon  Driving  Wheels  per  Ton. 

European  American  European  American 


Condition  of  Rails. 

European 

Practice. 

American 

Practice. 

Cdndition  of  Rails. 

European 

Practice. 

American 

Practice. 

Rails  very  dry 

Rails  very  wet 

Ordinary  working. . 

C. 

•3 

.27 

.2 

Lbs. 

670 

600 

450 

C. 

•33 

•25 

.222 

Lbs. 

667 

500 

444 

In  misty  weather  . 
In  frost  and  snow. 

c. 

.015 

.09 

Lbs. 

350 

200 

C. 
.2 
. 16 

Lbs. 

400 

333 

Adhesion  of  Locomotives , in  Lbs.  (.222  in  Summer  and  .2  in  Winter). 


Type  of  Locomotive. 

No.  of  Drivers. 

W ei{ 
Locomotive. 

jbt. 

On  Drivers. 

Adhe 

Summer. 

sion. 

Winter. 

A frmrinnn 

4 wheels  coupled 

6 “ connected.. 

6 “ 

8 “ “ 

6 “ “ 

4 “ “ •• 

Lbs. 
64000 
78000 
88  boo 
100  000 
68  000 
48  000 

Lbs. 
42000 
58  000 
72  000 
88  boo 
68  000 
48  000 

Lbs. 

9 350 
13  000 
16000 
X9  550 
15  10  O 
10650 

Lbs. 

8 400 
11  600 
14  000 
17  600 
13  600 

9 600 

Ten- wheeled 

\Togul 

Consolidation 

Tank  switching 

Tractive  Power. 

Traction  of  a locomotive  is  the  horizontal  resultant  on  the  track  of  the 
pressure  of  the  steam,  as  applied  in  the  cylinders. 

r)2  p 1,  — — T D representing  diameter  of  cylinder,  L length  of  stroke,  and 
diameter  of  driving  wheels,  all  in  ins.,  P mean  pressure  in  cylinder , m lbs.  per  sq. 
inch , and  T tractive  force  on  rails , in  lbs. 


Iilustratiov  —Assume  a locomotive,  cylinders  18  ins.  in  diarn.,  22  ins.  stroke, 
wheels  68  ins.  in  diam.,  and  average  steam  pressure  in  cylinders  50  lbs.  per  sq.  men. 


Then  18  X 18  X 50  X 22  -4-  68  = 5241  lbs. 


682 


RAILWAYS, 


Train  Resistances. 

Usual  formula  for  train  resistances,  on  a level  and  straight  line , is 

V2  V2 

1-  8 = R per  ton  of  train,  and p 6 = R per  ton  of  train  alone.  V repre- 

171  . 240 

senting  velocity  in  miles  per  hour , and  8 constant  axle  friction.  (D.  K.  Clark.) 

Note. — To  meet  the  unfavorable  conditions  of  quick  curves,  strong  winds,  and 
imperfection  of  road,  Mr.  Clark  estimates  results  as  obtained  by  above  formula 
should  be  increased  50  per  cent. 

Illustration.— At  20  miles  per  hour,  the  resistance  would  be: 

202  -T- 171  8 = 10.3  lbs.  per  ion  of  train. 

This  formula,  however,  is  empirical.  It  gives  results  which  are  too  large  for 
freight  trains  at  moderate  speeds,  and  too  small  for  passenger  trains  at  high  speeds. 

Engineers  are  not  agreed  as  to  exact  measure  and  value  of  each  of  the  elements 
of  train  resistances,  but  following  approximations  are  sufficient  for  practical  use: 

Analysis  of  Train  Resistances. 

Resistance  of  trains  to  traction  may  be  divided  into  four  principal  ele- 
ments: 1st.  Grades;  2d.  Curves;  3d.  Wheel  friction;  4th.  Atmosphere. 

1st.  Grades.  — Gradients  generally  oppose  largest  element  of  resistance 
to  trains.  Their  influence  is  entirely  independent  of  speed.  The  meas- 
ure of  this  resistance  is  equal  to  weight  of  train  multiplied  by  rate  of  in- 
clination or  per  cent,  of  grade.  Thus,  a gradient  of  .5  per  100  feet  (26.4 

feet  per  mile)  offers  a resistance  of  5 x 2000  — J12  ibs.  per  ton,  or  10  lbs. 
1 y 10  X 100 

per  2000  lbs.,  which  is  to  be  multiplied  by  weight  in  tons  of  entire  train. 

Following  table  shows  resistance,  due  to  gravity  alone,  for  the  most  usual  grades, 
in  lbs.  per  ton  of  train : 


1st.  Resistance  due  to  Grades. 


Rate  per  100  feet. 

. 1 

.2  , 

•3 

•4 

•5 

.6 

•7 

.8 

Lbs.  per  ton  of  2240  lbs. . . 

2.24 

4.48 

6.72 

8.96 

1 1. 2 

13-44 

15.68 

17.92 

Rate  per  mile 

5 

11 

16 

21 

26 

32 

37 

42 

Lbs.  per  ton  of  2000  lbs. . . 

2 

4 

6 

8 

10 

12 

16 

Rate  per  100  feet 

•9 

1 

1. 1 

1.2 

i-3 

1.4 

i-5 

1.6 

Lbs.  per  ton  of  2240  lbs. . . 

20.16 

22.4 

24.64 

26.88 

29.12 

3i-36 

33-6 

35-84 

Rate  per  mile 

47 

53 

58 

63 

68 

74 

79 

85 

Lbs.  per  ton  of  2000  lbs. . . 

18 

20 

22 

24 

26 

28 

30 

32 

2d.  Curves. — Recent  European  formula  is  that  given  by  Baron  von  Weber. 


. 6504  -4-  R — 55  = W.  R representing  radius  of  curve  in  metres. 

This  formula  assumes  that  resistance  due  to  curve  increases  faster  than  radius 
diminishes.  It  gives  results  varying  from  a resistance  of  .8  lb.  per  2000  lbs.  per 
degree  for  a curve  of  1000  metres  radius  (3310  feet,  or  i°  44')  to  a resistance  of  1.67 
lbs.  per  20C0  lbs.  per  degree  for  curves  of  100  metres  radius  (331  feet,  or  170  20')- 

Messrs.  Vuillemin,  Guebhard,  and  Dieudonne  found  curve-resistance  to  European 
rolling-stock  to  be  from  .8  to  1 lb.  per  2000  lbs.  per  degree,  on  a gauge  of  4 feet  8.5 
ins.,  while  Mr.  B.  H.  Latrobe,  in  1844,  found  that  with  American  cars  resistance  on 
a curve  of  400  feet  radius  did  not  exceed  .56  lb.  per  2000  lbs.  per  degree. 

Resistance  of  same  curve  varies  with  coning  given  tires  of  wheels,  elevation  of 
outer  rail,  and  speed  of  train  running  over  it,  but  both  reasoning  and  experiment 
indicate  that  the  general  resistance  of  curves  increases  very  nearly  in  direct  pro- 
portion to  degree  of  curvature,  or  inversely  to  the  radius. 

Recent  American  experiments  show  that  a safe  allowance  for  curve  resistance 
may  be  estimated  at  .125  of  a lb.  per  2000  lbs.  for  each  foot  in  width  of  gauge. 
Thus,  for  3 feet  gauge  resistance  would  be  .375  lb.  per  degree  of  curve;  for  standard 
gauge  of  4 feet  8.5  ins.  .589,  say  .60,  and  for  6 feet  gauge  .75  lb.  per  degree. 

For  standard  gauge,  when  radius  is  given  in  feet,  resistance  due  to  this  element  is: 
.60  X 5730  -r-  R = C in  lbs.  per  ton  of  train. 


RAILWAYS. 


683 

This  is  somewhat  reduced  when  curve  coincides  with  that  for  which  wheels  are 
coned  (generally  about  30),  and  when  train  runs  over  it,  at  precise  speed  for  which 
oX  rSns  elevated,  an  allowance  of  .5  lb.  per  ton  per  degree  is  found  to  give  good 
results  in  practice. 

2d.  Resistance  on  Curves . 

It  follows  from  above  estimate  of  curve  resistance  that,  in  order  to  have  the  same 
resistance  on  a curve  as  on  a straight  line,  the  gradient  should  be  diminished  by 
cn  per  100  feet  of  each  degree  of  curve.  Thus  a 30  curve  requires  an  easing  of  the 
grade  by  .09  per  100  feet,  a io°  curve  an  easing  of  .3  per  100,  etc. 

This  however  need  only  be  done  upon  the  limiting  gradients,  and  when  sum  of 
grade  and  curve’resistances  exceeds  resistance  which  has  been  assumed  as  limiting 

the  trains.  . , TTr,  7 ^ 

3d.  Resistance  due  to  Wheel  Friction. 

Experimenters  are  not  agreed  whether  friction  of  wheels  increases  simply  with 
weight  which  thev  carrv,  but  also  in  some  ratio  with  the  speed.  Originally  as- 
sumed as  a constant  at‘8  lbs.  per  ton,  improvements  in  condition  of  track  (steel 
rails  etc  ) and  in  construction  and  lubrication  of  rolling-stock  have  reduced  it  to 
- * and  4 lbs.  per  ton  for  well-oiled  trains.  Under  ordinary  circumstances,  m sum- 
mer it  will  be  safe  to  estimate  it  at  5 lbs.  per  ton  on  first-class  tracks,  and  6dbs. 
per  ton  on  fair  tracks.  It  may  run  up  to  7 or  8 lbs.  per  ton  on  bad  tracks  (iron 
rails)  in  summer,  and  all  these  amounts  should  be  increased  from  25  to  50  per  cent, 
in  cold  climates  in  winter,  to  allow  for  inferior  lubrication. 

4th.  Resistance  due  to  Atmosphere. 

Atmospheric  resistance  to  trains,  complicated  as  it  is  by  the  wind  which  may  be 
prevailing,  has  not  been  accurately  ascertained  by  experiment.  It  consists  of 
1st.  Head  resistance  of  first  car  of  train,  which  is  presumably  equal  to  its  exposed 
area,  in  sq.  feet,  multiplied  by  air  pressure  due  to  speed. 

2d  Head  resistance  of  each  subsequent  car.  This  varies  with  distance  they  are 
coupled  apart,  and  so  shield  each  other  from  end  air  pressure  due  to  speed. 

3d.  Friction  of  air  against  sides  of  each  car  depending  upon  the  speed.  This  is 
generally  so  small  that  it  may  be  neglected  altogether. 

4th.  Effect  due  to  prevailing  wind,  which  modifies  above  three  items  of  resistance. 
A head  wind  retards  the  train,  a rear  wind  aids  it,  while  a side  wind  increases  re- 
sistance by  pressing  flanges  of  wheels  against  one  rail,  and,  in  consequence  of  curves, 
a train  may  assume  all  of  these, positions  to  same  wind. 

Recent  experiments  on  Erie  Railway  seem  to  indicate  that  in  a dead  cairn  re- 
sistance of  first  car  of  a freight  train  may  be  assumed  at  an  exposed  surface  of  63 
sq.  feet*  multiplied  by  air  pressure  due  to  speed,  and  that  each  subsequent  car  may 
be  assumed  to  offer  a resistance  of  20  per  cent,  of  that  of  first  car,  while  in  aj}as- 
senger  train  first  car  may  be  assumed  at  an  area  of  90  sq.  feet,t  multiplied  by 
pressure  due  to  speed,  and  that  each  subsequent  car  adds  an  increment  equal  to  40 
per  cent,  that  of  first  car,  in  consequence  of  greater  distance  they  are  coupled  apart. 

This  resistance  is,  of  course,  entirely  independent  of  cars  being  loaded  or ■ e nipt y. 
In  practice  it  has  been  found  that  an  allowance  of  1.5  to  2 lbs.  per  ton  of  weight  of 
a freight  train  covers  atmospheric  resistance,  except  in  very  high  winds. 

In  consequence  of  complexity  of  elements above enumerated,  exact . formulas  can- 
not probably  be  now  given  for  train  resistances,  but  following,  if  applied  with  judg- 
ment (and  modified  to  fit  circumstances),  will  be  found  to  give  fairly  accurate  results 
i n practice.  They  are  for  standard  gauge,  and  in  making  them,  resist^ 

been  assumed  at  .5  lb.  per  degree,  wheel  friction  at  5 lbs.,  exposed  end  area  of  first 
car  at  90  sq.  feet  for  passenger  cars  and  63  feet  for  freight  cars,  and  increment  for 
succeeding  cars  at  .4  for  passenger  trains  and  .2  for  freight  trains. 

Passenger  Train.  W (G  + ^T+  s)  + 9°  P = R> 

Freight  Train.  W 5^  + ~ ^63P  = R. 


* This  is  less  than  area  of  car,  which  generally  measures  about  71  sq.feet ; but  part  Is  shielded  by 
tender  and  narts  beine  convex,  as  wheels,  bolts,  etc.,  offer  less  resistance  than  a flat  plane. 

t Not  only  is  end  area  of  passenger  cars  greater  than  that  of  freight  cars,  but  in  consequence  of  the 
projecting  roof  the  end  forms  a hood  in  nature  of  a concave  surface,  and  so  opposes  greater  resistance 
than  a flat  plane. 


684 


RAILWAYS. 


W representing  weight  of  train,  without  engine , in  tons  (2000  lbs.),  G resistance  of 
gradient  per  ton  (2000  lbs.;  see  table,  page  683),  C°  curve  in  degrees , n number  of  cars 
in  train , P pressure  per  sq.foot  due  to  speed,  to  which  an  allowance  must  be  made  for 
wind,  if  existing,  R resistance  of  train,  and  5,  wheel  friction,  both  in  lbs. 

Illustration  i. — Assume  a passenger  train  of  5 cars,  weighing  136  tons  (2000  lbs.), 
ascending  a grade  .5  per  100  (26.4  feet  per  mile),  with  curves  of  40,  at  a speed  of  60 
miles  per  hour  (for  which  the  pressure  is  18  lbs.  per  sq.  foot),  resistance  will  be: 

136  (10-f  2 -j-  5)  + (90  X iB)  ==  6524  lbs.,  of  which  2312  lbs.  are  due  to 

grade , curve,  and  wheels,  and  4212  lbs.  to  atmospheric  resistance. 

2.— Assume  a freight  train  of  31  cars,  weighing  620  tons  (2000  lbs.),  turning  a curve 
of  30,  up  a grade  of  52.8  feet  per  mile  (1  foot  per  100),  at  a speed  of  21  miles  per  hour 
(pressure  2 lbs.  per  sq.  foot),  resistance  Will  be: 

620  (20  -j-  1.5  + 5)  + + y)  (63  X 2)  = 17 312  lbs.,  requiring  a “Consolidation ” 

engine  to  haul  it,  allowance  being  made  for  possible  winds,  etc. 

Assume  conversely,  it  is  desired  to  know  how  many  tons  an  American  engine, 
with  an  adhesion  of  10650  lbs.,  will  draw  up  a grade  of  .9  per  100  (47  feet  per  mile), 
with  curves  of  40,  assuming  atmospheric  resistance  between  1.5  to  2 lbs.  per  ton  of 
train. 

Resistance  from  grade  .9  x 2000  -4-  100 =18  lbs. ) 

“ curve  4-4-2 = 2 “ [27  lbs. 

“ “ wheel  friction  5,  atmosphere  2 = 7 “ ) 

Hence,  10650-4-27  = 395  tons,  or  about  20  cars,  and  in  winter  same  engine  will 
haul  9600-4-27  ==  355  tons  (2000  lbs.),  or  about  18  cars. 

Following  table  approximates  to  best  modern  practice.  For  freight  trains  it  gives 
aggregate  resistance,  in  lbs.  per  ton  (2000  lbs  ),  for  various  grades  and  curves.  In 
using  it,  it  is  sufficient  to  divide  the  adhesion  in  lbs.  of  locomotive  used  by  number 
found  in  table,  in  order  to  obtain  number  of  tons  of  train  that  it  will  haul  at  or- 
dinary speeds  on  gradient  and  curve  selected.  Of  course,  if  grade  has  been  equated 
for  curves,  only  number  found  in  first  column  (for  straight  lines)  is  to  be  used  in 
computing  tons  of  train  on  limiting  gradient. 

Approximate  Rreiglit-train.  Resistances. 

Gauge  4 feet  8. 5 ins. 

In  Lbs.  per  2000  lbs.  at  Ordinary  Speeds. 

Curve  Resistance  assumed  at  .5  lbs.  per  °,  Wheel  Friction  at  5 lbs.,  Atmospheric  Re- 
sistance at  2 lbs.  per  Ton. 


Gra 

Per 

Cent. 

DE. 

Per 

Mile. 

j Straight. 

i° 

2° 

3° 

4° 

5° 

6° 

C 

7° 

URV] 

8° 

E. 

9° 

10° 

ii° 

12° 

*3° 

140 

*5° 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

Level. 

Feet. 

7 

7-5 

8 

8.5 

9 

9-5 

10 

10.5 

11 

IXi5 

12 

12.5 

13 

13-5 

14 

14-5 

.1 

5 

9 

9-5 

10 

10.5 

11 

II>5 

12 

12.5 

13 

13*5 

14 

14-5 

15 

*5-5 

16 

16.5 

.2 

11 

11 

ii-5 

12 

12.5 

13 

I3-5 

14 

i4- 5 

15 

i5- 5 

16 

16.5 

17 

17-5 

18 

18.5 

•3 

16 

13 

I3*5 

14 

14-5 

15 

I5-5 

16 

16.5 

17 

i7-5 

18 

18.5 

1 9 

i9-5 

20 

20.5 

•4 

21 

15 

i5-5 

16 

16.5 

17 

17-5 

18 

18.5 

l9 

i9-5 

20 

20.5 

21 

21.5 

22 

22.5 

•5 

26 

17 

i7-5 

18 

18.5 

*9 

*9-5 

20 

20.5 

21 

21.5 

22 

22.5 

23 

23-5 

24 

24- 5 

.6 

32 

J9 

i9-5 

20 

20.5 

21 

21.5 

22 

22.5 

23 

23-5 

24 

24-5 

25 

25-5 

26 

26.5 

•7 

37 

21 

21.5 

22 

22.5 

23 

23-5 

24 

24-5 

25 

25- 5 

26 

26.5 

27 

27-5 

28 

28.5 

.8 

42 

23 

23-5 

24 

24-5 

25 

25-5 

26 

26.5 

27 

27-5 

28 

28.5 

29 

29-5 

30 

30-  5 

•9 

47 

25 

25-5 

26 

26.5 

27 

275 

28 

28.5 

29 

29-5 

30 

3°-5 

31 

3i-5 

32 

32.5 

1 

53 

27 

27-5 

28 

28.5 

29 

29-5 

30 

30-5 

3i 

31-5 

32 

32-5 

33 

33-5 

34 

34-5 

1. 1 

58 

29 

29-5 

30 

30-5 

3i 

3i-5 

32 

32.5 

33 

33-5 

34 

34-5 

35 

35-5 

36 

3&-5 

1.2 

63 

31 

3i-5 

32 

32.5 

33 

33-5 

34 

34-5 

35 

35-5 

36 

36-5 

37 

37-5 

38 

38-5 

i-3 

68 

33 

33-5 

34 

34-3 

35 

35-5 

36 

36.5 

37 

37-5 

38 

38-5 

39 

39-5 

40 

40- 5 

i-4 

74 

35 

35-5 

36 

36-5 

37 

37-5 

38 

38.5 

39 

39-5 

40 

40-5 

41 

4^-5 

42 

42.5 

i-5 

79 

37 

37-5 

38 

38-5 

39 

39-5 

40 

4°  5 

4i 

4i-5 

42 

42-5 

43 

43*5 

44 

44-5 

1.6 

85 

39 

39-5 

40 

40-5 

41 

4i-5 

42 

42-5 

43 

43-5 

44 

44-5 

45 

45-5 

46 

46.5 

Illustration. — Assume  a “Mogul”  engine  to  have  an  adhesion  of  16000  lbs. ; 
what  weight  will  it  haul  up  a grade  of  74  feet  per  mile,  combiued  with  a curve  of  90  ? 
16  000  —4—  39. 5 = 405  tons  (2000  lbs. ). 


RAILWAYS. 


685 


Hence,  To  Compute  Adhesion  on  a Given  Grade  and  Curve , having  Weight 
of  Train. 

Rule— Multiply  tabular  number  by  weight  of  train  in  tons  (2000  lbs.), 
and  product  will  give  adhesion,  in  lbs. 

Example. — Assume  preceding  elements.  Then  39.5  X 4°5  = l(>  000  lbs. 

Note.— A “Consolidation”  engine,  by  its  superior  adhesion  (19550  lbs.)  would 
haul  up  a like  grade  and  curve  495  tons. 


Memoranda  011  English.  Railways. 

Regulations  [Board  of  Trade). 

Cast-iron  girders  to  have  a breaking  weight  = 3 times  permanent  load,  added  to 

6 times  moving  load.  , , 

WrouMit-iron  bridges  not  to  be  strained  to  more  than  5 tons  per  sq.  men. 
Minimum  distance  of  standing  work  from  outer  edge  of  rail  at  level  of  carnage 
steps,  3.5  feet  in  England  and  4 feet  in  Ireland. 

Minimum  distance  between  lines  of  railway,  6 feet. 

Stations  —Minimum  width  of  platform,  6 feet,  and  12  at  important  stations. 
Minimum  distance  of  columns  from  edge  of  platform,  6 feet,  feteepest  gradient  tor 
stations,  1 in  260.  Ends  of  platforms  to  be  ramped  (not  stepped).  Signals  and  dis- 
tant signals  in  both  directions. 

Carriages.  — Minimum  space  per  passenger  20  cube  feet.  Minimum  area  of  glass 
per  passenger,  60  sq.  ins.  Minimum  width  of  seats,  15  ins.  Minimum  breadth  of 
seat  per  passenger,  18  ins.  Minimum  number  of  lamps  per  cairiage,  2. 

Requirements. -Joints  of  rails  to  be  fished.  Chairs  to  be  secured  by  iron  spikes. 
Fang  bolts  to  be  used  at  the  joints  of  flat-bottomed  rails. 

Construction. 

Width,  single  line 18  24  6 

“ double  line 3°  38 

“ top  of  ballast,  single  line 13  6 15  ° 

“ “ “ double  line 24  6 29 

Slope  of  cuttings  from  centre,  1 in  30.  Width  of  land  beyond  bottom  of  slope, 
0 to  12  feet.  Ditch  with  slopes,  1 foot  at  bottom,  1 to  1.  Quick  mound,  18  ins.  in 
height.  Post  and  rail-fence  posts,  7 feet  6 ins.  X 6 ins.  X3.5  ins->  9 feet  apart,  3 : feet 
in  ground.  Intermediate  posts,  5 feet  6 ins.  X 4 ins.  X i-5  ins-?  3 feet  apait.  Rails 
4 of  4 X 1.5  i»s. 


Parliamentary  Regulations  for  Crossing  Pioads. 


Turnpike 

Road. 


Public  Occupation 

Road.  Road. 


Feet.  Ins. 


Feet.  Ins. 


Feet.  Ins. 


Clear  width  of  under  bridge,  or  approach 

Clear  height  of  under  bridge  for  a width  of  12  ft. 

a u a “ “ o “ 


“ “ “ at  springing 

Over  bridge,  height  of  parapets 

Approaches,  inclination  . . . 

14  height  of  fencing 


35 

16 


25 


12  — 


12  — 
4 — 
1 in  30 

3 — 


15  — 

12  — 

4 “ 
1 in  20 

3 ~ 


14  — 

4 — 
1 in  16 

3 — 


Limits  of  Deviation—  In  towns,  10  yards  each  side  of  centre  line.  In 
country,  100  yards,  or  5 chains  nearly. 

Level—  In  towns,  2 feet.  In  country,  5 feet. 

Gradient.  — Gradients  flatter  than  1 in  100,  deviation  10  feet  per  mile 
steeper.  Do.,  steeper,  3 feet  per  mile. 

Curve. — Curves  upwards  of  .5  a mile  radius,  may  be  sharpened  to  .5  mile 
radius.  Curves  of  less  than  .5  mile  radius  mav  not  be  sharpened. 

3 M 


686 


ROADS,  STREETS,  AND  PAVEMENTS. 


ROADS,  STREETS,  AND  PAVEMENTS. 

Classification  of  Roads. 

i.  Earth.  2.  Corduroy.  3.  Plank.  4.  Gravel.  5.  Broken  stone  (Mac- 
adam). 6.  Stone  sub-pavement  with  surface  of  broken  stone  (Telford). 
7.  Stone  sub-pavement  with  surface  of  broken  stone  and  gravel,  or  gravel 
alone.  8.  Rubble  stone  bottom  with  surface  of  broken  stone  or  gravel,  or 
both.  9.  Concrete  bottom  with  surface  of  broken  stone  or  gravel,  or  both. 

Oracle  of  Hoads. 

Limit  of  practicable  grade  varies  with  character  of  road  and  friction  of  ve- 
hicle. For  best  carriages  on  best  roads,  limit  is  1 in  35,  or  15  feet  in  a mile. 

Maximum  grade  of  a turnpike  road  is  1 in  30  feet.  An  ascent  is  easier 
for  draught  if  taken  in  alternate  ascents  and  levels,  than  in  one  continuous 
rise,  although  the  ascents  may  be  steeper  than  in  a uniform  grade. 

Ordinary  angle  of  repose  is  1 in  40  if  roads  are  bad,  and  1 in  30,  to  1 in  20. 

When  roads  have  a greater  grade  than  1 in  35,  time  is  lost  in  descending, 
in  order  to  avoid  unsafe  speed.  Grade  of  a road  should  be  less  than  its  angle 
of  repose.  Minimum  grade  of  a road  to  secure  effective  drainage  should  be 
1 in  80.  In  France  it  is  1 in  125. 

In  construction  of  roads  the  advantage  of  a level  road  over  that  of  an  in- 
clined one,  in  reduction  of  labor,  is  superior  to  cost  of  an  increased  length 
of  road  in  the  avoiding  of  a hill. 

Alpine  roads  over  the  Simplon  Pass  average  1 in  1 7 on  Swiss  side,  1 in  22 
on  Italian  side,  and  in  one  instance  1 in  13. 

In  deciding  upon  a grade,  the  motive  power  available  of  ascent  and  avoid- 
able of  waste  of  power  in  descending  are  to  be  first  considered. 

When  traffic  is  heavier  in  one  direction  than  the  other,  the  grade  in  as- 
cent of  lighter  traffic  may  be  greatest. 

When  axis  of  a road  is  upon  side  of  a hill,  and  road  is  made  in  parts  by 
excavation  and  by  embankment,  the  side  surface  should  be  cut  into  steps, 
in  order  to  afford  a secure  footing  to  embankment,  and  in  extreme  cases, 
sustaining  walls  should  be  erected. 

Constrvictioii. 

Estimate  of  Labor  in  Construction  of  Roads.  (M.  Ancelin.) 

A day’s  work  of  10  hours. of  an  average  laborer  is  estimated  as  follows: 

In  Cube  Yards. 


WOKK. 

Ordinary 

Earth. 

Loose 

Earth. 

Mud. 

Clay  and 
Earth. 

Gravel. 

Blasting 

Rock. 

Picking  and  digging 

18  to  23 

16 

_ 

9 

7 to  11 

2.4 

Excavation  and  pitching  \ 
6 to  12  feet ) 

8 to  12 

8 

7 to  16 

4 

2.2 

Loading  in  barrows 

22 

— 

8 

— 

19 

— 

Wheeling  in  barrows  per ) 
100  feet j 

20  tO  33 

- 

- 

- 

24  to  28 

Loading  in  carts 

16  to  48 

— 

— 

17  to  27 

— 

Spreading  and  levelling. . . 

44  to  88 

— 

25 

— 

30  to  80 

— 

Time  of  pitching  from  a shovel  is  one  third  of  that  of  digging. 


Ditches. — All  ditches  should  lead  to  a natural  water-course,  and  their  min- 
imum inclination  should  be  1 in  125. 

Depressions  and  elevations  in  surface  of  a roadway  involve  a material  loss  of 
power.  Thus,  if  elevation  is  1 inch,  under  a wheel  4 feet  in  diameter,  an  inclined 
plane  of  1 in  7 has  to  be  surmounted,  and,  as  a consequence,  one  seventh  of  weight 
has  to  be  raised  1 inch. 


ROADS,  STREETS,  AND  PAVEMENTS. 


687 

An  unyielding  foundation  and  surface  are  indispensable  for  a perfect  roadway. 

Earth  in  embankment  occupies  an  average  of  one  tenth  less  space  than  in  natural 
bank,  and  rock  about  one  third  more. 

Ruts.  — Surface  of  a roadway  should  be  maintained  as  intact  as  prac- 
ticable, as  the  rutting  of  it  not  only  tends  to  a rapid  destruction  of  it,  but 
involves  increased  traction. 

The  general  practice  of  rutting  a road  displays  a degree  of  ignorance  of 
physical  laws  and  mechanical  effects  that  is  as  inexplicable  as  it  is  injurious 
and  expensive. 

On  compressible  roadways,  as  earth,  sand,  etc.,  resistance  of  a wheel  decreases  as 
breadth  of  tire  increases. 

Depressing  of  axles  at  their  ends  increases  friction.  • Long  and  pliant  springs  de- 
crease effect  of  shock  in  passing  over  obstacles  in  a very  great  degree. 

Transverse  Section—  Best  profile  of  section  of  roadway  is  held  to  be  one 
formed  by  two  inclined  planes  meeting  in  centre  of  road  and  slightly 
rounded  otf  at  point  of  junction. 

Roads  having  a rough  surface  or  of  broken  stone  should  have  a rise  of 
1 in  24,  equal  to  a rise  on  crown  of  6 ins.,  and  on  a smooth  surface,  as  a 
block-stone  or  wood  pavement,  the  rise  may  be  reduced  to  1 in  48. 

On  roads,  when  longitudinal  inclination  is  great,  the  rise  of  transverse 
section  should  be  increased,  in  order  that  surface  water  may  more  readily 
run  off  to  sides  of  roadway,  instead  of  down  its  length,  and  consequently 
gullying  it. 

Stone  Breaking.  A steam  stone-breaking  machine  .will  break  a cube  yard 
of  stone  into  cubes  of  1.5  ins.  side,  at  rate  of  1 to  1.5  IP  per  hour. 

NT acadamiz e d.  Roads. 

In  construction  of  a Macadamized  road,  the  stones  (road  metal)  used 
should  be  hard  and  rough,  and  cubical  in  form,  the  longest  diameter  of  which 
exceed  2.5  ins.,  but  when  they  are  very  hard  this  may  be  reduced  to  1.25 
and  1.5  ins. 

The  best  stones  are  such  as  are  difficult  of  fracture,  as  basaltic  and  trap, 
and  especially  when  they  are  combined  with  hornblende.  Flint  and  sili- 
ceous stone  are  rendered  unfit  for  use  by  being  too  brittle.  Light  granites 
are  objectionable,  in  consequence  of  their  being  brittle  and  liable  to  disinte- 
gration ; dark  granites,  possessing  hornblende,  are  less  objectionable.  Lime- 
stones, sandstones,  and  slate  are  too  weak  and  friable. 

Dimensions  of  a hammer  for  breaking  the  stone  should  be,  head  6 ins.  in 
length,  weighing  1 lb.,  handle  18  ins.  in  length ; and  an  average  laborer  can 
break  from  1.5  to  2 cube  yards  per  day. 

Stones  broken  up  in  this  manner  have  a volume  twice  as  great  as  in  their 
original  form.  100  cube  feet  of  rock  will  make  190  of  1.5  ins.  dimension, 
182  of  2 ins.,  and  170  of  2.5  ins. 

A ton  of  hard  metal  has  a volume  of  1.185  cube  yards. 

Construction  of  a Roadway. — Excavate  and  level  to  a depth  of  1 foot, 
then  lay  a “bottom”  12  ins.  deep  of  brick  or  stone  spalls  or  chips,  clinker 
or  old  concrete,  etc.,  roll  down  to  9 ins,  then  add  a layer  of  coarse  gravel  or 
small  ballast  5 ins.  deep,  roll  down  to  3 ins.,  and  then  metal  in  2 equal  lay- 
ers of  3 ins.,  laid  at  an  interval,  enabling  first  layer  to  be  fully  consolidated 
before  second  is  laid  on  and  rolled  to  a depth  of  4 ins. ; a surface  or  “ blind’* 
of  .75  inch  of  sharp  sand  should  be  laid  over  last  layer  of  metal  and  rolled 
in  with  a free  supply  of  water. 


688  ROADS,  STREETS,  AND  PAVEMENTS. 

Proportion  of  Getters , Fillers , and  Wheelers  in  different  Soils.  Wheelers  computed 
at  a Run  of  50  Yards.  (Molesivorth. ) 


Getters. 

Fillers. 

Wheelers. 

Getters.1 

Fillers. 

Wheelers. 

Loose  earth, ) 

Hard  clay 

1 

1.25 

1.25 

Sand,  etc.  ) 

I 

1 

1 

Compact  ) 

2 

j. 

Compact  earth  . . . 

I 

2 

2 

gravel  j 

Marl 

1 

2 

2 

Rock 

3 

X 

1 

T'elford.  Roads. 

In  construction  of  a Telford  road,  metalling  is  set  upon  a bottom  course  of 
stones,  set  by  hand,  in  the  manner  of  an  ordinary  block  stone  pavement, 
which  course  is  composed  of  stones  running  progressively  from  3 inches  in 
depth  at  sides  of  road  to  4,  5,  and  7 inches  to  centre,  and  set  upon  their 
broadest  edge,  free  from  irregularities  in  their  upper  surface,  and  their  in- 
terstices filled  with  stone  spalls  or  chips,  firmly  wedged  in. 

Centre  portion  of  road  to  be  metalled  first  to  a depth  of  4 ins.,  to  which, 
after  being  used  for  a brief  period,  2 ins.  more  are  to  be  added,  and  entire 
surface  to^be  covered,  “blinded,”  with  clean  gravel  1.5  ins.  in  depth. 

Telford  assigned  a load  not  to  exceed  1 ton  upon  each  wheel  of  a vehicle, 
with  a tire  4 ins.  in  breadth. 

Gf ravel  or  Ecirtli  Roads. 

In  construction  of  a gravel  or  earth  road,  selection  should  be  made  between 
clean  round  gravel  that  will  not  pack,  and  sharp  gravel  intermixed  with 
earth  or  clay,  that  will  bind  or  compact  when  submitted  to  the  pressure  of 
traffic  or  a roll. 

Surface  of  an  ordinary  gravel  roadway  should  be  excavated  to  a depth  01 
from  8 to  12  ins.  for  full  width  of  road,  the  surface  of  excavation  conforming 
to  that  of  road  to  be  constructed. 

The  gravel  should  then  be  spread  in  layers,  and  each  layer  compacted  by 
the  gradual  pressure  due  to  travel  over  it,  or  by  a roller,  the  weight  of  it  in- 
creasing with  each  layer.  One  of  6 tons  will  suffice  for  limit  of  weight.. 

If  gravel  is  dry  and  will  not  readily  pack,  it  should  be  wet,  and  mixed 
with  a binding  material,  or  covered  with  a thin  layer  of  it,  as  clay  or  loam. 

In  rolling,  the  sides  of  road  should  be  first  rolled,  in  order  , to  arrest  the 
gravel,  when  the  centre  is  being  rolled,  from  spreading  at  the  side. 

To  re-form  a mile  of  gravel  or  earth  road,  30  feet  in  width  between  gutters, 
material  cast  up  from  sides,  there  will  be  required  1640  hours’  labor  of  men, 
and  20  of  a double  team. 

Corduroy  Roads. 

A Corduroy  road  is  one  in  which  timber  logs  are  laid  transversely  to  its  plane. 

Rlank  Roads. 

A single  plank  road  should  not  exceed  8 feet  in  width,  as  any  greater  width 
involves  an  expenditure  of  material,  without  any  equivalent  advantage. 

If  a double  track  is  required  it  should  consist  of  two  single  and  independ- 
ent tracks,  as  with  one  wide  track  the  wear  would  be  mostly  in  the  centre, 
and  consequently,  wear  would  be  restricted  to  one  portion  of  its  surface. 

Materials.—  Sleepers  should  be  as  long  as  practicable  of  attainment,  in  depth  3 or 
4 ins.,  according  to  requirements  of  the  soil,  and  they  should  have  a width  of  3 ms. 
for  each  foot  of  width  of  road. 

Pine,  oak,  maple,  or  beech  are  best  adapted  for  economy  and  wear. 

Planks  should  be  from  3 to  3.5  ins.  thick,  and  not  less  than  9 ins.  in  uidth,  or 
more  than  12  if  of  hard  wood,  or  15  if  of  soft. 

A plank  road  will  wear  from  7 to  12  years,  according  to  service,. material, 
and  location,  and  its  traction,  compared  with  an  ordinary  Macadamized  road, 
is  2.5  to  3 times  less,  and  Avith  a common  country  road  in  bad  order  7 tunes. 

For  other  elements,  see  Earth-work,  page  466. 


ROADS,  STREETS,  AND  PAVEMENTS.  689 


.A^spkalt. 

Asphalt  is  a bituminous  limestone,  and  is  synonymous  with  bitumen;  it 
consists  of  from  90  to  94  per  cent,  of  carbonate  of  lime  and  6 to  10  per  cent, 
of  bitumen. 

In  forming  a pavement  the  powder  is  heated  to  from  2120  to  2500,  and  its  par- 
ticles caused  to  adhere  by  pressure,  or  it  is  applied  as  a liquid  asphalt  or  asphaltic 
mastic,  which  is  thus  manufactured.  The  powder  is  heated  with  from  5 to  8 per 
cent,  of  free  bitumen  for  a flux,  and  the  mixture  when  melted  is  run  into  molds. 
To  be  remelted,  additional  bitumen  must  be  mixed  with  it,  without  which  it  would 
only  become  soft. 

For  paving  60  per  cent,  of  sand  or  gravel  must  be  mixed  with  it.  No  chemical 
union  takes  place  between  the  mastic  and  the  sand  or  gravel,  but  cohesion  is  so 
complete  that  gravel  will  fracture  with  the  mastic,  and  the  admixture  increases  the 
resistance  of  the  mass  to  heat  of  the  sun.  The  roadway  should  have  a convexity 
of  .01  of  its  breadth. 

Artificial  A sphait  — Pleated  limestone  and  gas  tar,  when  mixed,  possess 
some  of  the  proportions  of  alphalt  mastic,  but  it  is  very  inferior  for  the 
purposes  of  a pavement. 

To  repair  surface  of  roadway,  dissolve  bitumen  1 part  in  3 of  pitch  oil  or 
resin  oil,  apply  10  oz.  of  mixture  over  each  sq.  yard  of  roadway,  sprinkle  on 
it  2 lbs.  of  asphalt  powder,  and  then  cover  surface  with  sand. 

Wood  Pavement. 

Close-grained  and  hard  woods  only  are  suitable,  such  as  oak,  elm,  ash, 
beech,  and  yellow  pine,  and  they  should  be  laid  on  a foundation  of  concrete. 

Block  Stone  Pavement. 

Paving-blocks,  as  the  Belgian,  etc.,  where  crest  of  street  or  area  of  pave- 
ment does  not  exceed  1 inch  in  7.5  feet,  should  taper  slightly  toward  the 
top,  and  the  joints  be  well  filled,  “blinded,”  with  gravel.  The  common 
practice  of  tapering  them  downward  is  erroneous. 

The  foundation  or  bottoming  of  a stone  pavement  for  street  travel  should 
consist  either  of  hydraulic  concrete  or  rubble  masonry  in  hydraulic  mortar. 
The  practice  in  this  country  of  setting  the  stones  in  sand  alone  is  at  variance 
with  endurance  and  ultimate  economy,  but  when  resorted  to,  there  should  be 
a bed  of  12  ins.  of  gravel,  rammed  in  three  layers,  covered  with  an  inch  of 
sand.  Granite  or  Trap  blocks  should  beg  X 9 X 12  ins. 

Rubble  Stone  Pavement. 

Bowlders  or  Beach  stone  of  irregular  volumes  and  forms,  set  in  a bed  of 
sand,  involves  great  resistance  to  vehicles  and  frequent  repairs ; it  is  wholly 
at  variance  with  requirements  of  heavy  traffic  or  city  use. 

Concrete  Roads. 

Concrete  roads  are  constructed  of  broken  stones  (road  metal)  4 volumes, 
clean  sharp  sand  1.25  to  .33  volumes,  and  hydraulic  cement  1 volume.  The 
mass  is  laid  down  in  a layer  of  3 or  4 ins.  in  depth,  and  left  to  harden  during 
a period  of  3 days,  when  a second  and  like  layer  is  laid  on  and  well  rolled, 
and  then  left  to  harden  for  a period  of  from  10  to  20  days,  according  to 
temperature  and  moisture  of  the  weather. 

Roads.  ( Molesworth .) 

Ordinary  turnpike  roads . — 30  feet  wide,  centre  6 ins.  higher  than  sides; 
4 feet  from  centre,  .5  inch  below  centre ; 9 feet  from  centre,  2 ins.  below 
centre ; 15  feet  from  centre,  6 ins.  below  centre. 

Foot-paths— 6 feet  wide,  inclined  1 inch  towards  road,  of  fine  gravel,  or 
sifted  quarry  chippings,  3 ins.  thick. 

Cross-roads — 20  feet  wide.  Foot-paths — 5 feet. 

Side  drains — 3 feet  below  surface  of  road. 

Road  material — bottom  layer  gravel,  burned  clay  or  chalk,  8 ins.  deep. 
Top  layer,  broken  granite  not  larger  than  1.5  cube  ins.,  6 ins.  deep, 

3 M* 


690 


ROADS,  STREETS,  AND  PAVEMENTS. 


Miscellaneous  INTotes. 

Metalling  should  be  from  6 ins.  to  1 foot  in  depth,  and  in  cubes  of  1.5  to  1.75  ins. 

One  layer  of  material  of  a road  should  be  spread  and  submitted  to  traffic  or  roll- 
ing before  next  is  laid  down,  and  this  process  should  be  repeated  in  2 or  3 layers 
of  3 ins.  each. 

When  new  metal  is  laid  on  old,  the  surface  of  the  old  should  be  loosened  with  a 
pick.  Patching  is  termed  darning. 

Sand  and  Gravel,  Blinding , should  not  be  spread  over  a new  surface,  as  they  tend 
to  arrest  binding  of  metal.  Mud  should  be  scraped  oft'  of  surface. 

Hoggin  is  application  of  a binding  of  surface  of  a metal  road,  composed  of  loam, 
fine  gravel,  and  coarse  sand. 

Metalled  Roads  should  be  swept  wet. 

Rolling.  — Steam  rolls  are  most  effective  and  economical.  1000  sq.  yards  of  metal- 
ling will  require  24  hours’  rolling  at  1.5  miles  per  hour.  A roller  of  15  tons’  weight 
will  roll  1000  sq.  yards  of  Telford  or  Macadam  pavement  in  from  30  to  40  hours,  at 
a speed  of  1.5  miles  per  hour,  equal  .675  and  .9  ton  mile  per  sq.  yard. 

Sprinkling.— Go  cube  feet  of  water  with  one  cart  will  cover  850  sq.  yards.  100 
cube  feet  per  day  will  cover  1000  sq.  yards;  ordinarily  two  sprinklings  are  necessary. 

Granite  Pavement.—  The  wear  of  granite  pavement  of  London  Bridge  was  .22  inch 
per  year,  and  from  an  average  of  several  streets  in  London,  tbe  w ear  per  ioo\ehicles 
per  foot  of  width  per  day  is  equal  to  one  sixteenth  of  an  inch  per  year. 

Sweeping  and  Watering  of  granite  pavement  and  Macadam  road,  for  equal  areas 
and  under  alike  conditions  in  every  respect,  costs  as  1 for  former  to  7 of  latter. 

By  men,  \vith  cart,  horse,  and  driver,  costs  3.25  times  more  than  by  a machine, 
one  of  which  will  sweep  16000  sq.  yards  of  street  per  period  of  6 houis. 

Asphalt  Pavement.  — Average  cost  per  sq.  yard  in  London:  foundation,  50  cents; 
surface,  $3.25;  cost  of  maintenance  per  sq.  yard  per  year,  40  cents.  Wear  varies 
from  .2  to  .42  near  curb,  and  .17  to  .34  inch  on  general  surface  per  year. 

Washing.  — Surface  cleaning  of  stone  or  asphalt  pavement  by  a jet  can  be  effected 
at  from  1 to  2 gallons  per  sq.  yard. 

Wood  Pavement.  — Wfear  of  wood  pavement  in  London,  per  100  vehicles  per  day 
per  foot  of  width,  .083  inch  per  year. 

Macadamized  Roads.  — Annual  cost  of  maintenance  of  several  such  roads  in 
London  was  62  cents  per  sq.  yard. 

Block  Slone  Pavement—  Stones  should  be  set  with  their  tapered  or  least  ends  up- 
wards, wdtli  surface  joints  of  1 inch. 

Fascines , when  used,  should  be  in  two  layers,  laid  crosswise  to  each  other  and 
picketed  down. 

Bituminous  road  may  be  made  by  breaking  up  asphalt,  laying  it  2 ins  thick, 
covering  with  coal  tar,  and  ramming  it  with  a heavy  beetle.  To  repair  a bitumi- 
nous surface,  dissolve  one  part  of  bitumen  (mineral  tar)  in  three  of  pitch  oil  or  resin 
oil  spread  .625  of  a lb.  of  solution  over  each  sq.  yard  of  road,  sprinkle  2 lbs.  pow- 
dered asphalt  (bituminous  limestone)  and  then  sand,  and  sweep  off  the  surplus. 

Slipping. — Granite  safest  when  wet,  and  asphalt  and  wood  when  dry. 

Gravel , alike  to  that  of  Roa  Hook,  from  its  uniformity,  Will  bear  an  admixture 
of  from  .2  to  .25  of  ordinary  gravel  or  coarse  sand. 

Annual  cost  of  a Telford  pavement  4.2  cents  per  sq.  yard,  including  sprinkling, 
repairs,  and  supervision. 


Voids  in  a Cube  Yard  of  Stone. 

broken  to  a gauge  of  2.5  ins 10  cube  feet.  Shingle. 9 cube  feet. 

u u 2 u io.  66  “ “ Thames  ballast 4.5 

“ “ 1.5  “ ”-33  “ 

For  further  and  full  information,  see  Law  and  Clarke  on  Roads  and  Streets,  New 
fork  1867-  Weale’s  Series.  London.  1861  and  1877;  Roads,  Streets,  and  Pavements, 
>y  Brev.  Mai. -Gen.  Q.  A.  Gilmore,  U.  S.  A.,  New  York.  1876;  Engineering  Notes,  by 
<\  Robertson,  London  and  New  York,  1873;  and  Construction  and  Maintenance  of 
Soads,  by  Ed.  P.  North,  C.  E.,  see  Transactions  Am.  Soc:  of  C.  E.,  vol.  vm.,  May,  1879. 


SEWERS. 


69I 


SEWERS. 

Sewers  are  the  courses  from  a series  of  locations,  and  are  classed  as 
Drains,  Sewers,  and  Culverts. 

Drains  are  small  courses,  from  one  or  more  points  leading  to  a sewer. 
Culverts  are  courses  that  receive  the  discharge  of  sewers. 

Greatest  fall  of  rain  is  2 ins.  per  hour  = 54  308.6  galls,  per  acre. 
Inclination  of  sewers  should  not  be  less  than  1 foot  in  240,  and  for 
house  or  short  lateral  service  it  should  be  1 inch  in  5 feet.  * 


Fig.  1. 


Fig.  2. 


Circular.  55  V x 2 /=  v,  and  v a = V. 

„ Egg.  — = w,  — w',  and  D = r.  x representing 

3 3 

area  of  sewer  -f-  wetted  perimeter , f inclination  of  sewer 
per  mile , and  v velocity  of  flow  of  contents  in  feet  per 
minute  ; a area  of  flow,  in  sq.feet , V volume  of  discharge, 
in  cube  feet  per  minute;  D height  of  sewer,  w and  w' 
width  at  bottom  and  top,  and  r radius  of  sides , in  feet. 

For  diameter  of  sewer  exceeding  6 feet.  (T.  Ilawlcsley.) 

D — -z=.iv\  D diameter  of  a circular  sewer  of  area  required. 

9 

Elliptic. — Top  and  bottom  internal  should  be  of  equal  diam- 
eters. Diameter  .66  depth  of  culvert ; intersections  of  top 
and  bottom  circles  form  centres  for  striking  courses  connect- 
ing top  and  bottom  circles. 

Pipes  or  Small  Seicers. — Height  of  section  = 1 ; diameter 
of  arch  = .66 ; of  invert  = .33,  and  radius  of  sides  = 1. 

In  culverts  less  than  6 feet  internal  depth,  brickwork  should  be  9 ins.  thick ; 
when  they  are  above  6 feet  and  less  than  9 feet,  it  should  be  14  ins.  thick. 

If  diameter  of  top  arch  = 1,  diameter  of  inverted  arch  = .5,  and  total 
depth  =r  sum  of  the  two  diameters,  or  1.5 ; then  radius  of  the  arcs  which  are 
tangential  to  the  top,  and  inverted,  will  be  1.5. 

From  this  any  two  of  the  elements  can  be  deduced,  one  being  known. 

Drainage  of  Lands  by  Dipes. 


Soils. 

Depth 
of  Pipes. 

Distance 

apart. 

Soils. 

Depth 
of  Pipes. 

Distance 

apart. 

Coarse  gravel  sand 

Light  sand  with  gravel 
Light  loam 

Ft.  Ins. 
4 6 
A 

Feet. 

60 

eo 

Loam  with  gravel . . . 
Sandy  loam 

Ft.  Ins. 
3 3 

0 0 

Feet. 

27 

r 

3 6 

Soft  clay. 

3 y 

2 l 

2 6 

21 

Loam  with  clay 

3 2 1 

21 

Stiff  clay 

15 

Minimum  Velocity  and  GJ-rade  of  Sewers  and  Drains 
in  Cities.  (WicJcsteed.) 


Diam. 

Vel. 

per 

Minute. 

Grade, 
1 in 

Grade 

per 

Mile. 

Diam. 

Vel. 

per 

Minute. 

Grade, 
1 in 

Grade 

per 

Miie. 

Diam. 

Vel. 

per 

Minute. 

Grade, 
1 in 

Grade 

per 

Mile. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

4 

240 

36 

146.7 

15 

180 

244 

21.6 

42 

180 

686 

7-7 

6 

220 

65 

81.2 

18 

180 

294 

18 

48 

180 

784 

6.8 

8 

220 

87 

60.7 

24 

180 

392 

13-5 

54 

180 

882 

6 

IO 

210 

Ir9 

44.4 

30 

180 

490 

10.8 

60 

180 

980 

5-4 

12 

190 

r75 

30-2 

36 

180 

588 

9 

Area  of  Sewers  or  Pipes. — An  area  of  20  acres,  miles,  etc.,  will  not  re- 
quire 20  times  capacity  of  pipes  for  one  acre,  mile,  etc.,  as  the  discharge  from 
the  19  acres,  etc.,  will  not  flow  into  the  main  simultaneously  with  that  from 
one  acre,  etc.  Ordinarily  in  this  country  an  area  of  sewer  or  pipe  that  will 
discharge  a rainfall  of  1 inch  per  hour  (3630  cube  feet  per  acre)  is  sufficient. 


692  * SEWERS. 

Sewage . — The  excreta  per  annum  of  ioo  individuals  of  both  sexes  and 
all  ages  is  estimated  at  7250  lbs.  solid  matter  and  94  700  fluid,  equal  to  1020 
lbs.  per  capita , and  in  volume  16  cube  feet,  to  which  is  to  be  added  the 
volume  of  water  used  for  domestic  purposes.  A velocity  of  flow  of  from  2.5 
to  3 feet  per  second  will  discharge  a sewer  of  its  sewage  matter  and  prevent 
deposits.  The  minimum  velocity  should  not  be  less  than  1.3  feet  per  second. 

Surface  from  which.  Circular  Sewers  with  proper  Carves 
will  discharge  Water  equal  in.  Volume  to  One  Inch  in 
Depth  per  Hour,  including  City  Drainage.  {John  Roe.) 

Diameter  of  Sewers  in  Feet. 

Inclination  in  Feet.  2 


None 

1 in  480. . 
1 in  240. . 
1 in  160. , 
1 in  120.. 
1 in  80. . , 
1 in  60. . . 


Acres. 

38.75 

48 

50 

63 

78 

90 

125 


Acres. 

67.25 

75 

87 

113 

143 

i65 

182 


Acres. 

120 

135 

155 

203 

257 

295 


Acres. 

277 

308 

355 

460 

59° 

570 

730 


Acres. 

570 

630 

735 

950 

1200 

1388 

1500 


Acres. 

1020 

1117 

1318 

1692 

2180 

2486 

2675 


Acres. 

1725 

I925 

2225 

2875 

37°° 

4225 

4550 


Acres. 

2850 

3025 

3500 

4500 

5825 

6625 

7I25 


Surface  of  a Town  from  which  small  Circular  Drains 
will  discharge  Water  equal  in  Volume  to  Two  Inches 
in  Depth,  per  Hour.  {John  Roe.) 

n Ins.  Inclination.! 

7 I 8 Fall  of  1 Inch. 


rCLINATION. 

Diameter  of  I 

.11  of  1 Inch. 

3 

4 

5 

Acres. 

Feet. 

Feet. 

Feet.  ! 

.125 

120 

— 

— 

•25 

20 

120 

— 

•4375 

— 

40 

— ■ 

• 5 

— 

30 

80 

.6 

— 

20 

60 

1 

— 

— 

20 

1.2 

— 

— 

— 

i-5 

— 

— 

— 

1.8 

— 

— 

— 

2. 1 

— 

— 

Feet. 


Feet. 'Feet. 


60  120 

80 
— | 60 


Acres. 

2.1 

2-5 

2-75 

4- 5 

5- 3 
5-8 
7.8 

9 

10 

17 


Diameter  of  Drain  in  Ins. 

I 12  | 15 


Feet. 

120 

80 

60 


Feet. 


120 

80 

60 


Feet. 


240 

120 

80 

60 


Fe«t. 


240 

120 


Dimensions,  Areas,  and  Volume  of  Material  per  Lineal 
Loot  of  Egg-shaped  Sewers  of  different  Dimensions. 

Volume  of  Brick-work. 


Depth. 


Internal  Dimensions. 
Diam.  of 
Top  Arch. 


Feet. 

2.25 
3 

3- 75 

4- 5 

5- 5 

6 

6- 75 

7- 5 

8.25 

9 


Feet. 

1- 5 

2 

2- 5 

3 

3- 5 

4 

4- 5 

5 

5- 5 

6 


Diam.  of 
Invert. 


Feet. 

•75 

1 

1.25 
i-5 

1- 75 

2 

2.25 

2.5 

2- 75 

3 


Area. 


Sq.  Feet. 
2.53 
4-5 
7*°3 

10. 12 

13.78 
18 

22.78 

28.12 
34- °3 
4°-5 


4.5  Ins. 

thick. 


Cube  Feet. 

2. 81 

3- 56 

4- 31 
5.06 

5.81 

6. 56 
7-31 


9 Ins. 
thick. 

13  5 Ins. 
thick. 

ube  Feet. 

Cube  Feet. 

9-56 

— 

10.87 

— 

12.75 

— 

14.25 

— 

15-75 

24-75 

17.06 

27 

18 

28.41 

19.69 

3°-94 

Area  — product  of  mean  diameter  X height. 

Sewer  Pipes  should  have  a uniform  thickness  and  be  uniformly  glazed, 
both  internally  and  externally. 

Fire-clay  pipes  should  be  thicker  than  those  of  stone-clav. 


STABILITY. 


693 


STABILITY. 


Stability,  Strength , and  Stiffness  are  necessary  to  permanence  of  a 
structure,  under  all  variations  or  distributions  of  load  or  stress  to  which 
it  may  be  subjected. 

Stability  of  a Fixed  Body— Is  power  of  remaining  in  equilibrio  without 
sensible  deviation  of  position,  notwithstanding  load  or  stress  to  which  it 
may  be  submitted  may  have  certain  directions. 

Stability  of  a Floating  Body. — A body  in  a fluid  floats,  or  is  balanced, 
when  it  displaces  a volume  of  the  fluid,  weight  of  which  is  equal  to  weight 
of  body,  and  when  centre  of  gravity  of  body  and  that  of  volume  of  fluid  dis- 
placed "are  in  same  vertical  plane. 

When  a body  in  equilibrio  is  free  to  move,  and  is  caused  to  deviate  in  a 
small  decree  from  its  position  of  equilibrium,  if  it  tends  to  return  to  its 
original  position,  its  equilibrium  is  termed  Stable  ,*  if  it  does  not  tend  to  de- 
viate further,  or  to  recover  its  original  position,  its  equilibrium  is  termed 
Indifferent ; and  when  it  tends  to  deviate  further  from  its  original  position, 
its  equilibrium  is  Unstable. 

A body  in  equilibrio  may  be  stable  for  one  direction  of  stress,  and  unstable 
for  another. 

Moment  of  Stability  of  a body  or  structure  resting  upon  a plane  is  mo- 
ment or  couple  of  forces,  which  must  be  applied  in  a plane  vertically  inclined 
to  the  body  in  addition  to  its  weight,  in  order  to  remove  centre  of  resistance 
of  body  upon  plane,  or  of  the  joint,  to  its  extreme  position  consistent  with 
stability.  The  couple  generally  consists  of  the  thrust  of  an  adjoining  struct- 
ure, or  "an  arch  and  pressure  of  water,  or  of  a mass  of  earth  against  the 
structure,  together  with  the  equal  and  parallel,  but  not  directly  opposed,  re- 
sistance of  plane  of  foundation  or  joint  of  structure  to  that  lateral  thrust. 
It  may  differ  according  to  position  of  axis  of  applied  couple. 

Couple— Two  forces  of  equal  magnitude  applied  to  same  body  or  struct- 
ure in  parallel  and  opposite  directions,  but  not  in  same  line  of  action,  consti- 
tute a couple. 

Note.— For  Statical  and  Dynamical  Stability,  see  Naval  Architecture,  page  649. 

To  Ascertain  Stability  of  a Body  on  a Horizontal  Blaxie. 

— Big.  1. 


its  moment  would  be  5 X 4 = 20  tons , although  it  is  but  half  the  weight. 

To  Compute  "Weigh,  t of  a Griven.  Body  "to  Sastaiix  a 
Griveix  Thrust. 


and  l distance  of  fulcrum  from  centre  of  gravity  — as. 

T illustration. — Assume  figure  to  be  extended  to  a height  of  20  feet,  and  required 
to  be  capable  of  resisting  the  extreme  pressure  of  wind. 


b 


d 


0 


Illustration.  — Stability  of  a body,  A,  Fig.  1,  when  a 
thrust  is  applied  as  at  0,  to  turn  it  on  a , is  ascertained  by 
multiplying  its  weight  by  distance  as,  from  fulcrum  a to 
line  of  centre  of  gravity,  cs. 


Hence,  if  cubical  block  weighed  10  tons  and  its  base  is 


■e 


6 feet,  its  moment  would  be  10  X — = 30  tons. 

2 


If  upper  part,  abdc , was  removed,  remainder,  a e d, 


F h 

= W.  F representing  thrust  in  lbs. , h height  of  centre  of  gravity  of  body  = cs , 


STABILITY. KEVETMENT  WALLS. 


694 


Pressure  estimated  at  50  lbs.  F = 6 X 20  X 50  = 6000  lbs.  at  centre  of  gravity  of 
surface  of  body. 

6000  X 10  .. 

Then  = 20  000  lbs. 

3 

Note  i.— This  result  is  to  be  increased  proportionately  with  the  factor  of  safety 
due  to  character  of  its  material  and  structure. 

2.— If  form  of  body  has  a cylindrical  section,  as  a round  tower,  the  thrust  of  wind 
would  be  but  one  half  of  that  of  a plane  surface. 

When  the  Body  is  Tapered,  as  Frustum  of  Pyramid  or  Cone.  — Ascertain 
centres  of  gravity  of  surface  for  pressure  or  thrust,  and  of  body  for  its  sta- 
bility, and  proceed  as  before. 


Fig.  2. 


To  Ascertain  Stability  of  a Body  011 
air  Inclination.— Big.  S. 

Illustration.— Stability  of  body,  Fig.  2,  when  thrust 
is  applied  at  c,  is  ascertained  by  multiplying  its  weight 
by  distance  a b from  fulcrum,  b,  to  line  of  centre  of 
gravity,  a g. 

If  thrust  was  applied  at  0,  stability  would  be  ascer- 
tained by  distance  s r from  fulcrum  r. 


Angles  of  Eq^nilibrinm  at  "wliicli  various  Substances  ■will 
Repose,  as  determined  Toy  a Clinometer. 


Angle  measured  from  a Horizontal  Plane , and  falling  from  a spout. 


Degrees.  [ 

Degrees. 

Lime-dust 

45  i Sand,  less  dry. . . 

, ...  39.6 

Dry  sand 

40  1 Wheat 

Moist  sand. . . 1 

Degrees. 

Common  mold. . . 37 
Common  gravel. . 35  to  36 
Stones  or  Coal. . . 43 


Weiglrt  of  a Cifbe  Foot  of  Materials  of  Embankments, 
"Walls,  and  Barns. 


Concrete  in  cement. . . 137 

Stone  masonry 130 

Brick  “ 112 


Gravel 125 

Loam 126 

I Sand 120 


Clay. . 
Marl . 


120 

100 


Revetment  AW alls. 

When  a wall  sustains  a pressure  of  earth,  sand,  or  any  loose  material,  it 
is  termed  a Revetment  wall,  and  when  erected  to  arrest  the  fall  or  subsidence 
of  a natural  bank  of  earth,  it  is  termed  a Face  wall. 

When  earth  or  banking  is  level  with  top  of  wall,  it  is  termed  a Scarp  re- 
vetment, and  when  it  is  above  it,  or  surcharged,  a Counterscarp  revetment. 

When  face  of  wall  is  battered,  it  is  termed  Sloping,  and  when  back  is  bat- 
tered, Countersloping. 

Thrust  of  earth,  etc.,  upon  a wall  is  caused  by  a certain  portion,  in  shape 
of  a wedge,  tending  to  break  away  from  the  general  mass.  The  pressure 
thus  caused  is  similar  to  that  of  water,  but  weight  of  the  material  must  be 
reduced  by  a particular  ratio  dependent  upon  angle  of  natural  slope,  which 
varies  from  450  to  6o°  (measured  from  vertical)  in  earth  of  mean  density. 

Or,  natural  slope  of  earth  or  like  material  lessens  the  thrust,  as  the  cosine 
of  the  slope. 

Angle  which  line  of  rupture  makes  with  vertical  is  .5  of  angle  which  line 
of  natural  slope,  or  angle  of  repose,  makes  with  same  vertical  line,  y V hen 
earth  is  level  at  top,  its  pressure  may  be  ascertained  by  considering  it  as  a 
fluid,  weight  of  a cube  foot  of  which  is  equal  to  weight  of  a cube  foot  of  the 
earth,  multiplied  by  square  of  tangent  of  .5  angle  included  between  natural 
slope  and  vertical. 


STABILITY. — REVETMENT  WALLS. 


695 

Therefore  squares  of  the  tangents  of  .5  of  450  and  .5  of  6o°  = .iqi6  and 
which  are  the  multipliers  to  be  used  in  ordinary  cases  to  reduce  a 
cube  foot  of  material  to  a cube  foot  of  equivalent  fluid,  which  will  have 
same  effect  as  earth  by  its  pressure  upon  a wall. 

Pressure  of  Earth,  against  Revetment  "Walls. 

Let  ABCD,  Fig.  3,  be  vertical  section  of  a revetment 
wall,  behind  which  is  a bank  of  earth,  A D/e  ; let  Do 
represent  angle  of  repose,  line  of  rupture , or  natural  slope 
which  earth  would  assume  but  for  resistance  of  wall. 

In  sandy  or  loose  earth  angle  o D A is  generally  300; 
in  firmer  earth  it  is  36°;  and  in  some  instances  it  is  450. 

If  upper  surface  of  earth  and  wall  which  supports  it  are 
both  in  one  horizontal  plane,  then  the  resultant,  In,  of 
pressure  of  the  bank,  behind  a vertical  wall,  is  at  a dis- 
tance, D n,  of  one  third  A D. 

Line  of  Rupture  behind  a wall  supporting  a bank  of  vegetable  earth  is  at 
a distance  A 0 from  interior  face,  AD  = .618  height  of  it. 

When  bank  is  of  sand,  A 0 = .677  h ; when  of  earth  and  small  gravel  = 
.646  h ; and  when  of  earth  and  large  gravel  = .618  h. 

The  prism,  vertical  section  of  which  is  A D 0,  has  a tendencv  to  descend 
along  inclined  plane,  0 D,  by  its  gravity ; but  it  is  retained  in  its  plaoe  by 
resistance  of  wall,  and  by  its  cohesion  to  and  friction  upon  face  0 D.  Each 
of  these  forces  may  be  resolved  into  one  which  will  be  perpendicular  to  0 D, 
and  into  another  which  will  be  parallel  to  o D.  The  lines  c i , it  represent 
components  of  the  force  of  gravity,  which  is  represented  by  vertical  line  c l , 
drawn  from  centre  of  gravity,  c,  of  prism.  Lines  n r,lr  represent  compo- 
nents of  forces  of  cohesion  and  friction,  which  is  represented  by  horizontal 
line  n l.  Force  that  gives  the  prism  a tendency  to  descend  is  i l,  and  that 
opposed  to  this  is  r l , together  with  effects  of  cohesion  and  friction. 

Thus,  i l = r l + cohesion  + friction.  Consequently,  exact  solution  of  prob- 
lems of  this  nature  must  be  in  a great  measure  experimental. 

It  has  been  found,  however,  and  confirmed  experimentally,  that  angle 
formed  with  vertical,  by  prism  of  earth  that  exerts  greatest  horizontal  stress 
against  a wall,  is  half  the  angle  which  angle  of  repose  or  natural  slope  of 
earth  makes  with  vertical. 

Memoranda. 

Natural  slope  of  dry  sand  = 39°,  moist  soil  = 43°,  very  fine  sand  = 21°  wet  clay 
= 140,  and  gravel  = 350. 

In  setting  or  founding  of  retaining  walls,  if  earth  upon  which  wall  is  to  rest  is 
clayey  or  wet,  coefficient  of  friction  between  wall  and  earth  falls  to  . 3 ; hence  it  is 
necessary,  in  order  to  meet  this,  that  the  wall  should  be  set  to  such  a depth  in  the 
earth  that  the  passive  resistance  of  it  on  outer  face  of  wall,  combined  with  its  fric- 
ticn  on  its  bottom,  may  withstand  the  pressure  or  thrust  on  its  inner  face. 

Moment  of  a Retaining  Wall  is  its  weight  multiplied  by  distance  of  its  centre  of 
gravity  to  vertical  plane  passing  through  outer  edge  of  its  base. 

Moment  of  Pressure  of  Earth  against  a retaining  wall  is  pressure  multiplied  by 
distance  of  its  centre  of  pressure  to  horizontal  plane  passing  through  base  of  wall. 

Equilibrium  of  Retaining  Wall  is  when  respective  moments  of  wall  and  earth  are 
equal. 

Stability  of  a Retaining  Wall  should  be  in  excess  of  its  equilibrium,  according  to 
character  of  thrust  upon  it,  and  the  line  of  its  resistance  should  be  within  wall  and 
at  a distance  from  vertical  passing  through  centre  of  gravity  of  wall,  at  most  .44  of 
distance  of  exterior  axis  of  wall  from  this  line. 

Coefficient  of  Stability  varies  with  character  of  earth,  location,  exposure  to  vibra- 
tions,'floods,  etc. ; hence  thickness  of  base  of  wall  will  vary  from  1.4  to  2 6. 

Backs  of  retaining  walls  should  be  laid  rough,  in  order  to  arrest  lateral  subsidence 
of  the  filling. 


Fig.  3- 
T A 


s 


£4  c / 
f'i 


C ID 


STABILITY. REVETMENT  AVALLS. 


696 


When  filling  is  composed  of  bowlders  and  gravel,  the  thickness  of  wall  must  be 
increased,  and  contrariwise;  when  of  earth  in  layers  and  well  rammed,  it  may  be 

decreased. 

Courses  of  dry  wall  should  be  inclined  inwards,  in  order  to  arrest  the  flow  of 
water  of  subsidence  in  filling  from  running  out  upon  face  of  wall. 

Less  the  natural  slope,  greater  the  pressure  on  wall. 

Sea  walls  should  have  an  increased  proportion  of  breadth,  as  the  earth  backing 
is  not  only  subjected  to  being  flooded,  but  the  walls  have  at  times  to  sustain  the 
weight  of  heavy  merchandise. 

Buttress.—  An  increased  and  projecting  width  of  wall  on  its  front,  at  intervals  in 
its  length. 

Counterfort.— An  increased  and  projecting  width  of  wall  at  its  back  and  at  in- 
tervals. 

Coefficient  of  Friction  of  masonry  on  masonry  .67,  of  masonry  on  dry  clay  .51, 
and  on  wet  clay  .3. 

Face  of  wall  should  not  be  battered  to  exceed  1 to  1.25  ins.  in  a foot  of  height,  in 
consequence  of  the  facility  afforded  by  a greater  inclination  to  the  permeation  of 
rain  between  the  joints  of  the  courses. 

Footing  of  a wall,  projecting  beyond  its  faces,  is  not  included  in  its  width. 

Pressure.—  Limit  of  pressure  on  masonry  12  500  to  16  500  lbs.  per  sq.  foot  wall. 

Thickness  of  Walls,  in  Mortar , Faces  vertical.  For  Railways  or  Like  Stress. 

Cut  stone  or  Ranged  rubble .35  | Brick  or  Dressed  rubble 4 

When  laid  dry,  add  one  fourth. 

Friction  in  vegetable  earths  is  .5;  pressure  in  sand  .4. 

When  vegetable  earths  are  well  laid  in  courses,  the  thrust  is  reduced  .5. 

When  bank  is  liable  to  be  saturated  with  water,  thickness  of  wall  should  bo 
doubled. 

Centre  of  Pressure  of  earthwork,  etc.,  coincides  with  centre  of  pressure  of  water, 
and  hence,  when  surface  is  a rectangle,  it  is  at  .33  of  height  from  base. 

The  theory  of  required  thickness  of  a retaining  wall,  as  before  stated,  is,  that  the 
lateral  thrust  of  a bank  of  earth  with  a horizontal  surface  is  that  due  to  the  prism 
or  wedge-shaped  volume,  included  between  the  vertical  inner  face  of  the  wall  and 
a line  bisecting  the  angle  between  the  wall  and  the  angle  of  repose  of  the  material. 


To  Compute  Elements  of  Revetment  Weills. — If ig.  4r. 


Fig.  4. 

B A 


a 


p? 


Let  A D 0 represent  angle  of  repose  of  material,  resting 
against  a wall,  ABCD.  ADn  = .5  ADo.  Tan.  ADr 
h h 2 

Tan.  A D n h — , or  — tan.  A D n = V. 


= *492* 
wh 2 


321 


wh 2 


tan.  A D n = W 
h 


w h2 


tan.2  ADn- 


C D 
W h x'2 


W h X2  wh 3 


— — — tan. 2 A D n = E ; 


w A3 
6 

W hx2 


tan.2ADn  = P; 

W 

2 

tan.2  ADn  = S; 


/ w 

h tan.  AD  n / --== 
V 3 W 


: x,  and  h tan. 


. _ /2  IV  , 

•ADnViw  = x' 


tan.2  ADw  = M; 
w h 3 J 
~ 3 

h representing  height  of 


wall  in  feet,  V volume  of  section  of  prism  of  material  AD  n one  foot  in  length  in  cube 
feet , W and  w weights  of  a cube  foot  of  wall  and  of  material,  P lateral  pressure  of 
prism  of  earth  upon  wall , M and  m moments  of  pressure  and  weight  on  and  of  wall , 
E and  S equilibrium  and  stability  of  wall,  all  in  lbs. , and  x and  x,  C D for  weights 
of  wall  for  equilibrium  and  stability. 

Illustration. — A revetment  wall,  Fig.  4,  of  125  lbs.  per  cube  foot  and  40  feet  in 
height , sustains  a bank  of  earth  having  a natural  slope  of  52 0 24',  and  a weight  of 
89.25  lbs.  per  cube  foot;  what  is  pressure  or  thrust  against  it,  etc.  ? 


STABILITY. REVETMENT  WALLS. 


697 


Tan.2  A D n = .242.  Then  .492  X 40  X = 393-6  cube  feet. 

89.25  X 4g_  x 35  12g  3 lbs.  89.25  x ^ .4922  — 17  278.8  Z&s. 

2 2 

89-25  X 4°  a ^ 4922  x — = 230384  lbs.  125  X 40  X = 230400  lbs. 
23  2 

40  x. 492  a/— 1ffg  = 9-6/erf,  and  40  X -49 


For  Rubble  Walls  in  Mortar  or  Dry  Rubble , add  respectively  to  base  as  above 
obtained,  .14  and  .42  part. 

Note  i. — When  coefficient  of  friction  is  known,  use  it  for  tan.2  A D n. 

h X C = base  of  wall  for  stability.  ( Molesivorth .) 

2. — When  either  relative  weights  of  equal  volumes  of  wall  and  bank  of  earth  or 
their  specific  gravities  are  given,  S and  s may  be  taken  for  W and  w. 

These  equations  involve  simply  the  operation  of  a lever,  the  fulcrum  being  at 
the  outer  edge  of  wall  C.  The  moment  of  pressure  of  bank  is  product  of  lateral 
pressure  and  perpendicular  distance  from  fulcrum  to  line  of  direction  of  pressure. 

The  moment  of  weight  of  wall  is  product  of  weight  of  wall  and  perpendicular 
distance  from  fulcrum  to  vertical  line  drawn  through  centre  of  gravity  of  wall. 

When  Weights  of  Embankment  and  Wall  are  equal  per  Cube  Foot. 

C for  clay  = .336,  and  for  sand  .267. 

When  Weights  are  as  4/0  5.  C for  clay  = .3,  and  for  sand  .239. 

When  Wall  has  an  Exterior  Slope  or  Batter. — Fig.  5. 

5- -g  A — n_ p AA.  D -j-  E C — — M.  M representing 

J / moment  of  weight  of  wall  in  lbs. 

/ /'  Illustration. — Assume  weight  of  wall  120  lbs.  per 

cube  foot,  and  C D and  E C respectively  10  and  2. 5 feet, 
/'  and  all  other  elements  as  in  preceding  case. 


E C 


Hence,  x 

2 

w h 3 , 


/ 2 2-52\ 

^10  -j-  2-  5 — J ~ 37°  000  Z&S. 


W h( — 2 n2  h~\  wh 3 • 

z4 -nh ) = tan.2  A D n = 

2 V 3 / 3 


- — tan.  2ADn  — nh  = x.  x representing  A B or  C D.  n ratio  of 

3 3 W 

difference  of  widths  of  base  and  top  to  height.  In  absence  of  tan.2 *  A D n put  C,  co- 
efficient of  material. 

C = .0424  for  vegetable  or  clayey  earth,  mixed  with  large  gravel;  .0464  if  mixed 
with  small  gravel;  .1528  for  sand,  and  .166  for  semi-fluid  earths. 

Illustration. — Assume  elements  of  preceding  case.  n = one  fortieth,  and  tan. 
A D n — .492. 


40 


a / — — - + 1->^89  :25  x .492 2 — 1 — 12.6  feet. 
V 3 X 402  3 X 125  ^ 


Hence,  thickness  of  wall  at  base  = 12.6  -f-  1 (one  fortieth  of  height)  = 13.6  feet. 
Note. — If  n = one  twentieth, 


Hence,  wall  at  base  = 11.63-}-  2 (one  twentieth  of  height)  = 13. 63 /<?«£.  If  C was 
used,  n.32  feet. 


STABILITY. REVETMENT  WALLS. 


When  Wall  has  an  Interior  Slope  or  Batter , B E.— 
Fig.  6. 


10  h2  , ,oEr 
X tan.2  - = P. 


to  A3 


X tan. 


o E 7* 


i oh/  • — ; — ; CEA 

earth  for  equilibrium  ; ^DCx^C  + Ch — J = 


= M of 
CE*\ 

3 


M of  wall;  and 
bility. 


to  A3 


X tan.2  o E n = M of  earth  for  sta - 


Coefficients  for  Batter  of  following  Proportions. 

Base  = Height  X Tab.  number. 

Weight  of  Earth  to  Wall.  Weight  of  Earth  to  Wall. 


Batter  of 
Wall. 


i in  4 
i “ 5 

6 


As  4 to  5, 


Clay. 


.083 
. 122 
149 


Sand. 


.029 

.065 

.092 


As  1 to  1. 


Clay. 


•115 

•155 

•183- 


Sand. 


•054 

.092 

.118 


Batter  of 
Wall. 


1 in  8 

I “ 1_. 

Vertical. . 


As  4 to  5. 


Clay. 


.221 

•3 


Sand. 


.125 
. 16 
.239 


As  1 to  1. 


Clay. 


.218 

.256 

•336 


Sand. 


•iS3 
. 189 
.267 


I 

To  Compute  Pressure  Perpendicular  to  Back  of  Wall. 
— Eig.  V. 

Fig‘  7'  A n o p * = — or  i , and  f*  at  right  angle  to  back  of  wall, 

whether  vertical  or  inclined. 

LXA  n _ . ^ to  x A2  X tan.2ADn 

-/-/ 


, or  L X tan.  ADn,  or 


Fig.  8. 


-f  *.  L representing  weight  of  triangle  of  em- 

jy  bankment , as  A D n. 

This  is  pressure  independent  of  friction  between  surfaces  of  wall  and  earth. 

To  Ascertain  and  Compute  Amount  and.  Effect  of  Fric- 
tion of  Wall  and  Earth.- Eig.  8. 

Draw  / * by  scale  to  computed  pressure  at  right  angle 
n to  back  of  wall,  draw  angle  / * r = to  D o of  natural  slope 
of  earth  with  horizon,  draw/?*  at  right  angle  to/  *,  maKe 
r c —f  *,  then  c r will  represent  by  scale  effect  of  friction 
against  back  of  wall. 

Assume  friction  to  act  at  point  *,  then  r * will  give  by 
scale  resultant  of  the  two  forces  of  pressure  and  friction, 
equal  to  pressure  in  force  and  direction,  which  bears 
m against  wall. 

This  resultant  is  also  equal  to  / * X sec.  m D 0. 

= r *, or  x sec  m D 0>  or  L x tan.  ADn 


— 

\~fr/ 

J 

4-Ai 

D 

L X A n X sec.  m D 0 


JZ 


X sec.  mDo. 

To  Ascertain  Point  of  Moment  of  Pressure  of  a Wall. 

—Fig:-  9. 

lg'  By  its  resisting  lever  l a, added  to  its  weight. 

- Weight  of  wall  as  computed  assumed  as  concentrated  at  its 

/ centre  of  gravity  • 

Draw  a vertical  line  . 0 through  its  centre  of  gravity,  and  con- 
tinue line  of  pressure  P * to  l,  take  any  distance  r 0 by  scale  rep- 
resenting weight  of  wall,  and  r «,  by  same  scale  for  amount  of 
pressure  or  thrust  against  wall,  complete  parallelogram  1 o wm, 
then  diagonal  ru  will  give  resultant  of  pressure  in  amount  and 
direction  to  overturn  wall. 

For  stability  this  diagonal  should  fall  inside  of  base  at  a point 
not  less  than  one  third  of  its  breadth. 


E 


STABILITY. — REVETMENT  WALLS, 


699 


Surcharged.  Revetments. 


Fig.  10.  f r 7o 

as  t / 


D C 


When  the  earth  stands  above  a wall,  as  A B e, 
Fig.  10,  with  its  natural  slope,  Ay,  A B C is  termed 
a Surcharged  Revetment. 

If  C r is  line  of  rupture,  A fr  C is  the  part  of  earth 
that  presses  upon  wall,  which  part  must  he  taken  into 
the  computation,  with  exception  of  portion  A B e, 
which  rests  upon  wall;  that  is,  the  computation  must 
be  for  part  C efr,  which  must  be  reduced  by  multiply- 
ing weight  of  a cube  foot  of  it  by  square  of  tangent  of 
angle  e C r = angle  of  line  of  rupture,  or  half  angle 
e C 0,  which  natural  slope  makes  with  vertical,  and 
then  proceed  as  in  previous  cases  for  revetments. 


li'  / = breadth  or  CD.  W and  w representing  weights  of  wall  and 

V 3 hW 

embankment  in  lbs.  per  cube  foot,  and  h'  height  of  embankment,  as  C e. 

Illustration. — Height  of  a surcharged  revetment,  BC,  Fig.  10,  is  12  feet,  weight 
130  lbs.  per  cube  foot;  what  is  its  width  or  base  to  resist  pressure  of  earth  of  a weight 
of  100  lbs.  per  cube  foot,  and  a height,  C e,  of  15  feet,  angle  of  repose  45°*? 

Tan.* 2 * * * (45° -4- 2)  = .1716.  Then  15  V-°55  = 3-52  feet. 

v 3 a 12  A I 7 * *3° 


To  Ascertain  IPoint  of*  Moment  of  Pressure  of  a Sxir- 
cliarged  Wall.— Wig.  11. 

Fig.  11.  sj  Draw  a line,  P *,  parallel  to  slope,  C r,  through  centre 

of  gravity  of  sustained  backing,  BCr. 

When,  as  in  this  case,  this  section  is  that  of  a triangle, 
point  * will  be  at  .33  height  of  wall. 

When  natural  slope  is  1.5  in  length  to  1 in  height,  as 
with  gravel  or  sand,  w x .64  = pressure  P *. 

In  a surcharged  revetment,  as/B  0,  at  its  natural  slope, 
the  maximum  pressure  is  attaiued  when  the  backing 
reaches  to  r.  When  slope  of  maximum  pressure,  C nr, 
intersects  face  of  natural  slope,  Bf  so  that  if  backing  is 
raised  to /,  or  above  it,  there  is  theoretically  no  addi- 

tional stress  exerted  at  back  of  or  against  wall,  but  prac- 

tically there  is,  from  effect  of  impact  of  vibration  of  a 
passing  train,  proximity  to  percussive  actiQn,  alike  to  that  of  a trip-hammer,  etc. 

When  backing  rests  on  top  of  wall,  as  A B e.  Fig.  10,  small  triangle  of  it  is  omitted 

in  computations.  Direction  of  pressure  against  wall  is  same  as  when  wall  is  not 
surcharged. 

When  Wall  is  set  below  Surface  of  Earth. — Fig.  12. 


Fig.  12. 


1.4  tan.  45 


V -<■<  \/h*  45°  ~ 2 ) 

7 V — W 


^2 /V 


- — d. 


a representing  angle  of  repose  of  earth,  w and  W weights 
of  earth  and  wall  per  cube  foot,  f friction  of  wall  on  base 
A B,  and  V weight  of  wall. 

Illustration. — If  a wall  of  masonry,  Fig.  12,  8 feet  in  thickness 
and  13  in  height,  is  to  sustain  earth  ievel  with  its  upper  surface, 
earth  weighing  100  lbs.  per  cube  foot,  weight  of  wall  150  lbs.  per 
cube  foot  — 1 5 600  lbs.,  and  angle  of  repose  of  earth  300;  what 
should  be  the  depth  of  wall  below  surface  of  earth? 


Tan.  45  — 30  2 = . 5774,  and  /= 


'9360  <-05634. 3 

150 


Then  ,4  X 10°X-577f5;  2 X -3X  156°°  = -8°B4  xf 

= 4.027  feet. 

Note.— Coefficient  of  stability  is  assumed  by  French  engineers  for  walls  of  forti- 
fications 1.4  h,  and  if  ground  is  clayey  or  wet/=.3. 


STABILITY. EMBANKMENT  WALLS  AND  DAMS. 

In  Computing  Stability  of  a Surcharged  Waif  Fig.  13,  sub- 
stitute d for  h , as  in  following  illustration.  ( Molesworth .) 
d,  representing  depth  at  distance  l , = h. 

In  slopes  of  1 to  1,  d — 1.71  h\  of  1.5  to  1,=  1.55;  of  2 to  1,= 
1.45 ; of  3 to  ij==  1. 31,  and  4 to  1,==  1.24. 

To  Determine  Form  of  a,  Pier  to  Snstain 
eq.ti.al  Pressure  per  TJiait  of  Surface  at  all 
its  Horizontal  Sections,  or  any  Height. 

A nd  = a,  or  A N = a.  A and  a representing  areas  of  sections  at  summit  of  pier 
and  at  any  depth , d,  measured  from  summit , n a number  the  hyp.  log.  of  which  — 1 -4- 
height,  H,  of  a column  of  the  material  of  which  pier  is  constructed , due  to  required 

•4343  ^ 

pressure , and  N the  number , com.  log.  of  which  = — -g — . 

Illustration. — Height  of  a pier  is  20  feet,  and  area  of  section  of  its  summit  = 
1 foot;  what  should  be  its  areas  at  10  feet  and  base? 

1 -4- 20  — .05,  and  numbers  1. 0513;  i X 1.0513 10  = 1.649  feet;  and  1 X i.o5i320  = 
2.719  feet. 

Counterforts  are  increased  thicknesses  of  a wall  at  its  back,  at  intervals  of 
its  length. 

Enilo ankin e nt  "Walls  and.  Dams. 

Thrust  of  water  upon  inner  face  of  an  Embankment  wall  or  Dam  is 
horizontal. 

When  Both  Faces  are  Vertical,  Fig.  14. 

Assume  perpendicular  embankment  or  wall,  A BCD,  Fig.  14,  to  sustain 
pressure  of  water,  B C ef. 


Fig.  14- 


Let  h i be  a vertical  line  passing  through  0,  centre 
of  gravity  of  wall,  c centre  of  pressure  of  water,  dis- 
tance C c being  ==.33  B C.  Draw  c l perpendicular 
to  B C ; then,  since  section  A C of  wall  is  rectangular, 
centre  of  gravity,  0 , is  in  its  geometrical  centre,  and 
therefore  D i — .5  DC.  Now  l D i is  to  be  consid- 
ered as  a bent  lever,  fulcrum  of  which  is  D,  weight  of 
wall  acting  in  direction  Of  centre  of  gravity,  o,  on  arm 
D i,  and  pressure  of  water  on  arm  D /,  or  a force  equal 
to  that  pressure  thrusting  in  direction  c l. 

Then  P x D Z = P X — = W X — , or  P = 3 D . P representing  pressure 

3 2 2 b o 

of  water. 

Note.  — When  this  equation  holds,  a wall  or  embankment  will  just  be  on  the 
point  of  overturning;  but  in  order  that  they  may  have  complete  stability,  this 
equation  should  give  a much  larger  value  to  P than  its  actual  amount. 

The  following  formulas  are  for  walls  or  embankments  one  foot  in  length ; 
for  if  they  have  stability  for  that  length  they  will  be  stable  for  any  other 
length. 

P = — w,  also  W = h b W,  each  value  being  for  1 foot  in  length,  which,  being  sub- 
2 

stituted  in  the  equations,  there  will  result 

— w = 3 6 X W- , or  h2w  = 3 b2  W;  b /^-  = h,  and  h /-^  = b.  h rep - 
2 2 h ’ J V w V 3 w 

resenting  depth  of  water  and  ivall  or  embankment , which  are  here  assumed  to  be 
equal , b breadth  of  wall  or  embankment , and  W and  w weights  of  wall  and  water 
per  cube  foot  in  lbs. 

Which  gives  breadth  of  a wall  or  embankment  that  will  just  sustain 
pressure  of  the  water. 


STABILITY. — EMBANKMENT  WALLS  AND  DAMS.  J01 


To  Compute  Equilibrium.  h^J-^j—b. 

Illustration  i.— Height  of  a wall,  B C,  equal  to  depth  of  water,  is  12  feet,  and  re- 
spective weights  of  water  and  wall  are  62.5  lbs.  and  120  lbs.  per  cube  foot;  required 
breadth  of  wall,  so  that  it  may  have  complete  stability  to  sustain  the  pressure  of 
water. 


*7: 


62.5 


= 12  X .4166  = 5 feet,  breadth  that  will  just  sustain  pressure  of  the 


3 X 120 

water. 

Therefore  an  addition  should  be  made  to  this  to  give  the  wall  complete  stability, 
say  2 feet;  hence  54-2  = 7,  required  width  of  wall. 

2. —Width  of  a wall  is  3 feet,  and  weight  of  a cube  foot  of  it  is  150  lbs. ; required 
height  of  wall  to  resist  pressure  of  fresh  water  to  the  top. 

/2  W 

To  Compute  Stability,  h / = 0. 

Illustration.— Take  elements  of  preceding  case. 

12.  /2  *6^  = I2  X.589  = 7.07 feet. 

V 3 x 120 

Or,  Divide  1,  2,  or  3,  etc.,  according  as  the  nature  of  the  ground,  the  mate- 
rial, and  the  character  of  the  thrust  of  the  water  requires,  by  .05  weight  of 
material  of  wall,  per  cube  foot,  extract  the  square  root  of  quotient,  and  mul- 
tiply result  by  extreme  height  of  water. 

Example.  — What  should  be  the  thickness  of  a vertical  faced  wall  of  masonry, 
having  a weight  of  125  lbs.  per  cube  foot,  to  sustain  a head  of  water  of  40  feet,  and 

to  have  stability  ? . 

v'(2-r-.o5  X 125)  40  = V-32  X 4°  ==  22.63  fat. 

0r,  — 40  V-3472  = 23.56  feet. 

When  Dam  kas  an  Exterior  Slope  or  Batter , as  A D. — Fig.  15. 


Fig.  15.  a 


i E 0 n c 


Assume  prismoidal  wall,  A B C D,  to  sustain  press- 
jpjjjl  ure  of  water,  B C ef. 

jjggjg  Draw  A E perpendicular  to  D C ; h = B C,  the  top 
§§gg|§  breadth  A B = E C = b,  and  bottom  breadth,  D E, 

"ggj  of  sloping  part,  AED  = S. 

Ijlgf  Then  weights  of  portions  A C and  A E D respec- 
jflf  tively  for  one  foot  in  length  are  hb  W and  .5  W S 4, 
f these  weights  acting  at  points  n and  i respectively. 


To  Compute  Moment. 

, h S W 2 S 


hb  W X 4-  — ^ = moment  for  A C,  and  — - — X = moment  for  A E D. 

Hence,  — — l S 4-  b r j = moment  of  dam,  S representing  batter  or  base  E D. 

Illustration.— Height  of  a dam,  B C,  Fig.  15,  is  9 feet,  base  C E 3,  and  E D 4 feet ; 
what  is  its  moment  ? 

A C = 9 X 3 X 120  X (44-7)=? 324°  X 5- 5 = 17 820  Ms. 

ADE=  9 x 4 X 120  ^ 2 X 4 _ x 2-?-  = 5760  lbs. 

23  3 

Hence,  17  820  4-  5760  = 23  580  lbs.  moment.  Or,  - 20  * 9 ^44-3  — = 540  X 43^ 
= 23  580  lbs.  moment. 

3N* 


STABILITY. EMBANKMENT  WALLS  AND  DAMS. 


To  Compute  Elements  of  Walls  or  Dams  witb.  an 
Exterior  Batter.— Big.  1£5. 


To  Compute  WicLtb.  of  Top. 

-S  = &. 


. /2  h 2 

When  Width  of  Batter  is  Given.  -~ 

Illustration. — Assume  height  of  wall  9 and  batter  3 feet,  and  W and  w 120  and 
62. 5 lbs.  per  cube  foot. 


_ "+§! 
3 w T 3 


4 


^2  X 92  X 62.5  3f 


- x 3 = V28.125  + 3 — 3 = 2.s8/eei. 

3 X 120  3 


To  Compute  "Wid-tli  of  Base. 
When  Width  of  Batter  is  Given. 


j'z  h 

VT 


3 w 


W = B. 


+ — = 5-  58  feet  = S -f  b. 


72X9^X62.5  32 

V 3 X 120  3 

To  Compute  Widtli  of  Batter. 

3b 


. Ill 2 w 3 b 2 

When  Width  of  Top  is  Given.  + — 

j 


= S. 


_ l&lS  = V42.i8  + 4-99  ~ 3-87  = 3/^- 


lEAerc  IFiefcA  of  Bottom  is  Given.  3 B2  — - 


/t2  10 


W 


= S. 


To  Determine  Stability  of  a Detaining  Wall  or  Dam  "by 
Protraction.— Eig.  16. 

Assume  ABCD,  section  of  a wall.  On  horizontal 


line  01  centre  ui  miuou  u ~ ~ 

scale,  lay  off,  from  vertical  line  of  centre  of  gravity  • 
of  wall,  line  or  — thrust  against  wall,  and  on  vertical 
line  at  centre  of  gravity  of  wall,  at  its  intersection,  0, 
with  centre  of  thrust, let  fall  os  — weight  of  wall. 

Complete  parallelogram,  and  if  diagonal  0 u or  its 
prolongation  falls  within  C,  the  wall  is  stable,  and 
W X distance  from  line  os  — moment  of  wall. 

W representing  whole  weight  of  wall  in  lbs. 


To  Determine  Centre  of  G-ravity  of  a Wall  or  Dam.— 
Eig.  16. 


j / A B X C D\  .CD/2  A B + C D\  _ , 

By  Ordinates.  — (a  B + C D- - B + c — *.  and  3 Ub  + Ou) 


2 \ ' A B -J-  C D/  3 

To  Compute  Base  of  Dam. 

When  might , Rate  of  Balter , and  Weight  of  Materials  are  given.  Rule. 
—Multiply  square  of  Widtli  of  batter  by  .0166  weight  of  material  per  cube 
foot,  add  i,  2 or  3 times  square  of  depth  of  water,  according  as  resistance 
due  to  equilibrium  is  required,  divide  result  by  .05  weight  of  materia  pe 
cube  foot,  and  extract  square  root  of  quotient. 

nr  /xh2  + b2  X .0166  W _ x__  number  of  times  0f  resistance  required. 

Ur’V  .05  W 

rT.MPTF  Assume  a dam  40  feet  iu  height,  constructed  of  masonry  weighing 
I2olbs  per  *5  to  batter  3 ins.  per  foot  and  to  have  twice  the  resistance  due 
to  its  equilibrium ; what  should  be  its  breadth  at  its  base,  Dir 

40X3. 


^ — t/-i  — hnitor 


and  Ao*  X 2 + io°  X. 0166X120  _ ^ jrm  - aJ.  8 feet. 


STABILITY. EMBANKMENT  WALLS  AND  DAMS.  7°3 


When  Section  of  Dam  is  a Triangle , Fig.  17.  — As- 
PjggS  sume  dam,  A B C,* *to  sustain  a head  of  water,  ef. 

Rule.— Proceed  as  by  Rule  for  Fig.  14;  multiply  by 
|g|i  .033  instead  of  .05. 

Example. — As  before. 


f (2  -4-  .033  X 125)  40  = f .485  X 40  = 27.84  feet. 

Hi2  X w 

Or,  Formula  for  S (C  B),  Fig.  1 5.  / — = 28. 28  feet. 

To  Determine  Section  of  a Vertical  Wall  which  shall  have  Equal  Resist - 
ance  of  one  having  Section  of  a Triangle.  (See  J.  C.  Trautwine , Phila .,  1872.) 

To  Compute  Thickness  of  13ase  of  a Wall  or  Dain.- 
Dig.  18. 

Fig.  18.  Rule. — Divide  1,  2,  or  3 times  square  of  depth  of  water 

. by  .05  weight  of  material,  add  quotient  to  .5  batter  on  one 

jgsggg  face,  and  square  root  of  this  sum,  added  to  half  batter  on 

A|gg!jJ  other  side,  will  give  thickness. 

Or,  . / X-  4-  4-  — = Base.  b and  b'  representing 

it  f ’ V .05  W \ 2 / 2 

exterior  and  interior  batters , and  x,  as  before , number  of  times 
of  resistance  or  square  of  depth. 

^ Example. — Assume  a dam  40  feet  in  height,  to  batter  5 feet 

on  each  side,  constructed  of  masonry  weighing  120  lbs.  per  cube 
foot,  and  to  have  twice  the  resistance  due  to  its  equilibrium;  what  should  be 
breadth  of  base,  DC? 

/j°  -^X— -4-  (A)  7539.584-2.5  = 25.73/^. 

V .05  X 120  \2  / 2 


High.  Masonry  Dams. 

Rubble  Masonry,  well  laid  in  strong  cement,  will  bear  with  safety  a load 
equivalent  to  weight  of  a column  of  it  160  feet  in  height.  Assuming  such 
Fig.  I9.  masonry  as  twice  weight  of  water,  it  is  equivalent 

* to  a pressure  of  20  000  lbs.  per  sq.  foot. 

Log.  B 4-  .434  294  X ~-  — b.  B representing  width  of 

wall  at  top , and  d depth  at  any  desired  point  below  top, 
both  in  feet. 

Ordinarily,  B may  be  taken  at  18  feet,  and  in  cases 
of  extreme  and  exposed  heights  of  dam  at  20  and  more, 
and  when  b is  determined,  .9  of  it  is  to  be  on  outer  face 
of  wall,  as  A B,  and  . 1 on  inner  face. 

Illustration. — Determine  section  of  a dam,  Fig.  19, 
80  feet  in  height,  at  depths  of  10,  20,  40,  60,  and  80  feet. 
Log.  B ==  1.2553. 

Log.  1. 2553 4-. 4343  X^  = log.  1. 2553 4-. 0543  = 20. 4,  which  X .9  = 18.36. 

“ i- 2553 4- -4343  X ^ — log.  1. 2553 4-.  1086  = 23.11,  which  X .9  = 20.8. 

“ 1-2553 4- -4343  X log.  1. 2553 4-. 2172  = 29.68,  which  X .9  = 26.81. 

00 

x. 25534-. 4343  X ^ = log-  1-2553 4- -3257  = 38- 11,  which  X -9  = 34-3- 
“ 1-2553  + -4343  X ^ = !°g-  1-2553  4"  *4343  = 5°-°7,  which  X-9  = 45-o6. 


704 


STEAM. 


STEAM. 

Steam  is  generated  by  heating  of  water  until  it  attains  temperature 
of  ebullition  or  vaporization,  and  elevation  of  its  temperature  is  sensible 
to  indications  of  a thermometer  up  to  point  of  ebullition ; it  is  then 
converted  into  steam  by  additional  temperature,  which  cannot  be  in- 
dicated by  a thermometer,  and  is  termed  latent.  (See  Heat,  page  508.) 

Pressure  and  density  of  steam,  which  is  generated  in  free  contact  with  water, 
rises  with  the  temperature,  and  reciprocally  its  temperature  rises  with  the  press- 
ure and  density,  and  higher  the  temperature  more  rapid  the  pressure.  There  is 
but  one  and  a corresponding  pressure  and  density  for  each  temperature,  and  steam 
generated  in  free  contact  with  water  is  both  at  its  maximum  density  and  pressure 
for  its  temperature,  and  in  this  condition  it  is  termed  saturated , from  its  being  in- 
capable of  vaporizing  more  water  unless  its  temperature  is  raised. 

Saturated  Steam  is  the  normal  condition  of  steam  generated  in  free  contact  with 
water,  and  same  density  and  same  pressure  always  exist  in  conjunction  with  same 
temperature.  It  therefore  is  both  at  its  condensing  and  generating  points;  that 
is,  it  is  condensed  if  its  temperature  is  reduced,  and  more  water  is  evaporated  if 
its  temperature  is  raised. 

If,  however,  the  whole  of  the  water  is  evaporated,  or  a volume  of  saturated  steam 
is  isolated  from  water,  in  a confined  space,  and  an  additional  quantity  of  heat  is 
supplied  to  the  steam,  its  condition  of  saturation  is  changed,  the  steam  becomes 
superheated , and  both  temperature  and  pressure  are  increased,  while  its  density  is 
not  increased.  Steam,  when  thus  surcharged,  approaches  to  condition  of  a gas. 

With  saturated  steam,  pressure  does  not  rise  directly  with  the  temperature. 

Steam,  at  its  boiling-point,  is  equal  to  pressure  of  atmosphere,  which  is  14.723  307 
lbs,  (page  427),  at  6o°  upon  a sq.  inch. 

In  all  computations  concerning  steam,  it  is  necessary  to  have  some  or  all  of  fol- 
lowing elements,  viz. : 

Its  Pressure , which  is  termed  its  tension  or  elastic  force,  and  is  expressed  in  lbs. 
per  sq.  inch.  Its  Temperature , which  is  number  of  its  degrees  of  heat  indicated  by 
a thermometer.  Its  Density , which  is  weight  of  a unit  of  its  volume  compared 
with  that  of  water.  Its  Relative  volume , which  is  space  occupied  by  a given  weight 
or  volume  of  it,  compared  with  weight  or  volume  of  water  that  produced  it. 

Under  pressure  of  the  atmosphere  alone,  temperature  of  water  cannot  be  raised 
above  its  boiling-point. 

Expansive  force  of  steam  of  all  fluids  is  same  at  their  boiling-point. 

A cube  inch  of  water,  evaporated  under  ordinary  atmospheric  pressure,  is  convert- 
ed into  1642*  cube  ins.  of  steam,  or,  in  a unit  of  measure,  very  nearly  1 cube  foot, 
and  it  exerts  a mechanical  force  equal  to  raising  of  14.723307  X 144  = 2120.156208 
lbs.  1 foot  high. 

A pressure  of  1 lb.  upon  a sq.  inch  will  support  a column  of  mercury  at  a tem- 
perature of  6o°,  1 -4-. 4907769  (page  427)  = 2.037  586  *ns-  in  height;  hence  it  will 
raise  a mercurial  siphon  gauge  one  half  of  this,  or  1.018793  ins. 

Velocity  of  steam,  when  flowing  into  a vacuum,  is  about  1550  feet  per  second  when 
at  a pressure  equal  to  the  atmosphere ; when  at  10  atmospheres  velocity  is  increased 
to  but  1780  feet;  and  when  flowing  into  the  air  under  a similar  pressure  it  is  about 
650  feet  per  second,  increasing  to  1600  feet  for  a pressure  of  20  atmospheres. 

Boiling-points  of  Water,  corresponding  to  different  heights  of  barometer,  see 
Heat,  page  517. 

Volume  of  a cube  foot  of  water  evaporated  into  steam  at  2120  is  1642  cube  feet; 
hence  1 ~ 1642  = .000  609013,  which  represents  density  or  specific  gravity  of  steam 
at  pressure  of  atmosphere. 

Elasticity  of  vapor  of  alcohol,  at  all  temperatures,  is  about  2. 125  times  that  of  steam. 

Specific  Gravity , compared  with  air,  is  as  weight  of  a cube  foot  of  it  compared 
with  equal  volume  of  air.  Thus,  weight  of  a cube  foot  of  steam  at  2120  and  at 
pressure  of  atmosphere  is  266.124  grains;  weight  of  a like  volume  of  air  at  320  is 
565.096  grains,  and  at  62°  532.679  grains.  Hence  266. 124  =532. 679  = .499  59,  specific 
gravity  of  steam  compared  with  air  at  320,  and  with  water  it  is  .000609013. 


* Pole’s  Formuia  makes  it  1712. 


STEAM. 


705 


Total  Heat  of  Saturated  Steam. 

1081.4  -f-  .305  T = total  heat  T representing  initial  temperature  of  water. 
Illustration.— What  is  total  heat  of  steam  at  2120? 

1081. 4-}-. 305  x 212  = 1146.06. 

As  specific  heat  of  water  is  .9  greater  at  2120  than  at  320,  hence  the  2120  would 
be  212.9,  and  1146.33  the  result 

Total  Heat  of  Gaseous  Steam  1074.6  + 475  T =±±  total  heat 


Absorption  of  Heat  in  Generation  of  1 Lb.  of  Water  from  320  to  2120. 
Sensible  heat,  or  heat  to  raise  temperature  of  water  Units.  Force,  lbs. 

from  320  to  2120 180.9  X 772  = i39655 

Latent  heat  to  produce  steam 892.9 

“ tk  to  resist  atmospheric  pressure  14.7  lbs. 

per  sq.  inch 7^3  965-2  X 772  = 745  *34 

Total  or  constituent  heat 1146.1  884789 

This  number,  1146.1,  is  a Constant , and  expresses  units  of  heat  in  1 lb.  of  steam 
from  320  up  to  temperature  at  which  conversion  takes  place. 

Thus,  1 lb.  water  heated  from  32 0 to  332 °,  requires  as  much  heat  as 

would  raise  300  lbs.  i°.  Hence 3°°° 

And  1 lb.  water  converted  into  steam  at  3320  (=  106  lbs.  pressure),  ab- 
sorbs as  much  heat  for  its  conversion  as  would  raise  846.1  lbs.  water 

i°.  Hence 846.1° 

1146.1° 

Mechanical  Equivalent  of  Heat  contained  in  Steam. 

1 lb.  water  heated  from  32°  to  212°  requires  as  much  heat  as  would  raise 

180  lbs.  1°  Hence 180.9° 

1 lb.  water  at  2120,  converted  into  steam  at  212°  (=  14.7  lbs.  pressure), 
absorbs  as  much  heat  for  its  conversion  as  would  raise  966.6  lbs.  water 

1°  Hence 965-2° 

1146. 1° 

Mechanical  Equivalent , or  maximum  theoretical  duty  of  quantity  of  heat  in  1 lb. 
of  steam,  is  772  lbs.,  which  X 1146. 1 units  of  heat  •'■==■  884  789.2  lbs.  raised  1 foot  high. 

To  Compute  Pressure  of*  Steam. 

When  Height  of  Column  of  Mercury  it  will  Support  is  given.  . Rule. — Di- 
vide height  of  column  of  mercury  in  ins.  by  2.037  586,  and  quotient  will  give 
pressure  per  sq.  inch  in  lbs. 

Example.  — Height  of  a column  of  mercury  is  203.7586  ins. ; what  pressure  per 
sq.  inch  will  it  contain  ? 

203. 7586  -r--  2.037  586  = 100  lbs. 

To  Compute  Weiglit  of  a CiPfc>e  Foot  of  Steam. 
Rule. — Multiply  its  density  by  62.425. 

Example. — Density  of  a volume  of  steam  is  .000609013;  what  is  its  weight? 
.000609013  X 62.425 —-.038016  825  lbs. 


Note. — See  table,  page  708. 

1 atmosphere  or  14.723307  lbs.  per  sq.  inch  = 30  ins.  of  mercury. 

To  Compute  Temperature  of  Steam. 

Rule. — Multiply  6tli  root  of  its  force  in  ins.  of  mercury  by  177.2,  sub- 
tract 100  from  product,  and  remainder  will  give  temperature  in  degrees. 

Example. — When  elastic  force  of  steam  is  equal  to  a pressure  of  64  ins.  of  mer- 
cury, what  is  its  temperature? 

Note. — To  extract  6th  root  of  a number,  ascertain  cube  root  of  its  square  root. 
V64  = 8,  and  -^8  = 2.  Hence,  2 X 177-  2 — 100  = 254. 4°  t. 

Or, -93^' 3_r  85  — t.  p representing  pressure  in  lbs.  per  sq.  inch. 

6.1993544  — log.  p 


STEAM. 


706 

To  Compute  Y olume  of' Water  contained,  in  a given  'V'ol- 
nme  of*  Steam. 

When  its  Density  is  given.  Rule. — Multiply  volume  of  steam  in  cube 
feet  by  its  density,  and  product  will  give  volume  of  water  in  cube  feet. 

Example. — Density  of  a volume  of  16420  cube  feet  of  steam  is  .000609;  what  is 
the  weight  of  it  in  lbs.  ? 

16420  X .000609  = IO  — volume  of  water,  which  X 62.425  = 624.25  lbs. 

To  Compute  Pressure  of*  Steam  in  Ins.  of*  Mercury,  or 
X/bs.  per  Sq.  Incli. 

When  Temperature  is  given.  Rule  i. — Add  100  to  temperature,  divide 
sum  proportionally  by  177.2  for  temperature  of  2120,  and  by  160  for  tem- 
peratures up  to  4450  ; or,  177.6  for  sea-water,  and  185.6  for  sea-water  sat- 
urated with  salt,  and  6th  power  of  quotient  will  give  pressure. 

Example. — Temperature  of  steam  is  2540;  what  is  its  pressure? 

100 -f-254-r- 177.2  = 1.998,  and  1. 998s  = 63.62  ins. 

When  Ins.  of  Mercury  are  given.  2. — Divide  ins.  of  mercury  by  2.037  586, 
and  quotient  will  give  pressure. 

When  Pressure  in  Lbs.  is  given.  3. — Multiply  pressure  by  2.037  586. 

To  Compute  Specific  Gravity  of*  Steam  compared  with. 
.A.i  r. 

Rule.  — Divide  constant  number  829.05  (1642  X .5049)  by  volume  of 
steam  at  temperature  of  pressure  at  which  gravity  is  required. 

Example.— Pressure  of  steam  is  60  lbs.,  and  volume  437 ; what  its  specific  gravity? 
829.05  = 437  = 1.898. 

To  Compute  Volume  of*  a Cube  Foot  of  Water  in  Steam. 

When  Elastic  Force  and  Temperature  of  Steam  are  given.  Rule. — To 
430.25  for  temperature  of  2120,  and  332  for  temperatures  up  to  4450,  add 
temperature  in  degrees ; multiply  sum  by  76.5,  and  divide  product  by  elastic 
force  of  steam  in  ins.  of  mercury. 

Note. — When  force  in  ins.  of  mercury  is  not  given,  multiply  pressure  in  lbs.  per 
sq.  inch  by  2.037  586. 

Example.— Temperature  of  a cube  foot  of  water  evaporated  into  steam  is  386°, 
and  elastic  force  is  427.5  ins. ; what  is  its  volume? 

Assume  369  for  proportionate  factor.  369  -f  386  X 76. 5 = 427. 5 =;  135. 1 cube  feet. 

Or,  for  1 lb.  of  steam,  2.519  — .941  log.  j?  = log.  V in  cube  feet. 

Assume  p = 14.7  lbs.  2.519  — .941  log.  14.7  = 2.519  — 1.098  = 1. 421  = log.  26.34 
cube  feet,  which  X 62.425  = 164  feet. 

Or,  When  Density  is  given. — Divide  1 by  density,  and  quotient  will  give  volume 
in  cube  feet. 

To  Compute  Density-  or  Specific  (Gravity  of*  Steam. 
When  Volume  is  given.  Rule. — Divide  1 by  volume  in  cube  feet. 
Example. — Volume  is  210;  what  is  density? 

1 -7-  210  = .004  761.  Or,  for  1 lb.  of  steam,  .941  log.  p — 2. 5 19  = log.  D. 

When  Pressure  is  given. — Take  temperature  due  to  pressure,  and  proceed 
as  by  rule  to  compute  volume,  which,  when  obtained,  proceeds  as  above. 

To  Compute  Volume  of  Steam  required  to  raise  a (Given 
Volume  of*  Water  to  any  (Given  Temperature. 

Rule. — Multiply  water  to  be  heated  by  difference  of  temperatures  between 
it  and  that  to  which  it  is  to  be  raised,  for  a dividend ; then  to  temperature 
of  steam  add  965.2°,  from  that  sum  take  required  temperature  of  water  for 
a divisor,  and  quotient  will  give  volume  of  water. 


Example.  - 

tO  212° ? 


STEAM.  707 

-What  volume  of  steam  at  212°  will  raise  100  cube  feet  of  water  at  8o° 


iqq  X 212 80^  _ cube  feet  water;  or,  (13.68  X 1642  — 212)  = 22  463  of  steam. 

212+  965.2 — 212 

To  Compute  Volume  of  Water,  at  any-  Given  Temper- 
ature, that  must  he  Mixed  with  Steam  to  Raise  or  ne- 
dace  the  Mixture  to  any  Required  Temperatare. 
Rule. — From  required  temperature  subtract  temperature  of  water ; then 
ascertain  how  often  remainder  is  contained  in  required  temperature  sub- 
tracted from  sum  of  sensible  and  latent  heat  of  the  steam,  and  quotient  will 
give  volume  required. 

Sum  of  Sensible  and  Latent  Heats  for  a range  of  temperatures  will  be  found  under 
Heat,  pages  508  and  509. 

Example.— Temperature  of  condensing  water  of  an  engine  is  8o°,  and  required 
temperature  ioo°;  what  is  proportion  of  condensing  water  to  that  evaporated  at  a 
pressure  of  34  lbs.  per  sq.  inch  ? 

Sum  of  sensible  and  latent  heats  1190.4°. 

100  — 80  = 20.  Then,  1190.4  — ioo-=-2o  = 54.52  to  1. 

When  .Temperature  of  Steam  is  given.  1 representing  latent  heat , 

T and  t temperatures  of  steam  and  required  temperature , w temperature  of  condensing 
water , and  V volume  of  condensing  water  in  cube  feet. 

Illustration.— Temperature  of  steam  in  a cylinder  is  257.6°,  and  other  elements 
same  as  in  preceding  example;  required  volume  of  injection  water?  Latent  heat 
of  steam  at  230°  = 932.8°. 

_ *°9°-  4 _ ^ voiumes . 


932. 8 + 257. 6- 
100  — 80 


To  Compute  Temperature  of  Water  in  Condenser  or 
Reservoir  of  a Steam-engine. 


1 T-~^  x w — t.  Illustration.— Assume  elements  as  preceding. 

V + i 

932-8  + 257.6  + 54.52  X 80  __  5,552  __ 

54.52  + 1 55-52 

To  Compute  Latent  Heat  of  Saturated  Steam. 

1115.2  — .708  t = l.  Illustration.— Assume  temperature  257.6°  as  preceding. 
1115.2  — .708  X 257.6  = 932.8°. 

To  Compute  Total  Heat  of  Saturated  Steam. 

305  t-\-  1081.4  = H.  Illustration. — Assume  temperature  as  preceding. 

.305  X 257.6+  1081.4  = 1160. 

Elastic  Force  and  Temperature  of  Vapors  of  Alcohol, 
Ether,  Sulphuret  of  Carbon,  Petroleum,  and  Tur- 
pentine. 

Force  in  Ins.  of  Mercury. 

I Ins. 


0 1 

Ins. 

0 1 

1 Ins. 

0 1 

Ins. 

o | 

Ins. 

Alcohol. 

Alcohol. 

Ether. 

Sulphuret  of 

32 

•4 

140 

13-9 

34 

6.2 

Carbon. 

50 

.86 

160 

22.6 

54 

i5-3 

53-5 

7-4 

60 

1.23 

i73 

3° 

74 

16.2 

72-5 

12.55 

70 

1.76 

180 

34-73 

94 

24.7 

no 

3° 

80 

2-45 

200 

53 

96 1 

3° 

212 

126 

90 

3-4 

212 

67-5 

104) 

279-5 

300 

100 

4-5 

220 

78.5 

120 

39-47 

347 

606 

120 

8.1 

240 

111.24 

15° 

67.6 

130 

10.6 

264 

166. 1 

212 

178 

Petroleum. 
316  I 30 
345  44-i 

375  I 64 

Oil  of 
Turpentine. 
315  I 3° 
357  47-7^ 

370  | 62.4 


7 o8 


STEAM, 


Saturated.  Steam. 
Pressure,  Temperature , Volume , and  Density. 


Pressure 

per  I in 
Sq.  Mer- 

Inch.  cury. 

Temperature. 

Total  Heat 
from  Water 
at  320. 

Volume  of 
1 Lb. 

Lbs.  j 

Ins. 

0 

0 

Cub.  ft. 

1 | 

2. 04 

102. 1 

1 1 12. 5 

330.36 

2 

4.07 

126.3 

1119.7 

172. 08 

6. 11 

141.6 

1124.6 

117.52 

4 

8. 14 

*53* 

1128. 1 

89.62 

5 

10. 18 

162.3 

II30-9 

72.66 

6 

12.22 

170.2 

1*33-3 

61.21 

7 

14.25 

176.9 

i*35-3 

52-94 

8 

16.29 

182.9 

1137.2 

46.69 

9 

18.32 

188.3 

1138.8 

4*. 79 

10 

20.36 

*93-3 

1140-3 

37-84 

11 

22.39 

197.8 

**4*-7 

34-63 

12 

24- 43 

202 

**43 

31.88 

*3 

26.46 

205.9 

1144-2 

29-57 

*4 

28.51 

209.6 

1*45-3 

27.61 

14.7 

29.92 

212 

1146. 1 

26.36 

i5 

30.54 

213. 1 

1146.4 

25-85 

16 

32-57 

216.3 

1147.4 

24.32 

*7 

34.61 

219.6 

1148.3 

22.96 

18 

36.65 

222.4 

1149.2 

21.78 

*9 

38.68 

225.3 

1150-1 

20.7 

20 

40.72 

228 

1150.9 

19.72 

21 

42.75 

230.6 

**5*-7 

18.84 

22 

44-79 

233-* 

1152.5 

18.03 

23 

46.831235.5 

1*53-2 

17.26 

24 

48.86  | 237.8 

**53-9 

16.64 

25 

50.9 

240. 1 

ii54-6 

*5-99 

26 

52.93 

242.3 

*155-3 

*5-38 

27 

54-97 

; 244-4 

1*55-8 

14.86 

28 

57-o* 

j 246.4 

1156.4 

*4-37 

29 

59- °4 

248.4 

1*57-1 

*3-9 

30 

61.08 

250.4 

1157.8 

13.46 

3* 

63.11 

252.2 

1158.4 

*3-°5 

32 

65-15 

254.1 

1158.9 

12.67 

33 

67.19 

255-9 

**59-5 

12.31 

34 

69.22 

257.6 

1160 

11.97 

35 

7*. 26 

259-3 

1160.5 

11.65 

36 

73-29 

260.9 

1161 

**•34 

37 

75-33 

262.6 

1161.5 

11.04 

38 

77-37 

264.2 

1162 

10.76 

39 

79-4 

265.8 

1162.5 

10.51 

40 

81.43 

267.3 

1162.9 

10.27 

4* 

83-47 

268.7 

1163.4 

10.03 

42 

85-5 

270.2 

1163.8 

9.81 

43 

87-54 

271.6 

1164.2 

9-59 

44 

89.58 

273 

1164.6 

9-39 

45 

91.61 

274.4 

1165.1 

9. 18 

46 

93-65 

275.8 

*165-5 

9 0 

47 

95-69 

277.1 

11659 

8.82 

48 

97.72 

278.4 

1166.3 

8.65 

49 

99.76 

279.7 

1166.7 

8.48 

5° 

101.8 

281- 

1167. 1 

8.31 

51 

103.83 

282.3 

1167.5 

8.17 

52 

105.87 

283.5 

H67.9 

8.04 

53 

107.9 

284.7 

1168.3 

7.88 

54 

*09.94 

285.9 

1168. 6 

7-74 

55 

111.98 

287. 1 

1169 

7.61 

56 

114.01 

288.2 

1169.3 

7.48 

57 

116.05 

289.3 

1169.7 

1 7-36 

Lb. 
.003 
.005  8 
.008  5 
.011  2 
.013  8 
.016  3 
.018  9 
.021  4 
.0239 
.026  4 
.028  9 
031  4 
•°33’8 
. 036  2 
.038  02 
.0387 
.041  1 
•°4  35 
•045  9 
.0483 
.0507 
•053  1 
.055  5 
.058 
.060 1 
.062  5 
.065 
.0673 
.0696 

.071  9 

•°74  3 

.0766 

.0789 

.081  2 

.0835 

.085  8 

.088  1 

.0905 

.092  9 

.0952 

.0974 

.0996 

.102 

.104  2 

.1065 

.1089 

.111  1 

•1133 

.1156 
.1179 
. 120  2 
. 122  4 
. 124  6 
. 126  9 
. 129  1 
•*3*4 
•*336 
.1364 


Inc! 


Lbs. 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

8.3 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

96 

97 

98 

99 

100 

101 

102 

103 

104 

i°5 

106 

107 

108 

109 
no 
in 
112 
**3 

114 

115 


Mer- 

cury. 


Ins. 

118.08 
120. 12 

122. 16 
124.19 

126.23 

128.26 
I30-3 
I32-34 
134-37 
136.4 
138.44 

140.48 

142.52 
144-55 
*46-59 

148.62 

150. 66 
152.69 
154-73 

156.77 

158.8 

160.84 

162.87 
164.91 

166.95 
168.98 

171.02 
173-05 
175-09 
I77-I3 

179.16 

181.2 

183.23 

185.27 
187.31 
189.34 
191.38 
i93-4i 
195-45 

197.49 

199.52 
201.56 

203-59 

205.63 

207. 66 
209.7 
211.74 

213.77 
215.81 

217.84 

219. 88 
221.92 

223.95 

225.99 

228.02 
230.06 
232. 1 
234-  *3 


290.4 

291.6 

292.7 

293.8 

294.8 
295-9 

296.9 

298 

299 

3°o 

300.9 

301.9 

302.9 

303-9 

304.8 
305-7 

306.6 

307-5 

308.4 

309-3 

310.2 

311.1 
312 

312.8 

313- 6 

314- 5 

315- 3 

316.1 

316.9 

317-8 

318.6 
3I9-4 

320.2 
321 

321.7 

322.5 

323-3 

324.1 

324.8 
325-6 

326.3 

327-1 

327-9 

328.5 

329.1 

329-9 

330.6 
33i-3 

331- 9 

332- 6 

333- 3 
334 

334- 6 

335- 3 
336 

336- 7 

337- 4 
338 


53 

o w 


1170 

1170.4 

1 170. 7 

1171.1 

1171-4 

1171.7 
1172 

1172.3 

1 172. 6 
1172.9 
H73-2 
H73-5 

1173.8 
H74- 1 
**74-3 

1174.6 
H74.9 

1175.2 
H75-4 
ii75 -7 
1176 

1176.3 

1176.5 

1176.8 
1177.! 
1*77-4 

1177.6 

1*77-9 

1178. 1 

1178.4 

1178.6 

1178.9 

1179. 1 
i*79-3 
i*79-5 

1179.8 

1180 

1180.3 

1180.5 

1 180. 8 

1181 

1181.2 

1181.4 

1181.6 

1181.8 

1182 

1182.2 

1182.4 

1182.6 

1182.8 

1183 

1183-3 

1183.5 

1183.7 

1183.9 
! 1184.1 
I 1184.3 
| 1184.5 


Cub.  ft. 
7.24 
7. 12 
7.01 
6.9 
6.81 

6.7 
6.6 
6.49 

6.41 
6. 32 
6.23 

6.15 

6.07 
5-99 
5-83 

5-76 

5.68 
5.61 
5-54 
5-48 
5-4* 
5-35 

5-29 

5-23 
5- *7 
5-** 
5-05 
5 

4.94 

4.89 

4.84 

4-79 

4-74 

4.69 
4.64 

4.6 
4-55 
4-5* 
4.46 

4.42 
4-37 
4-33 
4.29 
4-25 
4.21 
4. 18 

4*4 

4. 11 

4.07 
4.04 
4 

3-97 

3-93 

3-9 

3.86 

3-83 

3-8 


1^ 


Lb. 

*38 

1403 

1425 

*447 

.1469 

*493 

.1516 

1538 

.156 

.*583 

.1605 
.1627 
. 1648 
.167 
. 1692 
1714 
.1736 
•*759 
.1782 
.1804 
.1826 
. 1848 
. 1869 
1891 
*9*3 
1935 
*957 
.198 
.2002 
2024 
2044 
.2067 
.2089 
.2111 

•2133 

•2155 

.2176 

.2198 

.2219 

.2241 

.2263 

.2285 

.2307 

.2329 

.235* 

•2373 

•2393 

.2414 

•2435 

.2456 

•2477 

.2499 

.2521 

-2543 

2564 

.2586 
. 2607 
. 2628 


STEAM.  709 


Pr 

e 

Inch. 

SSSURE 

in 

Mer- 

cury. 

Temperature. 

Total  Heat 
from  Water 
at  32°. 

Volume  of 
1 Lb. 

Density, 
or  Weight  of 
one  Cube  Foot. 

Pressure 

per  I in 

Sq.  Mer- 

Inch.  cury. 

Temperature. 

Total  Heat 
from  Water 
at  320. 

Volume  of 
1 Lb. 

Density, 
or  Weight  of 
one  Cube  Foot. 

Lbs. 

Ins. 

0 

O 

Cub.  ft. 

Lb. 

Lbs. 

Ins. 

0 

O 

Cub.  ft. 

Lbs. 

116 

236.17 

338.6 

1184.7 

3-77 

.2649 

149 

303-35 

357-8 

1 1 90. 5 

2.98 

•3357 

117 

238.2 

339-3 

1184.9 

3-74 

.2652 

150 

. 305-39 

358.3 

1 190. 7 

2.96 

-3377 

118 

240.24 

339-9 

1185.1 

3-7i 

.2674 

155 

3I5-57 

361 

1191.5 

2. 87 

•3484 

119 

00 

OJ 

c3 

cT 

340-5 

1185.3 

3-68 

.2696 

160 

325-75 

363-4 

1192. 2 

2.79 

•359 

120 

244.31 

341-1 

1185.4 

3-65 

•2738 

165 

335-93 

366 

1192.9 

2.71 

•3695 

121 

246.35 

341-8 

1185.6 

3.62 

•2759 

170 

346.11 

368.2 

XI93-7 

2.63 

•3798 

122 

248.38 

342-4 

1185.8 

3-  59 

.278 

175 

356.29 

370.8 

1x94.4 

2.56 

.3899 

123 

250.42 

343 

1186 

3-56 

.2801 

180 

366.47 

372-9 

1195-1 

2-49 

.4009 

124 

252.45 

343-6 

1186.2 

3-54 

.2822 

i85 

376.65 

375-3 

1195.8 

2-43 

.4117 

125 

254.49 

344-2 

1186.4 

3-5i 

.2845 

190 

386.83 

377-5 

1196.5 

2-37 

.4222 

126 

256.53 

344-8 

1186.6 

3-49 

.2867 

i95 

397-ox 

379-7 

1X97-2 

2.31 

•4327 

127 

258.56 

345-4 

1186.8 

3-46 

.2889 

200 

407. 19 

.381.7 

1197.8 

2.26 

•4431 

128 

260.6 

346 

1186.9 

3-44 

.2911 

210 

427-54 

386 

II99-I 

2. 16 

•4634 

129 

262.64 

346.6 

1187. 1 

3-4i 

•2933 

220 

447-9 

389-9 

1200.3 

2.06 

.4842 

130 

264.67 

347-2 

1x87.3 

3-38 

•2955 

230 

468.26 

393-8 

1201. 5 

1.98 

•5052 

131 

266.71 

347-8 

1187.5 

3-35 

•2977 

240 

488.62 

397-5 

1202.6 

1.9 

.5248 

132 

268.74 

348.3 

1187.6 

3-33 

•2999 

250 

508.98 

401. 1 

1203.7 

1.83 

•5464 

133 

270.78 

348-9 

1187.8 

3-3i 

.302 

260 

529-34 

404-5 

1204.8 

1.76 

.5669 

134 

272.81 

349-5 

1188 

3-29 

•304 

270 

549 -7 

407.9 

1205.8 : 

x-7 

.5868 

135 

274.85 

35o.i 

1188.2 

3-27 

.306 

280 

570. 06 

411.2 

1206.8 

1.64 

.6081 

136 

276. 89 

350.6 

1188.3 

3-25 

.308 

290 

590. 42 

4x4.4 

1207.8 

x-59 

.6273 

137 

278.92 

35i-2 

1188.5 

3.22 

.3101 

300 

610.78 

417-5 

1208.7  I 

i-54 

.6486 

138 

280. 96 

35i-8 

1188.7 

3-2 

• 3121 

350 

712-  57 

43o.i 

1212.6 

i-33 

.7498 

139 

282.99 

352.4 

1188.9 

3.18 

.3142 

400 

8x4-37 

444.9 

1217.1 

1. 18 

.8502 

140 

285.03 

352-9 

1189 

3. 16 

.3162 

45o 

916. 17 

456.7 

1220.7 

1.05 

•9499 

141 

287.07 

353-5 

1189.2 

3-i4 

.3184 

500 

1018 

467-5 

1224 

-95 

1.049 

142 

289.  j 

354 

1189.4 

3.12 

.3206 

550 

1 1 19. 8 

477-5 

1227 

•87 

1. 148 

143 

291.14 

354-5 

1189.6 

3-i 

.3228 

600 

1221.6 

487 

1229.9 

.8 

1.245 

144 

293-17 

355 

1189.7 

3.08 

-325 

650 

1323-4 

495-6 

1232.5 

•74 

1.342 

145 

295.21 

355-6 

1189.9 

3.06 

•3273 

700 

1425-8 

504.1 

1235- 1 

.69 

x-4395 

146 

297.25 

356.i 

1190 

3- °4 

•3294 

800 

1628. 7 

5I9-5 

1239.8 

.61 

1.6322 

147 

299.28 

356-7 

1 190. 2 

3.02 

•33i5 

900 

1832.3 

533-6 

1244.2 

•55 

1-8235 

148 

301.32 

357-2 

II90-3 

3- 

•3336 

1000  J 

2035-9 

546-5 

1248. 1 

•5 

2.014 

Saturated.  Steam  from  32°  to  STS0.  {Claudel.) 


Tem- 

pera- 

ture. 

PRE! 

Mercu- 

ry- 

5SURE. 

Per 

Sq. Inch. 

Weight 
of  100 
Cub.  Feet. 

Volume 

of 

1 Lb. 

Tem- 

pera- 

ture. 

Pre 

Mercu- 

ry- 

SSURE. 

Per 

Sq. Inch. 

Weight 
of  100 
Cub. Feet. 

Volume 

of 

1 Lb. 

0 

Ins. 

Lbs. 

Lb. 

Cub.  Feet. 

0 

Ins. 

Lbs. 

Lbs. 

Cub.  Feet. 

32 

.181 

.089 

.031 

3226 

125 

3-933 

x-932 

•554 

180.5 

35 

.204 

. 1 

•034 

2941 

130 

4-509 

2.215 

•63 

158-  7 

4° 

.248 

.122 

.041 

2439 

135 

5-174 

2.542 

•7x4 

140.  X 

45 

.299 

.147 

•049 

2041 

140 

5.86 

2.879 

.806 

124. 1 

50 

.362 

.178 

•059 

1695 

145 

6.662 

3-273 

.909 

IIO 

55 

.426 

.214 

•07 

X429 

150 

7-548 

3.708 

1.022 

97.8 

60 

•517 

•254 

.082 

1220 

155 

8-535 

4-193 

i- 145 

87-3 

65 

.619 

•304 

•097 

1031 

160 

9-63 

4-731 

x-333 

75 

70 

•733 

•36 

.114 

877.2 

165 

10.843 

5-327 

x.432 

69.8 

75 

.869 

•427 

■134 

746-3 

170 

12.183 

5-985 

1.602 

62.4 

80 

1.024 

•503 

.156 

641 

175 

13-654 

6. 708 

x-774 

56.4 

85 

1.205 

•592 

.182 

549-5 

180 

15.291 

7-5II 

x-97 

50.8 

90 

1.41 

•693 

.212 

47X-7 

185 

17.041 

8-375 

2.181 

45-9 

95 

1.647 

.809 

•245 

408.2 

190 

19.001 

9-335 

2.411 

4X-5 

100 

I-9I7 

•942 

.283 

353-4 

195 

21.139 

10.385 

2.662 

37-6 

105 

2.229 

1.095 

•325 

307-7 

200 

23.461 

11.526 

2-933 

34- x 

IIO 

2-579 

1.267 

•373 

268.1 

205 

25.994 

12.77 

3.225 

3i 

xx5 

2.976 

1.462 

.426 

234-7 

210 

28.753 

14.127 

3-543 

28.2 

120 

3-43 

1.685 

.488 

204.9 

212 

29.922 

14.7 

3.683 

27.2 

30 


7io 


STEAM. 


GASEOUS  STEAM. 

When  saturated  steam  is  surcharged  with  heat,  or  superheated,  it  is  termed 
gaseous  or  steam-gas.  The  distinguishing  feature  of  this  condition  of  steam 
is  its  uniformity  of  rate  of  expansion  above  230°,  with  the  rise  of  its  tem- 
perature, alike  to  the  expansion  of  permanent  gases. 

To  Compute  Total  Heat  of  Gaseous  Steam. 

1074.6 -475  £ = H.  t representing  temperature , and  H total  heat  in  degrees. 
Hence,  total  heat  at  2120,  and  at  atmospheric  pressure  = 1175.3°. 

Specific  gravity  = .622. 

To  Compute  'V'elocity  of  Steam. 

Into  a Vacuum.  Rule. — To  temperature  of  steam  add  constant  459,  and 
multiply  square  root  of  sum  by  60.2 ; product  will  give  velocity  in  feet  per 
second. 

Into  Atmosphere.  3.6  ffh  = V.  V representing  velocity  as  above , and  h height  in 
feet  of  a column  of  steam  of  given  pressure  and  uniform  density , weight  of  which  is 
equal  to  pressure  in  unit  of  base. 

Illustration. — Pressure  of  steam  100  lbs.  per  sq.  inch,  what  is  velocity  of  it# 
flow  into  the  air? 

Cube  foot  of  water  = 62. 5 lbs.,  density  of  steam  at  100  lbs.  = 270  cube  feet.  Hence, 

62. 5 : 100  : : 270  : 432  =1  volume  at  ico  lbs.  pressure , and  432  X 144  = 62  208  feet  = 
height  of  a column  of  steam  at  a pressure  0/100  lbs.  per  sq.  inch. 

Then  3. 6 y/62  208  = 898  feet. 

EXPANSION. 

To  Compute  Point  of  Cutting  off  to  Attain  Limit  of 
Expansion. 

b _p/  L p — point  of  cutting  off.  b representing  mean  bach  pressure  for  entire 

stroke , in  lbs.  per  sq.  inch , f friction  of  engine,  P initial  pressure  of  steam , all  in  lbs. 
per  sq.  inch , and  L length  of  stroke,  in  feet. 

Illustration.  — Assume  stroke  of  piston  9 feet,  pressure  30  lbs. . mean  back  press- 
ure 3 lbs.,  and  friction  2 lbs.  ’ ? 

3 + 2 X 9-^ 30  = 1-5  fMt 

To  Compute  Actual  Ratio  of  Expansion. 

I _1_  c 

-—A—  — R.  c representing  clearance  or  volume  of  space  between  valve  seat  and 

1 4~  c 

mean  surf  ace  of  piston,  at  one  or  each  end  in  feet  of  stroke,  l length  of  stroke  at  point 
of  cutting  off,  excluding  clearance  in  feet,  and  R actual  ratio  of  expansion. 

Illustration. — Assume  length  of  stroke  2 feet,  clearance  at  each  end  1.2  ins., 
and  point  of  cutting  off  1 foot. 

1.2  ins.  = . 1.  Then  ^4—  = !- 9 ratio.  t 

I+-.X 

To  Compute  Pressure  at  any  Point  of  Eeriod  of  Ex- 
pansion. 

When  Initial  Pressure  is  given.  P l-±-s=p.  p representing  pressure  at  period  t 
of  given  portion  of  stroke,  both  in  lbs.  per  sq.  inch , and  s any  greater  portion  of  stroke 
than  l. 

When  Final  Pressure  is  given.  P'xL'4-s  = j).  P'  representing  final  pressure, 
in  lbs.  per  sq.  inch,  and  L'  length  of  stroke,  including  clearance,  in  feet. 

Illustration  i. — Assume  length  of  stroke  6 feet,  clearance  at  each  end  1.2  ins., 
pressure  of  steam  60  lbs.,  point  of  cutting  off  one  third;  what  is  pressure  at  4 feet? 

1.2  ins.  .= . 1 foot.  60  X 2 -f- . 1 -4-  4 -f- . 1 = 30. 73  lbs. 

2.. — What  is  pressure  in  above  cylinder  at  2.8  feet,  when  final  pressure  is  21  lbs.  ? 

21  X 6 + .1  -r-  2.8  -f-  .1  = 44-17  lbs. 


STEAM. 


/II 


To  Compute  M.ean  or  Total  Average  Pressure. 


P P*  log~  R — — — p'  or  mean  or  average  pressure.  I ' length  of  stroke  at 

point  of  cutting  off,  including  clearance. 

Illustration. — Assume  elements  of  preceding  cases:  i -f  hyp.  log.  R = 2.065. 


60  (2. 1 X 2.065  — • 1) 254. 19 


: 42.365  ll)S. 


To  Compute  Pinal  Pressure. 

PXi'4-S  = P'. 

Illustration. — Assume  elements  of  preceding  cases,  steam  cut  off  at  2 feet. 
60  X 2 -f- . 1 -f-  6 -j- . 1 = 20. 65  lbs. 

To  Compute  Afean  Effective  Pressure. 


P (1 V 1 -fhyp.  log.  R — c 


— b,  or  (p'  — b). 


Illustration. — Assume  elements  of  preceding  cases,  b~  2 lbs.  per  sq.  inch. 

- — 2 = 40.  365  IbS . 


60  (2.1  X 2.065  — . 1) 


254- 19 


To  Compute  Initial  Pressure  to  Produce  a Given  .A.V— 
erage  Effective  or  I^et  Pressure. 

P'  L 


l'  ft  +■  hyp.  log.  R)  — c 

Illustration. — Assume  elements  of  case  1. 


= P. 


6 + -i 
2 + .1 


= 2.  g ratio. 


42-365  X 6 
(2.1  X 2.065)  — 


_ 254-19 
’4-2365" 


= 60  lbs. 


To  Compute  Point  of  Cutting  off  for  a Given  Patio  of 
Expansion. 

1/  -4-  R — c.  Or,  L + c -4-  R — c — l. 

Illustration. — Assume  elements  of  preceding  cases : R = — ==  2.9,  and  ~^~‘1 

2 -j- . 1 2.9 

— .1  = 2 feet. 

To  Compute  Pressure  in  a Cylinder,  at  any-  Point  of  Ex- 
pansion, or  at  End  of  Stroke. 

P r-r-H-7=  P,  or  P4-  R. 

Illustration. — Assume  elements, of  preceding  cases: 


60  X 2. 1 60 

- = 60  lbs.,  and  — = 20.69  lbs. 
2.9 


2-f  .1 


To  Compute  Initial  Pressure  for  a Required  3NTet  Effec- 
tive Pressure  for  a Given  Patio  of  Expansion. 
W+a&L  ^ p'  L 


Or,  - 


= P.  W representing  net- 


a (1/  1 — J—  hyp.  log.  k — c)  V 1 + hyp.  log.  R — c 

work  in  foot-lbs.  = aLp'  — b,  and  a area  of  piston,  in  sq.  ins. 

Illustration.— Assume  elements  of  preceding  cases:  area  of  piston  = 100  sq. 
ins.,  back  pressure  2 lbs.,  and  net  effective  pressure  = 42.365  lbs. 


100  X 6 X 42. 365  — 2 = 24  219  foot-lbs. 
24219  + 100X2X6  _ 25419 ^ 42.365X6 


_ 254+9 


= 60  lbs. 


712 


STEAM. 


Points  of  Expansion.  i 

Relative  points  of  expansion,  including  clearance  5 per  cent.,  assuming 
stroke  of  piston  to  be  divided  as  follows,  and  initial  pressure  = 1. 

Point 1 .75  -6875  .625  .5625  .5  .4375  -375  -333  -25  -2  .125  .1 

Ratio 1 1. 31  1.43  i-55  I-7I  *-9r  2-i5  2.43  2.74  3*5  4-4  6-  7- 

Hyp.  Log.  of  above  Ratios. 

.0  1.27  1.36  1.44  1.54  1-65  i-77  r-9  2 2-25  2-43  2-79  2-95 


Hyperbolic  Logarithms. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

No.  | 

Log. 

1.05 

i 

00 

00 

2.65 

•9746  . 

4-25 

1.447 

5-8 

x-758 

7-4 

2.001 

•0953 

2.66 

•9783 

4-3 

i-459 

5-85 

1.766 

7-45 

2.008 

1. 15 

.1398 

2.7 

•9933 

4-33 

1.466 

5-9 

1-775 

•7:5 

2.015 

1.2 

.1823 

2-75 

1. 0116 

4-35 

1.47 

5-95 

1-783 

7-55 

2.022 

1.25 

.2231 

2. 8 

1.0296 

4.4 

1.482 

6 

1.792 

7.6 

2.028 

1.3 

.2624 

2.85 

i-o473 

4-45 

i-493 

6.05 

1.8 

7-65 

2.035 

i-33 

.2852 

2.9 

1.0647 

4-5 

1.504 

6. 1 

1.808 

7.66 

2.036 

1.35* 

.3001 

2-95 

1. 0818 

4-55 

i-5i5 

6.15 

1. 816 

7-7 

2.041 

1.4 

•3365 

3 

1.0986 

4.6 

1.526 

6.2 

1.824 

7-75 

2.048 

i-45 

• 3716 

3°5 

1.1151 

4-65 

i-537 

6.25 

1-833 

7.8 

2.054 

1.5 

•4055 

3- 1 

1. 1314 

4.66 

i-539 

6-3 

1.841 

7-85 

2.061 

i-55 

-4383 

3^5 

I*I474 

4-7 

1.548 

6-33 

1.845 

7-9 

2. 067 

1.6 

•47 

3-2 

1.1632 

4-75 

i-558 

6-35 

1.848 

7-95 

2.073 

1.65 

.5008 

3-25 

1. 1787 

4.8 

1.569 

6.4 

1.856 

8 

2.079 

1.66 

.5068 

3-3 

*•*939 

4-85 

i-579 

6-45 

1.864 

8.05 

2.086 

1.7 

•53°6 

3-33 

1.203 

4.9 

1.589 

6-5 

1.872 

8. 1 

2.092 

1.75 

•5596 

3-35 

1.209 

4-95 

1-599 

6-55 

1.879 

8.15 

2.098 

1.8 

.5878 

3-4 

1.2238 

5 

1.609 

6.6 

1.887 

8.2 

2.104 

1.85 

.6152 

3-45 

1.2384 

5-05 

1.619 

6.65 

1.895 

8.25 

2.11 

1.9 

.6419 

3-5 

1.2528 

5-i 

1.629 

6.66 

1.896 

8-3 

2.116 

i-95 

.6678 

3-55 

1.2669 

5-I5 

1.639 

6.7 

1.902 

8-33 

2. 12 

2 

.6931 

3- 6 

1.2809 

5-2 

j.649 

6-75 

1.909 

8-35 

2.122 

2.05 

.7178 

3-65 

1.2947 

5-25 

1.658 

6.8 

1.917 

8.4 

2. 128 

2. 1 

.7419 

3.66 

1-2975 

5-3 

i.668 

6.85 

1.924 

8-45 

2.134 

2.15 

•7655 

3-7 

1-3083 

5-33 

1-673 

6.9 

I-93I 

8-5 

2.14 

2.2 

.7885 

3-75 

1.3218 

5-35 

1.677 

6-95 

1-939 

8-55 

2.146 

2.25 

.8109 

3-8 

i-335 

5-4 

1.686 

7 

1.946 

8.6 

2.152 

2-3 

.8329 

3-85 

1.3481 

5-45 

1.696 

7-05 

i-953 

8.65 

2.158 

2-33 

.8458 

3-9 

1.361 

5-5 

1-705 

7-i 

1.96 

8.66 

2.159 

2-35 

•B544 

3-95 

1-3737 

5-55 

i-7x4 

7-i5 

1.967 

8.7 

2. 163 

2.4 

.8755 

4 

1.3863 

5-6 

1-723 

7.2 

i-974 

8-75 

2. 169 

2-45 

.8961 

4-°5 

1.3987 

5-65 

1.732 

7-25 

1.981 

8.8 

2-175 

2-5 

•9i63 

4- 1 

1.411 

5.66 

i-733 

7-3 

1.988 

8.85 

2.18 

2-55 

•936 

4-  *5 

1. 4231 

5-7 

1.741 

7-33 

1.992 

8.9 

2.  IoO 

2.6 

•9555 

4.2 

I-435I 

5-75 

1.749 

7-35 

1-995 

8-95 

2.192 

To  Compute  Me  an  Pressure  of  Steam  upon  a Piston  ' 
"by-  Hyperbolic  Logarithms.  t: 

Rule.— Divide  length  of  stroke  of  a piston,  added  to  clearance  in  cylinder 
at  one  end,  by  length" of  stroke  at  which  steam  is  cut  off,  added  to  clearance  f 
at  that  end,  and  quotient  will  express  ratio  or  relative  expansion  of  steam  or 
number. 

Find  in  table  logarithm  of  number  nearest  to  that  of  quotient,  to  which 
add  1.  The  sum  is  ratio  of  the  gain. 

Multiply  ratio  thus  obtained  by  pressure  of  steam  (including  the  atmos- 
phere) as  it  enters  the  cylinder , divide  product  by  relative  expansion,  and 
quotient  will  give  mean  pressure. 

Note.— Hyp.  log.  of  any  number  not  in  tabic  may  be  found  by  multiplying  a 
common  log.  by  2.302*585,  usually  by  2.3. 


STEAM. 


713 


When  Relative  Expansion  or  Number  falls  between  two  Numbers  in  Table , 
proceed  as  follows : Take  difference  between  logs,  of  the  two  numbers. 
Then,  as  difference  between  the  numbers  is  to  difference  between  these  logs., 
so  is  excess  of  expansion  over  least  number,  which,  added  to  least  log.,  will 
give  log.  required. 

Illustration.— Expansion  is  4.84,  logs,  for  4.8  and  4.85  are  1.569  and  1.579,  and 
their  difference  .01.  Hence,  as  4.85  <x>  4.8  = .05  : 1.579  00  1.569  = . 01  ::  4.84  — 4.8  = 
.04  : .008,  and  1.569 -{-.008 = 1.577  = £0*7.  required. 

Example. — Assume  steam  to  enter  a cylinder  at  a pressure  of  50  lbs.  per  sq.  inch, 
and  to  be  cut  off  at  .25  length  of  stroke,  stroke  of  piston  being  10  feet;  what  will  be 
mean  pressure  ? Clearance  assumed  at  2 per  cent.  — . 2 feet. 

10  -f-  • 2 = 10. 2 feet , stroke  10  -r-  4 -f- . 2 = 2. 38  feet.  Then  10. 2 — 2.38  = 4. 29  rela- 
tive expansion. 

Hyp.  log.  4.29  = 1.456,  which  -f- 1 = 2.456,  and  ----- X 50  = 28.62  lbs. 

4,29 

Relative  Effect  of  steam  during  expansion  is  obtained  from  preceding  rule. 

Mechanical  Effect  of  steam  in  a cylinder  is  product  of  mean  pressure  in 
lbs.,  and  distance  through  which  it  has  passed  in  feet. 

Effects  of*  Expansion.  {Essentially  from  D.  K.  Clark.) 

Back  Pressure  is  force  of  the  uncondensed  steam  in  a cylinder,  consequent 
upon  impracticability  of  obtaining  a perfect  vacuum,  and  is  opposed  to  the 
course  of  a piston.  It  varies  from  2 to  5 lbs.  per  sq.  inch. 

It  must  be  deducted  from  average  pressure.  Thus:  assume  pressure  60  lbs., 
stroke  of  piston  as  in  preceding  case,  and  back  pressure  2 lbs. 


At  termination  of. ... . 1st,  2d,  3d,  4th,  5th,  and  6th  foot  of  stroke. 

Pressure 60  30  20  15  12  10  lbs.  per  inch. 

Back  pressure 2222  2 2 “ “ “ 


Effective  pressure 58  28  18  13  10  8 


Total  work  done  by  expansion  at  termination  of  each  foot  or  assumed 
division  of  stroke  of  piston  is  represented  by  hvp.  log.  of  ratio  of  expansion, 
initial  worker.  ' 1 


Thus,  for  a stroke  of  10  feet  and  a pressure  of  10  lbs. : 

At  end  of 1st,  2d,  3d,  4th,  5th,  6th,  7th,  8th,  9th,  and  ioth/oot. 

Steam  is  expanded  ) 

into  vols. , hyp.  V—  .69  1.1  1.39  1.61  1.79  1.95  2.o8«  2.2  2.3 

log.  of  which ) 


Initial  duty 

I 

1 

1 

1 

1 

1 

1 

1 

1 

Total  dutv 

1.69 

2.1 

2-  39 

2.61 

2.79 

2-95 

3.08 

3-2 

3-3 

Initial  duty  is  rep-  i 
resented  by  10. . J 

1 «o 

16.9 

21 

23-9 

26. 1 

27.9 

29-5 

30.8 

32 

33 

Resistance  for  each  | 

16 

18 

foot  of  stroke. . . ] 

['=■  2 

4 

6 

8 

10 

12 

14 

20 

Total  effective  ) 
duty j 

;==  8 

12.9 

i5 

15-9 

16. 1 

*5-9 

15-5 

14.8 

14 

13 

Gain  by  expansion  o 61.25  87.5  98.75  101.25  98.75  93.75  85  75  62.5 


The  same  results  would  be  produced  if  expansion  was  applied  to  a non-condens 
ing  engine,  exhausting  into  the  atmosphere. 

Again,  assume  total  initial  pressure  in  a non-condensing  cylinder  75  lbs.  per  sq. 
inch,  expanded  5 times,  or  down  to  15  lbs.,  and  then  exhausted  against  a back  press- 
ure of  atmosphere  and  friction  of  15  lbs. 


At  termination  of. 1st,  2d,  3d,  4th,  and  5th  foot  of  stroke 

Total  duty 1 1.69  2.1  2.39  2.6i 

“ “ performed...  75  126.75  157.5  179.25  195.75 foot-lbs. 

“ back  pressure 15  30  45  60  75  “ “ 

11  effective  duty....  60  96.75  112.5  119.25  120.75  u 


Gain  by  expansion o 61.25  87.5  98.75  101.25  per  cent. 


From  which  it  appears  that  the  total  duty  performed  by  expanding  steam  5 times 
its  initial  volume  is  full  2.5  times,  or  as  75  to  195.75. 

3 0* 


STEAM. 


7*4 


Relative  Effect  of*  Equal  Volumes  of  Steam. 

Relative  total  effect  or  work  of  steam  is  directly  as  its  mean  or  average  pressure 
(A)  and  inversely  as  its  final  pressure  (B),  or  volume  of  steam  condensed. 

If  former  is  divided  by  latter,  quotient  will  give  relative  total  effect  or  work  (C) 
of  a given  volume  of  steam  as  admitted  and  cut  off  at  different  points  of  stroke  of 
piston,  with  a clearance  of  3.125  per  cent. 

In  following  computations  resistance  of  back  pressure  is  omitted.  If  this  press- 
ure is  uniform  with  all  the  ratios  of  expansion,  it  is  a^uniform  pressure,  to  be  de- 
ducted from  the  total  mean  pressure  in  column  (A). 

(C) 

Relative 
Effect. 


Cut  off  at  | 

Press 

(A) 

Mean. 

iure. 

(B) 

Final. 

(C) 

Relative 

Effect. 

Cut  off  at 

Press 

(A) 

Mean. 

sure. 

(B) 

Final. 

1 

1 

I 

1 

•375 

.761 

•394 

•75 

.969 

.787 

1.28 

•33 

.702 

•335 

.6875 

.946 

.697 

i-35 

•25 

.628 

•273 

.625 

.924 

.636 

i-45 

.2 

•559 

.224 

•5625 

.889 

•576 

x-54 

.125 

•435 

•15 

•5 

•857 

.501 

I-7I 

-1 

.418 

•13 

1- 93 

2.09 

2- 3 
2.05 

2.9 
3.21 


To  Compute  Total  Effective  Work  iix  Oixe  Strolre  of  Ris- 
ton, or  as  Griven  Dy  an  Indicator  Diagram. 

a P a + hyp.  log.  R — c)  — w,  and  abh=z w'.  w representing  total  work , and 
iv'  back  pressure. 

Note Pressure  of  atmosphere  is  to  be  included  in  computations  of  expansion; 

it  is  therefore  to  be  deducted  from  result  obtained  in  non-condensing  engines.  In 
condensing  engines,  the  deduction  due  to  imperfect  vacuum  must  also  be  made, 
usually  2. 5 lbs.  per  sq.  inch. 

Illustration.— Assume  cylinder  of  a condensing  engine  26.1  ins.  in  diameter,  a 
stroke  of  2 feet,  pressure  of  steam  95  lbs.  (80.3  + 14-7)  Pe**  S(l-  ,inchi  cut  off  at  -5  stroke, 
with  an  average  back  pressure  of  2 lbs.  per  sq.  inch,  and  a clearance  of  5 per  cent. 

Area  of  piston*  deducting  half  area  of  rod  — 530  sq.  ins.  2 X 5-r-ioo  = .i  clear- 
ance, and  2-j-.i-ri  + .i='i.9  = ratio  of  expansion , and  1 -f  hyp.  log.  1. 9 — 1.642. 

Then  530  X 95  X i-i  X 1.642  — . 1 — 53°  X 2 X 2 = 50  35°  X 1. 706  - 2120  - 83  777  lbs. 
Illustration.— Assume  cylinder  of  a non-condensing  engine  having  an  area  of 
2000  sq.  ins.,  a stroke  of  8 feet,  steam  at  a pressure  of  50  lbs.  (35.3  + *4-7)»  cut  ott  at 
.25  of  stroke,  and  clearance  .25  foot. 

Ratio  of  expansion  3.66,  back  pressure  17  lbs.,  and  1 -f  hyp.  log.  3.66  — 2.297. 

2000  X 50  (2.25  1 + hyp.' log.  3.66  — .25)  = 100000  x 2.25  X 1 + 1-297  — c = 491 825 
foot- lbs. 

2000  x 17  X 8 = 272  oco  foot-lbs.  or  negative  effect , and  491  825  — 272  000  = 219  825 
foot-lbs. 

Total  Effect  of  One  ED.  of  Expanded  Steam. 

If  1 lb  of  water  is  converted  into  steam  of  atmospheric  pressure  = 14.7  lbs.  per 
sq  inch,' or  2116.8  lbs.  per  sq.  foot,  it  occupies  a volume  equal  to  26.36  pube  feet ; 
and  the  effect  of  this  volume  under  one  atmosphere  = 2116.8  lbs.  X 26.3 6Jeet  — 
«7Q0  foot-lbs.  Equivalent  quantity  of  heat  expended  is  x unit  per  772  foot- lbs  , 
51  55  7^9  7?2  — 72.3  units.  This  is  effect  of  1 lb.  of  steam  of  a pressure  of  one  at- 

mosphere on  a piston  without  expansion. 

Gross  effect  thus  attained  on  a piston  by  1 lb.  of  steam,  generated  at  pressures 
varying  from  15  to  100  lbs.  per  sq.  inch,  varies  from  56000  to  62000 Joot-lbs.  equiv- 
alent to  from  72  to  80  units  of  heat. 

Effect  of  1 lb.  of  steam,  without  expansion,  as  thus  exemplified,  is  reduced  by 
clearance  according  to  proportion  it  bears  to  volume  of  cylinder.  If _ clearance  is  5 
per  cent,  of  stroke,  then  105  parts  of  steam  are  consumed  in  the  work  of  a stroke, 
which  is  represented  by  100  parts,  and  effect  of  a given  weight  of  steam  vs  1 thou t ex- 
pansion, admitted  for  full  stroke,  is  reduced  in  ratio  of  105  to  100.  Having  deter- 
mined, by  this  ratio,  effect  of  work  by  1 lb.  of  steam  without  expansion,  as  reduced 
by  clearance,  effect  for  various  ratios  of  expansion  may  be  deduced  from  that,  in 
terms  of  relative  operation  of  equal  weights  of  steam. 


STEAM. 


715 


Volume  of  1 lb.  of  saturated  steam  of  100  lbs.  per  sq.  inch  is  4.33  cube  feet,  and 
pressure  per  sq.  foot  is  144X100=  14400  lbs.;  then  total  initial  work=  14400X4.33 
— 62  352  foot-lbs.  This  amount  is  to  be  reduced  for  clearance  assumed  at  7 per  cent. 

Then  62352  x 100 -r- 107  = 58273  foot-lbs.,  which,  divided  by  772  (Joule’s  equiva- 
lent), = 75.5  units  of  heat. 

Total  or  constituent  heat  of  steam  of  100  lbs.  pressure  per  sq.  inch,  computed  from 
a temperature  of  2120,  is  1001. 4 units;  and  from  1020  (temperature  of  condenser 
under  a pressure  of  1 lb.)  the  constituent  heat  is  1111.4  units. 

Equivalent,  then,  of  net  simple  effect  75.5  units  is  7.5  per  cent,  of  total  heat  from 
2120,  or  6.7  per  cent,  from  1020. 

When  steam  is  cut  off  at 

1 -75  -5  -33  -25  -2  .125  and  . 1 of  stroke, 

comparative  effects  are  as 

1 1.26  1. 616  1.92  2.14  2.27  2.51  and  2.6. 

Total  effects  as  given  in  table,  page  718. 

Effect  of  1 lb.  of  steam,  without  deduction  for  back  pressure  or  other  effects  varies 
from  about  60000  foot-lbs.,  without  expansion,  to  about  double  that,  or  120000  foot- 
lbs.,  when  expanded  3 times,  cutting  off  at  about  27  per  cent,  of  stroke-  and  to 
about  150000  foot-lbs.  when  expanded  about  6 times,  and  cut  off'  at  about  10  per 
cent,  of  stroke. 


Effect  of  Clearance. 

Clearance  varies  with  length  of  stroke  compared  with  diameter  of  cylinder, 
with  form  of  valve,  as  poppet,  slide,  etc. 

With  a diameter  of  cylinder  of  48  ins.,  and  a stroke  of  10  feet,  and  poppet 
valves,  clearance  is  but  3 per  cent.,  and  with  a diameter  of  34  ins.  and  a 
stroke  of  4.5  feet  and  slide  valves,  it  is  7 per  cent. 

Illustration  of  Effect.  — Assume  steam  admitted  to  a cylinder  for  .25  of  its 
stroke,  with  a clearance  of  7 per  cent. 

Mean  pressure  for  1 lb.  = .637,  and  loss  by  clearance  — 7 -f-  100  = .07,  which,  added 
to  .63 7,  = .707,  which  is  effect  of  a given  volume  of  steam,  if  there  was  not  any  loss 
by  clearance,  or  a gain  of  n per  cent. 


When  steam  is  cut  off  at 1 .71 

Loss  at  7 per  cent,  clearance.  . — 7 7.2 


•5  -33  *25  .125  and . 1 stroke. 

8.1  9.6  11  15.3  17  percent. 


To  Compute  Net  Volume  of  Cylinder  for  Given  Weight 
of  Steam,  Ratio  of  Expansion  and  One  Stroke. 

Rule.  Multiply  volume  of  1 lb.  of  steam,  by  given  weight  in  lbs.,  by 
ratio  of  expansion  and  by  100,  and  divide  product  by  100,  added  to  per  cent, 
of  clearance. 

Example. — Pressure  of  steam  95  lbs.,  cut  off  at  .5,  weight  .54  lbs.,  volume  of  1 lb. 
steam  4.55,  and  weight  = .2198  lbs.,  stroke  of  piston  2 feet,  and  clearance  7 per  cent.' 
Ratio  of  expansion  2 + . 14  -4-  1 -f- . 14  = 1. 88. 


4*55  X -54  X i-88  X 100  461.92  , _ 

; — - — = = 4.31  cube  feet. 

100+7  107  0 J 


To  Compute  Volume  of  Cylinder  for  Given  Effect  with 
a Given  Initial  Pressure  and  Ratio  of  Expansion. 

Rule.  — Divide  given  effect  or  work  by  total  effect  of  1 lb.  of  steam  of 
like  pressure  and  ratio  of  expansion,  and  quotient  will  give  weight  of  steam 
from  which  compute  volume  of  cylinder  by  preceding  rule. 

preceding E Assume  Siven  work  at  50766  foot-lbs.,  and  pressure  and  expansion  as 


Total  work  by  1 lb.,  100  lbs.  steam,  cut  off  at 
table  of  multipliers  for  95  lbs.  = .998,  which  x 


• 5»  =by  table  94  200  foot-lbs.,  and 
94  200  = 94  012  foot-lbs. 


Then  — -54  lbs.  weight  of  steam. 


by 


7 i6 


STEAM. 


Consumption  of  Expanded  Steam  per  IP  of  Effect  per 
Hour. 

~p~p  . — nm]  which  X 60^:  1980000  foot -lbs.  per  hour , which  — 1 lb. 

steam,  the  quotient  = weight  of  steam  or  water  required  per  IP  per  hour. 

Illustration.— Effect  of  1 lb.,  100  lbs.  steam,  without  expansion,  with  7 per  cent, 
of  clearance  = 58273  foot-lbs .,  and  1 = 34  lbs-  steam — weight  of  steam  con- 

sumed for  the  effect  per  IP  per  hour. 

When  steam  is  expanded,  the  weight  of  it  per  IP  is  less,  as  effect  of  1 lb.  of  steam 
is  greater  and  it  may  be  ascertained  by  dividing  1 980  oco  by  the  respective  effect, 
or  by  dividing  34  lbs.  by  quotient  of  total  mean  pressure  by  final  pressure,  as  given 
in  table,  page  718. 

When  steam  is  cut  off  at  1 .75  .5  -375  -33  .25  and  .2  of  stroke. 

Volumes  consumed  per  I . 2I  l8.5  I?.6  16  14.9  lbs. 

IP  per  hour j 

Hence,  assuming  10  lbs.  steam  are  generated  by  combustion  of  1 lb.  coal  per  IP 
of  total  effect  per  hour, 

The  coal  consumed  per)  6o  2<I  j.ge  1.76  1.6  1.49  lbs. 

IP  per  hour j J 9 y 

SATURATED  STEAM. 

To  Compute  Energy  and.  Efficiency  of  Saturated  Steam. 

-=C;  i-D; 


V = R’ 


S 


P 


, X XD 

-/Xa,  or-or  — = P; 


HD 

R 


: H'; 


1)  1 

K°rKS  = F; 


Jl)  ((  — «')  + L = HD; 


p — p'  xaRS  = X; 

h 


-XD 


h — Xz 
X 

— — e\ 


= *'■; 


= p" 


a p — ap 


R S * ’ P" 
nlap—p'  --x,  and 

cube  feet. 


3300° 

loooo 


— IIP;  Hp—p'a  = x;  FCX60  =/; 
— cube  feet  water  evaporated  per  hour  per  EP. 


33  000 

* - — = uwur,  J v 

pa — p a 02.5  a 

V and  v representing  volumes  of  mass  of  steam  entering  cylinder  and  of  it  at 
. • s nrnti.  s volumes  of  t lb.  steam  when  admitted  and 


V and  v representing  volumes  vj  muss  uj  'v.y  ~ 

termination  of  stroke  of  piston ; S and  s volumes  of  1 Zb.  sZeara  admitted  and 

when  at  termination  of  expansion ; C rotae  of  cylinder  per  minute,  for  each  IIP  ; 
R and  r ratios  of  expansion  and  effective  cut-off ; ¥ feed  water  per  cube  foot  qf  vol- 
ume of  cylinder  per  stroke  of  piston , and  f per  IIP  per  hour , all  tn  cube  feet.  D den- 
sity or  weight  of  1 cube  foot  of  steam  at  temperature  of  operation,  in  lbs.;  p mean 
pressure  * V mean  back  pressure  ; I initial  pressure  ; P mean  effective  pressure  or 
energy  per  cube  foot  of  volume  of  cylinder ; P 'pressure  per  sq.  inch  or  that  equivalent 
to  heat  expended,  and  P"  pressure  equivalent  to  expenditure  of  available  heat,  or  en- 
eray  all  in  lbs.  J Joule's  equivalent  = 772  foot-lbs. ; L as  per  following  table ; t and 
V absolute  temperatures  of  steam  at  initial  pressure  and  of  feed  water  in  degrees; 
HD  heat  expended  per  cube  foot  of  steam  admitted;  H'  heat  expended  per  cubefoo, 
of  volume  of  cylinder,  or  pressure  equivalent  to  heat  expended  per  sq.foot  ; H heat 
rejected  per  cube  foot  of  steam  admitted  ; H'"  heat  rejected  per  cube  foot  of  volume 
of cylinder  ; A available  heat  per  IIP  per  hour ; e energy  per  cube  foot  of  volume  of 
cylinder  to  point  of  cutting  off  or  of  steam  admitted;  h and  h heat  expended  and 
rejected  and  X energy  exerted , all  per  lb.  of  steam  and  m foot-lbs.  E efficiency ; x en- 
ergy exerted  per  minute  and  per  cube  foot  of  steam  admitted ; a area  of  piston  in 
sq  ins.  ; l length  of  stroke  of  piston  in  feet,  and  f feed  water  per  IIP  per  hour , in 
cube  feet. 

Illustrations.— Assume  volume  of  cylinder  and  clearance  (5  per  cent.  = .6  inch) 
1 cube  foot,  steam  (86.3  + 14-7)  1°°  lbs-  Per  ST  inch>  cut  off  at  -5,  mean  pressure  by 
rule  (page  711)  86  lbs.,  and  back  pressure  3 lbs. 

V — x. 


v = 2.  S = 4-33-  * = 8.31. 

t and  t'  — 327.9°  + 461. 20  and  ioo°  + 46i.2°. 


= 86.  p'  = 3- 
= 2 feet.  Ti  — i. 


a = 144  ins. 

L = IS7  748- 


STEAM. 


717 


86  — 3 X 144  = 11 952  lbs. 

.1x54  cube  feet. 


2 -4- 1 _ . 2 ratio.  4. 33  -f-  8. 31  = . 526  effective  cut-off. 

33  000  I 2-2T 

Sr— — ■ — -2.76  cube  feet.  = .231  lbs.  .. 

3X144  4-33  2 2X4.33 

772  X . 231  (789.  i°  — 561. 20)  + 157  748  = 198  389  foot-lbs. 

~ -f9  = 99  i95  foot-lbs.  59?389  __  g5g  g27  fQoUbs  99i£5  _ 6g  m 

•23i  144 

86  — 3 X 144  X 2 X 4. 33  = 103  504  foot-lbs. 

198 389 -4-. 231  — 103  504  x 2.31  =1x74479  foot-lbs.  1744794-2 — 87239  foot-lbs. 
T^5'5  X 100  X 144  X 4-33  = 966  456  foot-lbs.  9 66  456  — 103  504  = 862  952  foot-lbs. 


966  456 
2 X 4-33 


= hi  600  lbs. 


144X86  — 144X3 

tti6oo  ‘ 107 


1 080  000 


Or 


io7  =18  504  673  foot-lbs. 
23  9°4  . 


980  000  X = 18  504  673  foot-lbs'.  = . 725  IP, 

io3  5°4  33  000 

1 980  000 

r . . = • 306  cw&e  /ee£. 

62. 5 X 103  504  J J 


1 X 2 X 144  X 86  — 3 — 23  904  foot-lbs. 

2X86  — 3X144  = 23904  foot-lbs.  .1154X2.76X60  = 1 9.  n cube  feet. 


103  504 
2 X 4-33  " 


c 952  foot-lbs. 


33000 


- = 2.761  cube  feet. 


86X144  — 3X144 

0fft,^"““ff?nnKeCti0n.0f.e^™dit"re  of  available  heat  (A)  and  consumption 
resr  ond f J?  hiaTe, a total  heat  of  colnl'ustion  of  10000000*  foot-lbs.,  cor- 
f,®ap™d‘“g  an  equivalent  evaporative  power  under  i atmosphere  at  212°  of  13.4 
lbs  water  and  efficiency  of  furnace  .5;  then  available  heat  of  combustion  of  1 lb? 
coai  = 5 000  000  foot-lbs. 

tionew?th  th^r?ii0U  °f  COal,rir  IH>  in  an  engiue  of  like  dimensions  and  opera- 
tion with  that  here  given  would  be  19223000=5000000  = 3.8444  lbs. 

Properties  of  Steam  of  Maximum  Density.  ( RanJcine .) 
Per  Cube  Foot. 


Temp 

o 

32 

4i 

50 

59 

68 

77 

86 


248 

348 

481 

655 

881 

1171 

1538 


Temp. 


95 

104 

113 

122 

131 

140 

149 


X999 

257i 

3277 

4136 

0178 

6430 

7921 


Temp. 


158 

167 

176 

x85 

194 

203 

212 


9 687 
11  760 
14  200 
17  010 
20280 
24  020 
28  310 


Temp. 


221 
230 
239. 
248 
257 
266 
2 75 


L 


33  J8o 
38  700 
44  930 
51  920 
59720 
68  420 
78050 


Temp. 

L | 

Temp. 

L 

0 

0 

284 

88  740 

347 

197  700 

293 

100  500 

356 

219000 

302 

1 13  400 

365 

242  000 

3ii 

127  500 

374 

266  600 

320 

143  000 

383 

293  100 

329 

159  800 

392 

321  400 

338 

178000 

401 

351  600 

SUPERHEATED  STEAM. 

attuned  by  imparting  to  steam  a temperature  moderately  in 
excess  of  that  due  to  the  volume  or  density  of  saturated  steam  are : Y 

1.  An  increase  of  elasticity  without  a corresponding  increase  of  water  evaporated 

Both  of  these  results,  by  increasing  effect  of  the  steam,  economize  fuel. 
Superheated  steam  should  be  treated  as  a gas 

re^hM 

42140T  . t pv  85. 44  T.  T temperature  of  steam  + 46x.2°,  and  t 32°  + 46,  2° 
Iucstratiok— Assume  temperature  of  steam,  327.  9°,  superheated  to  341.1°. 

"T  32  + 461. 2°  — 68  549  foot-lbs. 

- 100  lbs.  per  sq.  inch,  and  at  341.1°  120. 


Then  42  140  X 461. 20  -f  34x.  1 
Hence,  as  pressure  of  steam  at  327. 90 
120  100  = 1.2  to  ; 


= a gain  of  one  fifth. 


* Coal  of  average  composition,  14 133  X 772 


7i8 


STEAM. 


To  Compute  Energy  and  Efficiency  of  Superheated 
Steam. 

In  following  illustrations  elements  are  same  as  those  in  preceding  cases  for  satu- 
rated steam,  with  addition  of  the  steam  being  superheated,  so  that 
I = n5  lbs.,  t = 338° -j- 461. 20  = 799. 20,  t'  = 290  + 461.2°  — 751-2°,  S = 3.8,  5 = 7-4- 


?AtlS— Raf>'S  = X; 

7, 


i5-5 


IaS  = /i; 


a p — ap 


F" 


: E;  h — X = 7i' ; ^ = P; 


- = H'"; 


- cw&e  ; 


1 980  000 


US  ’ RS  7 ap  — ap' 

Efficiency  of  saturated  steam  (p.  716)  .107,  and,  as  above, . 109;  hence  — ^ = 1.02  to  1. 

If  then,  available  heat  of  combustion  of  efficiency  of  furnace  is  assumed  at  5 000000 
foot-lbs .,  as  above,  consumption  of  coal  per  IEP  18  183486  = 5000000  = 3.637  lbs. 
Note.— For  further  illustrations  Rankine’s  “ Steam-engine,”  London,  1861, p.  436. 
Wire-drawing. 

Wire-drawing  of  steam  is  difference  between  pressure  in  boiler  and  pressure  in 

cylinder,  and  is  occasioned  as  follows:  ... 

Resistance  or  friction  in  steam-pipe  to  passage  of  steam  to  steam-chest  and  piston. 
Resistance  of  throttle-valve  to  passage  of  steam,  when  it  is  partly  closed  or  of  in- 
sufficient area  in  proportion  to  steam-pipe. 

Resistance  from  insufficient  area  of  valves  or  ports. 

Mr  Clark,  from  his  experimental  investigation,  declared,  that  resistance  in  a 
steam-pipe  is  inappreciable,  when  its  sectional  area  is  not  less  than  .1  area  of  piston, 
and  its  velocity  not  exceeding  600  feet  per  minute. 

When  velocity  of  a piston  is  from  200  to  240  feet  per  minute,  area  of  steam  may 
be  .04th  of  piston. 


Effect  of  Expansion  with  Equal  Volumes,  and  Effect  of 
One  Eh.  of  IOO  Lbs.  Pressure  per  Sq.  Inch. 
Clearance  at  each  End  of  Cylinder,  including  Volume  of  Steam  Openings,  7 per  cent, 
of  Stroke,  and  100  per  cent,  of  Admission  = 1. 


Ratio 
of  Ex- 
pansion. 

Initial 

Volume 

Point 

of 

Cut-off. 

Stroke 

Totai 

Final. 

Initial 

Pressure 

ra  Pressu 

Mean. 

Initial 

Pressure 

RES. 

Initial. 

Mean 

Pressure 

Weight 
of  Steam 
of  100  Lbs. 

for  one 
Stroke  per 
Cube  Foot. 

Actual 

By  1 Lb. 
of  100  Lbs. 
Steam. 

Effect. 

Per 

Sq. Inch 
per  Foot 
of  Stroke 
by  100  Lbs. 
Steam. 

Volume 
of  Steam 
expended 
per  IP 
of  Work 
per  Hour. 

Heat 

con- 

verted. 

Lbs. 

Foot-lbs. 

Foot-lbs. 

Lbs. 

Units. 

I 

j 

1 

X 

1 • 

•247 

58273 

IOO 

34 

75-5 

I.  I 

.9 

•9°9 

.996 

1.004 

.225 

63850 

99.6 

31 

82.7 

1. 18 

•83 

.847 

.986 

1. 014 

.209 

67  836 

98.6 

29.2 

87.9 

1.23 

.8 

..813 

.98 

1.02 

.201 

70  246 

28.2 

91 

1.3 

•75 

.769 

.969 

1.032 

,I9o 

73513 

96.9 

26.9 

95-2 

1.39 

.7 

■7I9 

•953 

1.049 

.178 

77  242 

95-3 

25.6 

IOO.  I 

1.45 

.66 

.69 

.942 

1.062 

• 17 

79  555 

94.2 

24.9 

102.9 

i-54 

.625 

.649 

•925 

1. 081 

.161 

83055 

92-5 

23.8 

107.6 

1.6 

.6 

.625 

•9I3 

1.095 

• 155 

85125 

9J-3 

23-3 

no.  3 

1.88 

.5 

53-2 

.86 

1.163 

• 131 

94  200 

86 

21 

122 

2.28 

•4 

•439 

•787 

1. 271 

. 108 

104  466 

787 

X9 

1325 

2.4 

•375 

.417 

.766 

I-3°5 

.103 

107  050 

76.6 

l8.5 

138.6 

2.65 

•33 

•377 

.726 

i-377 

•093 

1 12  220 

72.6 

17.7 

*45-4 

2.  Q 

.3 

•345 

.692 

i-445 

* .085 

1x6855 

69.2 

16.9 

i5i-4 

3.35 

.25 

.298 

•637 

i-57- 

.074 

124066 

63-7 

l6 

160.7 

4 

.2 

•25 

•567 

1.764 

.062 

132  770 

56.7 

I4.9 

171.9 

4.5 

.16 

.222 

.526 

1. 901 

•055 

138130 

52.6 

i4-  34 

178.8 

5 

• 14 

.2 

.488 

2.049 

.049 

142  180 

48.8 

13.92 

184.2 

5.5 

.125 

.182 

•457 

2.188 

•045 

146  325 

45-7 

i3- 53 

189.5 

5.9 

.11 

. 169 

•432 

2-3!5 

.042 

148  940 

43-2 

13.29 

192.9 

6.3 

. 1 

.159 

•4*3 

2.421 

•039 

15137° 

4i-3 

13.08 

196. 1 

6.6 

.09 

.152 

•398 

2.513 

•037 

152955 

39-8 

12.98 

197.7 

7 

.083 

•143 

, -381 

2.625 

•035 

155200 

38.1 

12.75 

201. 1 

7.8 

.066 

.128 

•348 

2.874 

.032 

158414 

34-8 

12.5 

205. 2 

8 

.0625 

; .125 

•342 

2.924 

.031 

159  433 

34-2 

11.83 

206.5 

STEAM. 


Alxiltipliers  for  Actual  Weight 


719 

and  Effect  for  other 


Pressure 

per 

Sq.  Inch. 

Multi 

Weight. 

pliers. 

Actual 

Effect. 

Pressure 

per 

Sq. Inch. 

Multi 

Weight. 

pliers. 

Actual 

Effect. 

Pressure 

per 

Sq. Inch. 

Mult 

Weight. 

ipliers. 

Actual 

Effect. 

Lbs. 

65 

70 

75 

80 

85 

.666 

.714 

•763 

.806 

.855 

•975 
.981 
.986 
.988 
.991  | 

Lbs. 

90 

95 

IOO 

IIO 

120 

.901 

•952 

1 

1.09 

1.17 

•995 

•998 

X.009 
1. on 

Lbs. 

130 

140 

150 

160 

170 

1.28 

I-37 

1.46 

!-55 

1.64 

1-015 

1.022 

1-025 

1.031 

*-°33 

iu  tuia  liiu&tiauou,  m connection  witn  preceding  table,  no  deductions  are  made 
pressure11011011  °f  temperature  of  steam  while  expanding,  or  for  loss  by  back 

When  steam  is  cut  off  at  .0625,  or  one  sixteenth,  its  expansion  is  16  times  but  as 
/ per  cent,  of  stioke  is  to  be  added  to  it  (.0625  - 1- .07)  =:  = 132  ? ner  ppnt  nr 

cedin'^  pa°elG  °f  ^ °r  °Dly  & liU1®  °Ver  7 times>  as  in  3d  column  of  table  on  pre- 

Column  7 is  product  of  58273  and  ratio  of  total  effect  of  equal  weights  of  steam 
when  expanded,  or  average  total  pressure  divided  by  average  final  pressure. 

Thus,  if  steam  is  cut  off  at  .5,  with  a clearance  of  7 per  ^rt  ./l'86  X 100  ~ 86 
1.6165,  aod  58273  X 1.6165  = 94  200  foot-lbs.  X ioo_  53.2 

Column  9 gives  volume  of  steam  consumed  per  IP  per  hour.  Thus  assume  cvl 
" ba^area  of  292  sq.  ins.,  a stroke  ot\  feet,  ind  pressure  o/steumToo gj 

_.2?k * 5%  400  foot-lbs.,  and  292  + 7 per  cent,  of  stroke  for  clearance  = 

' 4 ’ XQ  2-  M M4  = 4-  34  cube  feet,  and  weight  of  a cube  foot  of  such  steam 

Spe3r  table  5 4°°  : 4 34  X '23  " 33000  : '564>  which>X  60  minutes  = 33.84,  br  3” 

Tto  pressures  are  computed  on  premise  that  steam  is  maintained  at  a uniform 
W terminluonSof  s\rokTS'0n  l°  Cylinder>  and  that  expansion  is  operated  correctly 

Column  10  is  quotient  of  work  in  foot-lbs.,  divided  by  Joule's  equivalent  772. 
Thus,  94  200 -r-  772  = 122. 

For  percentage  of  constituent  heat,  converted  from  102°  and  21 20  assume 
122  as  in  last  case : ’ L 

X9^IOO  = 10,98  per  cent  for  ID2°  and  122  X 10-4-100  = 12.2  per  cent. 

“Wire-drawing”  will  cause  a reduction  of  pressure  during  admission  and  rlenr 
shoCrt"l'ide'ary  fr°”  3 t0  8 PCr  cent*  acco‘-di”8  10  *■>*»  ofvalve  as  po^et  ion+r 

qtTOkr^Snie^Wire‘-d?Win|0f  steara’  and  openinS  of  exhaust  b'efore  termination  of 
stroke,  involve  deviations  from  a normal  condition,  for  which  deductions  must  ho 
made,  added  to  which  there  is  the  back  pressure,  from  insufficient  condensation  in 
condensing  engines,  and  from  pressure  of  air  in  non-condensing  engines  and  com 
pression  of  exhaust  steam  at  termination  of  stroke.  s’  ana  com* 

rT  1,1  Feed  at  High  Temperature. 

rLi  ”iir  WTH:  T and  1 representing  total  heat  in  steam  and  temverature  of 

Illustration.  Assume  steam  at  248°,  feed  water  100°  in  one  case  and  150°  in 
another,  and  density  A,  and  total  heat  at  248°  = II57°;  what  is  gain? 


1 1 57  100  + 248  — 100  = 

IX57  — 150  + 248  — 150=  now 

Then  jL~H'  _ I2Q5  — 1x05 


1 205°  = total  heat  required  from  fuel. 

ieO=  “ u 

= .083  = 8.3  per  cent. 


H 


1205 


720 


STEAM. 


COMPOUND  EXPANSION. 


Compound  Expansion  is  effected  in  two  or  more  cylinders,  and  is  prac- 
tised in  three  forms. 

ist.  When  steam  in  one  cylinder  is  exhausted  into  a second,  pistons  of  the 
two  moving  in  unison  from  opposite  ends — that  is,  steam  from  top  or  for- 
ward-end of  first  cylinder  being  exhausted  into  bottom  or  after-end  of  the 
other,  and  contrariwise— this  is  known  as  the  Woolf*  engine. 

2d.  Steam  from  the  ist  cylinder  is  exhausted  into  an  intermediate  vessel, 
or  “ receiver,”  the  pistons  being  connected  at  right  angles  to  each  other. 

3d.  Steam  from  receiver  is  exhausted  into  a 3d  cylinder  of  like  volume 
with  2d,  pistons  of  all  being  connected  at  angles  usually  of  1200. 

The  two  latter  types  are  those  of  the  compound  engine  of  the  present  time. 

Expansion  from  Receiver.  The  receiver  is  filled  with  steam  exhausted 
from  ist  cylinder,  which  is  then  admitted  to  2d,  or  2d  and  3d.  in  which  it  is 
cut  off  and  expanded  to  termination  of  stroke. 

Initial  pressure  in  2d,  or  2d  and  3d  cylinders,  is  assumed  to  be  equal  to  final  press- 
ure in  ist  and  consequently  the  volume  cut  off  in  the  one  or  the  other  cylinders 
must  be  equal  in  volume  to  that  of  ist  cylinder,  for  its  full  volume  must  be  dis- 
charged therefrom. 

Inasmuch  as  3d  cylinder  is  but  a division  of  the  2d,  with  addition  of  receiver, 
this  engine,  in  following  illustrations,  will,  for  simplification,  be  treated  as  having 
but  two  cylinders. 

In  illustration,  assume  ist  and  2d  cylinders  to  have  volumes  as  1 to  2,  with  like 
lengths  of  stroke,  and  that  steam  is  cut  off  at  .5  stroke,  and  equally  expanded  in 
both  cylinders,  the  ratio  of  expansion  in  each  cylinder  being  thus  equal  to  their 
volumes. 

Volume  received  into  2d  cylinder  would  be  equal  to  that  exhausted  from  ist,  as- 
suming there  would  not  be  any  diminution  of  pressure  from  loss  of  heat  by  inter- 
mediate radiation,  etc.  This  is  based  upon  assumption  that  expansion  occurs  only 
upon  a moving  piston;  but  in  operation,  expansion  occurs  both  in  receiver  and  in 
intermediate  passages,  as  nozzles  and  clearances;  the  2d  cylinder,  therefore,  receives 
steam  at  a reduced  pressure,  increased  volume,  and  reduction  of  ratio  of  expansion. 
To  meet  this,  and  attain  like  effects,  volume  of  2d  cylinder  must  be  increased  m 
proportion  to  increased  volume  of  steam  and  its  ratio  of  expansion.  Consequently, 
there  is  no  loss  of  effect  aside  from  increased  volume  and  weight  of  parts  by  inter- 
mediate expansion,  provided  primitive  ratio  of  expansion  is  maintained  by  giving 
relative  increased  volume  to  2d  cylinder. 

Illustration.— Assume  cylinders  having  volumes  as  1 and  3,  initial  steam  of  ist 
cylinder  to  be  60  lbs.  per  sq.  inch,  stroke  of  piston  6 feet,  cut  oil'  at  one  third,  and 


clearance  7 per  cent.* 

Final  pressure,  as  per  rule,  page  711,  = 22.62  lbs.,  and  pressure  as  exhausted  into 
receiver,  reduced  one  fourth.  = 16.97  lbs.,  assuming  there  is  no  intermediate  fall  of 
pressure.  The  steam,  therefore,  is  expanded  to  1.33  times  volume  of  cylinder;  a 
like  volume,  therefore,  must  be  given  to  2d  cylinder,  to  admit  of  this  at  a like  press- 
ure If  therefore,  the  increased  terminal  volume  of  the  steam  in  the  ist  cylinder 
was  augmented,  including  a clearance  of  7 per  cent.,  the  effect  would  be  as  follows: 

Volume  admitted  to  2d  cylinder  is  equal  to  volume  of  ist  added  to  its  clearance, 
or  to  .33  volume  of  2d  cylinder  added  to  its  clearance;  that  is,  to  .33  of  107  per  cent 
or  33.66  per  cent.,  consisting  of  clearance,  and  35.66  — 7 = 28.66  per  cent  stroke  of 
2d  cylinder.  The  steam  exhausted  into  2d  cylinder  thus  fills  less  than  .33  of  its  stroke 
bv  4.67  (33.33  — 28.66).  As  steam  is  expanded  from  volume  of  ist  cylinder,  plus  its 
clearance,  to  2d  cylinder,  plus  its  clearance,  ratio  of  expansion  in  2d  cylinder  is  equal 
to  ratio  of  volume  of  both  cylinders,  which  is  3,  and 


100  (represent  ingfuU  sfrofce)  ± 7 = anU  flna,  ssure  22^2  = 7.54  lbs.  per  sq.  inch. 


28.66  + 7 


* Iii  1825-28  James  P.  Allaire,  of  New  York,  adopted  this  design  of  engine  in  the  steamboats  Henry 
Eckford,  Sun,  Commerce,  Swiftsure,  Post  Boy,  and  Pilot  Boy. 


STEAM. 


721 


Assuming  volume  of  receiver,  or  augmented  terminal  volume,  for  expansion  in  2d 
cylinder,  to  have  proportions  of  1,  1.25,  1.33,  and  1.5  times  volume  of  1st  cylinder 
plus  its  clearance,  the  relations  would  be  as  follows: 

Augmented  terminal  volumes) 


in  2d  cylinder  . 


Final  volumes  in  2d  cylinder) 

added  to  clearance J 

Ratio  of  expansion  in  2d  cyl’r. . 
Intermediate  reductions  of  \ 

pressure  

Equal  to 

Pressures  in  receiver  and  ini 
tial  pressure  in  2d  cylinder. 
Final  pressure  in  2d  cylinder 


I 

1.25 

i-33 

i-5 

( times  volume  of 
{ 1st  cylinder. 

1.07 

1-337 

1.427 

1.605 

( do.  do. 

J including  clear- 

3.21 

3.21 

3.21 

3-21 

( ance. 

l times  volume  of 
( 1st  cylinder. 

3 

2.4 

2.25 

2 

0 

.2 

•25 

•33 

(of  terminal  press- 
\ ure  in  1st  cyl’r. 

, 0 

4-52 

5-65 

11.31 

fbs.  per  sq.  inch. 

22.62 

18. 1 

16. 96 

ix- 3* 

do.  do. 

7-54 

7-54 

7-54 

7-54 

do.  do. 

To  Compute  Expansion  in.  a Compound  Engine. 
RECEIVER  ENGINE. 

Ratio  of  Expansion.  In  1st  cylinder , as  per  formula,  page  710.  In  2d  cylinder. 
— — - r — ratio.  Of  Intermediate  Expansion.  — ratio,  n representing  ratio 

of  intermediate  reduction  of  pressure  between  1st  and  2d  cylinder , to  final  pressure  in 
1st  cylinder,  and  r ratio  of  area  of  1st  cylinder  to  that  of  2d. 


Illustration. — Assume  n = 4,  and  r = 3. 

Then  - — - X 3 = 2.25  ratio , and  — - — = 1.33  ratio. 

4 4 — 1 

Total  or  Combined  Ratio  of  Expansion,  r R'  = product  of  ratio  of  1st  and  2d  cyl- 
inders by  ratio  of  expansion  in  1 st  cylinder.  As  when  r — 3,  and  R'  = 2.653,  then 
2-653  X 3 — - 7-959  tyfaZ  ratio. 

Hence,  Combined  Ratio  of  Expansion  in  both  cylinders.  - — - r R'=R".  R'  rep- 
resenting ratio  of  expansion  in  1 st  cylinder , and  R"  combined  ratio. 


Illustration.— Assume  as  preceding,  and  R'  = 2.653. 


Then  — — - X 3 X 2.653  1=1  5- 969  combined  ratio. 

• 4 

To  Compute  Effect  for  One  Stroke  and  a Given  Ratio 
of’  Expansion  in  Eirst  Cylinder. 


Without  Intermediate  Expansion.  Rule.  — Multiply  actual  ratio  of  ex- 
pansion in  1st  cylinder  by  ratio  of  both  cylinders,  and  to  hyp.  log.  of  com- 
bined ratio  add  1;  multiply  sum  by  period  of  admission  to  1st  cylinder  plus 
clearance,  and  term  product  A. 

Divide  ratio  of  both  cylinders,  less  1,  by  ratio  of  expansion  in  1st  cyl- 
inder ; to  quotient  add  1 ; multiply  sum  by  clearance,  and  term  product  B. 

Subtract  B from  A,  and  term  remainder  C.  Multiply  area  of  1st  cylinder 
in  sq.  ins.  by  total  initial  pressure  in  lbs.  per  sq.  inch,  and  by  remainder  C. 
Product  is  net  effect  in  foot-lbs.  for  one  stroke. 

With  Intermediate  Expansion.  Add  effect  thereof  to  result  obtained  above, 
and  by  following  formula : 

Or,  V 1 -J-  hyp.  log.  R"  — c ^1  -j-  aP  = E.  a representing  area  in  sq.  ins., 

P initial  pressure  in  lbs.  per  sq.  inch  of  1 st  cylinder , V length  of  admission  or  point 
of  cutting  off  plus  clearance , c clearance  in  feet , and  E effect  in  foot-lbs. 

3P 


722 


STEAM. 


Illustration.— Assume  areas  of  cylinders  i and  3 sq.  ins.,  length  of  stroke  6 feet, 
pressure  of  steam  60  lbs.  per  sq.  inch,  cut  off  at  2 feet,  clearance  7 per  cent.,  and 
area  of  intermediate  space,  as  receiver,  one  third  volume  of  1st  cylinder. 

R"=  ratio  of  expansion  in  2d  cylinder  -X.3  X 2.653  = 5.969  hyp.  log. 

4 

2.653  x 2.25  -fix  2.42— 3 — 1 = 2.653  4- 1 x .42  x i x 60  . ==  1.7865  4- 1 x 2.42  — 
2 = 2.6534-1  X .42  X 60  = 6.743  — .737  X 60  = 360.3 6 foot-lbs. 

1st  Cylinder . 

Effect  on  piston  60  lbs.  X 1 inch  x 2 feet = 120  foot-lbs. 

“ of  clearance  60  lbs.  x -42  foot = 25.2  “ 

Total  initial  effect  = 60  x 2 X .42 =145.2  foot-lbs. 


Then  145.2  X i4-hyp.  log-  2.653  or  1-976 . 

Less  effect  of  clearance 

Net  effect  on  piston  above  vacuum  line. 

Less  effect  of  back  pressure  60  = 2.653  = 22.61,  which,  x 3 sq. 

ins.  and  2 feet  stroke 

Net  effect  on  piston 


= 286.91  foot-lbs. 
= 25.2  “ 

= 261.71  foot-lbs. 

= 135.66  “ 

= 126.05  foot-lbs. 


2 d Cylinder. 

145.2  X 1 4~  hyp.  log.  2.25  or  1. 81 =262.81  foot-lbs. 

Effect  of  clearance  22.61  X 3 X -42 = 28.49  “ =234.32  foot-lbs. 

360. 37  foot-lbs. 


Intermediate  reduction  of  pressure,  as  given  at  page  721,  = .25  X 22.61  = 5.65  lbs. 
per  sq.  inch,  which,  x 3 sq.  ins.  and  by  2 per  foot  of  stroke, = 33.9  foot- lbs. 

Hence  360. 36  4-  33-  9 = 394-  26  foot-lbs. 

Or,  by  sum  of  the  three  results,  viz. : 

1st  cylinder 126.05  foot-lbs. 

Intermediate  expansion 33.9 

2d  cylinder 234.32  “ 

394.27  foot-lbs. 

WOOLF  ENGINE.  D.  K.  Clark. 


Ratio  of  Expansion.— In  1st  cylinder  as  per  formula,  page  710.  In  2 d cylinder , 
r y 4-35=i4-  x = ratio,  r representing  ratio  of  area  of  1st  cylinder  to  that  of  2d, 

l and  V lengths  of  stroke  and  of  stroke  added  to  clearance , in  ins.  or  feet,  and  x ratio 
value  of  intermediate  volume. 

Illustration. — Assume  1 = 6 feet,  V = 7 per  cent. = .42,  r = 3,  and  x = .333. 

6 , 

3Xg— 4--333 

Then  4 = 2.353,  ratio  of  expansion  in  2d  cylinder . 

1 4" -333 

Total  Actual  Ratio  of  Expansion.  R'  (r  -~\-xSJ=  ratio. 

Illustration. — Assume  preceding  elements,  R = 2.6s3. 

Then  2.653  (3  x 7— — h -333^  — 2-653  X 3. 137  = 8.322,  total  actual  ratio. 

\ 6.42  / 

Combined  Actual  Ratio  of  Expansion.  R'  [r  y 4-  »^  = 1 4-  x = ratio. 


Illustration.— Assume  preceding  elements. 

3 x 4-  -333  = 1 4~  -333  = ~~~  = 6,242>  combincd  actual  ratio. 


STEAM. 


723 


To  Attain.  Combined  Ratio  of  Expansion  and.  Einal 
JPressnre  in  Sd  Cylinder. 

Assuming  four  cases  as  taken  for  Receiver  Engine  with  a clearance  of 
7 per  cent.  The  relations  would  be  as  follows : 

Intermediate  spaces  are 


Add  to  these  1.07,  the  volume  of  1st ) 
cylinder  plus  its  clearance,  and. . . . J 
To  same  values  of  intermediate  space"! 
add  3,  the  volume  of  2d  cylinder,  I 
and  the  sums  are  the  final  volumes  j 

by  expansion  in  2d  cylinder J 

Ratios  of  expansion  in  2d  cyl’r  are  quo-  ) 
tients  of  final  by  initial  volumes..  ) 

Intermediate  falls  of  pressure  are,  in  ) 
parts  of  final  pressure  in  1st  cylinder  j 


The  initial  pressures  for  expansion  in  ' 
2d  cylinder  are : 


•333 

| part  of  volume  of  ist  cylin- 

•5 

1 

( der  plus  its  clearance,  or, 

, 0 

•357 

•535 

1.07 

f total  initial  volumes  for  ex- 

1.07 

1.427 

1.605 

2.14 

< pansion  in  2d  cylinder  or 
( times  volume  of  ist  cyl’r. 

3-357 

3-535 

4-°7 

f times  volume  of  ist  cyl- 

3 

( inder. 

2.804 

2-352 

2.202 

1.902  ratios  of  expansion. 

f of  final  pressure ; or,  as- 

.25 

•333 

j suming  initial  pressure  at 

•5 

| 63  lbs.,  and  final  pressure 

( at  23.75  lbs.,  they  are 

0 

5-94 

7.92 

11.87  lbs.  per  sq.  inch. 

1 

•75 

.66 

•5 

( of-final  pressure1  in  ist  eyl- 
( inder,  or 

23-75 

17.81 

15-83 

11.87  lbs.  per  sq.  inch. 

. 8.47 

7-57 

7.19 

6.24 

tbs.  per  sq.  inch. 

Hence,  final  pressures  in  2 d cyl'r  are . . 

Combined  Ratios  in  these  Four  Cases. 


1 St. 

ist  ratio  of  expansion 

1 to  2.653 

Combined  Ratio. 

2d 

do. 

do 

1 to  2. 804  = 

2.653  X 2.804  = 7.44. 

2d. 

ist 

do. 

do 

1 to  2.653 

2d 

do. 

do 

1 to  2.352  = 

2.653  X 2.352  = 6.24. 

3d- 

ist 

do. 

do 

1 to  2.653 

2d 

do. 

do 

1 to  2.202  = 

2.653  X 2.202  = 5.84. 

4th. 

ist 

do. 

do 

1 to  2.653 

2.653  X 1.905  = 5.05. 

2d 

do. 

do.  . . . . 

1 to  1.905  = 

1st  case. 
2d  case. 
3d  case. 
4th  case. 


Net  Effect. 
431.81  foot-lbs. 


Initial  effect  of  steam  at  63  lbs.  pressure,  admitted  to  1st  cylinder,  for  2 feet,  or  one 
third  of  stroke  of  piston,  and  with  a clearance  of  7 per  cent,  or  .42  feet,  is  as  follows : 
Effect  on  piston 63  X 2 feet  = 12 6 foot-lbs.  j Total  initial 

do.  in  clearance . . 63  X -42  foot  = 26. 46  = 63  X 2. 42  = 1 52. 4 6 foot-lbs.  ( effect. 

This  sum  is  initial  effect,  on  which  effect  by  expansion  is  computed,  while  it  is 

26.46  foot-lbs.  in  excess  of  the  initial  effect  on  the  piston. 

The  total  effect,  then,  is  computed  as  follows: 

152.46  X (i  + hyp.  log.  7.44)  or  3.0069  = 458.27 
Less  effect  of  clearance 26. 46 

152.46  X (i  + hyp.  log.  6.24)  or  2.831  =431.47 
Less  effect  of  clearance ,.  26.46  405.01  “ 

152.46  X (1  -f-hyp.  log.  5.84)  or  2.7647  = 421.35 
Less  effect  of  clearance 26. 46  394. 89  “ 

152.46  X (i  + hyp.  log.  5.05)  or  2.6294  = 399.29 
Less  effect  of  clearance 26. 46  372. 83  ‘ ‘ 

The  reductions  of  net  effect  in  2d,  3d,  and  4th  cases  are  6.2,  8.6,  and  13.7  per  cent, 
of  effect  in  1st  case. 

To  Compute  Effect  for  One  Stroke  and.  a G-iven  Com- 
bined Actual  Ratio  of  Expansion. 

Rule. — To  hyp.  log.  of  combined  actual  ratio  of  expansion  (behind  both 
pistons)  add  i ; multiply  sum  by  period  of  admission  of  steam  to  ist  cylin- 
der, added  to  clearance,  and  from  product  subtract  clearance. 

Multiply  area  of  ist  cylinder  in  sq.  ins.  by  initial  pressure  of  steam  in  lbs. 
per  sq.  inch  and  by  above  remainder.  Product  is  net  effect  in  foot-lbs.  for 
one  stroke. 


STEAM. 


724 


Example.— Assume  elements  of  1st  illustration  page  723. 

Hyp.  log.  6.24  + 1 = 2.831,  which,  X 2.42  = 6.85,  and  6.85  — .42  and  remainder 
X 60  = 385. 8 fool-lbs. 

Or,  V (1  +hyp.  log.  R')-CxaP  = E. 

Comparative  Effect  of  Steam  in  Receiver  and  Woolf 
Engines. 

The  effect  of  steam  in  a compound  engine,  without  clearance  and  without  any 
intermediate  reduction  of  pressure,  is  the  same  whether  operated  in  a receiver  or 
Woolf  engine. 

When,  however,  there  is  an  intermediate  space  between  the  two  cylinders,  as  a 
receiver,  there  is ’an  intermediate  reduction  of  the  pressure  of  the  steam,  conse- 
quent upon  the  increase  of  its  volume  in  the  receiver;  the  reduction  of  pressure, 
therefore,  being  less  rapid  than  with  the  Woolf  engine,  the  effect  is  greater. 

In  illustration,  the  following  comparative  elements  of  the  effect  of  both  engines 
is  furnished. 

Receiver.  (7  per  cent,  clearance.)  Woolf. 


Ratio  of  Expansion. 

Net  Effect. 

Ratio  of  Expansion. 

Net  Effect. 

1st  case. . . 

. • -7-96 

..422.3  foot-lbs. 

1st  case. . . 

. . .7.64 

. .431.71  foot-lbs. 

2d  kt  ... 

.••5-97 

..421.55  “ 

2d  “ .., 

. . .6.24. . . . . 

..405.11  “ 

3d  “ ... 

. . *5-  31 

..417.96  “ 

3d  “ ... 

• ••5.84 

..394.99  “ 

4th  “ ... 

•••3-98 

..402.78  “ 

4th  “ ... 

•••5;  05 

••372-93 

From  which  it  appears,  that  although  the  effect  of  a receiver  engine  is  the  great- 
est, its  ratio  of  expansion  is  less  than  with  the  Woolf  engine. 

Also  that  by  the  addition  of  clearance  to  the  pistons  of  each  engine,  the  actual 
ratios  of  expansion  are  sensibly  reduced,  as  compared  with  the  ratios  without 
clearance. 

INDICATOR. 


To  Compute  Nlean  Pressure  "by"  air  Indicator. 

in  „ m vo  Rule.— Divide  atmosphere  line,  o o in  fig- 
ure, into  any  convenient  number  of  parts,  as 
feet  of  stroke  of  piston,  and  erect  perpendic- 
ulars at  each  point.  Measure  by  scale  of 
parts  (alike  to  that  of  diagram)  the  actual 
mean  pressure,  as  defined  between  the  two 
lines  at  top  and  bottom  of  diagram,  add  the 
results,  divide  sum  by  number  of  points,  and 
123456789  10  quotient  will  give  mean  pressure  in  lbs.  per 
sq.  inch  upon  piston. 


Example.— Pressures,  as  above  given,  are: 

35  _j_  35  4.  35  + 34  4. 32  + 25  + 16  -f  10  + 8 -f  6 = 236,  which,  -4- 10,  = 23. 6 lbs. 
Note. — If  it  were  practicable  to  run  an  engine  without  any  load,  and  it  some- 
times is.  the  mean  pressure,  as  exhibited  by  an  indicator,  would  be  an  exact  meas- 
ure of  the  friction  of  the  engine. 


Conclusions  on  Actual  Efficiency  of  Steam. 

For  development  of  highest  efficiencies  of  steam,  as  used  in  an  engine,  means  for 
protecting  it  from  cooling  and  condensing  in  the  cylinder  must  be  employed.  Super- 
heating of  it  prior  to  its  introduction  into  a cylinder  is  probably  most  efficient 
means  that  may  be  employed  for  this  purpose.  Application  to  cylinder  of  gases 
hotter  than  it  is  next  best  means;  and  next  is  the  steam-jacket. 

In  cases  of  locomotive  and  portable  engines,  consumption  of  steam  per  IBP  per 
hour  is  less  than  for  that  of  single-cylinder  condensing  engines  for  like  ratios  of  ex- 
piinsion,  which  is  due  to  effect  of  temperature  of  non-condensing  cylinders,  always 
exceeding  2120. 

It  is  deduciblc  from  these  results  that  the  compound  engine  is  more  efficient  than 
the  single-cylinder,  and  that,  of  the  two  kinds  of  compound  engines,  the  receiver- 
engine  is  more  efficient  than  the  Woolf. 

Average  consumption  of  bituminous  coal  per  IIP  per  hour,  for  compound  engines 
in  long  voyages,  as  shown  by  Mr.  Bramwell,  ranged  from  1.7  to  2.8  lbs.  (Z>.  K.  Clark.) 


STEAM. 


725 


To  Compute  Volume  of  Water  Evaporated,  per  Lb. 
of  Coal. 

\ — v W __  VQjume  0y  waier^  {n  ibs%  y and  v representing  volume  of  steam  and 
F d 

relative  volume  of  water,  in  cube  feet,  W weight  of  cube  foot  of  water,  and  F weight  of 
fuel  consumed , both  in  lbs.,  and  d density  of  water,  in  degrees  of  saturation. 

Illustration.— Take  case  at  foot  of  page.  V = 449887  cube  feet,  ^ = 838  cube 
feet,  W = 64.3,  E = 1,  and  F = 4061  lbs. 

449  887  . 838  X 64. 3 _ g 2bs.  per  hour. 

4061  X 1 0 


Gain  in  Fuel , and  Initial  Pressure  of  Steam  required , when  Acting  Expan- 
sively, compared  with  Non-Expansion  or  Full  Stroke. 


Poiat  of 
Catting  off. 

Gain  in 
Fuel. 

Cutting 

off. 

Point  of 
Cutting  off. 

Gain  in 
Fuel. 

Cutting 

off. 

Point  of 
Cutting  off. 

Gain  in 
Fuel; 

Cutting 

off. 

Stroke. 

Per  Cent. 

Lbs. 

Stroke. 

Per  Cent. 

Lbs. 

Stroke. 

Per  Cent. 

Lbs. 

•75 

22.4 

1.03 

•5 

41 

1. 18 

•25 

58.2 

1.67 

.625 

32 

1.09 

•375 

49.6 

1.32 

.125 

67.6 

2.6 

Illustration.— What  must  be  initial  pressure  of  steam  cut  off  at  .5,  to  be  equiv- 
alent to  100  lbs.  per  sq.  inch  at  full  stroke  ? 

100  at  full  stroke  = 100,  and  100  X 1.18  = 118  lbs. 


To  Compute  Grain  in  Enel. 

Rule. — Divide  relative  effect  of  steam  by  number  of  times  the  steam  is 
expanded,  and  divide  1 by  quotient ; result  is  the  initial  pressure  of  steam 
required  to  be  expanded  to  produce  a like  effect  to  steam  at  full  stroke. 

Divide  this  pressure  by  number  of  times  the  steam  is  expanded,  and  sub- 
tract quotient  from  1,  remainder  will  give  gain  per  cent,  in  fuel. 

Example.— When  steam  is  cut  off  at  .5,  what  is  gain  in  fuel,  and  what  mechanical 
effect  ? 

Relative  effect,  including  clearance  of  5 per  cent.,=  1.64;  number  of  times  of  ex- 
pansion = 2. 

1. 64  -1-2  = .82,  and  1 -4-  . 82  = 1. 22  initial  pressure. 

1.22-4-2  = . 61’  and  1 —.61  = .39  per  cent. 

Mechanical  effects  of  steam  at  full  and  half  strokes  are  2 — 1.64  = .36  difference. 

Hence,  1.64  : .36  ::  50  (half  volume  of  steam  used)  : 10.97  per  cent,  more  fuel  to 
produce  same  effect  at  half  stroke,  compared  with  steam  at  full  stroke. 

To  Compuite  Consumption  of  Fuel  in.  a Furnace. 

When  Dimensions  of  Cylinder , Pressure  of  Steam , Point  of  Cut-off,  Revo- 
lutions, and  Evaporation  per  Lb.  of  Fuel  per  Minute  are  given. 

Rule.— Compute  volume  of  cylinder  to  point  of  cutting  off  steam,  in- 
cluding clearance.  Multiply  result  by  number  of  cylinders,  by  twice  number 
of  strokes  of  piston,  and  by  60  (minutes)  ; divide  product  by  density  of  steam 
at  its  pressure  in  cylinder,  and  quotient  will  give  number  of  cube  feet  of 
water  expended  in  steam. 

Multiply  number  of  cube  feet  by  64.3  for  salt  water  (62.425  for  fresh), 
divide  product  by  evaporation  per  lb.  of  fuel  consumed,  and  quotient  will 
give  consumption  in  lbs.  per  hour. 

Example.— Cylinder  of  a marine  engine  is  95  ins.  in  diameter  by  10  feet  stroke 
of  piston;  pressure  of  steam  in  steam-chest-  is  15.3  lbs.  per  sq.  inch,  cut  off  at  .5 
stroke;  number  of  revolutions  14.5,  and  evaporation  estimated  at  8.5  lbs.  of  salt 
water  per  lb.  of  coal;  what  is  consumption  of  coal  per  hour,  when  density  of  water 
is  maintained  at  2-32?  (See  Saturation,  page  726.) 

Volume  of  steam  at  above  pressure,  compared  with  wrater  (15.3  + I4-7)i  = 838- 
Area  of  95  ins.  4-2.5  per  cent,  for  clearance -4- 144  = 50.45  cube  feet.  Point  of  cut- 
ting  off  5 feet -f  2. 5 per  cent.  = 5 feet  1.5  ins.,  and  50.45  X 5 feet  *-5  ins-  X 14- 5 X 2 
X 60  = 449  887  cube  feet  steam  per  hour. 

3 P* 


yi6 


STEAM. 


Hence,  449  887  -4-  838  = 536. 86  cube  feet  water , which,  x 64. 3 = 34  520  lbs. , which, 
-4-  8. 5 = 4061  lbs.  coal  per  hour. 

Note. — Elements  given  are  those  of  one  engine  of  steamer  Arctic,  and  consump- 
tion of  clean  fuel  (selected)  for  a run  of  12  days  (one  engine)  was  3820  lbs.  per  hour. 

Utilization  of  Coal  in  a Marine  Boiler. 

Experiment  gives  from  .55  to  .8  per  cent,  of  the  heat  developed  in  the 
combustion  of  coal,  as  utilized  in  the  generation  of  steam.  Ordinarily  it 
may  be  safely  taken  at  .65. 

SALINE  SATURATION  IN  BOILERS. 

Average  sea-water  contains  per  100  parts : 

Chloride  of  sodium  (com.  salt)  . .2.5;  Chloride  of  magnesium  .33. . . . = 2.83 

Sulphuret  of  magnesium 53;  Sulphuret  of  lime = .54 

Carbonate  of  lime  and  of  magnesia 02 

Saline  matter 3. 39 

Water 96.61 


Hence,  sea-water  contains  .0339th  part  of  its  weight  of  solid  matter  in  solution, 
and  is  saturated  when  it  contains  36.37  parts. 

Mean  quantity  of  salts,  or  solid  matter,  in  solution,  is  3.39  per  cent.,  three  fourths 
of  which  is  common  salt. 

Removal  of  Incrustation  of  Scale  or  Sediment. 
Potatoes,  in  proportion  of  .033  weight  of  water.  Molasses,  in  proportion  of  1.6 
lbs.  per  IP  of  boiler.  Oalc,  suspended  in  the  water,  and  Mahogany  or  Oak  sawdust , 
and  Tanner's  and  Slippery  Elm  bark , renewed  frequently,  according  to  volume  of  it, 
and  the  evaporation  of  the  water.  Muriate  of  Ammonia  and  Carbonate  of  Soda,  in 
quantity  to  be  determined  by  observation. 

BLOWING  OFF. 

To  Compute  Loss  of  Heat  by  Blowing  Off  of  Saturated. 
Water  from  a Steam-boiler. 

- — * = proportion  of  heat  lost,  S — T X heat  required  from  fuel  for 

water  evaporated  in  degrees , and  - — ^ -■  ■ = loss  of  heat  per  cent.  S representing 

S — T E 1 

sum  of  sensible  and  latent  heats  of  water  evaporated , T temperature  of  feed  water, 
t difference  in  temperature  of  water  blown  off  and  that  supplied  to  boiler,  E volume 
of  water  evaporated,  proportionate  to  that  blown  off,  the  latter  being  a constant  quan- 
tity, and  represented  by  1. 

Values  of  E at  following  degrees  of  saturation,  and  volumes  to  be  blown  off: 


32. 

Value 

E. 

Volume 

to 

Blow  off. 

32. 

Value 

E. 

Volume 

to 

Blow  off. 

32. 

Value 

E. 

Volume 

to 

Blow  off. 

32. 

Value 

E. 

Volume 

to 

Blow  off. 

1.25 

•25 

4 

1.65 

•65 

i-54 

2 

1 

1 

2-75 

i-75 

•57 

i-35 

•35 

3 

1-75 

•75 

I-33 

2.25 

1.25 

.8 

3 

2 

•5 

i-5 

•5 

2 

1.85 

.85 

1. 18 

2-5 

i-5 

.66 

2.25 

2.25 

•45 

Thus,  when  water  in  a boiler  is  maintained  at  a density  of  — , 1 volume  of  it  is 

32 

evaporated,  and  an  equal  volume,  or  1,  is  to  be  blown  off. 

Hence  1 + 1 — 1 ==  1 — ratio  of  volume  evaporated  to  volume  blown  off. 
Illustration  i. — Point  of  blowing  off  is  2 (32),  pressure  of  steam  is  15.3  lbs.,  mer- 
curial gauge,  and  density  of  feed  water  1 (32)  ; what  is  proportion  of  heat  lost? 


S = 1157.8°. 

Then 


*=  *5-3+  *4- 7 = 250.4° 


= 150.4“ 


E = 1. 


1157.8  — 100  X 1 -f- 150.4 
150.4 


= 8.03  proportion  of  heat  lost. 


STEAM. — STEAM-ENGINE.  J2J 


2. — Assume  point  of  blowing  off  1.75  (32); 
preceding  case? 


E = *75* 


i5o-4 

1157.8  — 100  X .75+150.4 


what  would  be  loss  of  heat  per  cent,  in 
= 15.9  per  cent,  lost  by  blowing  off. 


3. — Assume  elements  of  preceding  case.  What  is  total  heat  required  from  fuel 
for  water  evaporated  ? 

1157.8  — 100  X -75  = 793-35° 


To  Compute  Volume  of  Water  Blown  Off  to  that 
Evaporated. 

When  Degree  of  Saturation  is  Given.  Rule. — Divide  1 by  proportionate 
volume  of  water  evaporated  to  that  blown  off,  or  value  of  E as  above,  for 
degree  of  saturation  given,  and  quotient  will  give  number  of  volumes  blown 
off  to  that  evaporated. 

Illustration. — Degree  of  saturation  in  a marine  boiler  is  — ^ ; what  is  volume 

32 

of  water  blown  off? 

E ==  1.25.  Then  1 -4-  1.25  — .8  blown  off. 


Proportional  Volumes  of  Saline  Matter  in  Sea-water . 


Baltic 1 in  152  | British  Channel.  . . .1  in  28  I Atlantic,  South.  .1  in  24 

Black  Sea 1 “ 46  Mediterranean 1 “ 25  “ North.  .1  “ 22 

Red  Sea 1 “ 131  | Atlantic,  Equator. . 1 “ 25  | Dead  Sea... 1 “ 2.59 

When  saline  matter  at  temperature  of  its  boiling-point  is  in  proportion  of  10  per 
cent.,  lime  will  be  deposited,  and  at  29.5  per  cent.  salt. 

Temperature  of  water  adds  much  to  extent  of  saline  deposits. 


STEAM-ENGINE. 

The  range  of  proportions  here  given  is  to  meet  the  requirements  of 
variations  in  speed,  pressure,  length  of  stroke,  draught  of  water,  etc., 
in  the  varied  purposes  of  Marine,  River,  and  Land  practice. 


CONDENSING. 

Eor  a Range  of  Pressures  of  from  30  to  SO  Tbs.  (IVIeren- 
rial  Grange)  per  Sq.  Inch.,  Cnt  Off  at  Half  Stroke. 

Piston-rod.  Cylinder  or  Air-pump  ( Wrought  /row),  .1  to  .14  of  its  diam. ; 
(Steel),  .8  diam. ; and  ( Copper  or  Brass),  .11  and  .125. 

Condenser  (Jet).  Volume,  .35  to  .6  of  cylinder.  (Surface.)  Brass  tubes, 
16  to  19  B W G,  .625  to  .75  in  diameter  by  from  5 to  10  feet  in  length,  and 
.75  to  1.25  in  pitch,  condensing  surfaces,  .55  to  .65  area  of  evaporating  sur- 
face of  boiler  with  a natural  draught;  .7  to  .8  with  a blower,  jet,  or  like 
draught.  Or,  for  a temperature  of  water  of  6o°,  1.5  to  3 sq.  feet  per  IIP. 

With  a very  effective  and  sufficient  circulating  pump,  areas  may  be  reduced  to 
.5  and  .6. 

Effect  of  vertical  tube  surface,  compared  to  horizontal,  is  as  10  to  7. 

Air-pump  (Single  acting  and  direct  connection).  Volume  from  .15  to  .2 
steam  cylinder.  Or,  2.75.  For  Double  acting  put  4 for  2.75.  V and  v 

representing  volumes  of  condensing  and  condensed  water  per  cube  foot,  and  n strokes 
of  piston  per  minute. 

Foot  and  Delivery  Valves.  Area,  .25  to  .5  area  of  air-pump. 

Delivei'y  Valve  (Out-board).  With  a Reservoir.  Area  from  .5  to  .8  Foot 
valve. 

Note.— Velocity  of  water  through  these  valves  should  not  exceed  12  feet  per 
second. 


STEAM-ENGINE. 


728 


Steam  and  Exhaust  Valves. — (Poppet').  - sn  — area  sor  steam.  - s n for 
v 1 c /7  24000  J ’ 20000  J 

exhaust;  (Slide),  for  steam,  and  ——  for  exhaust,  a representing 

jrea  of  steam  cylinder  in  sq.  ins.,  s stroke  of  piston  in  ins.,  and  n number  of  revolu- 
tions per  minute. 

Injection  Pipes. — One  each  Bottom  and  Side  to*  each  condenser;  area  of 
each  equal  to  supply  70  times  volume  of  water  evaporated  when  engine  is 
working  at  a maximum ; and  in  Marine  and  River  engines  the  addition  of 
a Bilge,  which  is  properly  a branch  of  bottom  pipe. 

Note  i.—  Proportions  given  wiil  admit  of  a sufficient  volume  of  water  when  en- 
gine is  in  operation  in  the  Gulf  Stream,  where  the  water  at  times  is  at  temperature 
of  84°,  and  volume  required  to  give  water  of  condensation  a temperature  of  ioo°  is 
70  times  that  of  volume  evaporated. 

2.  Velocity  of  flow  of  water  through  cock  or  valve  20  feet  per  second  in  river  or  at 
shallow  draught,  and  30  feet  in  sea  or  deep  draught  service. 

Feed  Pump* — (Single  acting,  Marine ),  Volume,  .006  to  .01  steam  cylinder. 
(River  and  Land),  or  when  fresh  water  alone  or  a surface  condenser  is  used, 
.004  to  .007. 

NON-CONDENSING. 

For  a Range  of  Pressures  of  from  SO  to  ISO  ll>s.  (Mercu- 
rial Gauge)  per  Sq.  Inch,  Cat  Off  at  Half  Stroke. 

Piston-rod. — (Wrought  Iron),  Diam.,  .125  to  .2  steam  cylinder.  (Steel), 
.8  diam. 


Steam  and  Exhaust  Valves . — Area  is  determined  by  rules  given  for  them 
in  a condensing  engine,  using  for  divisors  30  000  and  *22  750. 

A decrease  in  volume  of  cylinder  is  not  attended  with  a proportionate  decrease 
of  their  area,  it  being  greater  with  less  volume. 

Feed-pump.* — (Single  acting,  Marine),  Volume,  .008  to  .016  steam  cylin- 
der. (River  and  Land),  or  where  fresh  water  alone  is  used,  .005  to  .011. 

O-eTieral  IRrules. 

Engines. 

Cylinder.  Thickness . — (Vertical),  ^~  = t ; (Horizontal),^— =:t ; (In- 
clined), divide  by  2000  in  a ratio  inversely  as  sine  of  angle  of  inclination. 

D representing  diameter  of  cylinder,  p extreme  pressure  in  lbs.  per  sq.  inch  that  it 
may  be  subjected  to,  and  t thickness  in  ins. 

Shafts , Gudgeons,  Journals,  etc.  To  resist  Torsion.  See  rules,  pp.  790,  796. 
Coupling  Bolts.  ~\J n representing  number  of  bolts,  D diameter 

of  shaft,  d'  distance  of  centre  of  bolts  from  centre  of  shaft,  and  d diameter  of  bolts, 
all  in  ins. 

Cross-head,  Wrought-iron.  (Cylinder),  a~-  = S,  and  y/y  = d,  or  ~ =.  b. 

a representing  area  of  cylinder  in  sq.  ins.,  I length  of  cross-head  between  centres  of  its 
journals  in  feet,  and  S product  of  square  of  depth  d,  and  breadth,  b,  of  section,  both 

in  ins.  (Air-pump),  ^-=  S,  and  as  above  for  d and  b. 

If  section  of  either  of  them  is  cylindrical,  for  S put  ^Sx  1.7. 

Diam.  of  boss  twice,  and  of  end  journals  same  as  that  of  piston-rod.  Sec- 
tion at  ends  .5  that  of  centre. 


* See  Formulas,  pnge  736. 


STEAM-ENGINE.  729 

Steam-pipe. — Its  area  should  exceed  that  of  steam-valve,  proportionate  to 
its  length  and  exposure  to  the  air. 

Connecting-rod.  — Length,  2.25  times  stroke  of  piston  when  it  is  at  all 
practicable  to  afford  the  space ; when,  however,  it  is  imperative  to  reduce 
this  proportion,  it  may  be  twice  the  stroke. 

Comparative  friction  of  long  and  short  connecting-rods  is,  for  length  of  stroke  of 
piston,  12  per  cent,  additional;  twice,  3 per  cent. ; and  for  thrice,  1.33. 

Neck.  — Diam.  1 to  1.1  that  of  piston-rod.  Centre  of  body  ( Horizontal ), 
1.25  ins. ; (Vertical),  .06  inch  per  foot  of  length  of  rod. 

With  two  connecting-rods  or  links,  area  of  necks  .65  to  .75  area  of  attached  rod. 

When  a second  set  of  rods  is  used,  as  in  some  air-pump  connections,  area  of 
necks,  in  a ratio,  inversely  as  their  lengths  to  that  of  first  set. 

Straps  of  Connecting-rods , Links,  etc.  — (Strap),  area  at  its  least  section 
.65  neck  of  attached  rod ; (Gib  and  Keg),  .3  diam.  of  neck,  width,  1.25  times, 
(Slot)  1.35  times  (Draft)  of  keys  .6  to  .8  inch  per  foot.  Distance  of  Slot 
from  end  of  rod  .5  diam.  of  pin. 

/P  1 

Pins  (Cranks,  Beams,  etc.).  3/  — . 355  — d.  P representing  pressure  or  thrust 

of  rod  or  beam , l length  of  journal  in  ins.,  and  C,  for  Wrought  iron  = 6 40,  Cast,  560. 
Puddled  steel , 600,  and  Cast , 1200. 

Length,  1.3  to  1.5  times  their  diam.,  and  pressure  should  not  exceed  750 
lbs.  per  sq.  inch  for  propeller  engine,  and  1000  for  side-wheel. 

Cranks  (Wr  ought-iron). — Hub,  compared  with  neck  of  shaft,  1.75  diam., 
and  1 depth.  Eye,  compared  with  pin,  2 diam.,  and  1.5  depth.  Web,  at  pe- 
riphery of  hub,  width,  .7  width,  and  in  depth  .5  depth  of  hub  ; and  at  periph- 
ery of  eye,  width,  .8  width,  and  in  depth,  .6  depth  of  eye. 

(Cast-iron.)  Diameters  of  Hub  and  Eye  respectively,  twice  diam.  of  neck 
of  shaft,  and  2.25  times  that  of  crank  pin. 

Radii  for  fillets  of  sides  of  web  .5  width  of  web  at  end  for  which  fillet  is  designed; 
for  fillets  at  back  of  web,  .5  that  at  sides  of  their  respective  ends. 

Beams,  Open  or  Trussed. — Length  from  centres  1.8  to  2 stroke  of  piston, 
and  depth  .5  length.  If  strapped,  Strap  at  its  least  dimensions  .9  area  of 
piston-rod,  its  depth  equal  to  .5  its  breadth.  End  centre  journals  each  1,  and 
main  centre  journals  2.5  times  area  of  piston  or  driving-rod. 

This  proportion  for  strap  is  when  depth  of  beam  is  .5  length,  as  above;  conse- 
quently, when  its  depth  is  less,  area  of  strap  must  be  increased;  and  when  depth  of 
strap  is  greater  or  less  than  .5  width,  its  area  is  determined  by  product  of  its  bd2, 
being  same  as  if  its  depth  was  .5  its  width. 

(Cast-iron).  Area  of  Section  of  Centre.  = A.  p representing 

extreme  pressure  upon  piston  in  lbs.,  d depth  in  ins.,  and  l length  in  feet. 

Depth  at  centre  .5  to  .75  diam.  of  cylinder,  and,  when  of  uniform  thick- 
ness, a thickness  of  not  less  than  .1  of  depth. 

Vibration  of  End  Centres. — l -r-  2 — V {l  -r-  2) 2 — (s  4-  2)2  = vibration  at  each 
end  ; s representing  stroke  of  piston,  in  feet. 

Plumber  Blocks  (Shaft). — Binder  d \J  C ==  depth,  d representing  diam. 

of  bolts  when  two  to  binder , l distance  between  bolts,  b breadth  of  binder,  all  in  ins., 
and  C for  wrought  iron  1,  steel  .85,  and  cast  iron  .2. 

Holding-down  Bolts.  P -4-  3 C = area  at  base  of  thread  of  each  bolt.  C for  mild 
steel  for  small  and  large  bolts  6000  and  7000,  for  wrought  iron  4500  and  6000,  if  but 
two  are  used.  9 

Binder  (Brass) . y = depth. 


730 


STEAM-ENGINE. 


Codes. — Angles  of  sides  of  plug  from  70  to  8°  from  plane  of  it. 

Pumps. — Velocity  of  water  in  pump  openings  should  not  exceed  500  feet 
per  minute. 

Fly-wheels  and  Governors. — See  Rules,  pages  451  and  452. 

Water-wheels. 

Water-wheels  {Arms'). — Number  from  .75  to  .8  diam.  of  wheel  in  feet. 
( Blades ) Wood. — For  a distance  of  from  5 to  5.5  feet  between  arms,  thick- 
ness from  .09  to  .1  inch  for  each  foot  of  diam.  of  wheel. 

Area  of  blades,  compared  with  area  of  immersed  amidship  section  of  a 
vessel,  depends  upon  dip  of  wheels,  their  distance  apart,  model  and  rig  of 
vessel. 

In  River  service , area  of  a single  line  of  blade  surface  varies  from  .3  to  .4 
that  of  immersed  section;  in  Bay  or  Sound  service , it  varies  from  .15  to  .2 ; 
and  in  Sea  service , it  varies  from  .67  to  .1. 

Note.— A wrought-iron  blade  .625  inch  thick  bent  at  a stress  withstood  by  an 
oak  blade  3.5  ins.  thick. 

FLaclial  and.  Feathering. 

Radial. — Loss  of  effect  is  sum  of  loss  by  oblique  action  of  wheel  blades 
upon  the  water,  their  slip,  and  thrust  and  drag  of  arms  and  blades  as  they 
enter  and  leave  the  water. 


Loss  by  oblique  action  is  computed  by  taking  mean  of  square  of  sines  of 
angles  of  blades  when  fully  immersed  in  the  water. 

Loss  by  oblique  action  of  blades  of  wheel  of  steamer  Arctic , when  her  wheels 
were  immersed  7 feet  9 ins.  and  5 feet  9 ins.,  was  25.5  and  18.5  per  cent.,  which 
was  the  loss  of  useful  effect  of  the  portion  of  total  power  developed  by  engines, 
which  was  applied  to  wheels. 


Feathering. — Loss  of  effect  is  confined  to  thrust  and  drag  of  arms  and 
blades  as  they  enter  and  leave  the  water. 

Comparative  Effects. — In  two  wheels  of  a like  diameter  (26  feet,  and  6 feet  immer- 
sion), like  number  and  depth  of  blades,  etc.,  the  losses  are  as  follows  : 

Radial 26.6  per  cent.  | Feathering 15.4  per  cent. 

Loss  of  effect  by  thrust  and  drag  in  a feathering  wheel,  having  these  elements 
and  included  in  the  above  given  loss,  is  computed  at  2 per  cent. 

Relative  loss  of  effect  of  the  two  wheels  is,  approximately,  for  ordinary  immer- 
sions, 20  and,  15  per  cent,  from  circumference  of  wheel. 


— - ^ d=zc.  d and  df  representing  depths  of  blades 

3 d 2 — d'2 

below  surface  of  water,  and  c centre  of  pressure,  all  in  like  dimensions , from  bottom 


Centre  of  Pressure, 


In  the  cases  here  given,  centres  of  pressure  are  as  follows : 

Radial 6.41ns.  | Feathering 8.5  ins. 


IPropellers. 

Propellers  (Screw).  — Pitch  should  vary  with  area  of  circle  described  by 
screw  to  area  of  midship  section  of  vessel. 

AREA,  TWO-BLADED. 


Area  of  disk  of  propeller  to  mid- ) 
ship  section  being  1 to J 

6 1 5 

4-5 

4 I 3-5 

3 

2-5 

2 

Ratio  of  pitch  to  the  diameter  of  V 
propeller  is  1 to } 

.8  1.02 

1. 11 

1.2  j 1.27 

131 

1.4 

1.47 

For  Four-bladed  screws,  multiply  ratio  of  pitch  to  diam.  as  given  above, 
by  1.35.  Length , .166  diam. 


STEAM-ENGINE. 


731 


Slip.— Slip  of  a screw  propeller  is  directly  as  its  pitch,  and  economical 
effect  of  a screw  is  inversely  as  its  pitch ; greater  the  pitch  less  the  effect. 

An  expanding  pitch  has  less  slip  than  a uniform  pitch,  and,  consequently, 
is  more  effective. 

To  Compute  Thrust  of*  a Eropeller. 

]jp  2I1  — T.  S representing  speed  of  vessel  in  knots  per  hour. 


SLIDE  YALYES. 

All  Dimensions  in  Inches. 

To  Compute  Lap  required  on.  Steam  End,  to  Cnt-ofF  at 
any  given  Part  of  Stroke  of  Biston. 

Rule. — From  length  of  stroke  subtract  length  of  stroke  that  is  to  be  made 
before  steam  is  cut  off ; divide  remainder  by  stroke,  and  extract  square 
root  of  quotient. 

Multiply  this  root  by  half  throw  of  valve,  from  product  subtract  half  lead, 
and  remainder  will  give  lap  required. 

Example.— Having  stroke  of  piston  60  ins.,  stroke  of  valve  16  ins.,  lap  upon  ex- 
haust side  .5  in.  = one  thirty-second  of  valve  stroke,  lap  ppon  steam  side  3.25  ins., 
lead  2 ins.,  steam  to  be  cut  off  at  five  sixths  stroke;  what  is  the  lap? 

60  — — of  60  = 10.  = . 408.  . 408  X — = 3-  264,  and  3. 264  — — = 2. 264  ins. 

6 V 6°  2 2 

To  Ascertain  Lap  reqnirecL  on  Steam  End,  to  Cnt-ofF 
at  various  Portions  of  Stroke. 


Valve 


Distance  of  piston  from  end  of  its  stroke  when  steam  is  cut  off, 
in  parts  of  length  of  its  stroke. 


without  Lead. 

1 

5 

1 

7 

1 

5 

1 

1 

1 

1 

2 

12 

3 

24 

¥ 

t» 

TS 

2T 

Lap  in  parts  of) 
stroke j 

•354 

•323 

.286 

.27 

•25 

.228 

.204 

.177 

.144 

.102 

Illustration. — Take  elements  of  preceding  case. 
Under  i is  .204,  and  .204  X 16  = 3.264  ins.  lap. 


When  Valve  is  to  have  Lead. — Subtract  half  proposed  lead  from  lap  as- 
certained by  table,  and  remainder  will  give  proper  lap  to  give  to  valve. 

If,  then,  as  last  case,  valve  was  to  have  2 ins.  lead,  then  3.264  — 2 2 = 2. 264  ins. 


To  Compute  at  what  Part  of  Stroke  any-  given  Lap  on 
Steam  Side  will  Cut  off. 


Rule. — To  lap  on  steam  side,  as  determined  above,  add  lead ; divide  sum 
by  half  length  of  throw  of  valve.  From  a table  of  natural  sines  (pages  390- 
402)  find  the  arc,  sine  of  which  is  equal  to  quotient ; to  this  arc  add  90°, 
and  from  their  sum  subtract  arc,  cosine  of  which  is  equal  to  lap  on  steam 
side,  divided  by  half  throw  of  valve.  Find  cosine  of  remaining  arc,  add  1 
to  it,  and  multiply  sum  by  half  stroke,  and  product  will  give  length  of  that 
part  of  stroke  that  will  be  made  by  piston  before  steam  is  cut  off. 

Example. — Take  elements  of  preceding  case. 


Cos.  ^sin.  — + 900  — cos.  -f  1 X—=  cos.  (320  13'  -}-  900  — 730  34') 

60 

= 48°  39',  and  cos.  48°  39'  -f- 1 x — = 1. 66  X 30  = 49. 8 ins. 

2 

To  Ascertain  Breadth,  of  Ports. 

Half  throw  of  valve  should  be  at  least  equal  to  lap  on  steam  side,  added  to  breadth 
of  port.  If  this  breadth  does  not  give  required  area  of  port,  throw  of  valve  must  be 
increased  until  required  area  is  attained. 


7 32 


STEAM-ENGINE. 


IPortion  of  Stroke  at  -wliick  Exhausting  3?ort  is  Closed, 
and  Opened. 

Lap  on 
Exhaust 
Side  of 


Lap  on 
Exhaust 
Side  of 
Valve  in 
Parts  of 
its  Throw. 


Portion  of  Stroke  at  which  Steam 
is  cut  off. 


* tV 


Valve  in 
Parts  of 
its  Throw. 


Portion  of  Stroke  at  which  Steam 
is  cut  off. 


xV 


.001 

.008 

.013 

.022 


.125 
.062  5 
.031  25 


109 

071 

•053 

041 


• 093-074 

.058  .043 

-043-033 

.0331.022 


B 

.125 
.062  5 
•031  25 


.026 

.052 

.066 

.082 


.008 

.022 

•033 

.044 


.004 

.015 

•023 

•°33 


Units  in  columns  of  table  A express  distance  of  piston,  in  parts  of  its  stroke,  from 
end  of  stroke  when  exhaust  port  in  advance  of  it  is  closed;  and  those  in  columns 
of  table  B express  distance  of  piston,  in  parts  of  its  stroke,  from  end  of  its  stroke 
when  exhaust  port  behind  it  is  opened. 


Illustration.—  A slide  valve  is  to  be  cut  off  at  one  sixth  from  end  of  stroke,  lap 
on  exhaust  side  is  one  thirty-second  of  stroke  of  valve  (16  ins.),  and  stroke  of  piston 
is  60  ins.  At  what  point  of  stroke  of  piston  will  exhaust  port  in  advance  of  it  be 
closed  and  the  one  behind  it  open. 

Under  one  sixth  in  table  A,  opposite  to  one  thirty-second,  is  .053,  which  x 60, 
length  of  stroke  = 3. 18  ins.  ; and  under  one  sixth  in  table  B,  opposite  to  one  thirty- 
second,  is  .033,  which  X 60  ==  1.98  ins. 

If  lap  on  exhaust  side  of  this  valve  wTas  increased,  effect  would  be  to  cause  port  in 
advance  of  valve  to  be  closed  sooner  and  port  behind  it  opened  later.  And  if  lap  on 
exhaust  side  was  removed  entirely,  the  port  in  advance  of  piston  would  be  shut, 
and  the  one  behind  it  open,  at  same  time. 

Lap  on  steam  side  should  always  be  greater  than  that  on  exhaust  side,  and  differ- 
ence greater  the  higher  the  velocity  of  piston. 

In  fast-running  engines,  alike  to  locomotives,  it  is  necessary  to  open  exhaust  valve 
before  end  of  stroke  of  piston,  in  order  to  give  more  time  for  escape  of  the  steam. 


To  Compute  Stroke  of*  a Slide  Valve. 

Rule.— To  twice  lap  add  twice  width  of  a steam  port  in  ins.,  and  sum 
will  give  stroke  required. 

Expansion  by  lap,  with  a slide  valve  operated  by  an  eccentric  alone,  cannot  be 
extended  beyond  one  third  of  stroke  of  a piston  without  interfering  with  efficient 
operation  of  valve;  with  a link  motion,  however,  this  distortion  of  the  valve  is 
somewhat  compensated.  When  lap  is  increased,  throw  of  eccentric  should  also  be 
increased. 

When  low  expansion  is  required,  a cut-off  valve  should  be  resorted  to  in  addition 
to  main  valve. 


To  Compute  Distance  of*  a Diston  from  End  of  its 
Stroke,  when  Dead  produces  its  Effect. 

Rule.  — Divide  lead  by  width  of  steam  port,  both  in  ins.,  and  term  the 
quotient  sine ; multiply  its  corresponding  versed  sine  by  half  stroke,  and 
product  will  give  distance  of  piston  from  end  of  its  stroke,  when  steam  is  ad- 
mitted for  return  stroke  and  exhaustion  ceases. 

Example.— Stroke  of  piston  is  48  ins.,  width  of  port  2.5  ins.,  and  lead  .5  inch; 
what  will  be  distance  of  piston  from  end  of  stroke  when  exhaustion  commences? 

. 5 -4-  2. 5 == . 2 = sine , ver.  sin.  of . 2 = .0202,  and  .0202  x — = .4848  ins. 

2 


To  Compute  Dead,  when  Distance  of  a Diston  from 
tlie  End  of  Stroke  is  given. 

Rule. — Divide  distance  in  ins.  by  half  stroke  in  ins.,  and  term  quotient 
versed  sine ; multiply  corresponding  sine  by  width  of  steam  port,  and  prod- 
uct will  give  lead. 

Example. — Assume  elements  of  preceding  case. 

48 

4848  — = - 0202  = ver.  sin. , and  sine  of  ver.  sin.  .0202  == . 2,  and  . 2 X 2. 5 = . 5 inch. 


STEAM-EJSTGINE. 


733 


To  Compute  Distance  ofaPiston  from  End  of  its  Strolze, 
wlxen  Steam  is  admitted.  for  its  Return  Stroke.  ’ 

Rule.— Divide  width  of  steam  port,  and  also  that  width,  less  the  lead,  by 
.5  stroke  of  slide,  and  term  quotients  versed  sines  first  and  second.  Ascer- 
tain their  corresponding  arcs,  and  multiply  versed  sine  of  difference  between 
first  and  second  by  .5  stroke,  aild  product  will  give  distance. 

Example.— Assume  elements  of  preceding  case,  lap  = .5  inch , and  stroke  of 
slide  6 ins. 

and  = ■ 8333,  and  .667  and  ver.  sin.  8o°  24'  a,  7o°  33'  x ^ = . 3528  inch. 

To  Compute  Dap  and.  Lead  of  Locomotive  Valves. 

To  cut  off  at  .33,  .25,  and  .125  of  stroke  of  piston,  lap  = 289,  .25,  and  .177  t,  outside 
lead  = .07  t,  and  inside  lead  = .3  t.  t representing  stroke  of  valve,  all  in  ins. 


HORSE-POWER. 

Horse-power  is  designated  as  Nominal , Indicated,  and  Actual. 

Nominal,  is  adopted  and  referred  to  by  Manufacturers  of  steam-engines, 
in  order  to  express  capacity  of  an  engine,  elements  thereof  being  confined 
to  dimensions  of  steam  cylinder,  and  a conventional  pressure  of  steam  and 
speed  of  piston. 

. Indicated,  designates  full  capacity  in  the  cylinder,  as  developed  in  opera- 
tion, and  without  any  deductions  for  friction. 

Actual,  lefers  to  its  actual  power  as  developed  by  its  operation,  involving 
elements  of  mean  pressure  upon  piston,  its  velocity,  and  a just  deduction  for 
friction  of  operation  of  the  engine. 


To  Compute  Horse-power  of  aix  Engine. 

1ST ominal .—  Non- condensing,  —— , and  Condensing.  = B?  D revre- 

1000  u 1400  x 

senting  diameter  of  cylinder  in  ins.,  and  v velocity  of  piston  in  feet  per  minute. 

Non-condensing  is  based  upon  uniform  steam-pressure  of  60  lbs.  per  sq. 
inch  (steam-gauge),  cut  off  at  .5  stroke,  deducting  one  sixth  for  friction  and 
losses,  with  a mean  velocity  of  piston,  ranging  from  250  to  4=50  feet  per 
minute.  1 

Condensing  is  based  upon  uniform  steam-pressure  of  30  lbs.  per  sq.  inch 
(steam-gauge),  cut  off  at  .5  stroke,  deducting  one  fifth*  for  friction  and 
losses,  with  a mean  velocity  of  piston  of  300  feet  per  minute  for  an  engine 
of  short  stroke,  and  of  400  feet  for  one  of  long  stroke. 

Actual.— Non-condensing.  - Ft 2 8 r --  HP.  A representing 

area  of  cylinder  in  sq.  ins.,  P mean  effective  pressure  upon  cylinder  piston,  inclusive 
of  atmosphere,  f friction  of  engine  in  all  its  parts,  added  to  friction  of  load,  both  in 
lbs.  per  sq.  inch,  s stroke  of  piston  in  feet,  and  r number  of  revolutions  per  minute. 

Sum  of  these  resistances  is  from  12.5  to  20  per  cent.,  according  to  pressure  of 
steam,  being  least  with  highest  pressure. 


, * This  value  may  be  safely  estimated  at  2.5  lbs.  per  sq.  inch  for  friction  of  engine  in  all  its  parts,  and 
friction  of  load  may  be  taken  at  7.5  per  cent,  of  remaining  pressure. 

t This  value  is  best  obtained  by  an  Indicator',  when  one  is  not  used,  refer  to  rule  and  table,  op.  710-12 
In  estimating  value  of  P,  add  14.7  lbs.,  for  atmospheric  pressure,  to  that  indicated  by  steam  gauge  or 
safety-valve.  Clearance  of  piston  at  each  end  of  cylinder  is  included  in  this  estimate. 

1 This  value  may  be  safely  estimated  in  engines  of  magnitude  at  1.5  to  2 lbs.  per  sq.  inch,  for  friction 
©f  engine  in  all  its  parts,  and  friction  of  load  may  be  taken  at  5 to  7.5  per  cent,  of  remaining  pressure. 

bum  of  these  resistances  in  ordinary  marine  engines  is  from  10  to  20  per  cent.,  according  to  pressure 
of  steam,  exclusive  of  power  required  to  deliver  water  of  condensation  at  level  of  discharge  or  load-line 

di£fer“t  desi8"s  a,  „ti- 

3 Q 


734 


STEAM-ENGINE. 


Illustration.— Diameter  of  cylinder  of  anon-condensing  engine  is  loins.,  stroke 
of  piston  4 feet,  revolutions  45  per  minute,  and  mean  pressure  of  steam  (steam 
gauge)  60  lbs.  per  sq.  inch. 

A=78.54  sq.  ins.  P 60+14.7  = 74  7 lbs-  /=2*S+(6o+  *447-^2.5) X .075=7.92  lbs. 


Then  78-54  X (60+  '4-7~7-92+<4-7)  Xi  X 4 X 45  _ n w 
3300° 

Note  i. Power  of  a non-condensing  engine  is  sensibly  affected  by  character  of  its 

exhaust,  as  to  whether  it  is  into  a heater,  or  through  a contracted  pipe,  to  afford  a. 
blast  to  combustion. 

2.— If  an  indicator  is  not  used  to  determine  pressure  of  steam  in  a cylinder,  a 
safe  estimate  of  it,  when  acting  expansively,  is  .9  of  full  pressure,  and  when  at  full 
stroke  from  .75  to  .8. 

Condensing.  A Pt  ~f*  g. 

a 33  000 

Power  required  to  work  the  air-pump  of  an  engine  varies  from  .7  to  .9  lhs.  per  sq. 
inch  upon  cylinder  piston. 

Illustration.— Diameter  of  cylinder  of  a marine  steam-engine  is  60  ins.,  stroke 
of  piston  10  feet,  revolutions  15  per  minute,  pressure  of  steam  50  lbs.  per  sq.  inch, 
cut  off  at  .25  stroke,  and  clearance  2 per  cent. 

A = 2827. 4 sq.  ins.  P (per  Ex.,  page  713)  = 28.62  lbs.  f—  1.5  + 28.62  — 1.5  X .05 

==  2.467  tbs.  •_ 

Then  °827-4  X 28.66-2.856  X2X10X  15  = 662.2-5  TP. 

33  000 

From  which  is  to  be  deducted  in  marine  engines  power  necessary  to  discharge 
water  of  condensation  at  level  of  load-line,  which  is  determined  by  pressure  due  to 
elevation  of  water,  area  of  air-pump  piston,  and  velocity  of  its  discharge  in  feet  per 
second. 


Indicated. 


AP2s>:=H?and  33cooip 


33  000 


P 2 s r 


British.  Admiralty  Buile. — Nominal. 


7 Av 


D2  v 


33  000  6000 


= IP. 


French. — {Force  de  Cheval.)  1.695  D2Lr  = IP-  D and  L in  meters. 
Illustration.— Assume  a diameter  of  cylinder  of  .254  meters,  with  a stroke  of 
piston  of . 3 meters  and  250  revolutions  per  minute. 

1.695  X -2542  X .3  X 250  = 8.18  IP. 

A Force  de  Cheval  — 4500  kilometers  per  minute  = 32  549  foot-lbs.  = .987  57  IP. 
One  IP  ==  1. 0139  Force  de  Chevaux. 

Compound  Indicated.  ALr|ii  hyp.  log.  R"  — .000053  = IP. 

L representing  length  of  stroke  in  feet , R,/  combined  ratio  of  both  cylinders , and  b> 
back  pressure.  j 

Illustration. — Assume  area  of  cylinder  3 sq.  ins.,  stroke  6 feet,  one  stroke  of( 
piston,  and  steam  60  lbs.  per  sq.  inch,  cut  off  at  .25. 

A = 3 sq.  ins. , L = 6 feet , n = 1 stroke,  P = 60  lbs. , R"  = 5-  969>  b = 3 ' 

per  sq.  inch , and  r = . 5,  and  1 + hyp.  log.  R"  = 1 + 1.7865. 

Then  3X6X.5X  X 1 + 1.7865  — 3)  X-oooo53  = 9X  10.052X2.7865  — 3 

\5*  909  / _ , 

X .000053  = .on  93  IP,  which,  x 2 for  I revolution,  = .023  86  IP  per  revolution. 


To  Compute  Volume  of  Water  required  to  "be  Evapo- 
rated in  an  Engine. 

Rule.— Multiply  volume  of  steam  expended  in  cylinder  and  steam-chests 
by  twice  number  of  revolutions,  and  multiply  product  by  density  of  steam 
at  given  pressure.  


t X For  reference  see  2d  and  3d  foot-note  on  previous  paf?e. 


STEAM-ENGINE. 


735 


Example. — Wbat  volume  of  water  will  an  engine  require  to  be  evaporated  per 
revolution,  diam.  of  cylinder  being  70  ins.,  stroke  of  piston  10  feet,  and  pressure  of 
steam  34  lbs.  per  sq.  inch,  including  atmosphere,  cut  off  at  .5  of  stroke? 

Area  of  cylinder  =2  3848.5  ins.  10X12-7-2  = 60  ins.,  60  X 3848.5  =2  230910  cube  ins. 

Add,  for  clearance  at  one  end,  volume  of  nozzle,  steam-chest,  etc.,  17  317  cube  ins. 

Then  230  910  -f- 17  317  -4- 1728  X 2 = 287.3  cube  feet , which,  x .001336,  density  of 
steam  at  34  lbs.  pressure  (see  Note  2),  =2.3838  cube  feet. 

Note  r. — This  refers  to  expenditure  of  steam  alone;  in  practice,  however,  a large 
quantity  of  water  “ foaming,”  differing  in  different  cases,  is  carried  into  cylinder  in 
combination  with  the  steam;  to  which  is  to  be  added  loss  by  leaks,  gauges,  etc. 

2. — Volume  of  steam  is  readily  computed  by  aid  of  table,  pp.  708-9.  Thus,  den- 
sity or  weight  of  one  cube  foot  of  steam  at  above  pressure  =2 .0835  lbs.  Hence,  as 
62.5  lbs.  : 1 cube  foot  .0835  lbs.  : .001  336  cube  foot. 

To  Compute  Volume  of  Circulating  Water  required.  Toy 
an  Engine. 

1114  + .3T  — t „ m 

^ — -p = v . T representing  temperature  of  steam  upon  entering  the  con- 
denser, t,  t\  and  t"  temperatures  of  feed  water , of  water  of  condensation  discharged , 
and  of  circulating  water , all  in  degrees. 

Illustration.— Assume  exhaust  steam  at  8 lbs.  per  sq.  inch,  temperatures  of  dis- 
charge ioo°,  feed  water  1200,  and  sea-water  750. 

1114-f  .3  X 183— 120 

_ 41.05  times. 

100  — 75  * y 


Temperature  at  8 lbs.  pressure  = 183°. 


To  Compute  Volume  ofFlow  through  an  Injection  IPipe. 

Rule.— Multiply  square  root  of  product,  of  64.33  and  depth  of  centre  of 
opening  i to  condenser,  from  surface  of  external  water,  added  to  height  of  a 
column  of  water  due  to  vacuum,  in  condenser,  all  in  feet,  by  area  of  opening 
in  sq.  ins. ; and  .6  product,  divided  by  2.4  (144  -h  60)  will  give  volume  in 
cube  feet  per  minute. 

Example. — Diameter  of  an  injection  pipe  is  5.375  ins.,  height  of  external  water 
above  condenser  6.13  feet,  and  vacuum  24.45  ^s. ; what  is  volume  of  flow  per  min.? 

Area  of  5.375  ins.  = 22.69  ins.,  c = .6.  Vacuum  ^---45  inS’  ==  J2  lbs. ; 12X2.24 
feet  (sea- water)  = 26. 88  feet,  and  26. 88  -f  6. 13  — 33. 1 jeet.  °4 


Then 


^64.33  X 33.1  X 22.69  x ,6  628. 15 


2=  261.73  cube  feet. 


To  Compute  Area  of  an  Injection  Eipe. 

Rule.— Ascertain  volume  of  water  required  by  rule,  page  706,  in  cube  ins. 
per  second,  multiply  it  by  number  of  volumes  of  water  required  for  con- 
densation, by  rule,  page  707,  divide  it  by  velocity  due  to  flow  in  feet  per 
second,  and  again  by  12,  and  quotient  will  give  area  in  sq.  ins. 

Example.— An  engine  having  a cylinder  70  ins.  diam.,  stroke  of  piston  10  feet 
per  minute  15,  and  steam  19.3  lbs.,  mercurial  gauge  cut  off  at  .5;  what 
should  be  area  of  its  injection  pipe  at  its  maximum  operation? 

Volume  of  cylinder  267.25  cube  feet,  cut  off  at  .5  = 133.625  ins. 

wot p r nm rm t ‘lt  34  lbS'  I4’7)  = -001  336.  Velocity  of  flow  of  injected 

watei  (computed  from  vacuum  and  elevation  of  condensing  water)  33  feet  per  second. 

Then  133.625  X 15  X 2 X 1728 -h  60  = 115452  cube  ins.  steam  per  second,  and 
115  452  X .001  336  =2  154.24  cube  ins.  water  per  second. 

onhaaT^nn^lTe  L • required  t0  condense  steam  is  about  70  times  volume 
is  about  4oPtimesd’  WhlCh  °U  7 °CCUrS  m the  Gulf  of  Mexico5  ordinary  requirement 

154  24  -f-  11.59  (=  7-5  Per  cent,  for  leakage  of  valves,  etc.)  = 165.83,  which,  x 70 
as  above,  = u 608. 1 cube  ins.,  and  n 608. 1 -4-  33  x 12  2=  29. 31  sq.  ins. 


STEAM-ENGINE. 


7 36 


Coefficient  of  velocity  for  flow  under  like  conditions  ==  .6;  hence,  29.31^.6  = 
48.85  sq.  ins. 

■njote This  is  required  capacity  for  one  pipe.  It  is  proper  and  customary  that 

there  should  be  two  pipes,  to  meet  contingency  of  operation  of  one  being  arrested. 

To  Compute  Area  of*  a Deed  Dump.  {Sea-water.) 

Rule. — Divide  volume  of  water  required  in  cube  ins.  by  number  of  single 
strokes  of  piston,  both  per  minute,  and  divide  quotient  by  stroke  of  pump,  in 
ins. ; multiply  this  quotient  by  6 (for  waste,  leaks,  “running  up,  etc.),  and 
product  will  give  area  of  pump  in  sq.  ins. 

Example— Assume  volume  to  be  5 cube  feet  and  revolutions  of  engine  15  per 
minute,  with  a stroke  of  pump  of  3.5  feet. 

5 X i72j  _ ^ which -r-  3.5  X 12  = 13  72,  and  13.72  X 6 = 82.32  sq.  ins. 

15 

Note.— In  fresh  water,  this  proportion  may  be  reduced  one  half. 


STEAM-IN JECTOlt.  William  Sellers  & Co. 
Self-adjusting. 

Volume  of*  Water  Discharged  per  Hour. 


Pressure  of  Steam  in  Lbs. 

No. 

No. 

60 

80 

100 

120 

Cub.  feet. 

Cub.  feet. 

Cub.  feet. 

Cub.  feet. 

3 

28.12 

31.66 

35-2 

38-75 

7 

4 

52.16 

58.44 

64.72 

71 

8 

5 

82.18 

92.02 

101.86 

111.7 

9 

6 

119.09 

133-33 

147-57 

161.82 

10 

Pressure  of  Steam  in  Lbs. 

60 

80 

100 

120 

Cub.  feet. 

Cub.  feet. 

Cub.  feet. 

Cub.  feet. 

162.65 

182.1 

201.55 

221 

213.2 

238.8 

264.4 

290 

269.97 

302.28 

334-  59 

366.9 

333-64 

373-57 

4*3-49 

453-41 

Highest  temperature  admissible  of  feed  water  1350. 

To  Compute  Size  of  Injector  re<qnired. 


One  nominal  IP  per  hour  will  require  one  cube  foot  of  water  per  hour. 
When  the  lbs.  of  coal  burned  per  hour  can  be  ascertained,  divide  them  by 
7.5,  and  quotient  will  give  the  volume  of  water  in  cube  feetpei  hour. 

When  the  area  of  grate-surface  is  known,  multiply  it  by  1.6  for  IP. 

In  case  of  plain  cylindrical  boilers,  divide  the  number  of  sq.  feet  of  heat- 
ing-surface by  10  for  the  IP.  In  case  of  flue  boilers,  divide  by  12,  and  with 
multi-tubular  boilers,  by  15,  for  the  nominal  IP. 

Minimum  capacity  of  Injectors,  about  50  per  cent,  of  Tabular  capacity. 


To  Compute  Volume  of  Injection  Water  required  per 
IIP  per  Hour. 

Operation.— Assume  temperature  of  water  8o°,  and  of  condensation  ioo°.  I ol- 
ume  of  cylinder  per  IIP  as  per  formula,  page  716,  and  illustration,  page  717,  — 2.70 
feet  per  minute. 

Then,  as  per  rule  page  707,  1 gQ — = 52-3  cu^e  ins.  per  cube  foot  of  steam. 

2.76  X 52- 3 X 62^5  _ ^ 22  > which,  X 60,  = 313.2  lbs. 

1728 

To  Compute  Net  Volume  of  Feed  Water  required  per 
I DP  per  Hour. 

Operation.— Assume  elements  of  formula,  page  716,  and  illustration,  page  717. 
Then  .1154  X 2.76  X 60  = 19.11  lbs. 

Feed  Pipes . — fv  = diameter  for  small,  and  ^-Vv,  for  large  pumps. 
d representing  diameter  of  plunger  in  ins.,  and  v its  velocity  in  feet  per  minute. 


STEAM-ENGINE. 


737 


Itesvilts  of  Operations  of  Steam-engines.  ( D . K.  Clark.) 


Condensing  Engine. 

Actual 
Ratio  of 
Expan- 
sion. 

Steam 
per 
IIP  as 
cut-off. 

Coal 

per 

IIP. 

Initial 

Pressure 

at 

cut-off. 

Steam 
per  IIP 
per  hour. 

SINGLE. 

5-2 

6.07 

Lbs. 

I4-5I 

14.27 

12.92 

Lbs. 

2-5 

2.2 

Lbs. 

34-5 

46 

23.25 

50 

Lbs. 

17.4 

18.7 

20.72 

19.6 

18.62 

3.62 

Sulzer  Corliss  valves. 

10 

4. 122 

3.3 

Simpi-hpntpfl  Dim 



60 

* ’ 

J 

COMPOUND 

' Receiver 

I.85 

I.852 

4.OI 

1-857 

2.486 

o.  221 

14-45 
14.85 
10.94 
13-34 
13. 18 

13-87 
actual 
22. 21 

1. 61 

56 

85.5 

J.  Elder  & Co J 

[ Marine,  jacketed 
1 Receiver 

J.  & E.  Wood j 

[ stationary 

2.14 

— 

Donkin j 

[ Woolf,  stationary 
j jacketed 

5o-5 

22.51 

15-37 

American,  Woolf j 

[ 1st  cylinder 

[ both 

O'  ** x 

2.3I 

5-63 

3-77 

Q.  IQ 

_ 

90 

“ “ jacketed  < 

f 1st  cylinder 

iboth 

-j 

20.71 

— 

90 

14.1 

NON-CONDENSING. 

Marshall,  Sons,  & Co 

4.8 

16.87 

76 

25-9 

29.6 

3i-36 

21.24 

Davev,  Paxman,  & Co 

5 

T,nnnmnt,ive  “Great  Britain” 

l.AC 

31-36 

21.24 

/ 5 

“ “ 

2.94 

— 

T7 

[Practical  Efficiency  of  Steam-engines.  Initial  Volume  =.  i. 


Cylinders. 

Most  Efficient 
Ratio  of  Ex- 
pansion. 

Steam  * per  IH? 
per  hour. 

Cylinders. 

Most  Efficient 
Ratio  of  Ex- 
pansion. 

Steam  * per  IH* 
per  hour. 

CONDENSING. 

Single  cylinder,  jacketed. . . 

6 

Lbs. 

19-5 

Compound,  jacketed,  Woolf 
Compound,  Woolf. 

IO 

7 

Lbs. 

20.5 

23 

Single  cylinder 

A 

“ “ superheated 

Compound,  jacketed,  Re- 1 
ceiver j 

T 

4 

6 

18.5 
1 9 

NON- CONDENSING. 

Single  cylinder,  + jacketed. . 
Single  cylinder,  $ 

4 

3 

to  to 

M 

* From  boiler.  t 70  lbs.  pressure.  \ 90  lbs.  pressure. 


Standard  Operation  of  a Portable  Engine . 

Grate 5.5  sq.  feet. 

Heating  surface 220  “ “ 

Coal  per  EP  per  hour 6.25  lbs. 

“ “ sq.  foot  of  grate.  9 “ 

“ “ hour 50  “ 


Water  evaporated  from 1 

and  at  212°  per  hour. j 45° 

“ “ per  H?  per  hour  62.5 

“ “ “ sq.  foot  of] 

grate ; 


lbs. 


81.8 


Ratio  of  heating  surface  of  grate 40  to  x. 


MIXTURE  OF  AIR  AND  STEAM. 

Water  contains  a portion  of  air  or  other  uncondensable  gaseous  matter,  and  when 
it  is  converted  into  steam,  this  air  is  mixed  with  it,  and  when  steam  is  condensed 
it  is  left  in  a gaseous  state.  If  means  were  not  taken  to  remove  this  air  or  gaseous 
matter  from  condenser  of  a steam-engine,  it  would  fill  it  and  cylinder,  and  obstruct 
their  operation;  but,  notwithstanding  the  ordinary  means  of  removing  it  (by  air- 
pump),  a certain  quantity  of  it  always  remains  in  condenser. 

20  volumes  of  water  absorb  1 volume  of  air. 

3 Q* 


738 


STEAM-ENGINE. 


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for  like  Volume 
and  Speed. 

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STEAM-ENGINE. — BOILER. 


739 


BOILER. 

Its  efficiency  is  determined  by  proportional  quantity  of  heat  of  com- 
bustion of  fuel  used,  which  it  applies  to  the  conversion  of  water  into 
steam,  or  it  may  be  determined  by  weight  of  water  evaporated  per  lb. 
of  fuel. 

In  following  results  and  computations,  water  is  held  to  be  evaporated  from  stand- 
ard temperature  of  2120. 

Proportion  of  surplus  air,  in  operation  of  a furnace,  in  excess  of  that  which  is 
chemically  required  for  combustion  of  the  fuel,  is  diminished  as  rate  of  combustion 
is  increased;  and  this  diminution  is  one  of  the  causes  why  the  temperature  in  a 
furnace  is  increased  with  rapidity  of  combustion. 

When  combustion  is  rapid,  some  air  should  be  introduced  in  a furnace 
above  the  grates,  in  order  the  better  to  consume  the  gases  evolved. 

Natural  Draught. 

Grate  {Coal)  should  have  a surface  area  of  1 sq.  foot  for  a combustion  of 
15  lbs.  of  coal  per  hour,  length  not  to  exceed  1.5  times  width  of  furnace,  and 
set  at  an  inclination  toward  bridge-wall  of  1 to  1.5  ins.  in  every  foot  of  length. 

When,  however,  rate  of  combustion  is  not  high,  in  consequence  Of  low  ve- 
locity of  draught  of  furnace,  or  fuel  being  insufficient,  this  proportion  of  area 
must  be  increased  to  one  sq.  foot  for  every  12  lbs.  of  fuel. 

Width  of  bars  the  least  practicable,  spaces  between  them  being  from  .5  to 
.75  of  an  inch,  according  to  fuel  used.  Anthracite  requiring  less  space  than 
bituminous.  Short  grates  are  most  economical  in  combustion,  but  generate 
steam  less  rapidly  than  long. 

Level  of  grate  under  a plain  cylindrical  boiler  gives  best  effect  with  a fire 
5 ins.  deep,  when  grate  is  but  7.5  ins.  from  lowest  point. 

Depth,  Cast-iron,  .6  square  root  of  length  in  ins. 

{Wood),  their  area  should  be  1.25  to  1.4  that  for  coal. 

Automatic  (Vicar’s).  — Its  operation  effects  increased  rapidity  in  firing 
and  more  effective  evaporation. 

Ash-pit. — Transverse  area  of  it,  for  a combustion  of  15  lbs.  of  cogl  per 
hour,  2 to  .25  area  of  grate  surface  for  bituminous  coal,  and  .25  to  .3  for 
anthracite.  Or  15  to  20  ins.  in  depth  for  a width  of  furnace  of  42  ins. 

Furnace  or  Combustion  Chamber. — {Coal)  Volume  of  it  from  2.75  to  3 cube 
feet  per  sq.  foot  of  grate  surface.  {Wood)  4.6  to  5 cube  feet. 

The  higher  the  rate  of  combustion  the  greater  the  volume,  bituminous 
coal  requiring  more  than  anthracite.  Velocity  of  current  of  air  entering 
an  ash-pit  may  be  estimated  at  12  feet  per  second. 

Volume  of  air  and  smoke  for  each  cube  foot  of  water  converted  into  steam  is, 
from  coal,  1780  to  1950  cube  feet,  and  for  wood,  3900. 

Rate  of  Combustion.  — In  lbs.  of  coal  per  sq.  foot  of  grate  per  hour. 
Cornish  Boilers,  slowest,  4 ; ordinary,  10.  Stationary , 12  to  16.  Marine , 
16  to  24.  Quickest:  complete  combustion  of  dry  coal,  20  to  23;  of  caking 
coal,  24  to  27 ; Blast  or  Fan  and  Locomotive , 40  to  120. 

Bridge-wall  {Calorimeter).— Cross-section  of  an  area  of  1.2  to  1.6  sq.  ins. 
for  each  lb.  of  bituminous  coal  consumed  per  hour,  or  from  18  to  24  sq.  ins. 
for  each  sq.  foot  of  grate,  for  a combustion  of  15  lbs.  of  coal  per  hour. 

Temperature  of  a furnace  is  assumed  to  range  from  1500°  to  2000°,  and 
volume  of  air  required  for  combustion  of  1 lb.  of  bituminous  coal,  together 
with  products  of  combustion,  is  154.81  cube  feet,  which,  when  exposed  to 
above  temperatures,  makes  volume  of  heated  air  at  bridge- wall  from  600  to 
750  cube  feet  for  each  lb.  of  coal  consumed  upon  grate. 


740 


STEAM-ENGINE. BOILER. 


Hence,  at  a velocity  of  draught  of  about  12  feet  per  second,  area  at  bridge- 
wall,  required  to  admit  of  this  volume  being  passed  off  in  an  hour,  is  2 to  2.5 
sq.  ins.,  and  proportionately  for  increased  velocity,  but  in  practice  it  may  be 
1. a to  1.6  ins. 

When  20  lbs.  of  coal  per  hour  are  consumed  upon  a sq.  foot  of  grate,  20  x 1.2  or 
1.6  = 24  or  32  sq.  ins.  are  required,  and  in  a like  proportion  for  other  quantities. 

Or,  When  area  of  flues  is  determined  upon,  and  area  over  bridge-wall  is 
required,  it  should  be  taken  at  from  .7  to  .8  area  of  lower  flues  for  a natural 
draught,  and  from  .5  to  .6  for  a blast. 

When  one  half  of  tubes  were  closed  in  a fire-tubular  marine  boiler,  the  evapora- 
tion per  lb.  of  coal  was  reduced  but  1. 5 per  cent. 

Firing. — Coal  of  a depth  up  to  12  ins.  is  more  effective  than  at  a less 
depth.  Admission  of  air  above  the  grate  increases  evaporative  effect,  but 
diminishes  the  rapidity  of  it. 

Air  admitted  at  bridge-wall  effects  a better  result  than  when  admitted  at 
door,  and  when  in  small  volumes,  and  in  streams  or  currents,  it  arrests  or  pre- 
vents smoke.  It  may  be  admitted  by  an  area  of  4 sq.  ins.  per  sq.  foot  of  grate. 

Combustion  is  the  most  complete  with  firings  or  charges  at  intervals  of 
from  1 5 to  20  minutes. 

With  a fuel  economizer  (Green’s)  an  increased  evaporative  effect  of  9 per 
cent,  has  been  obtained. 

When  external  flues  of  a Lancashire  boiler  were  closed,  evaporative  power  was 
slightly  increased,  but  evaporative  efficiency  was  decreased;  and  when  25  per  cent, 
of  like  surface  in  setting  of  a plain  cylindrical  boiler  was  cut  off,  evaporation  was 
reduced  but  1.5  per  cent.  When  temperature  at  base  of  chimney  was  630°,  with  a 
fire  12  ins.  in  depth,  it  was  decreased  to  556°  with  one  9 ins.  in  depth,  and  to  5390 
with  one  6 ins. 


High  wind  increases  evaporative  effect  of  a furnace;. 

Stationary  or  Land.— Set  at  an  inclination  downward  of  .5  inch  in  10  feet. 

Smoke  Preventing.  —A  test  of  C.  Wye  Williams's  design  of  preventing  smoke,  at 
Newcastle,  1857,  as  reported  by  Messrs.  Longridge,  Armstrong,  and  Richardson, 
gave  an  increased  evaporative  effect  with  the  “practical  prevention  of  smoke.” 
Hence  it  was  concluded,  “ That  by  an  easy  method  of  firing,  combined  with  a due 
admission  of  air  in  front  of  furnace,  and  a proper  arrangement  of  grate,  emission 
of  smoke  may  be  effectually  prevented  in  ordinary  marine  multi-tubular  boilers, 
with  suitable  coals.  2d.  That  prevention  of  smoke  increases  economic  value  of  fuel 
and  evaporative  power  of  boiler.  3d.  That  coals  from  the  Hartley  district  have  an 
evaporative  power  fully  equal  to  that  of  the  best  Welsh  steam-coals.” 


Heating  Surfaces. 

Marine  ( Sea-water ).  — Grate  and  heating  surfaces  should  be  increased 
about  .07  over  that  for  fresh  water. 

Relative  Value  of  Heating  Surfaces. 

Horizontal  surface  above  the  flame  = 1 I Horizontal  beneath  the  flame = . 1 

Vertical = -5  1 Tubes  and  flues. = .56 

Minimum  Volumes  of*  Fuel  Consumed  per  Sq.  Foot  of* 
Grrate  per  Hour,  for  given  Surface-ratios.  (D.  K.  Clark.) 

Surface-ratios  of  Heating  Surface  to  Grate. 


Stationary 

Marine 

Portable 

Locomotive  (coal) 
(coke) 


20  30 


Lbs. 

12. 1 

11. 2 
3-2 
5-2 

7 


50 


Lbs. 

18.9 

i7-5 

5 


Lbs. 

26 

24 

n.7 

16 


18.3 

25 


26.3 

36 


32.5 

44 


At  extreme  consumption  of  fuel  (120  lbs.)  coke  will  withstand  disturbing  effect 
of  a blast  better  than  coal. 


STEAM-ENGINE. — BOILER. 


741 


A scale  of  sediment  one  sixteenth  of  an  inch  thick  will  effect  a loss  of  14.7  per 
cent,  of  fuel. 

One  sq.  toot' of  fire  surface  is  held  to  be  as  effective  as  three  of  heating. 

Relation  of  Grrate,  Heating  Surface,  and  Fuel. 

When  Grate  and  Heating  Surface  are  constant,  greater  the  weight  of  fuel 
consumed  per  hour,  greater  the  volume  of  water  evaporated ; but  the  volume 
is  in  a decreased  proportion  to  fuel  consumed. 

In  treating  of  relations  of  grate,  surface,  and  fuel,  D.  K.  Clark,  in  his  valuable 
treatise,  submits,  that  in  1852  he  investigated  the  question  of  evaporative  perform- 
ance of  locomotive-boilers,  using  coke;  and  he  deduced  from  them,  that  assuming 
a constant  efficiency  of  fuel,  or  proportion  of  water  evaporated  to  fuel,  evaporative 
effect,  or  volume  of  water  which  a boiler  evaporates  per  hour,  decreases  directly  as 
grate-area  is  increased;  that  is  to  say,  larger  the  grate,  less  the  evaporation  of  water 
at  same  rate  of  efficiency  of  fuel,  even  with  same  heating  surface. 

2d.  That  evaporative  effect  increases  directly  as  square  of  heating  surface  with 
same  area  of  grate  and  efficiency  of  fuel. 

3d.  Necessary  heating  surface  increases  directly  as  square  root  of  effect— viz  for 
four  times  effect,  with  same  efficiency,  twice  heating  surface  only  is  required.  ’ 

4th.  Necessary  heating  surface  increases  directly  as  square  root  of  grate,  with  same 
efficiency;  that  is,  for  instance,  if  grate  is  enlarged  to  four  times  its  first’area  twice 
heating  surface  would  be  required,  and  would  be. sufficient,  to  evaporate  same  vol- 
ume of  water  per  hour  with  same  efficiency  of  fuel. 

Result  of  40  experiments  with  a stationary  boiler  (fresh  water),  with  an 
evaporation  of  9 lbs.  water,  per  lb.  of  fuel  consumed,  the  coefficient  .002  22 
was  deduced. 

(ji  \ 2 

-)  .00222  ==W.  W representing  volume  of  water  in  cube  feet,  and  g 

and  h areas  of  grate  and  heating  surfaces  in  sq.  feet. 

Illustration. — Assume  a heating  surface  of  90  feet,  and  a grate  of  v what  will 
be  the  evaporation  1 5 ’ 


To  Compute  Areas  ofGrate  and  Heating  Surfaces 
Volume  of  Water,  and  Weight  of  EPnel. 

For  a Temperature  of  281°,  or  Pressure  of  50  lbs.  per  Sq.  Inch. 


Eo  Compute  Ratio  of  Heating  Surface  to  -Area  of  Grate 
and.  to  inflect  a Given  Evaporation. 


Then  90-^3  x .002  22  = 1.998  cube  feet. 


When  Water  per  Sq.  Foot  of  Grate  per  Hour  and  Surface  Ratio  are  Given. 
W — X R2 


To  Compute  Weight  of  Fuel. 


— x R2  ^ 

==  F,  and  x R2  = (E  — C)  F. 


F,  and  x R2  — (E  — C)  F. 


Illustration. — Assume  elements  as  preceding. 


.02 


.02 


STEAM-ENGINE. — BOILER. 


742 


When  Efficiency  of  Fuel  and  Fuel  consumed  per  Sq.  Foot  of  Grate  per 
Hour  are  given . = E or  efficiency  of  fuel  or  weight  of  water  evaporated  per  lb. 

/(E  — C)F 

Of  fad.  f— y—  = R- 

To  Compute  Fuel  tliat  may  fee  consumed  per  Sq.  Foot 
of  Grate  per  Hour,  corresponding  to  a driven  Effi- 
ciency. 

When  Efficiency  of  Fuel,  that  is,  Weight  of  Water  evaporated  per  Lb.  of 
Fuel,  and  the  Surface  Ratio , are  given. 

,R!±CF  c + ^K!  = Eiand  £*l  = r. 


F ’ ' ' F 

Illustration. — Assume  elements  as  preceding. 


E— C" 


.02  x 5°2~f~  IQ  X_^5  _ I3  33;  ,0  + 


.02  X 5° 


- = i3-33, 


and 


.02  X 5°2 


= 15  lbs. 


is  *^3  ; — 

Combustion  of  Coal  per  sq.  foot  grate.  -Natural  Draught , from  20  to  25  lbs.  can 

be  consumed  per  hour. — Steam-jet , 30  lbs.,  and  Exhaust-blast  65  to  80  lbs. 

From  Results  of  Experiments  upon  Marine  Boilers,  see  Manual  of  D.  K.  Clark, 
page  808;  he  deduced  the  following  formula,  as  applicable  to  all  surface  ratios  in 
such  boilers. 

Newcastle  .021  56  R2-}~9-71  hnd  for  Wigan  .01  R2-j-  10.75  F = W in  lbs. 

And  the  general  formulas  he  deduced  from  all  the  various  experiments  are  as 

follows.  , . 

From  and  at  2120. 


Portable 008  R2-}“8.6  F = W. 

Stationary...  0222  R2-j- 9.56  F ==  W. 

Locomotive,  coke 


Marine 016  R2-f- 10.25  F = W. 

Locomotive,  coal,  .009  R2-f  9-7  F = W. 
..  0178  R2-}- 7.94  F = W. 


As  the  maximum  evaporative  power  of  fuel  is  a fixed  quantity,  the  preceding 
formulas  are  not  fully  applicable  in  minimum  rates  of  its  consumption  and  e\apo- 
rative  quality. 

With  coal  and  coke  the  limits  of  evaporative  efficiency  may  be  taken  respectively 

at  12.5  and  12  lbs.  water  from  and  at  2120. 

Illustration  l— Assume  a marine  fire-tubular  boiler  with  a surface  ratio  of  heat- 
ing surface  to  grate  of  30  and  a consumption  of  coal  of  15  lbs.  per  sq.  foot  ot  grate 
per  hour,  what  will  be  its  evaporation  per  sq.  foot  of  grate? 

.016  X 3P2-f*  10.25  X 15  = 168.15  lbs. 

2.— Assume  a like  boiler,  using  fresh  water,  to  have  a ratio  of  heating  surface  to 
grate  of  30  and  an  evaporation  of  165  lbs.  water  per  sq.  foot  of  grate  per  hour,  what 
would  be  consumption  of  coal  per  sq.  foot  of  grate  per  hour? 


165— .016  xjp;  m 

jo.  25 

Tube  Surface  (Iron)  per  lb.  of  coal  1.58,  per  sq.  foot  of  grate  32,  and  per  IIP  4-  27 
sq.  feet. 

Locomotive  Boiler  has  from  60  to  90  sq.  feet  per  foot  of  grate,  and  consumes  65 
lbs.  coal  per  sq.  foot  per  hour. 

Evaporative  Capacity-  of  Tubes  of  Varying  Length. 
Total  Length  of  Tubes  12  Feet  3 ins.  (M.  Paul  Hevrer,  1874.) 


Furnace  and 

TUI 

1 E S. 

Surface  and  Water. 

3 ins.  in  Length 
of  Tubes. 

3.02 

Feet. 

3.02 

Feet. 

3.02 

Feet. 

3.02 

Feet. 

Surface  in  sq.  feet 

76.43  ■ 

179 

179 

179 

179 

Water  evaporated  per  sq. ) 
foot  per  hour  in  lbs j 

24- 5 

8.72 

4.42 

2.52 

1.68 

STEAM-ENGINE. BOILER. 


743 


Results  of  Operation  of  Boilers  under  Varying  Propor- 
tions of  Grate,  Area,  and.  Length,  of  Heating  Surface, 
Draught  of  Furnace,  and  Rate  of  Combustion. 


Fire-tubular. 

Agricultural  and  Hoisting 

tl  U U 

Locomotive 

English 


Marine1 


Stationary  4. 


Area  of 
Grate. 

Heating 

Surface. 

Grate  to 
Heating 
Surface. 

Coal  per 
Sq.  Foot 
of  Grate 
per  Hour. 

Evapor 
Water  fi 
per  sq.  ft. 
of  grate. 

ation  of 
rom  2120 
per  lb. 
of  Coal. 

Fuel. 

Sq.  Feet. 

Sq.  Feet. 

Ratio. 

Lbs. 

Lbs. 

Lbs. 

4-7 

158 

34 

13 

1 19 

9-33 

Welsh. 

, 3-2 

220 

69 

12.8 

151 

11.83 

( 26. 25 

9^3-5 

36-7 

30.86 

327 

10.6 

(16 

8l8 

51 

38 

375 

10.47 

10.5 

788 

75 

45 

419 

11.04 

10.6 

IO56 

100 

i57 

1401 

10.41 

22 

748 

34 

24- 3 

265 

10.7 

18 

749 

41.6 

23.6 

264 

11. 2 

10.3 

9J5 

50 

41.25 

468 

11.36 

10.3 

508 

49-3 

27.63 

309.8 

n-54 

Lanc’r 

10.8 

151.2 

14 

27.76 

205 

7-39 

Anth’e 

3.1-5 

945 

30 

28.87 

293-7 

10.17 

Welsh. 

3i-5 

767 

24.4 

14 

141.4 

10. 1 

“ 

2 and  4 Wigan. 

3 Experimented  at  New  York. 

i New  Castle. 

* Effect  of  reducing  the  tube-surfaces  was  tried  by  stopping  one  half  the  number  of  tubes  in  alter- 
nate diagonal  rows,  so  that  the  tube  surface  was  reduced  206.5  sq.  feet.  The  results  with  fires  12  ins. 
deep  were  as  follows : 

Tubes  open.  Tubes  half  closed. 

Coal  per  sq.  foot  of  grate  per  hour 25  lbs.  24  lbs. 

Water  from  2120  per  lb.  of  coal 12.41  “ 12.23  “ 

Smoke  per  hour,  very  light 2.8  minutes.  8 minutes. 

Evaporative  Effects  of  Boilers  for  Different  Rates  of 
Combustion,  and  Surface  Ratios.  {D.  K.  Clark.) 

Water  from  arid  at  212 0 per  Hoar. 

Surface  Ratio  30. 


Fuel  per 
Sq.  Foot 
of  Grate 
per  Hour. 

Statio 

Wa 

per 

Sq.  foot. 

NARY. 

ter 

per  lb. 
of  Coal . 

Mar 

Wa 

per 

Sq.  foot. 

INE. 

ter 

per  lb. 
of  Coal  . 

Port. 

Wa 

per 

Sq.  foot. 

1.BLE. 

ter 

per  lb. 
of  Coal . 

C01 

Wa 

per 

Sq.  foot. 

Locos/ 

al. 

ter 

per  lb. 
of  Coal . 

IOTIVE. 

Co 

Wt 

per 

Sq.  foot. 

ke. 
iter 
per  lb. 
of  Coal. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

10 

116 

11. 6 

117 

n-7 

93 

9-3 

io5 

10.5 

95 

9.5 

15 

163 

10.9 

168 

11. 2 

136 

9 

154 

10.3 

135 

9 

20 

211 

10.6 

219 

10.9 

W9 

9 o 

202 

IO.  1 

175 

8.7 

30 

307 

10.2 

322 

10.7 

265 

8.8 

299 

10 

254 

8-5 

Surface  Ratio  SO. 


15 

187 

12.5 

187-5 

12.5 

149 

9.9 

168 

rr.2 

164 

20 

247 

12.3 

248 

12.5 

192 

9.6 

217 

10.9 

203 

30 

342 

11.4 

348 

11. 6 

278 

9-3 

3i4 

10.4 

283 

40 

438 

10.9 

450 

*i-3 

364 

9.1 

411 

10.3 

362 

50 

534 

10.7 

552 

11 

450 

9 

508 

10. 1 

442 

10.9 

10.2 

9.4 

9 

8.8 


Surface  Ratio  75. 


Locomotive,  coal. . 
“ coke . 


Water. 

Fuel  per 

Sq.  Foot 

30 

40 

50 

Per  sq.  foot. 
“ lb.  coal. 

Lbs. 

Lbs. 

Lbs. 

342 

11.4 

439 

11 

536 

10.7 

“ sq.  foot. 
V lb.  coal. 

338 

XI-3 

-k 

p 00 

A 

497 

9.9 

Lbs. 

633 

10.7 

576 

9.6 


Lbs. 
77  8 
10.4 
695 
9-3 


90 


Lbs. 

927 

l°.3 

8i5 

9 


Lbs. 

1020 

10.2 

894 


L „ y • ’ . 01  ieea-water  is  raised  above  that  ob 

reduced  condensinS  engine,  the  proportions  of  surfaces  may  be  correspondingly 


744 


STEAM-ENGINE. BOILEE. 


Results  of  Operation,  of  various  Designs  of  Boiler,  nu- 
clei’ varying  Proportions  of  Grate,  Calorimeter,  Area 
and.  Length  of  Beating  Surface,  Draught,  Firing,  and 
Bate  of  Combustion. 


Stationary. 

Area 

of 

Grate. 

Heat- 

ing 

Surface. 

Grate 

to 

Heating 

Surface. 

Circuit  of 
Heating 
Surface. 

Temperature 
of  Chimney. 

Coal  per 
S(j.  Foot 
of  Grate 
per  Hour. 

Water  E 

from  2120 
per  lb.  of 
Coal. 

lyaporated 

per  Sq.  Foot 
of  Grate 
per  Hour. 

Sq.  Ft. 

Sq.  Ft. 

Ratio. 

Feet. 

0 

Lbs. 

Lbs. 

Lbs. 

Lancashire  double 1 

internal  and  ex-  > 

20.5 

612 

29.8 

79 

51 1 

15-35 

8.32 

125  4 

ternal  Sued1. . . ) 

U U 2 

21 

767 

36.5 

80 

505 

21.5 

10.88 

204  4 

Galloway  vertical ) 

water-tubular2.  ( 

21 

719 

22.8 

79 

505 

22.7 

10.77 

212  4 

U .1  2 

3i-5 

719 

34-3 

80 

630 

18.3 

IQ.  17 

162  4 

Fairbairn1 

20.5. 

1017 

49-5 

• \ 

387 

15.27 

8.67 

133 

French1 

20.1 

607 

3°-3 

: ■ — > 

16.42 

8.12 

133  „ 

Cylindrical  flued3. . . 

14.2 

377-5 

26.8 

56 

292 

7-43 

9.08 

59  5 

Marine. 

At  Pressure  of  Atmosphere. 

Horizontal  fire-tub.2 

10.3 

5°8 

49-3 

— 

_ 

27-5 

II.92 

328  ? 

U U 2 

10.3 

508 

49-3 

— 

— 

41.25 

II.36 

469  8 

U U 2 

10.3 

3°2 

30 

— 

— 

24 

12.23 

268  9 

a u 

i9-3 

749 

39 

— 

21 

IO 

182  10 

u u 

28.5 

749 

26.3 

— 

— 

21.15 

8.94 

164  10 

u u 

28.5 

749 

26.3 

— 

— 

19 

II. 13 

335  11  , 

(C.  Wye  Williams)  j 

i5-5, 

749 

48.3 

— 

600 

37-4 

IO.63 

398 

“ U } 

22 

749 

34 

- 

600  - 

17.27 

11.7 

202  12 

::  } 

42 

749 

17.6 

— 

— 

16 

9-65 

154  13  1 

n u 3 

10.8 

150 

13-9 

8.5 

j 

10.99 

8-95 

88  14  ' 

“ “ 3 

4-32 

i47 

34 

8.5 

— 

27.58 

7.24 

40  J4 

1 Trial  in  France.  2 At  Wigan,  1866-68,  height  of  chimneys  100  feet.  3 Navy- 
yard,  Washington,  U.  S.,  chimney  61  feet.  4 At  pressure  of  atmosphere,  fires  12  ins.  ; 
deep,  at  40  lbs.  pressure^  evaporation  was  reduced  12  per  cent.  5 Bituminous  coals.  , 
6 Anthracite,  at  pressure  of  6.5  lbs.  above  atmosphere.  7 Fires  14  ins.  deep,  air  ad- 
mitted through  furnace  - doors.  8 Ditto  do.,  jet  blast.  9- Half  tubes  closed  up. 

10  Air  through  grate  only.  11  Air  through  grate  and  door,  no  smoke.  12  One  open- 
ing in  door,  temp.  625°,  with  two  633°,  with  four  638°,  and  with  six  6oo°.  *3  Long 

grates,  air  spaces  fully  open,  no  smoke.  I4  0ne  furnace,  anthracite  coal,  5 ins.  deep. 


Dranglit. 

Draught  of  Furnace. —Volume  of  gas  varies  directly  as  its  absolute  tem- 
perature, and  draught  is  best  when  absolute  temperature  of  gas  in  chimney 
is  to  that  of  external  air  as  25  to  12. 

r ~f~  461-2 — _ — v".  V,  V',  and  V"  representing  absolute  temperatures  at  T 

32°  + 461.2°  V'  ’ ’ 1 J 

or  temperature  given , and  at  32  °,  in  degrees  and  volume  of  furnace  gas  at  tempera- 
ture T in  cube  feet. 

Illustration. — Assume  temperature  of  furnace  or  T = 15000,  and  12  lbs.  air  per 
lb.  of  fuel. 

-522 — ~h4^r;2_—  3,q8  and  as  150  cube  feet  is  volume  of  gas  per  lb.  of  fuel  at  12 
32°  + 461.2° 

lbs.  supply  of  air , 150  X 3-98  = 597  cube  feet. 


— c.  W representing  weight  of  fuel  consumed  in  furnace  per  second  in  lbs., 

v volume  of  air  at  320  supplied  per  lb.  of  fuel  in  cube  feet,  t absolute  temperature  of 
gas  disch  arged  by  chimney  in  degrees , a area  of  chimney  in  sq.feet,  and  C velocity  of 
current  in  chimney  in  feet  per  second. 


STEAM-ENGINE. — BOILER. 


'45 


.USTRATION.—  Assume  W = .16,  V = 150,  t 

.16  X 150  X 1000 24000 

2466 


1000 °,  and  a: 
= 9-73  feet. 


5 X 493-2° 

.084  to  .087  = D.  D representing  weight  of  a cube  foot  of  gas  discharged  by 


ley,  in  lbs.  Illustration. 


493-2 


X -086  3=  .0424  lb. 


(1  -(-  G -}-  — ^ = H.  G representing  a coefficient  of  resistance  and  friction  of 


rough  grate  and  fuel  * f coefficient  of  friction  of  gas  through  flues  and  over 
surfaces,  1 1 length  of  flues  and  chimney , m hydraulic  mean  depth, % and  H height 
oj  chimney , all  in  feet. 

Illustration  i. — Assume  C = 9.73,  l = 60,  and  m = .72,  all  in  feet. 

64-: 


g.732  / . .012  X 6o\  94-67  ■ _ . C a V' 

+ ) = X 14  = 20.6 feeL  — = W. 

64.33  V -72  / 64.33  vt 


'.—Assume  preceding  elements.  ?-73  X 5 X 493-2  _ _ # l6  ^ 
150  X xooo 


When.  H is  given.  <J(ll  2 g- 4- 1 + G + ^ = C 

Illustration.— Assume  preceding  elements.  V20.6  X 64.33 *4  = 9-73  feet. 

.192  X pressure  in  lbs.  per  sq.  foot  = head  in  ins.  of  water. 

Temperature  at  base  of  smoke-pipe  or  chimney,  or  termination  of  flues  or 
tubes,  is  estimated  at  500° ; and  base  of  chimney,  or  its  calorimeter , should 
have  an  area  of  1.3  to  1.6  sq.  ins.  for  every  lb.  of  coal  consumed  per  hour. 
With  tubes  of  small  diameter,  compared  to  their  length,  this  proportion  may 
be  reduced  to  1 and  1.2  ins. 

Admission  of  air  behind  a bridge-wall  increases  temperature  of  the  gases, 
but  it  must  be  at  a point  where  their  temperature  is  not  below  8oo°. 

Loss  of  DPresscire  by  Flow  of  .Air  in  2?ipes. 

Length  3280  Feet , or  1000  Meters. 


Velocity  at 
Pi] 

Feet 

per  Second. 

Entrance  of 

pe. 

Meter 
per  Second. 

4 1 

Diameter  of  Pipe  in  Ins. 

6 | 8 J iO  J 12  | 

Loss  of  Pressure  in  Lbs.  per  Sq.  Inch. 

14 

3.28 

1 

.114 

.076 

•057 

•057 

.038 

.038 

6. 56 

2 

•5 

•343 

•25 

.21 

. 172 

•153 

9.84 

3 

1.183 

.8 

•592 

•477 

•394 

•343 

13.12 

4 

2.06 

r-374 

1.03 

.84 

.687 

• 6 

16.4 

5 

3-2 

2.16 

1. 61 

1. 29 

1. 1 

•923 

19.68 

6 

4.446 

2.964 

2.223 

1.778 

1.482 

1.28 

At  Mount  Cenis  Tunnel,  the  loss  of  pressure  from  84  lbs.  per  sq.  inch,  in  a pipe 
7.625  ins.  in  diameter  and  1 mile  15  yards  in  length,  was  but  3.5  per  cent. 


^Artificial  Draught. 

In  production  of  draught  in  an  ordinary  marine  boiler,  from  20  to  33  per 
cent,  of  total  heat  of  combustion  of  fuel  is  expended. 

Blast.— By  experiments  of  D.  K.  Clark  and  others  it  was  deduced  that  the  vacuum 
in  back  connection  is  about  .7  of  blast  pressure,  and  in  the  furnace  from  .33  to  .5 
of  that  in  back  connection ; that  rate  of  evaporation  varies  nearly  as  square  root  of 
vacuum  in  back  connection;  that  best  proportions  of  chimney  and  passages  thereto 
are  those  which  enable  a given  draught  to  be  produced  with  greatest  diameter  of 
blast- pipe;  for  the  manifest  reason,  that  the  greater  that  diameter,  the  less  the  back- 
pressure due  to  resistance  of  orifice,  and  that  these  proportions  are  best  at  all  rates 
of  expansion  and  speeds. 


* Which,  in  furnaces  consuming  from  20  to  24  lbs.  coal  per  sq.  foot  of  grate  per  hour,  is  assigned  by 
Peclet  at  12.  f Estimated  by  same  authority  at*  .012. 

t For  a square  or  circular  flue  is  .25  its  diameter. 


74 ^ STEAM-ENGINE. DRAUGHT. SAFETY  YALYES. 


Velocity  of  Draught.  Locomotive.  36. 5 VH  (T  — t)  = V.  H represents 
height  of  chimney  or  pipe  in  feet,  T and  t temperatures  of  air  at  base  and  ton  of  chim- 
ney, and  V velocity  in  feet  per  second.  J 

Sectional  area  of  tubes  within  ferrules grate 

“ “ of  smoke-pipe o66&  u 

Area  of  blast-pipe  (below  base  of  smoke-pipe) .*.*.*.*!!  .’015  “ 

Volume  of  back  connection 3 feet  x area  of  grate 

Height  of  smoke-pipe  4 times  its  diameter.  8 

Steam-jet— Rings  set  above  base  of  smoke-pipe, and  should  equally  divide 
the  area ; jets  .06  to  .1  inch  in  diameter,  3 ins.  apart  at  centres. 

A Steam-jet,  involving  50  per  cent,  increased  combustion  of  coal,  produced 
45  Per  cent.  more  evaporation  at  nearly  same  evaporation  per  lb.  of  coal. 
Fan  Blowers. — See  page  447. 

Comparative  Result  of  Experiments  with  a Steam -jet  in  a Marine  Boiler , 
with  Bituminous  Coal . (Nicoll  and  Lynn , Eng. ) 

Without  Jet.  With  Jet. 

Area  of  grate. sq.  feet 10.3  10.3 

Coal  per  sq.  foot  of  grate  per  hour lbs 27.5  41-25 

Water  “ “ u “ 293.1  419.37 

“ from  2120  per  lb.  of  coal  “ X1.9  1I>36 

Duration  of  smoke  in  an  hour, ) 

very  light minutes 1.1  - 

Comparative  K fleet  of  Draught  and  Blasts. 

By  late  experiments  in  England,  with  boilers  of  two  steamers,  to  deter- 
mine relative  effects  of  the  different  methods  of  combustion,  the  results  were: 
Natural  draught  1,  Jet  1.25,  and  Blast  1.6. 

Flow  of  -Adr.  (HawJcsley.) 

In  Cylindrical  Pipes.  3g6ff  = V,  = 3„  /i£6=Q, 

If?L\aud 

135  21200  000 

In  Conduits  of  Various  Sections.  796  /—  — v,  -v  —h 

V Gl  633 000  a ’ 

, /a 3 h Yah  Q h 3 V3  C l 

= ^ ~^r  = ^6’and  6^000  000  = IP-  In  which  x inch  water  is 

taken  as  equivalent  to  a pressure  of  5.2  lbs.  per  sq.  inch  for  any  passage. 

V representing  velocity  in  feet  per  second , h head  of  water  in  ins. , d diameter  of 
pipe,  t length , and  C perimeter,  all  in  feet,  a area  of  section  in  sq.feet,  Q (Va)  volume 
of  air  discharged  per  second  in  cube  feet,  and  BP  horse-power. 

Safety  Yalves. 

Up  to  a pressure  of  100  lbs.  per  sq.  inch,  area  in  sq.  ins.  equal  product  of 
tveight  of  water  evaporated  in  lbs.  per  hour  by  .006. 

Act  of  Congress  (V.  S.). — For  boilers  having  flat  or  stayed  surfaces,  30  ins.  for 
every  500  sq.  feet  of  effective  heating  surface;  for  cylindrical  boilers,  or  cylindrical 
flued,  24  sq.  ins. 

Board  of  Trade,  Eng. — Two  of  .5  inch  area  per  sq.  foot  of  grate.  Or,  /— = 

V 452 

diameter.  G representing  area  of  grate  in  sq.  ins. 

Locked  Safety-valves. — Effective  heating  surface,  less  than  700  sq.  feet,  valve  2 ins. 
in  diameter;  less  than  1500,  3 ins.  in  diameter;  less  than  2000,  4 ins.  in  diameter; 
less  than  2500,  5 ins.  in  diameter;  and  above  2500,  6 ins.  in  diameter. 

Or,  (.05  G -j-  .005  S)  ~ = area  of  each  of  two  valves.  G representing  sq.  inch, 
per  sq.foot  of  grate,  and  S sq.  inch,  per  sq.foot  of  heating  surface. 


STEAM-ENGINE. FLUES  AND  TUBES. 


747 


Illustration. — Assume  G = 50  sq.  feet,  S = 1600  sq.  feet,  and  P = 80  lbs.  (m.  g.) 

’hen,  (.05  X 50 -f-- 005  X 1600)  X Vioo-r-8o  = 2.5  + 8 X 1.118  = 11.73  sq.  ins. 

IPipes. 

Area.  .25  G -f-  .01  S G representing  area  of  grate  and  S area  of  heat- 

surface,  both  in  sq.  feet,  and  P pressure  per  mercurial  gauge  in  lbs. 

Copper),  Thickness.  Steam,  . 125  -f  ; Feed, . 125  -f  ; Blow  ( Bottom 
Surface ),  .125  — : Supply , .1  4-—  ; Discharge,  .1  -f  — ; Feed,  Suction, 

9OOO  300  2CO 

Bilge  discharge,  .09  and  Steam  Blow-off,  .05-] . d representing 

200  500 

internal  diam.  of  pipe,  and  p internal  pressure  per  sq.  inch  in  lbs. 

Flanges.  — Of  brass,  thickness  4 times  that  of  pipe;  breadth,  2.25  times 
diam.  of  bolt ; bolls , diam.  equal  to  and  pitch  5 times  thickness  of  flange. 

For  lower  pressure  or  stress,  pitch  of  bolts  6 times. 

Fines  and  Tubes. 

Flues  and  Tubes.— Cross  section,  for  15  lbs.  of  coal  consumed  per  hour, 
an  area  of  from  .18  to  .2  area  of  grate,  area  being  measurably  inverse  to 
diameter,  and  directly  increased  with  length.  Thus,  in  Horizontal  Tubular 
Boilers,  area  .18  to  .2  area  per  sq.  foot  of  grate,  and  in  Vertical  Tubular  .22 
teo  .25,  area  decreasing  with  their  length,  but  not  in  proportion  to  reduction 
of  temperature  of  the  heated  air,  area  at  their  termination  being  from  .7 
to  .8  that  of  calorimeter  or  area  immediately  at  bridge-wall. 

Large  flues  absorb  more  heat  than  small,  as  both  volume  and  intensity  of  heat  is 
greater  with  equal  surfaces. 

Tubes. — Surface  1 sq.  foot,  if  brass,  and  1.33,  if  iron,  for  each  lb.  of  coal 
consumed  per  hour ; or  20  of  brass  and  27  of  iron  for  eacli  sq.  foot  of  grate, 
and  2.6  sq.  feet  of  brass  and  3.7  of  iron  per  JH?. 

Set  in  vertical  rows,  and  spaces  between  them  increased  in  width  with 
number  of  the  rows. 

Temperature  of  base  of  Chimney  or  Smoke-pipe,  or  termination  of  the 
flues  or  tubes,  is  estimated  at  500°  ; and  base  of  chimney,  or  its  calorimeter , 
with  natural  draught,  should  have  an  area  of  1.33  sq.  ins.  for  every  lb.  of 
coal  consumed  per  hour.  With  tubes  of  small  diameter,  compared  to  their 
length,  this  proportion  may  be  reduced  to  1 and  1.2  ins. 

When  combustion  in  a furnace  is  very  complete,  the  flues  and  tubes  may 
be  shorter  than  when  it  is  incomplete. 

Evaporation. 

1 sq.  foot  of  grate  surface,  at  a combustion  of  15  lbs.  coal  per  hour,  will 
evaporate  2.3  cube  feet  of  salt  water  per  hour. 

A sq.  foot  of  heating  surface,  at  a like  combustion  of  fuel,  will  evaporate 
from  5 to  6.2  lbs.  of  salt  water  per  hour ; and  at  a combustion  of  40  lbs.  coal 
per  hour  (as  upon  Western  rivers  of  U.  S.),  from  10  to  11  lbs.  fresh  water, 
exclusive  of  that  lost  by  being  blown  out  from  boilers. 

13.8  to  17.2  sq.  feet  of  surface  will  evaporate  1 cube  foot  of  salt  water  per 
hour,  at  a combustion  of  15  lbs.  coal  per  hour  per  sq.  foot  of  grate. 

Relative  evaporating  powers  of  Iron,  Brass,  and  Copper  are  as  1,  1.32,  and  1.56. 

Note. — Boilers  of  Steamer  Arctic,  of  N.  Y.,  vertical  tubular,  having  a surface  of 
33.5  to  1 of  grate,  consuming  13  lbs.  of  coal  per  sq.  foot  of  grate  per  hour,  evapo- 
rated 8.56  lbs.  of  salt  water  per  lb.  of  coal,  including  that  lost  by  blowing  out  of 
saturated  water. 


748  STEAM-ENGINE. SMOKE-PIPES  AND  CHIMNEYS. 


Water  Surface . 

At  low  evaporations,  3 sq.  feet  are  required  for  each  sq.  foot  of  grate  sur- 
face, and  at  high  evaporation  4 to  5 sq.  feet. 


Steam  liooin. 

From  15  to  18  times  volume  that  there  are  cube  feet  of  steam  expended 
for  each  single  stroke  of  piston  for  25  revolutions  per  minute,  increasing 
directly  with  their  number.  Or,  .8  cube  feet  per  IIP  for  a side-wheel  engine, 
and  .65  for  an  ordinary  and  .55  for  a fast-running  screw-propeller. 

Space  is  required  proportionate  to  volume  of  steam  per  stroke  of  piston. 
Thus,  with  like  boilers,  the  space  may  be  inversely  as  the  pressures. 

Steam-drums  and  steam-chimneys,  by  their  height,  add  to  the  effect  of 
their  volume,  by  furnishing  space  for  water  that  is  drawn  up  mechanically 
by  the  current  of  steam,  to  gravitate  before  reaching  the  steam-pipe. 

Grate.  — Area  in  sq.  feet  per  lb.  of  coal  per  hour  for  following  boilers. 
Width,  1.5  diameter  of  furnace: 

Cornish  and  Lancashire,  slow  I Portable,  moderate  forced  . . .03  sq.  foot. 

combustion 2 sq.  foot.  Locomotive  and  like,  strong 

Marine,  tubular 05  to  .066  “ “ | blast 01  “ “ 

Thickness  of  Tubes  per  B W G. 

External  diameter  in  ins 2 2.25  2.5  2.75  3 3.25  3.5  3.75  4 

Thickness  for  pressure  of  50  lbs.,  number.  .12  12  n 11  mo  10  10  9 

“ “ “ 100  u u ..11  10  99  98  88  7 


Smoke-pipes  and.  Chimneys. 

Area  at  their  base  should  exceed  that  of  extremity  of  flue  or  flues,  to 
which  they  are  connected. 

In  Marine  service  smoke-pipe  should  be  from  .16  to  .2  area  of  grate.  In 
Locomotive,  it  should  be  .1  to  .083.  -r 
Intensity  of  their  draught  is  as  square  root  of  their  height.  Hence,  rela- 
tive volumes  of  their  draught  is  determined  by  formula : 
fh  .ia  — volume  in  sq.  feet,  h representing  height  of  pipe  or  chimney  in  feet,  and 
a its  area  in  sq.  feet. 

When  wood  is  consumed  their  area  should  be  1.6  times  that  of  coal. 


Chimneys  {Masonry). — Diameter  at  their  base  should  not  be  less  than  from 
. 1 to  .11  of  their  height. 

Batter  or  inclination  of  their  external  surface  .35  inch  to  a foot,  which  is 
about  equal  to  1 brick  (.5  brick  each  side)  in  25  feet. 

Diameter  of  base  should  be  determined  by  internal  diameter  at  top,  and 
necessary  batter  due  to  height. 

Thickness  of  walls  should  be  determined  by  internal  diameter  at  top ; 
thus,  for  a diameter  of  4 feet  and  less,  thickness  may  be  1 brick,  but  for  a 
diameter  in  excess  of  that  1.5  bricks. 


A rea. 


15  C 
fh 


— a.  C representing  weight  of  coal  consumed  per  hour  in  lbs.,  and 


a area  of  ditto  at  top , in  sq.  ins. 


{Brick  masom'y.) — 25  tons  weight  per  sq.  foot  of  brickwork  in  height  is 
safe  if  laid  in  hydraulic  mortar. 

Less  the  height  of  a smoke-pipe  or  chimney,  the  higher  the  temperature  of 
its  gases  is  required. 


; 

t 

; 


STEAM-ENGINE.— PUMPS.— PLATES  AND  BOLTS.  749 


Velocities  of  Current  of  Heated  Air  in  a Chimney  100  Feet  in,  Height. 
In  Feet  per  Second. 

Air  at  Base  of  Chimney.  , A._  Air  at  Base  of  Chimney. 


32o 

5o° 


250°  | 350° 


Feet. 

24 


Feet. 

30 

28 

27 


Feet. 

33 

3i 

3° 


450 

Feet. 

35 

34 

33 


6o° 

7°o 

8o° 


Feet. 


250 

Feet. 

26 

25 

24 


35o 

Feet. 

29 

29 


450 

Feet. 

33 

32 

32 


hen  Height  of  Chimney  is  less  than  ioo  feet. — Multiply  velocity  as  ob- 
d for  temperature  by  .1  square  root  of  height  of  chimney  in  feet. 

Draught  consequent  upon  a steam -jet  in  a smoke-pipe  or  chimney  is 
nearly  equal  to  that  of  a moderate  blast. 

The  most  effective  draught  is  when  absolute  temperature  of  heated  air  or 
gas  is  to  that  of  external  air  as  25  to  12,  or  nearly  equal  to  temperature  of 
melting  lead. 

In  chimneys  of  gas  retorts,  ovens,  and  like  furnaces,  the  draught  is  more 
intense  for  a like  height  of  chimney  than  in  ordinary  furnaces,  in  con- 
sequence of  the  great  mass  of  brick  masonry,  which,  becoming  heated,  adds 
to  intensity  of  draught. 


Chimneys.  Lawrence  Manufacturing  Co.,  Mass.  Octagonal. 

Height  above  ground  21 1 feet.  Diameters  15,  and  10  feet  1.5  ins.  Wall  at  base 
23.5,  and  at  top  11.5  ins.  Shell  at  base  15  ins.,  at  top  3.75  ins. 

Foundation  22  feet  deep. 


England.  —Square.— Height 190  feet.  Diameter  at  base. 

kt  “ 3°°  “ “ “ 

Round.  “ 312  “ “ “ 

“ “ 300  “ “ “ 


Diameter  at  base  usually  . 1 of  height  above  ground. 


20  feet. 

29  “ 

30  “ 
20  “ 


Vacuum  at  base  of  chimney  ranges  from  .375  to  .43  ins.  of  water. 


Circulating  Pumps. 

Single-acting.  — .6  volume  of  single-acting  air-pump  and  .32  of  double- 
acting. 

Double-acting.  — .53  volume  of  double-acting  air-pump. 


Volume  of  Pump  compared  to  Steam  Cylinder  or  Cylinders. 

Engine.  Pump.  Volume. 

Expansive,  1.5  to  5 times Single-acting 08  to  .045. 

Compound do.  04510.035. 

Expansive,  1.5  to  5 times Doubleracting 045  to  .025. 

Compound do.  025  to  .02. 


Voices.— Area  such  as  to  restrict  the  mean  velocity  of  the  flow  to  450  feet 
per  minute. 


PLATES  AND  BOLTS. 


Wrought-iron. — Tensile  strength  ranges  from  45500  to  70000  lbs. 
per  sq.  inch  for  plates,  and  60000  to  65000  lbs.  for  bolts,  being  increased 
when  subjected  to  a moderate  temperature. 

English  plates  range  from  45000  to  56000  lbs.,  and  bolts  from  55000  to 
59000  lbs. 

D.  K.  Clark  gives  best  quality  of  Yorkshire  56000  lbs.,  of  Staffordshire  44  800  lbs. 

Test  of  IPlates.  ( XJ . S.)  — All  plates  to  be  stamped  at  diagonal  corners  at 
about  four  ins.  from  edge,  and  also  in  or  near  to  their  centre,  with  name  of  manu- 
facturer, his  location,  and  tensile  stress  they  will  bear. 

Plates  subjected  to  a tensile  stress  under  45000  lbs.  per  sq.  inch,  should  contract 
in  area  of  section  12  per  cent.,  45000  and  under  50000,  15,  and  50000  and  over,  25, 
at  point  of  rupture. 

3 R* 


750 


STEAM-ENGINE. PLATES. 


Brands.  (C  No.  i)  Charcoal  No.  i. — Plates , will  sustain  a stress  of  40000  lbs.  per 
sq.  inch ; hard  and  unsuited  for  flanging  or  bending. 

(C  No.  1 R H)  Reheated , hard  and  durable,  suited  for  furnaces,  unsuited  for  con- 
tinued bending. 

(C  H No.  1 S)  Shell,  will  sustain  a stress  of  50000  to  54  000  lbs.  in  direction  of  fibre, 
and  34000  to  44000,  across  it:  hard  and  unsuited  for  flanging  or  even  bending  with 
a short  radius. 

(C  H No.  1 F)  Flange,  will  sustain  a stress  of  50000  to  54000  lbs.,  soft  and  suited 
tor  flanging. 

(C  H No.  1 F B)  Furnace  and  (C  H No.  1 FFB)  Flange  Furnace.  The  first  is 
hard,  but  capable  of  being  flanged,  the  other  is  hard,  and  suited  for  flanging. 

The  especial  brands  are  Sligo,  Eureka , Pine , etc. 

The  best  English  plates  known  are  the  Yorkshire , as  Low  Moor , Bowling , Farnley, 
Monk  Bridge,  Cooper  db  Co.,  etc.  (See  Steam-boilers,  W.  H.  Shock,  U.  S.  N.,  1880.) 

Steel.— Tensile  strength  ranges  from  75000  to  96000  lbs.  Mr.  Kirkaldy 
gives  85  966  lbs.  as  a mean. 

When  used  in  construction  of  boiler-plates  should  be  mild  in  quality,  containing 
but  about  .25  to  .33  per  cent,  of  carbon;  for  when  it  contains  a greater  proportion, 
although  of  greater  tensile  strength,  it  is  unsuited  for  boilers,  from  its  hardness  and 
consequent  shortness  in  its  resistance  to  bending. 

Crucible  steel  may  be  used,  but  that  obtained  by  the  Bessemer  or  Siemens-Martin 
process  is  best  adapted  for  boiler-plates.  Its  strength  becomes  impaired  by  the 
processes  of  punching  and  shearing,  rendering  it  proper  thereafter  to  submit  it  to 
annealing. 

Steel  rivets,  when  of  a very  mild  character  and  uniformly  heated  to  a bright  red, 
are  superior  to  iron  in  their  resistance  to  concussion  and  stress. 

Copper. — Tensile  strength  is  33000  lbs.,  being  reduced  when  subjected 
to  a temperature  exceeding  120°.  At  2120  being  32  000,  and  at  550°  25  000  lbs. 

"W roiiglit-iron  Shell  IPlates. 

Pressure  and.  Thickness.  (U.  S.  Law.) 

Based  upon  a Standard  of  One  Sixth  of  Tensile  Strength  of  Plates.  Iron  or  Steel. 

Results  with  a Tensile  Strength  of  50000  Lbs. 

Thick-  Diameters  in  Ins. 


ness. 

36 

38 

40 

42 

44 

46 

48 

54 

60 

66 

72  | 

1 78 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•25 

116 

no 

104 

99 

95 

91 

87 

77 

69 

63 

58 

53 

•3125 

145 

137 

130 

124 

118 

IX3 

109 

96 

87 

79 

72 

67 

•375 

174 

165 

156 

149 

142 

136 

130 

116 

104 

95 

87 

80 

•5 

232 

220 

208 

198 

190 

182 

174 

154 

138 

126 

116 

106 

1 84 

90 

96 

102 

108 

| 114 

120 

126 

132 

i35 

140 

144 

Inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

•375 

74 

69 

65 

61 

58 

55 

52 

49 

47 

46 

44 

43 

•4375 

86 

80 

76 

71 

68 

64 

61 

57 

55 

53 

5i 

50 

•5 

99 

92 

87 

81 

77' 

73 

69 

65 

*>3 

61 

39 

57 

•5625 

hi 

103 

98 

91 

87 

82 

78 

73  . 

7* 

69 

67 

64 

•75 

148 

138 

130 

121 

115 

109 

103 

97 

94 

9l 

88 

85 

•875 

172 

160 

152 

142 

136 

128 

122 

no 

106 

102 

100 

1 

198 

184 

174 

162 

i54 

146 

138 

130 

126 

122 

118 

114 

To  which  20  per  c6nt.  is  to  be  added  for  double  riveting  and  drilled  holes. 

Iron  plates  .375  inch  in  thickness  will  bear,  with  stay  bolts  at  4,  5,  and  6 ins. 
apart  from  centres,  respectively  170,  150,  and  120  lbs.  per  sq.  inch. 

Iron  plates,  as  tested  by  Mr.  Phillips  at  Plymouth  Dockyard,  .4375  inch  in  thick- 
ness, with  screw  stay  bolts  1.375  ins.  in  diameter  riveted  over  heads,  15.75  and  15.25 
ins.  from  centre  = 240  sq.  ins.  of  surface  for  each  bolt;  bulged  between  bolts  and 
drew  from  bolts  at  a pressure  of  105  lbs.  per  sq.  inch  of  plate. 

Iron  plates  .5  inch  in  thickness,  under  like  conditions  with  preceding  case,  bulged 
and  drew  from  bolts  at  a pressure  of  140  lbs.  per  sq.  inch  of  plate.  Hence,  it  ap- 
pears. resistances  of  plates  are  as  squares  of  their  thickness. 

When  nuts  were  applied  to  ends  of  bolt  through  .4375  inch  plate,  its  resistance  in- 
creased to  165  lbs.  per  sq.  inch  of  plate. 


STEAM-ENGINE. — SHELLS. — PLATES. 


75i 


Cylindrical  Shelly.  (U.  S.  Law.) 

To  Compute  Pressure  for  a Given  Thickness  and. 

Diameter,  or  Thickness  for  a Given  Pressure  and 

Diameter. 

For  Pressure . Rule.— Multiply  thickness  of  plate  in  ins.  by  one  sixth 
of  tensile  strength  of  metal,  and  divide  product  by  radius  or  half  diameter 
of  shell  in  ins.  . . 

When  rivet-holes  are  drilled,  and  longitudinal  courses  are  double  riveted, 
add  one  fifth  to  result  as  above  attained. 

Example.  — Assume  boiler  8 feet  in  diam.,  and  plates  .5  inch  thick;  what  work- 
ing pressure  will  it  sustain,  tensile  strength  of  plates  equal  to  a stress  of  60000  lbs.? 

8 X 12  5000  , ,, 

. c y 60  000  -r-  one  sixth  -4 — = — Q-  = 104. 16  lbs. 

3 2 48 

For  Thickness.  Rule.— Multiply  pressure  by  radius  of  shell,  and  divide 
product  by  one  sixth  of  tensile  strength  of  metal. 

Example.— Assume  pressure,  radius,  and  tensile  strength  as  preceding. 

104.  r6  X 96  -r-  2 = jooo^  = 5 inch 
. 60 000  -j-  one  sixth  10  000 

For  Evaporation  of  Salt  Water.—  Add  one  sixth  to  thickness  of  plates  and  sec- 
tional area  of  stay  bolts. 

IPor  Freight  and.  Fiver  Steamboats. 

Standard.  — 150  lbs.  pressure  for  a boiler  42  ins.  in  diameter  and  plates 
.25  inch  thick. 

For  Pressure.  Rule— Multiply  thickness  of  plate  by  12 600,  and  divide 
result  by  radius  of  boiler  in  ins. 

Example.— Assume  a boiler  42  ins.  in  diameter,  and  plates  .25  inch  in  thickness; 
what  working  pressure  will  it  sustain  ? _ 

.25  X 12 600-7- 42 -T- 2 = 150  lbs. 

Proof— All  boilers  by  U.  S.  Law  to  be  tested  to  a hydrostatic  pressure  of  50  per 
cent,  above  that  of  their  working  pressure. 

Relative  Mean  Strength  of  Riveted  Joints  compared 
to  that  of  Dlates. 

Allowances  being  made  for  Imperfections  of  Rivets,  etc. 

Plates , 100;  Triple,  .72  to  .75;  Double  or  Square,  .68  to  .72;  Double 
with  double  abut  straps,  .7  to  .75 ; Staggered,  .65 ; Single,  .56  to  .6. 

Board  of  Trade , England. 

Coefficient  or  Factor  of  Safety.  — When  shells  are  of  best  material  and 
workmanship,  rivet-holes  drilled  when  plates  are  in  place,  abut  strapped, 
plates  at  least  .625  inch  in  thickness  and  double  riveted,  with  rivets  com- 
puted at  a resistance  not  to  exceed  75  per  cent,  over  the  single  shear,*  the 
coefficient  is  taken  at  5.  Boilers  must  be  tested  by  hydrostatic  pressure  to 
twice  that  of  working  pressure. 

Tensile  strengths  of  plates  are  taken,  with  fibre  47000  lbs.  per  sq.  inch, 
across  it  40  000  lbs.,  and  when  in  superheaters  from  30  000  to  22  400  lbs. 

47  000  ^ ^ f __  p an(j  D P C — _ ^ p repregenting  pressure  that  shell  will  sus- 
D G 47  000  B 2 

tain  per  sq.  inch  in  lbs. , B least  per  cent,  of  strength  of  rivet  or  plate  (■ whichever  is 
least)  at  lap , D diam.  of  shell  and  t thickness  of  plate,  both  in  ins.,  and  C coefficient 
of  safety. 


Shearing  or  detrusive  resistance  of  wrought  iron  is  from  70  to  80  per  cent,  of  its  tensile  strength. 


752 


STEAM-ENGINE. SHELLS. PLATES. 


Illustration.— Assume  T = 50000  lbs.  tensile  strength  of  plate,  B — 75  per  cen 
D — 120  ins.,  C = 5,  and  < = .5.  What  pressure  will  shell  sustain,  and  what  shoe 
be  thickness  of  plates  for  such  pressure  and  diameter? 

50000  X .75  X .5  X 2 __  . 120X62.5  xs' 

• — = 62. 5 lbs. , and — — . e inch. 

120  X 5 50000  X .75  X 2 5 

For  all  practicable  deficiencies  in  drilling,  punching,  and  riveting  in  tran 
verse,  courses,  if  existing,  this  coefficient  is  increased  up  to  6.75,  and  in  lo 
gitudinal  courses  to  8.75,  and  when  courses  are  not  properly  broken,  ; 
addition  is  made  to  above  of  .4. 

Diameter  of  rivets  should  not  be  less  than  thickness  of  plates. 

Molesworth. 

P d_  cttG  -Yd 

— Pj  ancl  — t.  d representing  diameter  and  t thickness 
metal,  both  in  ins.,  P working  pressure  in  lbs.  per  sq.  inch,  and  C as  follows  : 

Single  riveted.  Double  riveted. 

Best  Yorkshire  plates ) rC  — 6200  and  2800 

u Staffordshire  plates i one  ninth  of  tensile  («  u 7 

Ordinary  plates. strenSth-  j « = 35^  a 

Working  stress  not  to  exceed  .2  tensile  strength  of  joint  or  riveted  plate. 

Then  for  a pressure  of  no  lbs.,  and  a diameter  of  42  ins.,  as  given  for  a standard 
U.  S.  boiler. 

Taking  C as  above  for  best  single-riveted  plate  at  6200,  — ° X 42  = . 272  4-  ins 

.....  ...  2 X 6200 

m thickness,  or  . 122  inch  m excess  of  U.  S.  Law  for  a plain  cylindrical  boiler,  single 
r iveted. 

Lloyd's. 

Thickness  of  shells  to  be  computed  from  strength  of  longitudinal  joints. 

*Tri  ^ < J C ^ — d n a 

~~p  — — ~ — ==  xi  an(t  — z.  t representing  thick- 

ness of  plate,  D diameter  of  shell,  p pitch  and  d diameter  of  rivets , all  in  ins.  ; J per 
cent,  of  strength  of  joint  or  rivets,  the  least  to  be  taken ; C a constant  as  per  table  ; 
Y working  pressure  in  lbs.  per  sq.  inch  / n number  and  a area  of  rivet  / x per  cent, 
of  strength  of  plate  at  joint  compared  with  solid  plate,  and  z per  cent,  of  strength  of 
tivets  compared  with  solid  plate. 

When  plates  are  drilled,  take  .9  of  z,  and  when  rivets  are  in  double  shear, 
put  1.75  a for  a. 

Constants. 


*JC_p  PD 

D “ ’ C J ~ 


Joint. 

Ik 

.5  inch 
and 
under. 

on  Plate 
.75  inch 
and 
under. 

s. 

Above 
.75 inch. 

.375  inch 
and 
under. 

Steel  Plates. 
.3625  I '.‘75  inch 
inch  and  and 

under.  | under. 

Above 

•75inch. 

t ( punched  holes 

155 

170 

165 
1 80 

170 

200 

21^  I 

230 

240 

ljap  j drilled  do 

I9O 

I90 

200 

Double  abut  j punched  holes 
strap  (drilled  do. 

I70 

180 

180 

190 

215 

230 

250 

260 

When  plates,  as  in  steam-chimneys,  superheaters,  etc.,  are  exposed  to  direct  ac- 
tion of  the  flame,  these  constants  are  to  be  reduced  .33. 

Illustrations.  — Assume  pitch  4 ins.,  diam.  of  rivet  1.375  ins.,  and  thickness  of 
plate  1 inch,  both  single  and  double  riveted.  Area  1.375  — 1.48  sq.  ins. 

- — * °75  = .656  per  cent,  strength  of  joint  compared  to  solid  plate.  — *^’4-  — .37 

per  cent,  strength  of  rivet  to  solid  plate  when  single  riveted,  and  — — . 647 

4X1 

per  cent,  when  double  riveted.  Rivets  at  Joint.  ^ X 100  with  punched  holes  and 


by  90  with  drilled. 


p t 


STEAM-ENGINE. PLATES. — ABUT  STRAPS,  ETC.  753 


IPlates. 


To  Compute  Thickness  of  Flates  for  a Griven  Pressure 
and.  Fitch,  and.  Pressure  and  Fitch,  for  Griven  Thick- 
ness. 


sixteenths  of  an  inch , p pitch  of  stays  or  distance  apart  at  centres  in  ins.,  P working 
pressure  in  lbs.  per  sq.  inch,  and  C a constant,  as  follows  : 


For  a Tensile  Strength  of  Metal  of  50000  Lbs.  per  Sq.  Inch. 

Screw  Stay-bolts  with  Riveted  Heads. — Plates  up  to  .4375  inch  in  thickness  C = 90, 
and  above  that  100. 

Screw  Stay-bolts  with  Nuts.  — Plates  up  to  .4375  inch  in  thickness  C = iio,  and 
above  that  120. 

Screw  Stay-bolts  with  Double  Nuts  and  Washers. — Up  to  4.375  ins.  in  thickness 
C = 140,  and  above  that  160. 

When  stay-bolts  are  not  exposed  to  corrosion,  these  constants  may  be  reduced  .2. 

Resistance  of  a flat  surface  decreases  in  a higher  ratio  than  space  between 
stays.  Hence,  C must  be  decreased  in  proportion  to  increase  of  pitch  above 
that  of  ordinary  boiler-plates. 

Illustration  i.— Assume  pressure  no  lbs.  per  sq.  inch,  and  pitch  of  stays  5 ins. ; 
what  should  be  thickness  of  plate  for  screw-bolts  and  riveted  heads? 


2.  — Assume  thickness  of  metal  5 sixteenths  inch  thick,  stay-bolts  screwed  and 
riveted  over  its  threads,  and  working  pressure  of  steam  80  lbs.  per  sq.  inch. 


Double  Abuts  should  be  at  least  .625  thickness  of  plate  covered.  Single, 
.125  thicker  than  plate  covered,  and  Double , .625. 


Direct.  — Tensile  stress  should  not  exceed  5000  lbs.  per  sq.  inch  for  Iron, 
and  7000  for  Steel. 

Diagonal  or  Oblique.  — Ascertain  area  of  direct  stay  required  to  sustain 
the  surface ; multiply  it  by  length  of  diagonal  stay,  and  divide  product  by 
length  of  a line  drawn  at  a right  angle  to  surface  stayed,  to  end  of  diagonal 
stay,  and  quotient  will  give  area  of  stay  increased  to  that  which  is  required. 

Stress  upon  an  oblique  stay  is  also  equal  to  stress  which  a perpendicular 
stay  supporting  a like  surface  would  sustain,  divided  by  cosine  of  angle 
which  it  forms  with  perpendicular  to  surface  to  be  supported. 

Illustration.  — Assume  pressure  no  lbs.  per  sq.  inch,  area  of  supported  surface 
36  sq.  ins.,  and  angle  of  stay  450;  what  would  be  pressure  or  stress  upon  stay? 


t representing  thickness  of  metal  in 


fno  X 52 
95 


=f 


-LA-  — 5. 38—  sixteenth. 


95 


>c  2 X Q5 

— -A-  = 5.45  ins.  pitch. 


Abut  Straps. 


Stays. 


Cosine  450  = .7O7  n.  Then  no  X 36  -4-.  707 11  = 5600  lbs. 


754  STEAM-ENGINE. — GIRDERS. — FLUES,  ETC. 


.66  neck  of  rod. 


Stay-Dolts. — Iron,  are  not  to  be  subjected  to  a greater  stress  than 
6000  lbs.  per  sq.  inch  of  section ; Steel,  8000  lbs.,  both  areas  computed  from 
weakest  part  of  rod,  and  when  of  steel  they  are  not  to  be  welded. 


To  Compute  Diameter  and.  IPitcli  of  Stay  - Dolts,  and 
Resistance  they  -will  Sustain. 


Screwed, 
d 95 


WP 


d 70 


70 


= d,  — p,  and 


Vp 


r-.p,  and 


VP 


m 


Socket.  — — a 
95 


P.  d representing  diameter  in  ins. 


Illustration. — Assume  pitch  of  stay  bolts  6 ins.,  and  working  pressure  100  lbs. 
per  sq.  inch;  what  should  be  diameters  of  bolts,  both  screw  and  socket? 


6 X V 100  o -7 , o ^ , 6 X V 100 

= . 857  inch  Screwed , and 


70 


95 


: 6 2, inch  Socket. 


C d2  t 


= P, 


Grirders.  (Lloyd's.) 

P (L  —p)  DL_^  /P  (L—  p)  D L 


= d.  L representing 


(L  — p)  D L ’ C d2  *’  V Ot 

length  of  girder,  d its  depth,  t its  thickness  at  centre  or  sum  of  its  thicknesses , D its 
distance  apart  from  centre  to  centre , and  p pitch  of  stays,  all  in  ins.,  and  C a constant 
as  per  following : 

One  stay  to  each  girder,  C = 6000.  If  two  or  three  — 9000.  If  four  = 10  200. 

Illustration. — Assume  triple  stayed  girder,  24  ins.  in  length,  3 ins.  in  depth,  1 
inch  thick,  and  stayed  at  intervals  of  6 ins. ; what  working  pressure  will  it  sustain  ? 

^ ooooX62X  1 324000 

C = 9000.  Then  .w,  = — — = >25  U>s. 


(24  — 6)  X 6 X 24  2592 

ITlm.es,  -A^rclied.  or  Circmlai?  Furnaces.  U.  S.  Law. 
.3125  inch  for  each  16  ins.  of  diameter.  English  iron,  being  harder  than 


American,  is  better  constructed  to  resist  compression,  and  consequently  a 
less  thickness  of  metal  is  required  for  like  stress. 


89  6< 


>o  t2  _ /PLD_ 
D “ ’ V 89  600  “ 


Lloyd's. 


89  600  t2 


: D,  and  ^ — L.  D representing 


L D “ ’ V 89  600  “ ’ PL 
external  diameter  of  flue  or  furnace , and  t thickness  of  plate,  both  in  ins..  L length 
of  flue  or  furnace  between  its  ends  or  between  its  rings,  in  feet , and  P working  press- 
ure in  lbs.  per  sq.  inch. 

Illustration.— Assume  diameter  of  flue  16  ins.,  length  6 feet,  and  working  press- 
use  of  steam  80  lbs.  per  sq.  inch. 


Then 


80  X 6 X 16 


89  600 


= V'°%57  — *29  Furnace. — P not  to  exceed 


8000  t 


STEAM-ENGINE. — RIVETING. 


755 

IllustratMn.-— Assume  diameter  ot  ST  ^ 

one  48  ins.,  working  pressure  of  steam  80  lbs.,  ana  iengm  o 

ZO~  V A V ifi  , ,,  • .7 ... 


Then 


/80  X 6 X 48  __  / ' ^ __  __  507  inch  thickness 
\ 89600 


RIVETING. 

-D,  . The  strength  of  a joint  is  determined  by  ascertaining  which 
of  *e  Uvo"’the  plate  or  fhe  rivets,  has  the  least  resistance;  the  stress  on  the 
first  being  tensile  and  the  latter  detrusrve. 

*1  * in  he  taken  from  that  of  the  article  under  consider- 

The  for  construction  and  location  of  the  joint,  and 

A^c^^ttr^rontress,  as  with  or  across  the  fibre  of  the  metal, 
or  exposed  to  high  heat  as  in  a superheater. 

IVith  or  Across  the  Fibre.— From  experiments  of  Mr.  D.  Kirkaldy  and 
othersfthe  difference  in  strength  of  Iron  plates  is  ascertained  to  be  from  6.5 
to  18  per  cent.,  the  average  10  per  cent. 

a,  ; ~oinioa  The  relative  strength  of  plates  with  or  across  the  fibre,  as 
detenni^dby  MnKiAaldy^for  ‘Fagersta^’  is  9 per  cent.,  and  for  “Siemens” 
it  is  without  material  difference. 

puSedlo™^ 
to  be  15  per  cent. 

In  Riveted  Joint  exposed  to  a tensile  stress,  area  of  rivets  should  be 
to  area  of  section  of  plates  through  line  of  rivets,  running  a little  in i excess 
up  to  .5625  inch  diameter  of  rivet,  and  somewhat  less  beyond  that>  d 

ing  determined  by  relative  shearing  and  tensile  resistances  of  met  and 

plate. 

Note.— For  Riveting  of  Hulls  of  Vessels,  see  pp.  828-30. 

Essentially  by  Nelson  Foley. 

Single  Lap  Riveting. 

S— A = b for  plate,  for  rivets,  ptb'  = a,  and 

P PL 

*-27  *>'  f _ d p representing  pitch , t thickness  of  plate,  and  d diameter  oj  rivets, 
being  punched. 

Illustration.— Assume j)  = 3 in*-,  d= i inch,  a = .7854  «,  and  t=.Sinch. 

3 1 .66  per  cent,  strength  of  lap,  =,  -5^3  P*  '««**•  °frimt  to  ^id,PMe’ 


i— .66' 


Z3ins.,and  X .5  = 7 <ncfc  3 X -5  X .523+  = -7854  area. 


— .00  A 

When  Shearinq  Strength  of  Rivet  is  not  Equal  to  Tensile  Strength  of  Plate. 
—Then  diamete/of  rivet  must  be  increased  in  ratio  of  excess  of  strength  o 
plate  over  rivet. 


Or, 


1.27  b'  T 


r t and  S representing  tensile  and  shearing  strengths , which  may 

/ j ^ ^ ^ 

be  taken  at  5 and  4 for  Iron  and  7 and  6 for  Steel. 


|<Z  lUKtn  ui  5 U.IM  Otj”’  ' 

When  full  value  of  rivet  section  is  not  allowed  as  by  Lloyd’s  rules  for  dnlle 
holes,  b'  = b'  X -9- 


756 


STEAM-ENGINE. RIVETING. 


Pitches  as  Determined  Djr  Diameter  of*  Rivets. 

Plate  — • 

between 
Edges 
of  Holes. 


Per  Cent. 
50 
52 
55 


Pitch  = 
Diam.  of 
Rivet  X 


2.08 

2.22 


Plate 
between 
Edges 
of  Holes. 


Per  Cent. 
58 
60 
62 


Pitch  = 
Diam.  of 
Rivet  X 

Plate 
between 
Edges 
of  Holes. 

Pitch  = 
Diam.  of. 
Rivet  X 

Plate 
between 
Edges 
of  Holes. 

2.38 

Per  Cent. 

65 

2.86 

Per  Cent. 
72 

2-5 

2.63 

68 

3-43 

75 

70 

3-33 

78 

Pitch  = 
Diam.  of 
Rivet  X 


Operation. —If  distance  between  edges  of  holes,  or  p — d,=6z  per  cent 
plate,  and  diam.  of  rivet  r inch,  then  2*86  x 1 = 2.  86  im.piicK  5 P 


3- 57 
4 

4- 55 

. of  solid 


When  Plate  and  Rivets  are  of  equal  strength  in  ultimate  tension , b'  = b = B. 
it  I 27  D 

Hence,  t 4=  d.  In  illustration  of  B,  assume  p = 3,  d = 1. 1,  and  t .-= . 5. 

l-9 


Then  3 1.1  _ 1.9,  and  -y=  .633  ==  b,  or  per  cent  of  strength  of  punched  to 

solid  plate.  Area  1. 1 = .95,  and  = .633  =±  b ',  or  per  cent  of  section  of  rivet  to 
solid  plate.  Hence,  B = .633. 


Illustration.— Assume  as  shown,  B = .633. 


TiirtM  I,27  X *633 

en  — j 633  x -5  = i-o95  or  1. 1 ms.  diam. 


Diameter  of  Rivets  as  Determined  Tby-  IPlate. 


D 

Or  Strength 
at  Joint. 

Diam.  = Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  = Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Per  Cent. 

T = S. 

.9  per  cent, 
of  Section 

Per  Cent. 

T = S. 

.9  per  cent, 
of  Section 

Per  Cent. 

1.38 

of  Rivet. 

of  Rivet. 

52 

x-53 

55 

1.56 

*•73 

58 

53 

x-44 

x-59 

56 

1.62 

1.8 

60 

54 

x-5 

1.66 

57 

X.69 

1,87 

62 

Diam.  = Thickness 
of  Plate  X 


1.76 

1.91 

2.08 


of  Rivet. 
I-95 
2.12 
2.31 


Operation.-  If  thickness  of  plate  = .5  inch  and  plate  and  rivet  have  equal  resist- 
ance, or  B ==  62  per  cent.,  then  .5  x 2.08  = 1.04  ins.  diameter. 


T>  on  Die  Lap  Riveting. 

Preceding  formulas  for  single  lap  riveting  apply  to  this,  wit'll  substitution 
of  2 a for  a and  .64  for  1.27. 

Illustration. — Assume p = 3 ins.,  t = .5  inch , and  6'  = .589. 

3X.5X.589  „ .64X.58Q  3 — <7  e 

= .4418  area  ofd,  4jtfx.sa.B  d,  2— l”  = £5d 

•4418X2  . ,,  ' 3 

-=.5896'. 


3X-5 

Diameter  of  Rivets  as  Determined  by  IPlate. 


B 

Or  Strength 
at  Joint. 

Diam.  = Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  = Thickness 
of  Plate  X 

B 

Or  Strength 
at  Joint. 

Diam.  = Thickness 
of  Plate  X 

Per  Cent. 

T = S. 

.9  per  cent, 
of  Section 

Per  Cent. 

T = S. 

.9  per  cent, 
of  Section 

Per  Cent. 

T = S. 

.9  per  cent, 
of  Section 

68 

of  Rivet. 

of  Rivet. 

of  Rivet. 

x-35 

x-5 

7X 

1.56 

x-73 

74 

1. 81 

2 

69 

1.42 

x-57 

72 

1.64 

1.82 

75 

1.91 

2.12 

70 

1.48 

1.65 

73 

1.72 

1.91 

76 

•2  . 

2.25 

1 = .5  men  ana  h = 7o  per  cent.,  tensile  stren, 
to  shearing  being  as  7 to  6.  What  should  be  diameter  of  the  rivets? 


.5  X 1.48  X -|*  = .863  inch.  When  rivets  are  in  double  shear,  put  1.9  a for  a. 


STEAM-ENGINE. — DUTY. — EVAPORATION.  757 

Triple  Lap  Riveting. 

Preceding  formulas  for  single  lap  riveting  apply  to  this,  with  substitution 

of  3 a for  a and  .42  fof  1.27.  

lLLUSTRATiox.-Assumep  = 3^^-5^,and&  -.883. 

4417  area  ofd,  X -5  = -74  . diam. , J ' “ 

and  i44^X  3 = .883  &'• 

3 X -5 


2 = -75  6, 


Diameter  of  Rivets  as  Determined  by  IPlate. 


B 

Or  Strength 
at  Joint. 


Per  Cent. 


70 

71 

72 


Diam.  = Thickness 
of  Plate  X 


•99 

1.04 

1.09 


.9  per  cent 
of  Section 
of  Rivet. 
1. 1 
i-i5 


Or  Strength 
at  Joint. 


Per  Cent. 

73 

74 

75 


Diam.  = Thickness 
of  Plate  X 


Or  Strength 
at  Joint. 


t: 15 

1. 21 
1.27 


9 per  cent, 
of  Section 
of  Rivet. 
1.27 
1-34 
1. 41 


Per  Cent. 

76 

77 

78 


Diam.  = Thickness 
of  Plate  X 


T = S. 

1-34 

1.42 

i-5 


.9  per  cent, 
of  Section 
of  Rivet. 

1.67 


Operation.— As  shown  by  preceding  tables. 

Greneral  Formtilas  and  Illnstrations. 

1.27  BT^  and  y. 

i (1  — iB)  S.  ’ p tT 
i.75  (1  — B)  S P*T 


Tftrefc  tw  Single  Shear. 
Rivets  in  Double  Shear. 
Rivets  in  Triple  Shear . 
Zigzag  Riveting 


.27  B T 
2-5  (1 


, a 2.5  S _ , 

t — d,  and  — 77=-  = 0 . 

B)  S ’ ptT 


7-  „„„  Riveting.  Strength  of  plate  between  holes  diagonally  is 
equafto  that  horizontally  between  holes,  when  diagonal  pitch  _ .6  and  or- 

izontal  = diameter  of  rivet  + .4. 

Thus,  .6  p -}-  .4  p — diagonal  pitch. 

Duty  of  Steam-engines?. 

The  conventional  duty  of  an  engine  is  the  number  of  lbs.  raised  by  it  1 
foot  in  height  by  a bushel  of  bituminous  coal  (112  lbs.). 

Cornish  Engine.— Axe  rage  duty,  70000000  lbs. ; the  highest  duty  ranging 
from  47  000  000  to  101  900  000  lbs.  ■ 

A condensing  marine  engine,  working  with  steam  at  .75  lbs.  (mercurial 
gauge),  cut  offat  -5  stroke,  will  require  from  1.75  to  2 lbs.  bituminous  coal 
per  TP  per  hour. 

Relative  Cost  of  Steam-engines  for  Eqnal  Effects. 

In  Lbs.  of  Coal  per  IP  per  Hour.  Lbs. 

A theoretically  perfect  engine 

A Cornish  condensing  engine 2 

A marine  condensing  engine J 

Evaporative  Rower  of  Boilers. 

The  Evaporative  power  of  a boiler,  in  lbs.  of  water  per  lb.  of  fuel  consumed, 
is  ascertained  approximately  by  formula 

x 833  ( S \ e = lbs.  S representing  total  heating  surface  in  sq.  feet , F fuel 
consumed  bilf .per  hour,  and  e theoretical  evaporative  power  of  the  fuel. 

Illustration.  — Assume  evaporative  power  of  the  fuel  at  15,  consumption  per 
hour  800  lbs.,  and  heating  surface  1600. 

Then  {u^TfsSo)  x 15  = IO'448  lbs' 

3S 


STE  AM-EN  GINE. — WEIGHTS, 


Efficiency  of  boiler.  1.833 


1600 


:-7 33- 


\1600  X 2 -f  800/ 

.nT^eoeiT.ap0.?Uve  power  of  different  fuels,  from  and  at  2I2°  is  for  coals  fr, 
to  16.8  lbs.,  the  average  of  Newcastle  being  is.,  fornatent  f , 
cohe  I3.3,  Peat  IO,3,  and  Woods,  when 

INTotes  on  Horse-power 

horizontal  zection  r-ir  cube  r«,t  of  water,  and  ,s  sj.  reel  of  graLara’a^r  ?ub 

^^Sing'siSS^nnd  T^foot  of  grate-area!  k°ri 

, SUfHh  !!0llerS  WlH  range  from  3 4 times  that  of  the  nominal. 

Multitubular  £oilers.-.75  sq.  foot  of  grate-area  and  2.5  of  heating  surface. 

^ iglits  of1  Steam-engines. 

Side-wheels.- American  Marine 


Engine. 


Vertical  beam. 


Oscillating. 


Inclined 

* Without  frame. 


Frame. 

Water- 

wheels. 

( 

No. 

Cylinders. 

Volume. 

Weight  per 
Cube  Foot. 

Wood.* 
Wood.  * 
Wood.* 
Wood.* 
Wood.* 
Iron. 
Iron. 
Iron. 

Wood. 

Wood. 

Wood. 

Wood. 

Iron. 

Iron. 

Iron. 

Iron. 

1 

2 

1 

2 

1 

2 
2 
2 

Cube  Feet. 

63 

216 

430 

253 

725 

540 

1502 

535 

Lbs. 

1100 

1040! 

1225 

1480$ 

io89f 

850 

55°§ 

1100 

t With  frame  1109.  % Including  boilers. 


Service. 


Screw  Propellers. — American  Marine  ( Condensing ). 


River. 
Coast. 
Coast. 
Coast. 
Sea. 
Sea. 
Sea. 
Sea. 

§ Single  frame. 


Engine. 


Cylinders. 


Vertical  direct,  Jet  Condens’g . 
“ “ Surface  Cond’g 

“ Jet  “ 


Horizontal  back-action . 

“ direct... 
Vertical  compound. 


No. 


“ direct . 


:i!l 


o 

l O 


“ Non-Condensing. 


Volume. 


CubeFeet. 

4 

12.5 

12.5 

33 

506 

68 

67 

4.8 

24-3 

425 

3-6 

35 
1.86 
2.77  | 


Weights. 


Engine. 


Lbs. 
22040 
59006 
48  130 
120450 
1 523  o6q 
289  680 
201 000 
24  7<>5 
94196 
1 022  400 
30  534 
172028 
14  410 
*4  759, 


Boilers. 


Description. 


English  Marine  ( Condensing ). 

Weights. 


Trunk 

Horizontal  direct. 
Vertical  direct  . 

Oscillating 

Vertical  compound 


Horizontal  compound. 


Engines. 

I Propeller 
and 

Shafting. 

Boilers 

and 

Water. 

Total. 

Tons. 

Tons. 

Tons. 

Tons. 

121 

47 

257 

425 

223 

85 

303 

611 

165 

48 

144 

357 

11 7 

43 

135 

295 

4-25 

, *75 

7-25 

12.25 

497 

167 

656 

1320 

55 

i5 

1 10 

180 

130 

27 

162 

3i9 

STEAM-ENGINE. — WEIGHT  OF  BOILERS. 


759 


Land-engines. — (Non- condensing. ) 


Engine. 

Volume 

of 

Cyl’r. 

Engine. 

Spur-wheel 

and 

Connections. 

Sugar-Mill 

Complete. 

Boilers, 
Grates,  etc. 

Engine  per 
Cube  Foot 
of  Cylinder. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Vertical! 

1 18  ins.  X4  feet 

7 

67  200 

37800 

89  600 

26  880 

9600 

beam 

f 30  ins.  X5  feet 

24-5 

105000 

137 179 

265  879 

75060 

4290 

Horizon' 

1, 14  ins.  X 2 feet 

.2.2 

10  914 

— 

— 

8 200 

5100 

22  ins.  X 4 feet 

10.6 

56  000 

— 

30140 

5600 

To  Oompxite  Weiglit  of  a,  Vertical  13 earn  and  Side-wlieel 
Jet  Condensing  Engine.  (T.  F.  Rowland,  A.S.C.E.) 

Including  all  Metals,  Boiler  and  A ffackmenfs , Smoke-pipe,  Grates , Iron  Floors , 
and  Iron  in  Wooden  Water-wheels,  omitting  Coal-biinJcers. 

For  a Pressure  per  Mercurial  Gauge  of  40  tbs.  per  Sq.  Inch. 

For  surface  condenser  add  10  to  15  per  cent. 

Rule. — Multiply  volume  of  cylinder  in  cube  feet  by  Coefficient  in  follow- 
ing table  corresponding  to  length  of  stroke,  and  product  will  give  rough 
weight  in  lbs.  For  finished  weight  deduct  6 per  cent. 


Stroke. 

Coefficient. 

Stroke. 

Coefficient. 

Stroke. 

Coefficient. 

Stroke. 

Coefficient. 

Feet. 

5 

2467 

Feet. 

7 

2213 

Feet. 

9 

1865 

Feet. 

11 

1619 

6 

2340 

8 

• 2000 

10 

W30 

12 

1546 

Example  1.— What  are  the  rough  and  finished  weights  of  a vertical  beam  engine 
cylinder  80  ins.  in  diameter  and  12  feet  stroke  of  piston? 

Area  of  80  ins.  = 5026. 56,  which  x 12  feet  = 419  cube  feet , and  X 1546  for  12  feet 
stroke  = 647  774  lbs.  rough  weight. 

Then  647  774  X -06  = 38  866,  and  647  774  — 38  866  = 608  908  lbs.  finished  weight. 


WEIGHTS  OF  BOILERS. 

Weights  of  Iron  Boilers  (including  Doors  and  Plates , and  exclusive  of  Smoke- 
pipes  and  Grates)  per  Sq.  Foot  of  Heating  Surface. 

Surface  Measured  from  Grates  to  Base  of  Smoke-pipe  or  Top  of  Steam  Chimney. 


Boiler.  For  a Working  Pressure  of  40  Lbs. 


Weight. 


Single  return,  Flue* water  bottom 

u it  a 

“ “ ‘ | Multi-flue  * ! water  bottom 

Horizontal  return,  Tubular f water  bottom 

“ “ “ t 

u “ “ * 

Vertical  “ “ f 

Horizontal  direct,  Tubular *. 


.water  bottom 


Lbs. 

25.6  to  32.9 

24  tO  30 
27  to  45 

25  tO  43 

22.5  to  35 
21 ' to  33 

17.7  to  26.7 

18.5  to  26.5 

19.8  to  23.8 
17  to  21 

23.5  to  24 
18. 1 to  18.6 
16.3  to  17.3 
24  to  26 

Weight  of  Cylindrical  Furnace  .and  Shell  Boilers,  all  complete  for  Sea  Service  and 
for  a pressure  of  60  lbs.  steam,  200  lbs.  per  IIP. 

* w^!?l?f:f',rnLCeiqUare--  Swu-  71Ln,irical-  . , t Section  of  furnace  and  shell  square. 

♦ v\  rought-iron  heads,  .375  inch  thick,  flues,  .25  inch,  and  surface  computed  to  half  diameter  of  shell. 


Cylindrical,  external  furnace, t 36  ins.  in  diam.,  .25’inch  thick.. 
“ Flue  “ 4361042  “ .25  “ “ 

Horizontal  direct,  Tubular Locomotive. 

Vertical  Cylinder  direct,  Tubular 


Notes,  i.  The  range  in  the  units  of  weight  arises  from  peculiarities  of  construc- 
tion, consequent  upon  proportionate  number  of  furnaces,  thicknesses  of  metal  vol- 
ume of  shell  compared  with  heating  surface,  character  of  staying,  etc. 

2.  If  pressure  is  increased  the  above  units  must  be  proportionately  increased. 


76O  STEAM-ENGINE—  BOILER-BOW  EE,  COMBUSTION. 


Boiler-power. 

The  power  of  a boiler  is  the  volume  or  weight  of  steam  alone  (indepen- 
dent oflany  water  that  it  may  hold  in  suspension)  that  it  will  generate  at  its 
operating  pressure  in  a unit  of  time. 

\r*» rinp  boilers  of  the  ordinary  type  and  proportions,  with  natural  draught,  burn- 
cA  produce  3-5  *5.5  Iff  per  s^oot  of  grate  pertour^  w.th  a 

iSAScS' hour.  8 to  10  ik. 

cr^pa^ 

expansion,  under  a pressure  of  70  lbs.,  20  lbs.  steam.  . 

With  a blast  draught  and  consuming  30  to  40  lbs.  of  a fair  quality  of  coal  pel  sq. 
foot  of  grate  per  hour,  7 to  10  IP  per  hour  can  be  attained, 
in  locomotive  boilers  having  from  50  to  90  sq.  feet  of  heating  surface  per  sq.  foot 

and  pressure. 

To  Compute  Volume  of  Air  and  Gas  in  a Furnace. 

When  Volume  at  a Given  Temperature  is  known.  Rule— Multiply  given 
volume  by  its  absolute  temperature,  and  divide  product  by  the  given  abso- 
lute temperature. 

Note. Absolute  temperature  is  obtained  by  adding  461°  to  given  or  acquired 

temperature. 

Fvamptf  —Assume  volume  of  air  entering  a furnace  at  1 cube  foot,  its  tempera- 
ture 6o°,  and  temperature  of  furnace  1623°;  what  would  be  the  increase  of  volume . 

1 X 1623°  -f-  461° 2084 

~ 521 


= 4 times. 


6o°  461° 

Volume  of  Furnace  Gas  per  ITb.  of  Coal.  ( Ranlane .] 


Tempera- 

ture. 

12  Lbs. 

\ir  Supplied. 
18  Lbs. 

24  Lbs. 

320 

150 

225 

300 

68 

161 

241 

322 

104 

172 

258 

344 

212 

205 

3°7 

409 

572 

3H 

47i 

628 

Tempera- 

ture. 

12  Lbs. 

k\r  Supplied. 
18  Lbs. 

24  Lbs. 

752° 

369 

553 

738 

1112 

479 

718 

957 

1472 

588 

882 

1176 

1832 

697 

1046 

1395 

2500 

906 

1 J357 

1812 

Temperature  of  ordinary  boiler  furnaces  ranges  from  15000  to  2500° 

The  opening  of  a furnace  door  to  clean  the  fire  involves  a loss  of  from  4 to  7 per 
cent,  of  fuel. 

For  other  illustrations,  see  ante , page  744-6. 

Rate  of  Combustion. 

The  rate  of  combustion  in  a furnace  is  computed  by  the  lbs.  of  fuel  consumed  per 
sq.  foot  of  grate  per  hour.  ■ . 

Tn  .rpnpnl  practice  the  rate  for  a natural  draught  is,  for  anthracite  coal  from  7 to 
16  lbs.,  for  bituminous,  from  10  to  25  lbs.,  and  with  artificial  orfonced  raugi ^ y 
a blower,  exhaust-blast,  or  steam-jet,  the  rate  may  be  increased  from  30  to  120  lbs. 

The  dimensions  or  size  of  coal  must  he  reduced  and  the  depth  of  the  fire  increased 
directly,  as  the  intensity  of  the  draught  is  increased. 

Temperature  of  gases  at  base  of  chimney  or  pipe 
resistance  of  surface  of  chimney  is  as  square  of  velocity  of  current  ot  &ases. 

Ordinarily  from  20  to  32  per  cent,  of  total  heat  of  combustion  is  1 “S’be  added  ‘the 
production  of  the  chimney  draught  m a mmebote,  ^f^‘' me\  «nd  the  di'utioS 
losses  bv  incomplete  combustion  of  the  gaseous  portion  of  the  tuel  and  tnc  anuiiou 
of  the  gLes  by  an  excess  of  air,  making  a total  of  fully  60  per  cent.  ( Steam-boilers, 
Wm.  H.  Shock,  XI.  S.  N.,  i83i.) 


STRENGTH  OF  MATERIALS. — ELASTICITY.  j6l 


STRENGTH  OF  MATERIALS. 

Strength  of  a material  is  measured  by  its  resistance  to  alteration  of 
form,  when  subjected  to  stress  and  to  rupture,  which  is  designated  as 
Crushing,  Detrusive,  Tensile,  Torsion,  and  Transverse,  although  trans- 
verse is  a combination  of  tensile  and  crushing,  and  detrusive  is  a form 
of  torsion  at  short  lengths  of  application. 

ELASTICITY  AND  STRENGTH. 

Strength  of  a material  is  resistance  which  a body  opposes  to  a per- 
manent separation  of  its  parts,  and  is  measured  by  its  resistance  to 
alteration  of  form,  or  to  stress. 

Cohesion  is  force  with  which  component  parts  of  a rigid  body  adhere  to 
each  other. 


Elasticity  is  resistance  which  a body  opposes  to  a change  of  form. 

Elasticity  and  Strength , according  to  manner  in  which  a force  is  exerted 
upon  a body,  are  distinguished  as  Crushing  Strength , or  Resistance  to  Com- 
pression ; Detrusive  Strength , or  Resistance  to  Shearing ; Tensile  Strength, 
or  Absolute  Resistance ; Torsional  Strength , or  Resistance  to  Torsion ; and 
Transverse  Strength , of  Resistance  to  Flexure. 

Limit  of  Stiffness  is  flexure,  and  limit  of  Resistance  is  fracture. 

Neutral  Axis , or  Line  of  Equilibrium,  is  the  line  at  which  extension  ter- 
minates and  compression  begins. 

Resilience , or  toughness  of  bodies,  is  strength  and  flexibility  combined ; 
hence,  any  material  or  body  which  bears  greatest  load,  and  bends  most  at 
time  of  fracture,  is  toughest. 

Stiffest  bar  or  beam  that  can  be  cut  out  of  a cylinder  is  that  of  which 
depth  is  to  breadth  as  square  root  of  3 to  i ; strongest , as  square  root  of  2 to 
1 ; and  most  resilient , that  which  has  breadth  and  depth  equal. 

Stress  expresses  condition  of  a material  when  it  is  loaded,  or  extended  in 
excess  of  its  elastic  limit. 


General  law  regarding  deflection  is,  that  it  increases,  cceteris  paribus,  di- 
rectly as  cube  of  length  of  beam,  bar,  etc.,  and  inversely  as  breadth  and  cube 
of  depth. 

Resistance  of  Flexure  of  a body  at  its  cross-section  is  very  nearly  .9  of  its 
tensile  resistance. 


Coefficient  of  Elasticity. 


Elasticity  of  any  material  subjected  to  a tensile  or  compressive  force, 
within  its  limits,  is  measured  by  a fraction  of  the  length,  per  unit  of  force 
per  unit  of  sectional  area,  termed  a constant,  and  coefficient  of  elasticity  is 
usually  defined  as  the  weight  which  would  stretch  a perfectly  elastic  bar  of 
uniform  section  to  double  its  length. 

Unit  of  force  and  area  is  usually  taken  at  one  lb.  per  sq.  inch.  E represent - 
ing  denominator  of  fraction. 

Example.— If  a bar  of  iron  is  extended  one  12000000th  part  of  its  length  per  lb. 
of  stress  per  sq.  inch  of  section,  x r 

12000000  E * 

The  bar  would,  therefore,  be  stretched  to  double  its  normal  length  by  a force  of 
12000000  lbs.  per  sq.  inch,  if  the  material  were  perfectly  elastic. 

s* 


762 


STRENGTH  OF  MATERIALS. ELASTICITY. 


The  same  method  of  expressing  coefficient  of  elasticity  is  applied  to  re- 
sistance to  compression.  That  is,  coefficient,  in  weight,  is  expressed  by  de- 
nominator of  fraction  of  its  length  by  which  a bar  is  compressed  per  unit  of 
weight  per  sq.  inch  of  section. 

Ultimate  extension  of  cast  iron  is  500th  part  of  its  length. 

Extension  of  Cast-iron  Bars,when  suspended  V ertically. 

1 Inch  Square  and  10  Feet  in  Length.  Weight  applied  at  one  End. 


Weight.  | 

Extension. 

Set. 

Weight. 

Extension. 

Set. 

Weight. 

Extension. 

| Set. 

Lba. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

529 

.0044 

— 

21x7 

.0190 

.000059 

8468 

.0871 

.00855 

1058 

.0092 

.000015 

4234 

•0397 . 

.O0265 

14820 

. 1829 

•02555 

Woods. — MM.  Chevaudier  and  Wertheim  deduced  that  there  was  no 
limit  of  elasticity  in  woods,  there  being  a permanent  set  for  every  extension. 
They,  however,  adopted  a set  of  .00005  of  length  as  limit  of  elasticity. 
This  is  empirical. 

MODULUS  OF  ELASTICITY. 


Modulus  or  Coefficient  of  Elasticity  of  any  material  is  measure  of  its 
elastic  reaction  or  force,  and  is  height  of  a column  of  the  material, 
pressing  on  its  base,  which  is  to  the  weight  causing  a certain  degree  of 
compression  as  length  of  material  is  to  the  diminution  of  its  length. 

It  is  computed  by  this  analogy  : As  extension  or  diminution  of  length 
of  any  given  material  is  to  its  length  in  inches,  so  is  the  force  that  pro- 
duced that  extension  or  diminution  to  the  modulus  of  its  elasticity. 

P 

Or  x : P ::  l : w = — . x representing  length  a substance  1 inch  square  and  ifoot 
’ ’ * * " x 

in  length  would  be  extended  or  diminished  by  force  P,  and  w weight  of  modulus  in  lbs. 

To  Compute  "WeigHt  of  Modulus  of  Elasticity. 

Rule. — As  extension  or  compression  of  length  of  any  material  1 inch 
square,  is  to  its  length,  so  is  the  weight  that  produced  that  extension  or  com- 
pression, to  modulus  of  elasticity  in  lbs. 

Example.— If  a bar  of  cast  iron,  1 inch  square  and  10  feet  in  length,  is  extended 
.008  inch,  with  a weight  of  1000  lbs.,  what  is  the  weight  of  its  modulus  of  elasticity? 

.008  : 120.(10  X 12)  ::  1000  : 15000000  lbs. 

To  Compute  Moduli  of  Elasticity. 

When  a Bar  or  Beam  is  Supported  at  Both  Ends  and  Loaded  in  Centre. 
Rule.— Multiply  weight  or  stress  per  sq.  inch  in  lbs.  by  length  of  material 
in  ins.,  and  divide  product  by  modulus  of  weight. 


I W 


= M; 


E M 

~T~ 


= W. 


I representing  length  in  ins.,  M modulus , 


W weight  in  lbs.  per  sq.  inch , and  E compression  or  extension , 

Example  i. — If  a wrought- Won  rod,  60  feet  in  length  and  .2  inch  in  diameter,  is 
subjected  to  a stress  of  150  lbs.,  what  will  it  be  extended? 

Modulus  of  elasticity  of  iron  wire  is  28  230  500  lbs.  (see  following  table),  and  area 
of  it . 2 2 X .7854  = . 314  16. 


.31416 


= 477-46  lbs.  per  sq.  inch , and  60  X 12  = 720  ins. 


Then  477-46  X 


720 


. 34J  771:2  „ OJ2  jg  inch. 


“ 28  230  500  28  230  500 

2.— Take  elements  of  preceding  case  under  rule  for  weight  of  modulus. 

120  X 'OOQ  _ ocg  frlc>l  .008  X 15000000  = IOOO  lbs 
15  000000  120 


STRENGTH  OF  MATERIALS. — COHESION.  763 


Modulus  of  Elasticity-  and  "Weight  of  Various  Materials. 


Substances. 


Height. 


Weight. 


Substances. 


Height. 


Feet. 

Lbs. 

Ash 

4 970  000 

1 656  670 

Beech 

4 600000 

I 345  DOO 

Brass,  yellow 

2 460000 

8 464  DOO 

“ wire 

4 1 12  000 

14632  720 

Copper,  cast 

4 800000 

18  240  DOO 

Elm 

5 680  000 

1499  500 

Fir,  red 

8 330  000 

2 Ol6  DOO 

Glass 

4 440  000 

5 550  DOO 

Gun-metal 

2 790  000 

8 844  300 

Hempen  fibres. . . . 

5 000000 

1 70  DOO 

Ice 

6 000000 

2 37O  DOO 

Iron,  cast 

5750000 

17  968  500 

11  wrought 

7 550000 

25  820  DOO 

“ wire 

8377000 

28  230  500 

Feet. 


Larch 

Lead,  cast 

Lignum-vitse 

Limestone 

Mahogany 

Marble,  white 

Oak 

Pine,  pitch 

“ white., 

Steel,  cast 

“ wire 

Stone,  Portland  . . . 

Tin,  cast 

Zinc 


4 415  000 
146000 

1 850000 

2 400  000 
6 570  000 
2 150000 
4 750  000 
8 700  000 
8 970  000 

8 530000 

9 000  000 
1 672  coo 
1 053  000 
4 480  000 


Weight. 

Lbs. 

1 074  000 
720000 

1 080  400 
3 300  000 

2 071  060 
2 508  000 

1 710000 

2 430  000 
1 830  000 

26  650000 
28  689  000 
1 718  800 

3 510  000 
13  440000 


Weight  a Material  will  hear  per  Sq.  Inch,  without 
Permanent  Alteration  of  its  Length. 


Material. 

| Lbs. 

Material. 

Lbs. 

Material.  ] 

Lbs. 

Metals. 

Brass 

6 00 

Stones , etc. 
Marble 

Woods. 

Beech  

2360 

3240 

4290 

2060 

Gun -metal 

IO  OOO 

Limestone* 

4QOO 

2000 

Elm 

Tron  cast 

15  000 
17  800 
I ^OO 

Portland 

1500 

Fir,  red 

“ wrought... 
Lead 

Larch 

Woods. 

Mahogany  

3000 

3960 

Steel 

45  000 

Ash 

1 3540 

Oak 

* Tensile  strength  2800.. 


Comparative  Resilience  of  Woods. 


Ash 

Chestnut 

| Larch 

..  .84  I 

| Spruce 

Beech 

86 

Elm 

Oak 

. . .63 

Teak 

Cedar 

66 

Fir 

..  .4  | 

| Pitch  Pine. . . 

••  -57 

| Yellow  Pine. 

MODULUS  OF  COHESION. 

To  Compute  Length  of  a Prism  of  a Miaterial  which  would 
he  Severed  hy  its  own  "Weight  when  Suspended. 

Rule. — Divide  tensile  resistance  of  material  per  sq.  inch  hy  weight  of  a 
foot  of  it  in  length,  and  quotient  will  give  length  in  feet. 

Illustration. — Assume  tensile  resistance  of  a wrought-iron  rod  to  be  60000  lbs. 
per  sq.  inch.  Weight  of  1 foot  = 3.4  lbs. 

Then  60000  -4-  3.4  = 17  647.06  feet. 

Length  in  Feet  required  to  Tear  Asunder  the  following  Substances : 

Rawhide 15  375  feet.  | Hemp  twine. ..  75000  feet.  | Catgut 25  000  feet. 

Elasticity  of  Ivory  as  compared  with  Glass  is  as  .95  to  i. 

When  Height  is  given.  Rule. — Multiply  weight  of  1 foot  in  length  and 
1 inch  square  of  material  by  height  of  its  modulus  in  feet,  and  product  will 
give  weight. 

To  Compute  Height  of  Modulus  of  Elasticity. 

Rule. — Divide  weight  of  modulus  of  elasticity  of  material  by  weight  of 
1 foot  of  it,  and  quotient  will  give  height  in  feet. " 

Example.  — Take  elements  of  preceding  case  (page  762I,  weight  of  1 foot  being 
3 lbs. ; what  is  height  of  its  modulus  of  elasticity  ? 

15  000  000  -r-  3 = 5 000  000  feet. 


76 4 


STRENGTH  OF  MATERIALS. CRUSHING. 


From  a series  of  elaborate  experiments  by  Mr.  E.  Hodgkinson,  for  the 
Railway  Structure  Commission  of  England,  he  deduced  following  formulas 
for  extension  and  compression  of  Cast  Iron  : 

Extension:  13934040  — 290743200 


c c2 

Compression : 12  931  560  — — 522  979  200  — = W.  e and  c representing  extension 
and  compression , and  l length  in  ins. 

Illustration. — What  weight  will  extend  a bar  of  cast  iron,  4 ins.  square  and  10 
feet  in  length,  to  extent  of  .2  inch? 

2 2 ^ 

13  934  040  X 290  743  200  -1— ^ ==  23  223. 4 — 8076. 2 = 15  147. 2,  which  x 4 ins- 

120  120 

= 60  588. 8 lbs. 


CRUSHING  STRENGTH. 

Crushing  Strength  of  any  body  is  in  proportion  to  area  of  its  section, 
and  inversely  as  its  height. 

In  tapered  columns,  it  is  determined  by  the  least  diameter. 

When  height  of  a column  is  not  5 times  its  side  or  diameter,  crushing 
strength  is  at  its  maximum. 

Cast  Iron. — Experiments  upon  bars  give  a mean  crushing  strength  of 
100000  lbs.  per  sq.  inch  of  section,  and  5000  lbs.  per  sq.  inch  as  just  sufficient 
to  overcome  elasticity  of  metal ; and  when  height  exceeds  3 times  diameter, 
the  iron  yields  by  flexure.  When  it  is  10  times,  it  is  reduced  as  1 to  1.75 ; 
when  it  is  15  times,  as  1 to  2 ; when  it  is  20  times,  as  1 to  3 ; when  it  is  30 
times,  as  1 to  4 ; and  when  it  is  40  times,  as  1 to  6. 

Experiments  of  Mr.  Hodgkinson  have  determined  that  an  increase  of 
strength  of  about  one  eighth  of  destructive  weight  is  obtained  by  enlarging 
diameter  of  a column  in  its  middle. 

In  columns  of  same  thickness,  strength  is  inversely  proportional  to  the 
1*fi3  power  of  length  nearly. 

A hollow  column,  having  a greater  diameter  at  one  end^  than  the  other, 
has  not  any  additional  strength  over  that  of  an  uniform  cylinder. 

Wrought  Iron—  Experiments  give  a mean  crushing  stress  of  47 000  lbs. 
per  sq.  inch,  and  it  will  yield  to  any  extent  with  27  000  lbs.  per  sq.  inch, 
while  cast  iron  will  bear  80000  lbs.  to  produce  same  effect. 

Effects. — A wrought  bar  will  bear  a compression  of  g of  its  length,  with- 
out its  utility  being  destroyed. 

With  cast  iron,  a pressure  beyond  27000  lbs.  per  sq.  inch  is  of  little,  if 
any,  use  in  practice.  j 

Glass  and  hard  Slones  have  a crushing  strength  from  7 to  9 times  greater 
than  tensile ; hence  an  approximate  value  of  their  crushing  strength  may  be  ; 
obtained  from  their  tensile,  and  contrariwise. 

Various  experiments  show  that  the  capacity  of  stones,  etc.,  to  resist  effects 
of  freezing  is  a fair  exponent  of  that  to  resist  compression. 

Seasoning. — Seasoned  woods  have  nearly  twice  crushing  strength  of  un- 
seasoned. 

Elastic  Limit  compared  to  Crushing  Resistance. 

Wrought-iron  Commerce 545  Cast  steel 

Bessemer  steel 615  Fagersta  steei 

Cast  steel 473 


STRENGTH  OF  MATERIALS. CRUSHING. 


765 


Crvisliing  Strength,  of  various  Materials,  deduced  from 
Experiments  ofMaj.  Wade,  Hodgkinson,  Capt.  Nleigs, 
TJ.  S.  .A..,  Stevens  Institute,  and.  lay  Gr.  JLi.  Yose. 

Reduced  to  a Uniform  Measure  of  One  Sq.  Inch. 

Cast  Iron. 


Figures  and  Material. 


Gun-metal,  American. 


“ mean 

Low  Moor,  No.  1,  English. 

“ No.  2,  “ 

Clyde,  No.  3,  “ 


Crushing 

Weight. 


Lbs. 
174803 
85  000 
125  000 
100  000 
62  450 
92330 
106  039 


Figures  and  Material. 


Clyde,  average,  English 

Stirling,  mean  of  all,  English  . . 
“ extreme,  English 

Extreme,  English { 


Average  (Hodgkinson),  English 
Blaenavon  No.  2 


Crushing 

Weight. 


Lbs. 
82  000 
122393 
134  400 
53  760 
153  200 
84  240 
109  700 


Wrought  Iron. 


Aluminium  bronze,  95  cop. . 

Fine  brass. 

Cast  copper 

Steel,  cast 


Fagersta  . 


1 127  720  1 
83500 

I English || 

1 ,47040  1 

1 ‘ ‘ average | 

Various  Metals. 

129  920 
164  800 
1 17  000 
105  000 
250000 
154  5oo 

Steel,  Bessemer 

“ “ soft 

“ tempered 

“ Siemens 

Tin,  cast 

Lead 

65  200 
40  000 
37  850 


50000 
66  200 
335ooo 


15500 
7 730 


Elastic  Crushing  Strength  of  Wrought  Iron  and  Crucible  Steel  is  equal  to  its  ten* 
sile,  of  Bessemer  Steel,  50  per  cent,  of  its  transverse  strength. 


Woods. 


Ash 

Beech 

Birch 

Box 

Cedar,  red 

“ seasoned.  ... 
Chestnut....  ... 

Elm 

“ seasoned 

“ English 

Hickory,  white 

Larch 

Locust 

Mahogany,  Spanish. 


6 663 
6963 
3300 
7900 
10513 
5968 
6500 

5 350 

6 831 
10000 
10300 

8925 
3 200 
5 5oo 
9ll3 
8198 


Maple 

Oak,  American  white.. 
“ Canadian  white. . 
“ “ live . . . 

“ English 

“ Dantzic,  dry 

Pine,  pitch 

“ white.... ;.. 

“ yellow 

“ Deal,  Christiana. 

Spruce,  white 

Teak 

Walnut 

Willow,  seasoned 


O IOO 
IOOOO 

7500 

5982 

6850 

9500 

6 484 

7 700 
8947 
5 775 

8 200 
5850 

5 950 
12  100 

6645 

6 000 


Oak. 


Crosswise  of  Fibre. 
Larch 1300 


Pines . 


55o 

Increase  in  Strength  of  Cubes  of  Sandstone , per  Sq.  Inch  {under  Blocks 
of  Wood),  as  Area  of  Surface  is  increased.  (Gen' l Gillmore , U.  S.  A.) 

Inches. 

•5  I 1 


Yellow  Berea  sandstone  . . 
Blue  “ “ 


Lbs. 

6080 


Lba. 

6990 

9500 


Lbs. 

8 226 
10  730 


Lbs. 

8 955 
12000 


2.25 

2.75 

3 

4 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

9 r 3° 

9838 

10125 

11  720 

12  500 

13  200 

— 

— 

766  STRENGTH  OF  MATERIALS.— CRUSHING. 

Stones,  Cements,  etc.  (Per  Sq.  Inch.) 


Figures  and  Material. 


Basalt,  Scotch 

“ Welsh 

Beton,  N.  Y.  S.  ConcretiDg  Co. 


Brick,  pressed 

“ Gloucester,  Mass. . 
“ hard  burned 


Crushing 

Weight. 


“ common | 

“ yellow-faced  burned,  Eng. 

“ Stourbridge  fire-clay, 

“ Staffordshire  blue,  “ 

“ stock,  English 

“ Fareham,  English 

44  red,  English 

“ Sydney,  N.  S 

Caen,  France 

Cement,  Hydraulic,  pure,  Eng.  | 

44  Portland,  sand  i 

44  44  sand  3 

44  44  3 mos 

44  44  i sand,  3 mos. . . . , 

44  44  9 mos 

44  44  i sand,  9 mos 

44  44  12  inch  cubes, ) 

12  mos.  > 
i sand  and  gravel ) 

« « 3 ^ 

“ Roman 

44  “ pure,  Eng 

44  Rosendale 

“ Sheppey,  Eng 

Concrete,  lime  i,  gravel  3 j 

Freestone,  Belleville,  N.  J 

“ Connecticut 

44  Dorchester,  Mass 

“ Little  Falls,  N.  Y. .. . 

Glass,  crown 

Gneiss 

Granite,  Aberdeen,  Eng 

44  Cornish,  “ 

“ Dublin,  ’ “ 

44  Newry,  “ 

* Tested  by  author  at  Stevens’  Institute,  N.  J. 

r,  • , 1 m n + O 


Lbs. 

8300 

16  800 
800 

1 400 

6 222 
10  219 
14  216* 

363° 

800 

4000 

1440 

1 650 

7 200 

2 250 
5 boo 

808 

2 228 
1543 

17  000 
32  000 

1 280 
600 

3 800 
2464 
5980 

2 33° 

2650 

1 800 
342 
750 
3270 
1 280 
460 
775 

3 522 

33*9 
3069 
2991 
31  000 
19600 
10  760 
6 339 
10450 
12  850 


Figures  and  Material. 


Granite,  Patapsco,  Md. . 
‘ Portland,  Eng. 
* Quincy,  Mass. . 

Greenstone,  Irish 

Limestone 


44  compact,  Eng. .. . 

44  Magnesian, “ .... 

44  Anglesea  “ 

44  Irish  “ 

Marble,  Baltimore,  Md 

‘ 4 East  Chester,  N.  Y.  t 

44  Hastings,  N.  Y 

44  Irish 

“ Italian 

44  44  white,..,. 

“ Lee,  Mass 

44  Montgomery  Co.,  Pa.. . . 

44  Statuary 

44  Stockbridge,  Mass.$. ... 
41  Symington,  large  .. . 

44  “ fine  crystal . . 

44  “ strata  horizontal 

Masonry,  brick,  common. . . . 

44  44  in  cement. , 

Mortar,  good 

44  lime  and  sand 

44  “ beaten... 

44  common 

Oolite,  Portland 

Pottery  pipe,  Chelsea 

Sandstone,  Aquia  Creek  . 

44  Arbroath,  Eng. . . 

44  Connecticut 

44  Craigleth,  Eng.. . 

“ Derby  grit  “ ... 

44  Holyh’d  quartz,  Eng. 

44  Seneca  II.... 

44  Yorkshire,  Eng. . 


Slate,  Irish 

Terra  Cotta 

Whinstone,  Scotch. 


+ Post-office,  Wash.  $ City  Hall,  New  York. 

\ WasMu^n,  D.  C.  I Smitbsoaian  Institute. 

Safe  Load  of  Hollow,  Cylindrical,  and  Solid  Columns, 
Arches,  CHords,  etc-.,  of  Cast  Iron. 

Hollow  Columns.  Per  Sq.  Inch.  (F.  W.  Shields,  M.  I.  C.  E.) 


Length. 

Thick- 

ness. 

Load. 

Length. 

ThicH  Load, 
ness. 

Length. 

Thick- 

ness. 

Load. 

Length. 

Thick- 

ness. 

Load. 

20  to  24 
diam’s. 

Inch. 

•375 

•5 

Lbs. 

2800 

336o 

20  to  24 
| diam’s. 

Inch.  Lbs. 
.625  3920 

•75  1 44So 

25  to  30 
diam’s. 

Inch. 

•375 

•5 

Lbs. 

2240 

2800 

25  to  30 
diam’s. 

Inch. 

.625 

•75 

Lbs. 

336o 

3920 

Solid  Columns,  etc.-336°  lbs-  per  sq.  inch.  (Brunei.) 
Arclies.— 5600  lbs.  per  sq.  inch. 


STRENGTH  OF  MATERIALS. CRUSHING. 


767 


Chords  and  Posts.— 1 inch  diameter  and  not  more  than  15  diameters  in 
Jength  .2  of  breaking  weight  of  metal. 

.625  inch  diameter  and  not  more  than  25  diameters  in  length  .5  of  breaking  weight 
of  metal,  and  when  more  than  25  diameters  in  length  from  .1  to  .025  of  breaking 
weight  of  metal.  (. Baltimore  Bridge  Go.) 


Wrought-iron  Cylinders  and  Rectangular  Tubes. 


External 

Internal 

Thickness. 

Area. 

Crushing  Weight 

Length. 

Diameter. 

Diameter. 

per  Sq.  Inch. 

Cylinders. 

Ins. 

Ins. 

Ins. 

Sq.  Ins. 

Lbs. 

10 

feet 

1-495 

1.292 

. 1 

•444 

14661 

10 

2.49 

2.275 

.107 

.804 

29  779 

10 

6.366 

6. 106 

•13 

2.547 

35886 

Rectangular  Tubes. 

10  feet 

4-1 

X 

4.1 

•03 

•504 

10980 

5 “ 

a! 

4.1 

X 

4.1 

•°3 

•504 

n 514 

10  “ 

4-1 

X 

4.1 

.06 

1.02 

19  261 

10  “ 

0 

4-25 

X 

4-25 

•134 

2-395 

21585 

7-5  “ 

4-25 

X 

4-25 

•134 

2-395 

23  202 

10  “ 

’ O. 

8.4 

X 

4-25 

J.26 

i.126 

6.89 

29981 

10  “ 

8.1 

X 

8.1 

.06 

2.07 

13276 

7.66  “ 

8.1 

X 

8.1 

.06 

2.07 

133OO 

10  “ 

) internal 

8.1 

X 

8.1 

.0637 

3-55i 

19  732 

5 “ . 

j diaphrag’s 

8. 1 

X 

8.1 

.0637 

3-551 

23  208 

Strength  per  Sq.  Inch  of  2-Inch  Cubes  under  Blochs 
of  Wood.  ( GenH  Gillmore , U.  S.  A.) 

Surfaces  Worked  to  a Clear  Bed. 


Granite. 


Lbs. 


Staten  Island  blue 22250 

Maine 15000 

Quincy,  dark 17750 

“ light 14750 

Westchester  Co.,  N.  Y 18250 

Millstone  Point,  Conn 16 187 

New  London,  Conn 12  500 

Richmond,  Ya 21  250 

“ “gray 14 100 


Cape  Ann,  Mass, 


Westerly,  R.  L,  gray 14937 

Fall  RiVer,  Mass.,  gray 1593 7 

Garrisons,  Hudson  River,  gray. . 13  370 

Duluth,  Minn.,  dark 17750 

Keene,  N.  H.,  bluish  gray 12  875 

Used  in  Central  Park,  N.  Y.,  red  17  500 

Jersey  City,  N.  J.,  soap 20750 

Passaic  Co. , “ gray 24  040 


Limestone. 

Glen’s  Falls,  N.  Y 11475 

Lake  Champlain,  N.  Y 25000 

Canajoharie,  N.  Y 20700 

Kingston,  “ 13900 

Garrisons,  “ 18500 

Marblehead,  0.,  white 12600 

Joliet,  111.,  white 16900 

Lime  Island,  Mich.,  drab \ 18000 

1 2 1:  000 


Sturgeon  Bay,  Wis.,  bluish  drab  21  500 


Limestone.  Lbs. 

Bardstown,  Ky.,  dark 16250 

Cooper  Co.,  Mo.,  dark  drab 6650 

Erie  Co. , N.  Y. , blue 12  250 

Caen,  France 3 650 

Marble. 

East  Chester,  N.  Y 13  504 

Italian,  common 13062 

Dorset,  Vt 7612 

Mill  Creek,  111.,  drab 9687 

North  Bay,  Wis.,  drab 20025 

Sandstone. 

Little  Falls,  N.  Y.,  brown 9850 

Belleville,  N.  J.,  gray. n 700 

Middletown,  Conn.,  brown 6950 

Haverstraw,  N.  Y. , red 4 350 

Medina,  N.  Y.,  pink 17725 

Berea,  0. , drab { 7250 

’ ’ l 10250 

Vermillion,  0.,  drab 8850 

Fond  du  Lac,  Wis.,  purple 6250 

Marquette,  Mich.,  “ 7450 

Seneca,  0. , red  brown 9 687 

Cleveland,  O. , olive  green 6 800 

Albion,  N.  Y.,  brown 13  500 

Kasota,  Minn.,  pink 10700 

Fontenac,  Minn.,  light  buff. ....  6 250 

Craigleth,  Edinburgh 12000 

Dorchester,  N.  B.,  freestone 9 150 

Massillon,  0.,  yellow  drab 8 750 

AY arrensb  u rg,  Mo. , bl  u i sh  d rab . 5 000 


CRUSHING. 


STRENGTH  OF  MATERIALS. 


To  Compute  Crushing  Weight  of  Columns. 
Deduced  by  Mr.  L.  D.  B.  Gordon  from  Results  of  Experiments  of  various  Authors. 


a representing  area  of  metal  in  sq.  ins.,  r ratio  of  length  to  least  external  diameter 
or  side , and  W crushing  weight  in  tons. 

Illustration.— What  is  the  crushing  weight  of  a hollow  cylindrical  column  of 
cast  iron  io  ins.  in  diameter,  24  feet  in  length,  and  1 inch  in  thickness? 


Length. 

Feet. 

2 

Ins. 

3 

Ins. 

4 

Ins. 

5 

Ins. 

6 

Ins. 

7 

Ins. 

8 

Ins. 

9 

Ins. 

10 

Ins. 

11 

Ins. 

12 

Ins. 

13 

Ins. 

14 

Ins. 

Tr5 

Ins. 

5 

12.4 

44 

102 

184 

288 

414 

560 

728 

916 

1126 

1354 

— 

— 

— 

6 

9.4 

36 

88 

164 

264 

386 

532 

698 

884 

1082 

1320 

I57° 

• — 

— 

7 

7.2 

30 

76 

146 

242 

360 

502 

660 

850 

1056 

1282 

1530 

1798 

2086 

8 

24 

66 

130 

218 

332 

470 

630 

812 

1016 

1240 

i486 

• 1754 

2040 

9 

— 

20 

56 

114 

198 

306 

440 

596 

774 

974 

1196 

1440 

1706 

1992 

10 



18 

48 

102 

180 

282 

410 

560 

739 

932 

1152 

1392 

1656 

1940 

12 

— 

— 

38 

80 

136 

238 

354 

494 

658 

846 

1056 

1292 

1550 

1828 

14 





28 

64 

122 

200 

304 

432 

586 

774 

966 

1192 

1440 

1712 

16 







52 

100 

I70 

262 

378 

520 

686 

878 

1094 

1332 

i596 

18 





— 

44 

84 

144 

226 

332 

462 

616 

796 

1000 

1228 

1482 

20 

— 

— 

— 

72 

124 

196 

292 

410 

552 

720 

912 

1130 

1372 

Subtract  weight  that  may  be  borne  by  a column,  of  diameter  of  internal 
diameter  of  tube  from  external  diameter,  and  remainder  will  give  weight 
that  may  be  borne.  Thickness  of  metal  should  not  be  less  than  one  twelfth 
diameter  of  colufnn. 

Illustration. — Required  the  safe  load  of  a solid  cast-iron  column  6 ins.  in  diam- 
eter and  20  feet  in  length. 

Under  6 and  in  a line  with  20  is  72,  which  X 1000  = 72 000  lbs. 

Note. — This  is  about  one  sixth  of  destructive  weight. 


Cast  Iron.  ( Hodgkinson .) 


Round  Solid  or  Hollow.  ---  a - — W.  For  rectangular  put  500. 
r2 


Rectangular  Solid  or  Hollow.  - — = W.  For  L,  T,  U,  etc.,  put  — - — 


Wrought  Iron.  [Stoney.) 


Round  Solid.  — 16  a - =2  W.  Rectangular  Solid, 
r 2 


2400 


3000 


Steel.  [Baker.) 


Round  Solid.  —Strong  steel,  - 51  a - = W ; m ild  steel,  — ' - -2  - =±  W. 


1400 


Rectangular  Solid. — Strong  steel,  — - — mild  steel, 


1 1600 


Area  of  10  ins.  =78.54.  — — 2 = 28, 8,  and  28. 82  = 829.44.  Area  of  10  ins — 


Weight  home  with  Safety  by  Solid  Cast-iron  Columns. 


In  1000  Lbs.— [New  Jersey  Steel  and  Iron  Co.) 


Diameter. 


For  Tubes  or  Hollow  Columns. 


STRENGTH  OF  MATERIALS 


CRUSHING. 


S„ft  L»*  „ *»•««»  b »„ 

« /«,  ...  .<*  » «»  “*  •**  “ " 

TFoods,  one  seventh  to  one  tenth. 

WOODS. 

TO  compute  Destructive  Weight  of  Colour. 


d4  .6 

Cylinder.  -75-  o = 


than  30  diameters  in  length. 


. w.  Rectangle. 


C = W. 


£/iort  Columns , or  less 


Wr  a S __  w.  d representing  diameter  and  s 


column  of  Wee  dimensions  in  lbs. 


Ash 22000 

“ Canadian 17000 

Beech I5r  5°° 

Cedar *4  000 

Elm 


Red  Pine 17  5°° 

Yellow  pine 12000 

White  u 9 000 

Spruce 14000 

Walnut 12500 


Coefficients 

Elm,  rock 26000 

Fir,  Dantzic 22000 

Oak,  white 20000 

“ Eng 23000 

- ~ Pitch  Pine 2000 

Illustr  \tioh.  —What  Is  destructive  weight  of  a column  of  yellow  pine  ,o  ina 
souare  and  “feet  in  length  or  height? 

^ x 12000  = X 12  000  = 833  333  tbs. 

RectangulI-oJ^ ^0y  nniun~coCm 

Sti^ronen^h  of  .column,  divided  by  this  least  dimension 
and  by  wfdth  of  column,  all  dimensions  m ms. 

L ’ T i L | 


T 


.36 

•35 

•33 

•31 

.29 

•27 


.26 

.24 

•23 


19 


.098 

.097 


093 

089 

084 

08 

077 

073 


•43 
•43 
•42 
•4 

Illustration.— Assume  a'whiteiakLoJLn,  secured  at  both  ends,  12  by  8 ins., 

and  20  feet  in  length  which  = . Hence,  12  X 8 X .097  = 9-3™  tons- 

strerrsth  axrd  Staffiie. mUex  - YeUow  pine . . . 

Cast  Iron 1000  1 Cast  Steel. . . . 2518  | Oak 108.8  | Pine.. . . . . . 7 -5 

.■■^^^l^irm^orVit-e^im^s^s^nd^f  f^Alak^d^vid^by^amHbr^Mim^^ie^ 

* All  ton.,  except  when  otherwi.e  designated,  arc  2240  lbs. 

a T 


STRENGTH  OP  MATERIALS.— DEFLECTION. 
DEFLECTION". 

Deflection  of  Bars,  Beams,  GHrders,  etc. 
and  wilC  lik«  l'“1"’ 

S^^®Mfc-'srfssSSsi 
swwr**  **"-  - 

In  experiments  of  Hodgkinson,  it  was  further  shown  that  sets  from  dp 
flections  were  very  nearly  as  squares  of  deflections.  1 d 

In  a rectanguiar  bar  beam,  etc.,  position  of  neutral  axis  is  in  its  centro 
Vs  110t  SfenS1i  y alteref  by  variations  in  amount  of  strain  applied  In 
bars,  beams,  etc.,  oi  cast  and  wrought  iron,  position  of  neutral  axisPvaries  in 
rLbeam’tand  °nly  fixed  whiIe  elasticity  of  beam  is  perfect.  When  a 
bar,  beam,  etc.,  is  bent  so  as  to  injure  its  elasticity,  neutral  line  changes  and 
continues  t°  change  during  loading  of  beam,  until  its  elasticity  is  destroyed 

susnendlrt  oi  beamS’  SS’’ are  of  same  lenSth,  deflection  of  one,  weight  being 
is  as  R tnd,f  nL°riK  eI1K’  C°Tare<l  Wlth  that  of  a beam  Uniformly  Loaded, 

case  is  as \’to  8 Whence^  I ^ stuPP.or?edcat  both  ends,  deflection  in  like 
j . Whence,  if  a bar,  etc.,  is  in  first  case  supported  in  middip 

nermhted  Pe^mitte<^  *?  deflect,  and  in  second,  ends  supported,  and  middle 
permitted  to  descend,  deflection  in  the  two  cases  is  as  3 to  5 

“ mi;  tat™  U«”  0'“  “ ****  *nd  ““  “*  totoMJ 

When  a bar,  beam,  etc.,  is  Uniformly  Loaded,  deflection  is  as  weight  and 
approximately  as  cube  of  length  or  as  square  of  length ; and  element  of  de 
flection  and  strain  upon  beam,  weight  being  the  sam<?  will  be  but  half  of  that 

when  weight  is  suspended  from  one  end.  mac 

Deflection  of  a bar,  beam,  etc.,  Fixed  at  one  End, , and  Loaded  at  other 

Z^dedeVlf^7  °fia  .bea^.of  twice  \en8th>  Supported  at  both  Ends  and 
Loaded  in  Middle , strain  being  same,  is  as  2 to  1 ; and  when  length  and 
loads  are  same,  deflection  will  be  as  16  to  1,  for  strain  will  be  four  times 

fore  all  °other  Xed  h*  ^ ^ °n  °ne  suPPorted  at  both  ends ; there- 

lore  an  other  things  being  same,  element  of  deflection  will  be  four  times 

greater  ; also,  as  deflection  is  as  element  of  deflection  into  square  of  length 
then,  as  lengths  at  which  weights  are  borne  in  their  cases  are  as  1 to  2 ^de- 
flection is  as  1 : 22  x 4 = 1 to  16. 

^fleC(J101?  of  a bar’  beam>  etc.,  having  section  of  a triangle,  and  supported 
at  its  ends,  is  .33  greater  when  edge  of  angle  is  up  than  when  it  is  down? 

whfrp^f  co.uatera^  deflection  of  a beam,  etc.,  under  stress  of  its  load, 
where  a horizontal  surface  is  required,  it  should  be  cambered  on  its  unner 
surface,  equal  to  computed  deflection.  upper 


STRENGTH  OF  MATERIALS. — DEFLECTION. 


Safe  Deflection.— One  fortieth  of  an  inch  for  each  foot  of  span,  with  a 
factor  of  safety  for  load  of  .33  of  destructive  weight  = xfVch  but  for  ordinary 
loads  and  purposes, 

Cast  Iron , Wnr  to  Wo  0 i and  Wrought  Iron , WffTF  to  Woo  or  WiFT>> 
after  beam,  etc.,  has  become  set. 

When  Length  is  uniform , with  same  weight,  deflection  is  inversely  as 
breadth  and  square  of  depth  into  element  of  deflection,  which  is  inversely  as 
depth.  Hence,  other  things  being  equal,  deflection  will  vary  inversely  as 
breadth  and  cube  of  depth. 

Illustration.— Deflections  of  two  pine  battens,  of  uniform  breadth  and  depth,  and 
equally  loaded,  but  of  lengths  of  3 and  6 feet,  were  as  1 to  7.8. 

Deflection  of  different  bars,  beams,  etc.,  arising  from  their  own  weight, 
having  their  several  dimensions  proportional,  will  be  as  square  of  either  of 
their  like  dimensions. 

jjj  construction  of  models  on  a scale  Intended  to  be  executed  in  full  di- 
mensions, this  result  should  be  kept  in  view. 

When  a continuous  girder,  uniformly  loaded,  is  supported  at  three  points 
by  two  equal  spans,  middle  portion  is  deflected  downwards  over  middle  bear- 
ing and  it  sustains  by  suspension  the  extreme  portions,  which  also  have  a 
bearing  on  outer  bearings.  Middle  portion  is,  by  deflection,  convex  up- 
wards and  outer  portions  are  concave  upwards;  and  there  is  a point  of 
“contrary  flexure,”  where  curvature  is  reversed,  being  at  junction  of  con- 
vex and  concave  curves,  at  each  side  of  middle  bearing.  This  point  is  dis- 
tant from  middle  bearing,  on  each  side,  one  fourth  of  span.  Of  remaining 
three  fourths  of  each  span,  a half  is  borne  by  suspension  by  middle  portion, 
and  a half  is  supported  by  abutment.  Hence,  distribution  of  load  on  bear- 
ings is  easily  computed,  as  given  above.  Deflection  of  each  span  is  to  that 
of  an  independent  beam  of  same  length  of  span  as  2 to  5. 

In  a beam  of  three  equal  spans,  deflection  at  middle  of  either  of  side  spans 
is  to  that  of  an  independent  beam  as  13  to  25.  . 

In  a long  continuous  beam,  supported  at  regular  intervals,  deflection  ot 
each  span  is  to  that  of  an  independent  beam  of  one  span  as  1 to  5. 

Cylinder.— If  a bar  or  beam  is  cylindrical,  deflection  is  1.7  times  that  of  a 
square  beam,  other  things  being  equal. 


Formulas  for  Deflection  of  Beams  of  Rectangular  Sec- 
tion, etc. 

r Z3  w &d3CD 

Fixed  \ Loaded  at  One  End.  ^ - = D ; ana 


= W. 


at 

One  End. 


Fixed  \ Loaded  in  Middle, 
at  -< 

Uniformly. 


bd 3C  ,7’  ” "■  l 3 

3 Z3  w 8 6d3CD  _ 

“ Uniformly.  ^ = D ; and  = V. 

Z3  w 


Doth  Ends. 


' < 

Ends.  I 


24  b d3  C 
5Z3W 


n . 24H3CD  _ 
— D ; and  — — — = W. 

Z3 


= D;and8X2467CD  = W. 


Supported  | 
at  -( 
Both  Ends. 


f Loaded  in  Middle. 

Uniformly. 


8 X 24  Z>  d3  C ’ 5 I3 

Z3  W _ , x6H3CD 

___  = D;and  — 75—  = 


W. 


5 Z3  W _ 8 X 16  5 ds  C D 

2 — = D ; and  =■  W. 


“ at  any  one  Point. 


8 X 16  b d3  C 
n2  W 


5 I3 
Z 6 d3  C D 


Z 6 d3  C 


= W. 


Supported  in  Middle. 

3 Z3  W _ . 5 X 16  b d3  C 

Ends  Uniformly  loaded.  5 x l6  6 d3  q ~ D ’ and ^73  — w* 

l representing  length , b breadth , and  d depth , all  in  ins. , W weight  or  stress  in  lbs.  or 
tons , m n distances  of  weight  from  supports , C a constant , and  D deflection , m ms. 


772  STRENGTH  OF  MATERIALS. DEFLECTION, 


Fixed  at 
Both  Ends. 


Deflection  of  Beams  of  Rectangular  Section. 

, = D;  Uniformly , = 


Fixed  at  fr  . . . _ — . 

One  End  '[  Loaded  at  0ne  End- 


{ 


Loaded  in  Middle. 


Z3  w 
b cZ3  C 
Z3  W 

24  6 ^3  (J 

Z3  W 


Supported  { Loaded  in  Middle.  jo  ^ 

ni  I i6  6d3C 


's.|  “ 


, = D ; Uniformly, 
==  D ; Uniformly, 


5W 

8 X 24  b cZ3  C 
5 Z3  w 


m2  w2  W 
Z 6 tZ3  (j 


= D. 


8 X 16  b rf3  c 
W weight  in  tons. 


- = D. 
= D. 


i?oZ/i  Ends.  / a aZ  any  one  Point. 

C a Constant  as  follows. 

Cast  Iron.  Rectangular  Bars.—  Loaded  at  One  End 875. 

“ at  the  Middle 28000. 

Round  Bars. — Loaded  at  One  End 594. 

“ at  the  Middle 19000. 

Wrought  Iron.  Cast  Iron. 

Rectangular.—  For  tons  and  l in  ins.  put  C = 47 000.  28  000. 

“ “ l in  feet  “ C = 27.  16. 

Round.  “ “ Zinins.  “ 0 = 32000.  19000. 

“ “ l in  feet  “ C=  18.  10.7. 

Hence,  in  order  to  preserve  same  stiffness  in  bars,  beams,  etc.,  depth  must 

be  increased  in  same  proportion  as  length,  breadth  remaining  constant. 

Woods. 

Mean  of  LasleWs,  Barlow,  etc.  ( D . K.  Clark.) 

/3  W 


Supported  at  Both  Ends.  Loaded  in  Middle. 


Ash,  Canadian 1476 

u Eng 2722 

Beech 2418 

Blue  Gum 2559 

Elm 1227 

Fir,  Dantzic 2490 

u Memel 3630 

“ Riga 2920 

Greenheart 1888 

Iron  Bark 4378 


C 

Iron-wood 4228 

Larch 2100 

Mahogany,  Honduras  2118 
“ Mexican.  3608 
“ Spanish. . 3360 

Norway  spar 2465 

Oak,  Baltimore 2761 

“ Canadian 3445 

“ Dantzic 2080 

“ Eng 1848 


bd 3 C 


= D. 


Oak,  French 2656 

“ white 2114 

Pitch  pine 2968 

Bed  “ 2434 

Rock  elm 2319 

Spruce 3300 

“ Amer 2669 

“ Scotch 1583 

Teak 1804 

Yellow  pine 2084 

Application  of  Table  : To  Compute  Deflection  of  a Rectangular  Beam  of  Wood. 
Illustration. — What  is  the  deflection  of  a floor  beam  of  yellow  pine,  3 by  12  ins., 
12  feet  between  its  supports,  under  a uniformly  distributed  load  of  3000  lbs.  ? 

5 X 12 3 x 3000  15000  . 

— — . 299  inch. 


C = 2084. 


8X3X128x2084  50016 

Hence , To  compute  weight  that  may  be  borne  by  a given  deflection  of  such  a beam, 

8 X 3 X 123  X 2084  X .299  14  955  t. 

1 — - 3 • = =2991  lbs. 

5X123  5 

Deflection,  of  Continuous  Grirders  or  Beams. 
Beams  of  Uniform  Dimensions,  Supported  at  Three  or  More  Bearinqs, 
(D.  K.  Clark. ) 


1.  Two  Equal  Spans  or  3 Bearings. 
Weight  on  1st  and  3d  bearing  = .375  W l 
4k  “ 2d  bearing = 1.25  W l 


2.  Three  Equal  Spans  or  4 Bearings. 
Weight  on  1st  and  4th  bearing  = .4  W l 
“ a 2d  “>  3d  = 1.1  W l 


3.  Four  Equal  Spans  or  5 Bearings. 

Weight  on  1st  and  5th  bearing  = .39  W l j Weight  on  2d  and  4th  bearing  = 1.14  W l 
Weight  on  3d  bearing  = .93  W l. 

Z3  W 

Cylindrical  Beam.  -rz-r.  = D ; 


d*  C 


Z3 


STRENGTH  OF  MATERIALS. DEFLECTION.  773 


To  Compute  Maximum  Load,  that  may  he  Dome  hy  a 
Rectangular  Beam. 

Deflection  not  to  exceed  Assigned  Limit  of  one  hundred  and  twentieth  of  an 
Inch  for  each  Foot  of  Span. 

Supported  at  Both  Ends.  Loaded  in  Middle. 

d3  _ w b and  d representing  breadth  and  depth  in  ins. , l length  in  feet,  C con- 
l2C  ’ 

slant,  and  W weight  or  load  in  lbs. 

Constants 


Cast  Iron 0003 

Wrought  Iron .0021 

Hickory 018 

Teak 024 


Oak,  red 039 

Hemlock 039 

Pine,  white 039 

Chestnut,  horse. 051 


Oak,  white 027 

Ash,  white 03 

Pine,  pitch 033 

( “ yellow ...r.  .036 

Illustration.— What  is  maximum  load  that  may  be  borne  by  a beam  of  white 
pine,  3 by  12  ins.,  20  feet  between  its  supports,  and  loaded  in  its  middle? 

A> <i!i.=  5^4  = 332.3  »«. 

20 2 X -039 


C = .039. 


Then 


WROUGHT  IRON. 

Deflection  of  NMuoxiglit-iron  Bars. 


No.  Form. 

Length  of  Bear- 
ing. 

Breadth. 

Depth. 

Weij 

by  Actual 
Observation. 

?ht  and  Deflect 

at  one  sixth 
of  Destruc- 
tive Weight. 

ion 

at  yi-g^ th  of 
an  Inch  for 
each  Foot  of 
Span. 

Constant  at 
Reduced  Weight 
and  Deflection. 

> 

1 

Q 

-0 

1 

2 

Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

c 

1.  American.  |j| 

1.83 

I 

I 

600 

.06 

266 

.027 

148 

.015 

I 

2.  English...  “ 

2-75 

2 

2 

4480 

.08 

131° 

.022 

1310 

.022 

1.29 

3.  “ ....  j 

2-75 

i-5 

2-5 

8960 

. 104 

2128 

.025 

00 

.022 

1.25 

4.  * “ ....  “ 

2-75 

i-5 

3 

8960 

.q88 

co 

0 

0 

•O37 

2259 

.022 

.88 

To  Compute  Deflection  of,  and  Weight  Chat  may  he  home 
hy,  a Rectangular  Bar  or  Beam  of  Wrought  Iron. 
\VJ3  W l3  _ ^ 60000  b d3  CD_y 

60 000  b d ^ D 60  000  b d3  C l^ 


OOOOO  U U,~  ^ w ~ 

Illustration. — What  weight  will  a beam  2 ins.  in  breadth,  5 ins.  in  depth,  and 
15  feet  between  its  supports,  bear  with  safe  deflection  of  yiy  of  an  inch  for  each 
foot  of  space,  or  T of  its  length  ? 

C from  table  = .88.  D = of  15  = .12  inch. 

60000  X 2 X 5 3 X -88  X . 12  _ 1800000  _ 533  30  jbs 
153  ~ 3375  “ 


*3  33/3 

D.  K.  Clark  gives  for  Elastic  deflection,  47  000  for  Rectangular  bars,  and  32  000  for 
Cylindrical. 

Note.— Deflection  of  to  of  the  length  may  be  allowed  under  special  cir- 
cumstances; but  under  ordinary  loads  the  deflection  should  not  exceed  one  fourth 
of  these,  as  yg1^  to  ^y1^. 

Practice  in  U.  S.  is  to  allow  y^1^  after  girder  has  taken  its  permanent  set. 

In  small  bridges  there  is  a slight  increase  in  deflection  from  high  speeds,  about 
.166  or  .144  of  the  normal  deflection,  with  the  same  load  moving  at  slow  speed. 

In  large  girders  there  is  no  perceptible  difference  between  the  deflection  at  high 
and  low  speeds. 

3T* 


774  STRENGTH  OF  MATERIALS. DEFLECTION. 


Deflection  of  Wronght-iron  Rolled  Beams. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 

W n 

- C at  Reduced  Weight  and  Deflection. 


70000  d2  (4  a + 1.155  «')  D 


No.  Form. 

Length. 

Fla 

Width, 

nges. 

Mean 

Thick- 

ness. 

Web. 

Depth . 

Weight  and 

by  Actual 
Observation. 

Deflection 

at  one  sixth 
of  Destructive 
Weight. 

C 

Feet. 

Ins. 

Inch. 

Inch. 

InS. 

Lbs. 

Ins. 

Lbs. 

Inch. 

*•  X 

10 

3 

*485 

•5 

7 

12000 

•4 

3800 

.127 

1.05 

2.  “ 

20 

4.6 

.8 

•5 

9-85 

16000 

i-i5 

6300 

•453 

.92 

3- 

20 

5-7 

•643 

.6 

“•75 

20  000 

.85 

8000 

■34 

.98 

To  Compute  Deflection  of,  and  Weigh,  t that  may  De  Dome 
hy,  a W r o vigli  t— iron  Rolled  Ream  of  Uniform  and  Sym- 
metrical Section. 


Supported  at  Both  Ends. 
W Z3 


Weight  applied  in  Middle. 
— D. 


(D.  K.  Clark.) 


70000  d2  (4  a -)-  1 . 155  a')  D 


: W. 


70000  d2  (4  a- f- 1. 155  a')  l3 

l representing  span  in  feet,  d reputed  depth , or  depth  less  thickness  of  lower  flange 
in  ins.,  a area  of  section  of  lower  flange,  a ' area  of  section  of  web  for  reputed  depth 
of  beam,  both  in  sq.  ins.,  and  W weight  or  stress  in  lbs. 

Illustration'. — What  is  deflection  of  a wrought-iron  rolled  beam  of  New  Jersey 
Steel  and  Iron  Co.,  10.5  ins.  in  depth,  flanges  5 by  .5  ins.,  and  width  of  web  .47 
inch,  when  loaded  in  its  middle  with  8000  lbs.,  and  supported  over  a span  of  20  feet? 

q.  ins.,  and  a'  = 10  X .47  = 4.7  sq.  ins . 
64000000 


d — 10. 5 - 
Then  — 


.5  = 10  ms.,  a = 5 X- 5 = 2.5 
8000  X 20 3 


107  899  500 


'±= . 59  inch. 


70000  X io2  X (4  X 2.5  -f  1. 155  X 4-7) 

If  weight  is  uniformly  distributed,  divide  by  112  500  instead  of  70000. 

A like  beam  6 ins.  in  depth,  loaded  with  2608  lbs.,  and  supported  over  a span  of 
12  feet,  gave  by  actual  test  a deflection  of  .3  inch,  and  by  above  formula  it  is  also 
.3  inch. 

Note. — Deflection  for  such  a beam,  for  a statical  weight  or  stress  of  17  100  lbs., 
uniformly  distributed , by  rules  of  N.  J.  Steel  and  Iron  Co.,  would  be  .54  inch,  which, 
with  difference  in  weights,  will  make  deflections  alike. 

Deflection  of  Wrought-iron  Riveted.  Reams. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 

W 1 3 

- = C at  Reduced  Weight  and  Deflection. 


168000  ( - 


(~+f) 


d 2 D 


No.  Form. 

Length. 

Flanges. 

Angles. 

Web. 

Depth. 

Weight  and 

by  Actual 
Observation. 

Deflection 
at  one  sixth 
of  Destructive 
Weight. 

c 

Feet. 

Ins. 

IllS. 

Iuch. 

Ins. 

Lbs. 

Inch. 

Lbs. 

! Inch. 

2.125X2 

I 

'•  T 

7 i 

\ 

“ 

X-28 

2.125X2 

[•25 

, 7 

4 216 

. 1 

4062 

.096 

•63 

JL 

1 

l 

X.29 

J 

i 

1 

[ 

4-5X 

2X2 

I 

2 T 

n.66-^ 

•5 

4-5X 

X.3125 

2X2 

1 

f.25 

12.5 

77  280 

.46 

12  880 

•075 

1.96 

t 

•375 

X.3125 

J 

1 

4-5X 

2X2 

I 

3.  “ 

22.5  \ 

•5 

7 X 

X-375 

3X3 

1 

^•375 

16.5 

115584 

•875 

19265 

4* 

OO 

3-86 

\ 

c 

•5 

X-4375 

J 

STRENGTH  OF  MATERIALS. DEFLECTION.  775 


To  Compnte  Deflection  of,  and.  Weight  tLat  may  "be  "borne 
by,  a Riveted.  Beam  of  Wrought  Iron. 

w*3  - = D.  ^ooo /£+£  + £)  a.  CD 

ld.0  ■ 2 4/ W. 


(il  -f-  (l'  . & \ 
168  OOO  ^ — J C 


a,  a',  and  a"  representing  areas  of  upper  and  lower  flanges  with  their  angle  pieces, 
and  of  web  for  its  entire  depth , all  in  sq.  ins. 

Note.— If  there  are  not  any  flanges,  as  in  No.  1,  angle  pieces  alone  are  to  be  computed  for  flange 
area. 

Illustration.— What  weight  will  a riveted  and  flanged  beam  of  following  dimen- 
sions sustain,  at  a distance  between  its  supports  of  25  feet,  and  at  a safe  deflection 
of . 2 inch  or  of  its  length  ? 

Top  flange 6 X - 5 ins-  I Web 5 

Bottom  flange 6X-5  “ I Depth 17  “ 

Angles 2.25  X 2.25  X .5  ins. 

a and  a'  each  = 6 X .5  — 3 + 2.25  -f  2.25  — .5  X .5  X 2 = 7 sq.  ins. 

a"  — . 5 x 17  = 8. 5 sq.  ins.  C,  as  per  No.  2 .43,  but  inasmuch  as  flanges  in  this 
case  are  much  heavier,  assume  .5. 


168000 


Then  - 


172  x 2 x .5 


= 44  303120 

15625 


Strength  of  a Riveted  beam  compared  to  a Solid  beam  is  as  1 to  1.5,  while  for 
equal  weights  its  deflection  is  1.5  to  1. 

Tubular  Girders.  Wrought  Iron. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 


No. 

Section. 

Length 

of 

Bearing. 

Breadth. 

Dej 

Inter- 

nal. 

>th. 

Ex- 

ternal. 

Weight. 

Deflection. 

Deflection  at 
.008  inch  for 
each  Foot  of 
j Span. 

1 1 
£ 

1 

ft 

•o 

VO 

Feet. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Inch. 

( 

cT 

I.  □ 

Thickness  .03  inch 

3*75 

1.9 

2.94 

3 

448 

.1 

•°3 

288 

2.  “ 

“ .525  “ 

3° 

i5-5 

22.95 

24 

33  685 

•56 

.24 

473 

top  .372  “ ) 

3-  “ 

bottom  .244  “ / 

30 

16 

23.28 

24 

32  538 

1. 11 

.24 

224 

sides  .125  “ ) 

4-  “ 

Thickness  .75  “ 

45 

24 

34-25 

35-75 

128  850 

1.85 

•36 

362 

O 

Thickness  .0375“ 

17 

12 

11.925 

12 

2 755 

•65 

.136 

62.8 

7 • 0 

“ .0416“ 

*7 

9-25 

13-535 

13.62 

2 262 

.62* 

.136 

47-9 

“ -i43  “ 

17 

9-25 

14.714 

15 

l6  800 

1.39* 

.136 

119 

* Destructive  weight. 

To  Compute  Deflection  of,  and  Weiglit  tLat  may  L>e 
"borne  by,  a "Wronglit-iron  Tubular  Girder. 


16  b d*  C D 


— W. 


- =D. 


1 3 16  b d*  C~ 

Illustration.— What  weight  may  be  safely  borne  by  a wrought-iron  tube,  alike 
to  No.  3 in  preceding  table,  for  a length  of  40  feet,  and  a deflection  of  .32  inch? 

16  X 16  X 24s  X 224  X .24  __  190253629  _ 6 lbs 

303  27000 

Flanged  Rivets. 

Deflection  of  Iron  and  Steel  Flanged  Rails  within  their  elastic  limit,  compared 
with  their  transverse  strength,  is  as  17  to  20,  and  with  double-headed  it  is  as  n to  23. 


STRENGTH  OF  MATERIALS. DEFLECTION, 


RAILS. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 
Iron . 


No. 

Fokm. 

Length 

of 

Bearing. 

Head. 

Bottom. 

Weight 

per 

Yard. 

| Depth. 

Area. 

Observed 
Weight  and 
Deflection. 

Destructive 

Weight 

and 

Deflection. 

Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Sq.  Ins. 

Lbs. 

Ins. 

Lbs. 

In3. 

1. 

1 

2-75 

2.25X1 

2.25X1 

60 

4-5 

6.166 

13  440 

•°34 

26  680 

.065 

2. 

u 

4-5 

2 3 XI 

2.3  Xi 

65 

4-5 

6.68 

11  200 

.11 

24640 

.204 

3- 

u 

5 

2.3  Xi 

2.3  Xi 

82 

5-4 

8.25 

25  760 

.2 

51520 

•378 

4- 

T 

2-75 

3-5  X .8 

2.25X1 

60 

4 

6.7 

11  200 

•035 

26  680 

.065 

5- 

A 

2.58 

2.23X1 

3-5  X -6 

57 

3-5 

5.85 

II  200 

•°97 

20160 

.128 

Steel. 


-d  ti 

Depth. 

Observed 

Destructive 

No.  Form. 

tr<~  .5 
e 0 

^ n 

a 

Bottom. 

Weig] 

per 

Yard 

Web. 

Centres 

of 

Heads. 

Total. 

Area. 

Weight  and 
Deflection. 

Weight 

and 

Deflection. 

Feet. 

Ins. 

Ins. 

Lbs. 

Inch. 

Ins. 

Ins. 

S.IU3. 

Lbs. 

In. 

Lbs. 

Inch. 

«•  I 

5 

- 

78 

•75 

4.2 

5-4 

7.67 

36  086 

•25 

80  192 

•55 

7-  “ 

Bessemer. 

3.62 

- 

86 

- 

- 

5-5 

8-43 

22  400 

.14 

26  680 

.165 

8-X 

5 

2-5 

6-375X;|7 

1 

84 

.65 

3-37 

4-5 

8. 24 

27  290 

.24 

27  290 

.24 

To  Compute  Deflection  of  Double-headed  Hails  within. 

Elastic  Limit.  (D.  K.  Clarlc.)  t 

Supported  at  Both  Ends.  Weight  applied  at  Middle. 

IRON. 

W 1 3 

— --  = D.  a representing  area  of  one  head , less  portion  per - 

57000  (4  ad'2- f-  1. 155  td 3) 

taining  to  web , d whole  depth  of  rail , d'  vertical  distance  between  centres  of  headsf 
t thickness  of  web,  all  in  ins.,  I length  in  feet,  and  W weight  in  lbs. 

STEEL.  \ 


For  57  000  put  67  400. 

Illustration. — Take  case  No.  3 (Iron),  in  preceding  table,  with  a weight  of  26000 


lbs. ; what  will  be  its  deflection  between  bearings  5 feet  apart? 

a = 1.911.  ^'  = 4.2.  d = 5.4.  t~.  82. 

26000  X 53  3 250 000 


Then 


57000  (4  X 1-911  X 4-22+  1. 155  X .82  X 5-43)  57  000 X 284 


.2  inch. 


To  Compute  Deflection  of  Iron  and  Steel  Hails  of*  TU n- 
symmetrical  Section  within  Elastic  Limits. 

Elastic  Deflection  of  Steel  Flanged  Rails  of  Metropolitan  Railway  of  London,  as 
determined  by  Mr.  Kirkaldy,  at  a span  of  5 feet,  and  loaded  in  middle,  was  ,02  inch 
per  ton.  ( See  Manual  of  D.  K.  Clark,  pp.  667-670.) 


STRENGTH  OF  MATERIALS. DEFLECTION.  777 


CAST  IRON. 

Deflection  of  Bectangnlar  Bars  and  Beams  of  various 
Sections,  etc.,  by  TJ.  S.  Ordnance  Corps,  Barlow, 
Hodgkinson,  and  CnLitt. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 

. 6 


©*gl 

2 


x.  American. 

2.  English. . . 

3.  “ ... 

4.  “ ... 

5-  “ ••• 


Length  of  Bear- 
ing. 

Breadth. 

Depth. 

Wei* 

By  Actual 
Observation. 

;ht  and  Deflect 

At  one  sixth 
of  Breaking 
Weight. 

don. 

At  1th  of 
an  inch  for 
each  foot  of 
span. 

Feet. 

Ins. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

1.66 

2 

2 

5000 

.036 

1666 

.012 

1805 

.013 

4 

I 

1 

212 

•32 

80 

. 12 

22 

•033 

16 

4 

4 

1008 

•4 

5333 

2.  II. 

337° 

x-33 

4-5 

3 

1 

1120 

1.42 

2x5 

.27 

30 

•037 

4-5 

1 

l2:55 

2231 

•51 

422 

.1 

156 

•037 

3.81 
4. 11 
3-89 
2-37 

2-33 


To  Compute  Deflection  of,  and  "Weiglit  tliat  may  t>e 
■borne  V>y,  a Rectangular  Bar  or  Beam  of  Cast  Iron. 

10  400  b d3  C D 


W 1 3 


- = C. 


W l3 


- — D. 


zW. 


10400  b cZ 3 D 10400  6 d3  C l 3 

Illustration.  — What  weight  will  a beam  2 ins.  in  breadth,  5 ins.  in  depth,  and 
16  feet  between  its  supports,  bear  with  safe  deflection  of  y^  of  an  inch  for  each 
foot  of  span,  or  yy1^  of  its  length  ? 

C from  table  = 3. 89.  D = yU  of  16  = 1. 33  ins. 

10400  X 2 X 5s  X 3.89  X t.3S  _ 13  451  620  _ ^ 

163  — " 4096 

Clark  gives  C uniform  for  Rectangular  bars  of  2.69,  and  1.85  for  Cylindrical. 

FLANGED  BEAMS.  Cast  Iron. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 

To  Compute  Deflection  of,  and  W eiglit  tliat.  may  "be 
"borne  Toy,  a Flanged  Beam  of  Cast  Iron  of  "Uniform 
and  Symmetrical  Section. 

Wl3 

:I). 


27  000  d2  (4  a -{-  1. 155  a'2)  D 


= W. 


27  000  d2  (4  a -f- 1. 155  a'2)  l3 

Illustration. — What  is  deflection  of  a cast-iron  beam  (Hodgkinson’s)  7.15  ins., 
flanges  2.6  X -86  ins.  and  5X1-6  ins.,  and  width  of  web  1 inch,  when  loaded  in  its 
middle  with  n 200  lbs.,  over  a span  of  15  feet? 

^ = 7.15 — 1-6  = 5.55  ins.,  <1  = 5 X 1.6  = 8 ins.,  and  a'  = 7.15  — 1.6  = 5.55  ins. 
11200X  153  __  37800000 


Then 


: 1.48  ins. 


27000  5.55^  (4  X 8 + ..155  X 5-55=)  27000  3a8  (32  + 35-57) 

Note  i. — The  observed  deflection  of  this  beam  was  1.28  ins.,  at  one  sixth  of  its  de- 
structive weight  it  was  .3,  and  at  y|-^  of  an  inch  for  each  foot  of  span  it  was 
.125  inch. 

2.— The  mean  ratio  of  elastic  to  destructive  stress  is  73  per  cent. 

Formulas  for  value  of  deflection  signify  that  deflection  varies  directly  as  weight, 
and  as  cube  of  length;  and  inversely  as  breadth,  cube  of  depth,  and  coefficient  of 
elasticity. 


77 8 STRENGTH  OF  MATERIALS. DEFLECTION. 


Elastic  Strength  of  Beams  of  Unsymmetrical  Section.— Elastic  strength  is 
approximately  Reducible  from  ultimate  strength,  according  to  ordinary  ratio 
of  one  to  the  other,  ascertained  experimentally.  Elastic  strength  and  de- 
flection of  a homogeneous  beam  of  any  section  is  same,  whether  in  its  nor- 
mal position  or  turned  upside  down. 


Comparative  Strength,  a nd  Deflection  of  Cast-iron 
Flanged.  Beams. 


Description  of  Beam. 

Comp. 

Strength. 

Description  of  Beam. 

Comp. 

Strength, 

Beam  of  equal  flanges 

.58 

Beam  with  flanges  as  1 to  4.5. . 

.78 

“ with  only  bottom  flange. 

.72 

“ “ u 1 to  5.5. . 

.82 

“ “ flanges  as  1 to  2 

.b3 

“ “ “ 1 to  6 . . 

“ “ “ 1 to  4 

•73 

“ “ “ 1 to  6. 73  •' 

.92 

SHAFTS. 

To  Compute  Deflection  and  Distributed  Weight  for 
Limit  of  Deflection. 


Wrought  Iron. 

Deflection. 


Supported  at  Ends. 


Fixed  at  Ends. 


Weight. 


Supported  at  Ends.  Fixed  at  Ends. 

= w. 
= w. 


Round. 

W Z3 

and 

W Z 3 

= D. 

664  cZ4 

and 

1330  <Z4 

66  400  d 4 

133000  cZ4 

Z2 

z 2 

Square. 

W Z3 

and 

W Z3 

==  D. 

975  «4 

and 

1950  $4 

97  500  s4 

195  OOO  s4 

Z2 

Z 2 

Cast 

Iron. 

Round. 

WZ3 

and 

W Z 3 

= D. 

394 

and 

790  d4 

39  400  d 4 

79000  (Z4  _ 

Z2 

1 2 

Square. 

W Z 3 

and 

W Z 3 

= D. 

580  S4 

and 

1160  S4 

58000  S4 

116000  s4 

Z 2 

Z2 

Steel. 

Round. 

WZ3 

and 

W 1 3 

==  D. 

788  <Z4 

and 

1576  <Z4 

78  800  cZ4 

158000  cZ4 

Z 2 

Z2 

Square. 

WZ3 

and 

WZ3 

= D. 

1160  b4 

and 

2320  64 

1 16  000  $4 

232  OOO  S4 

Z 2 

Z2 

= w. 
= w. 


= w. 
= w. 


d representing  diameter  and  s side  of  shaft , in  ins.,  I length  between  centres  of  bear- 
ings, in  feet,  and  W weight  in  lbs. 


Deflection  of  a Cylindrical  Shaft  from  its  Weight  alone, 
when  Snpported  at  Both  Ends. 

Z4 

.007318  q = D.  Z representing  length  in  feet,  d diameter  in  ins.,  and  C con- 
stant, ranging  from  475  to  550. 

The  greatest  admissible  deflection  for  any  diameter  is  .001 67  —=  D. 


Admissible  Distances  "between  Bearings.  ^.9128  dC  — l. 


Diam. 
of  Shaft. 

Disti 

Wrought 

Iron. 

ance. 

Steel. 

Diam. 
of  Shaft. 

Dist; 

Wrought 

Iron. 

ance. 

Steel. 

Diam. 
of  Shaft. 

Dish 

Wrought 

Iron. 

jnce. 

Steel. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

Ins. 

Feet. 

Feet. 

1 

12.27 

12.61 

5 

20. 99 

21-57 

9 

25-53 

26.24 

2 

15.46 

15.84 

6 

22.3 

22.92 

10 

26.44 

27.18 

3 

17.7 

18.19 

7 

23.48 

24.13 

11 

27-3 

28.05 

4 

19.48 

20.02 

8 

24-55 

2523 

12 

28.1 

28.88 

When  Ends  of  Shaft  are  rigidly  connected  at  Ends. 

Barlow  gives  D=z.66  of  results  obtained  by  above  formula;  but  when  deflection 
of  attached  length  is  considerable,  Navier  gives  D = .25  of  above. 


STRENGTH  OF  MATERIALS. DEFLECTION.  779 


1 3 W 


Deflection,  of*  NTill  and.  Factory  SHafts. 

= D.  I representing  length  between  supports  in  ins.,  W weight  at  middle 


6 7rd*C 

in  lbs.,  d diameter  of  shaft  in  ins.,  and  C as  follows : 

Bessemer  steel 3800000  | Wrought  iron 3500000 


To  Compute  Deflection  of  a Cylindrical  Shaft. 


Rule. — Divide  square  of  three  times  length  in  feet  by  product  of  follow- 
ing Constants  and  square  of  diameter  in  ins.,  and  quotient  will  give  deflection. 


Cast  iron,  cylindrical 1500  I Wrought  iron,  cylindrical . 1980 

“ “ square 2560 1 “ u square 3360 

Example.— Length  of  a cast-iron  cylindrical  shaft  is  30  feet,  and  its  diameter  in 
centre  15  ins. ; what  is  its  deflection  ? 


30  X 3 
1500  X 15  2 


8100 

** — — — .024  ms. 
337  5oo 


SPRINGS. 

Flexure  of  a spring  is  proportional  to  its  load  and  to  cube  of  its  length. 


Deflection  of  a Carriage  Spring. 

A railway-carriage  spring,  consisting  of  10  plates  .3125  inch  thick,  and  2 
of  .375  inch,  length  2 feet  8 ins.,  width  3 ins.,  and  camber  or  spring  6 ins., 
deflected  as  follows,  without  any  permanent  set : 

.5  ton 5 inch.  I 1.5  tons 1.5  ins.  I 3 tons 3 ins. 

' “ 1 “ I 2 u 2 “ 14  “ 4 “ 


Compression  of  an  India-rubber  Buffer  of  3 Ins.  Stroke. 

1 ton 1.3  ins.  I 2 tons 2 ins.  I 5 tons 

1.5  tons 1.75  u I 3 “ 2.375  “ I 10  “ 


2.75  ins. 
3 “ 


Gfeneral  Deductions. 

Deflection  depends  essentially  upon  form  of  Girder,  Beam,  etc. 

A continuous  weight,,  equal  to  that  a beam,  etc.,  is  suited  to  sustain,  will 
not  cause  deflection  of  it  to  increase  unless  it  is  subjected  to  considerable 
changes  of  temperature. 

Heaviest  load  on  a railway  girder  should  not  exceed  .16  of  that  of  de- 
structive weight  of  girder  when  laid  on  at  rest. 

Semi-girders  or  Beams.— Deflection  of  a beam,  etc.,  fixed  at  one  end  and 
loaded  at  other,  is  32  times  that  of  same  beam  supported  at  both  ends  and 
loaded  in  middle. 

Deflection.  consequent  upon  Velocity  of  Load. — Deflection  is  very  much  in- 
creased by  instantaneous  loading ; by  some  authorities  it  is  estimated  to  be 
doubled. 

Momentum  of  a railway  train  in  deflecting  girders,  etc.,  is  greater  than 
effect  from  dead  weight  of  it,  and  deflection  increases  with  velocity. 

When  motion  is  given  to  load  on  a beam,  etc.,  point  of  greatest  deflection 
does  not  remain  in  centre  of  beam,  etc.,  as  beams  broken  by  a travelling  load 
are  always  fractured  at  points  beyond  their  centres,  and  often  into  several 
pieces. 

Heaviest  running  weight  that  a bridge  is  subjected  to  is  that  of  a loco- 
motive and  tender,  which  is  equal  to  2 tons  per  lineal  foot. 

. Girders  should  not,  under  any  circumstances,  be  deflected  to  exceed  .025 
mch  to  a foot  in  length. 


780 


STRENGTH  OF  MATERIALS. DEFLECTION. 


A carriage  was  moved  at  a velocity  of  10  miles  per  hour  ; deflection  was 
.8  inch,  and  when  at  a velocity  of  30  miles  deflection  was  1.5  ins. 

In  this  case,  4150  lbs.  would  have  been  destructive  weight  of  bars  if  ap- 
plied in  their  middle,  but  1778  lbs.  would  have  broken  them  if  passed  over 
them  with  a velocity  of  30  miles  per  hour. 


Relative  Elasticity 

Ash 2.9  I Cast  Iron. 

Beech 2.1  | Elm  and  Oak  . . 


of  various  ^Materials.  (Trumbull.) 

. . . t I Pine,  white. ...  2.4  I Pine, pitch. ...  2.9 
9 I “ yellow.. . 2.6  I Wrought  Iron.  .86 


Cast  Iron. — Permanent  deflection  is  from  .33  to  .5  of  its  breaking  weight, 
and  deflection  should  never  exceed  .125  of  ultimate  deflection,  and  it  is  not 
permanently  affected  but  by  a stress  approaching  its  destructive  weight. 


- j X A.  W 

By  experiments  of  U.  S.  Ordnance  Corps  (Report,  1852),  set  or  permanent  deflec- 
tion was  .38  of  its  breaking  weight,  ultimate  deflection  .133  ins.  Deflection  for 
1 of  ultimate  deflection. 


of  span  = .013,  or  . 

By  experiments  of  Mr.  Hodgkinson  (See  Rep.  of  Commas  on  Railway  Structures. 
London,  1849),  set  for  English  iron  bore  a much  greater  proportion  to  its  breaking 
weight. 

A beam,  etc.,  will  bend  to  .33  of  its  ultimate  deflection  with  less  than  .33 
of  its  breaking  weight,  if  it  is  laid  on  gradually,  and  but  .16  if  laid  on 
rapidly. 

Chilled  bars  deflect  more  readily  than  unchilled. 


Results  of'  Experiments  011  tlie  Six  Ejection  of  Cast-iron 
Bars  to  continued  Strains. 


(Rep.  of  Commas  on  Railway  Structures,  London,  1849.) 

Cast-iron  bars  subjected  to  a regular  depression,  equal  to  deflection  due  to 
a load  of  .33  of  their  statical  breaking  weight,  bore  10000  successive  de- 
pressions, and  when  broken  by  statical  weight,  gave  as  great  a resistance  as 
like  bars  subjected  to  a like  deflection  by  statical  weight. 

Of  two  bars  subjected  to  a deflection  equal  to  that  carried  by  half  of  their 
statical  breaking  weight,  one  broke  with  28602  depressions,  and  the  other 
bore  30000,  and  did  not  appear  weakened  to  resist  statical  pressure. 

Hence,  Cast-iron  bars  will  not  bear  continual  applications  of  .33  of  their 
breaking  weight. 


Mr  Tredgold,  in  his  experiments  upon  Cast  Iron,  has  shown  that  a load  of  300 
lbs.,  suspended  from  middle  of  a bar  1 inch  square  and  34  ins  between  its  sup- 
ports, gave  a deflection  of  .16  of  an  inch,  while  elasticity  of  metal  remained  unim- 
paired. Hence  a bar  1 inch  square  and  1 foot  in  length  will  sustain  850  lbs.,  and 
retain  its  elasticity 


Wrought  Iron.— All  rectangular  bars,  having  same  bearing,  length,  and 
loaded  in  their  centre  to  full  extent  of  their  elastic  power,  will  be  so  deflect- 
ed that  their  deflection,  being  multiplied  by  their  depth,  product  will  be  a 
constant  quantity,  whatever  may  be  their  breadth  or  other  dimensions,  pro- 
vided their  lengths  are  same. 

A bar  of  Wrought  Iron,  2 ins.  square  and  9 feet  in  length  between  its  sup- 
ports, was  subjected  to  100000  vibratory  depressions,  each  equal  to  deflec- 
tion due  to  a load  of  .55  of  that  which  permanently  injured  a similar  bar, 
and  their  depressions  only  produced  a permanent  set  of  .015  inc  1. 

Greatest  deflection  which  did  not  produce  any  permanent  set  was  due  to 
rather  more  than  .5  statical  weight,  which  permanently  injured  1 . 

A wrought-iron  box  girder,  6x6  ins.  and  9 feet  in  length,  was  subjected 
to  vibratory  depressions,  and  a strain  corresponding  to  3762  lbs.,  repea 
43  37°  times,  did  not  produce  any  appreciable  effect  on  the  rivets. 


STRENGTH  OF  MATERIALS. — DEFLECTION.  78  I 

Deflection  of  Solid  rolled  beams  compared  to  Riveted  beams  is  as  1 to  1.5. 

Wrought-iron  Girders  of  ordinary  construction  are  not  safe  when  sub- 
jected to  violent  impacts  or  disturbances,  with  a load  equal  to  .33  of  their 
destructive  weight. 

Wood. — In  consequence  of  wood  not  being  subjected  to  weakening  bv  the 
effect  of  impact,  a factor  of  safety  of  5 for  single  pieces  is  held  to  be  suffi- 
cient, but  for  structures,  in  consequence  of  loss  of  strength  in  its  connections, 
a factor  of  from  8 to  10  becomes  necessary. 

Working  Strength,  or  Factors  of  Safety.* 

Elastic  strength  of  materials  is,  in  general  terms,  half  of  its  ultimate  de- 
structive or  breaking  strength.  If  a working  load  of  .5  elastic  strength,  or 
.25  of  ultimate  strength,  be  accepted,  equal  range  for  fluctuation  within 
elastic  limit  is  provided.  But,  as  bodies  of  same  material  are  not  all  uni- 
form in  strength,  it  is  necessary  to  observe  a lower  limit  than  .25  where 
material  is  exposed  to  great  or  to  sudden  variations  of  load  or  stress. 

Cast  Iron.  Mr.  Stoney  recommends  .25  of  ultimate  tensile  strength,  for 
dead  weights  ; .16  for  bridge  girders ; and  .125  for  crane  posts  and  machin- 
ery. In  compression,  free  from  flexure,  cast  iron  will  bear  8 tons  (17920 
lbs.)  per  sq.  inch  ; for  arches,  3 tons  (6720  lbs.)  per  sq.  inch ; for  pillars, 
supporting  dead  loads,  .16  of  ultimate  strength;  for  pillars  subject  to 
vibration  from  machinery,  .125  ; and  for  pillars  subject  to  shocks  from 
heavy-loaded  wagons  and  like,  .1,  or  even  less,  where  strength  is  exerted  in 
resistance  to  flexure. 

Wrought  Tim.— For  bars  and  plates,  5 tons  (11  200  lbs.)  per  sq.  inch  of 
net  section  is  taken  as  safe  working  tensile  stress ; for  bar  iron  of  extra 
quality,  6 tons  (13440  lbs.).  In  compression,  where  flexure  is  prevented, 
4 tons  (8960  lbs.)  is  safe  limit ; in  small  sizes,  3 tons  (6720  lbs.).  For  col- 
umns subject  to  shocks,  Mr.  Stoney  allows  .16  of  calculated  breaking  weight ; 
with  quiescent  loads,  .25.  For  machinery,  .125  to  .1  is  usually  practised; 
and  for  steam-boilers,  .25  to  .125. 

Mr.  Roebling  claims  that  long  experience  has  proved,  beyond  shadow  of 
a doubt,  that  good  iron,  exposed  to  a tensile  strain  not  above  .2  of  its  ulti- 
mate strength,  and  not  subject  to  strong  vibration  or  torsion,  may  be  de- 
pended upon  for  a thousand  years. 

Steel. — A committee  of  British  Association  recommended  a maximum 
working  tensile  stress  of  9 tons  (20  160  lbs.)  per  sq.  inch.  Mr.  Stoney  rec- 
ommends, for  mild  steel,  .25  of  ultimate  strength,  or  8 tons  (17920  lbs.)  per 
sq.  inch.  Limit  for  compression  must  be  regulated  very  much  by  nature  of 
steel,  and  whether  it  be  annealed  or  unannealed.  Probablv  a limit  of  9 tons 
(20  160  lbs.)  per  sq.  inch,  same  as  limit  for  tension,  would  be  safe  max- 
imum for  general  purposes.  In  absence  of  experience,  Mr.  Stoney  further 
i recommends  that,  for  steel  pillars,  an  addition  not  exceeding  50  per  cent, 
should  be  made  to  safe  load  for  wrought-iron  pillars  of  same  dimensions. 

Wood.— Owe  tenth  of  ultimate  stress  is  an  accepted  limit.  Piles  have,  in 
some  situations,  borne  permanently  .2  of  their  ultimate  compressive  strength. 

Foundations. — According  to  Professor  Rankine,  maximum  pressure  on 
foundations  in  firm  earth  Is  from  17  to  23  lbs.  per  sq.  inch;  and,  on  rock,  it 
should  not  exceed  .125  of  its  crushing  load. 

Masonry.  Mr.  Stoney  asserts  that  working  load  on  rubble  masonry, 
brick- vTork,  or  concrete  rarely  exceeds  .16  of  crushing  weight  of  aggregate 
mass ; and  that  this  seems  to  be  a safe  limit.  In  an  arch,  calculated  pressure 
should  not  exceed  .05  of  crushing  pressure  of  stone. 


Eseentially  from  Manual  of  D.  K.  Clark,  l.ondou,  1877. 


782 


STRENGTH  OF  MATERIALS. DETRUSIYE. 


Ropes. — For  round,  working  load  should  not  exceed  .14  of  ultimate  strength, 
and  for  flat  .11. 

Dead  Load.  Live  Load. 


Perfect  material  and  workmanship. .....  2 

Dr.  Rankine  gives  ( Good  ordinary  material  \ ‘ t3n 

following  factors : ) and  workmanship [ Masonry  !.**.!!”  4 4 


4 

6 

8 to  :.o 
8 


A Dead  Load  is  one  that  is  laid  on  very  gradually  and  remains  fixed. 

A Live  Load  is  one  that  is  laid  on  suddenly,  as  a loaded  vehicle  or  train 
passing  swiftly  over  a bridge. 


DETRUSIYE  OR  SHEARING  STRENGTH. 

Detrusive  or  Shearing  Strength  of  any  body  is  directly  as  its  strength, 
or  thickness,  or  area  of  shearing  surface. 


Results  of  Experiments  upon  Detrusive  Strength,  of 
NEetals  -with  a Punch. 


Metals. 

Diameter 

of 

Punch. 

Thickness 

of 

Metal. 

Power 

exerted. 

Power  required  for  a 
Surface  of  Metal  of  One 
Sq. Inch. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Brass 

1 

•045 

5 448 

37000 

' 2. 

Cast  iron 

— 

— 

. t — 

30000 

^ 5*  % 

Copper j 

•5 

1 

.08 

•3 

3 983 
21  250 

30000 
22  3OO 

® ® 2 
p*  p.  3. 
s .M 

< CD  1 

Steel 

.5 

• 25 

3472° 

90  OOO 

CD  ® I 

“ Bessemer 

•875 

•75 

(103  600 

51  800 

3 02 

(184  800 

92  4OO 

si® 

( 

•5 

•17 

11  950 

45000 

g.2  m 

Wrought  iron t 

1 

.615 

82  870 

43  9°° 

CD  ^ © 

( 

2 

1.06 

297  400 

44  3°° 

7. 2 2, 

To  Compute  Power  to  Punch  Iron,  Brass,  or  Copper. 

Rule. — Multiply  product  of  diameter  of  punch  and  thickness  of  metal  by 
150000  if  for  wrought  iron,  by  128000  if  for  brass,  and  by  96000  if  for 
cast  iron  or  copper,  and  product  will  give  power  required,  in  lbs.  ; 

Example.— What  power  is  required  to  punch  a hole  .5  inch  in  diameter  in  a plate  ■ 
of  brass  ,2S  inch  thick?  ,5  x .25  X 128000  = 16000  lbs. 

Comparison  "between  Detrusive  and.  Transverse 
Strengths. 

Assuming  compression  and  abrasion  of  metal  in  application  of  a punch  of 
one  inch  in  diameter  to  extend  to  .125  of  an  inch  beyond  diameter  of  punch, 
comparative  resistance  of  wrought  iron  to  detrusive  and  transverse  strain, 
latter  estimated  at  600  lbs.  per  sq.  inch,  for  a bar  1 foot  in  length,  is  as  3 to  1. 


WOODS. 


Detrusive  Strength  of  Woods.  Per  Sq.  Inch.  & 


Lbs.  | 

Lbs.  1 

Lbs.  | 

Lbs.  < 

Spruce  

. . . 470 

Pine,  pitch. . 

• 5-io 

Ash 

Oak 

Pine,  white. . . 

. . . 490  | 

Hemlock 

• 540  1 

Chestnut. . . 

| Locust — 

To  Compute  Length  of  Surface  of  Resistance  of  Wood 
to  Horizontal  Thrust. 

Rule. — Divide  4 times  horizontal  thrust  in  lbs.  by  product  of  breadth  of 
wood  in  ins.,  and  detrusive  resistance  per  sq.  inch  in  lbs.  in  direction  of  fibre, 
and  quotient  will  give  length  required. 

Example. — Thrust  of  a rafter  is  5600  lbs.,  breadth  of  tie  beam,  of  pitch  or  Georgia 
pine,  is  6 ins. ; what  should  be  length  of  beyond  score  for  rafter? 

4 X 5600  22  400 

Assume  strength  510  as  aboye.  Then  6 x 5IQ~  = 3o6(y  = 7-32  ms> 


STRENGTH  OF  MATERIALS. — DETRUSIVE.  783 


Sh.ea.x*ing. 

Wrought  Iron. 

Resistance  to  shearing  of  American  is  about  75  per  cent.,  and  of  English 
80  per  cent.,  of  its  tensile  strength. 

Resistance  to  shearing  of  plates  and  bolts  is  not  in  a direct  ratio.  It  ap- 
proximates to  that  of  square  of  depth  of  former,  and  to  square  of  diameter 
of  latter. 

Results  of  Experiments  upon.  Shearing  Strength  of 
"Various  Metals  Toy  Barallel  Cutters. 

Wrought  Iron.— Thickness  from  .5  to  1 inch,  50000  lbs.  per  sq.  inch. 

Made  by  Inclined  Cutters , angle  = 70. 


Plates. 

Thickness. 

Power. 

Bolts. 

Diam. 

Power. 

Ins. 

•05 

.297 

.24 

•51 

I 

Lbs. 

54° 
11  196 

14  93° 
39  1 5° 
44  800 

Brass 

Ins. 
1. 11 
•775 
•775 
.32 
1. 142 

Lbs. 
29  700 
ji  310 
28  720 
3°93 
35  4i° 

Copper 

Cf  np] 

Steel 

Wrought  iron j 

Wrought  iron j 

Result  of  Experiments  in  Shearing,  made  at  \J . S.  NTavy 
Yard,  Washington,  on  Wrought-iron  Bolts. 


Diam. 

Minimum. 

Stress. 

Maximum. 

Per  Sq.Inch. 

Diam. 

Minimum. 

Stress. 

Maximum. 

Per  Sq.Inch. 

Inch. 

•5 

•75 

Lbs. 
8900 
18  400 

Lbs. 

9400 

19650 

Lbs. 
44149 
39  553 

Inch. 

•875 

1 

Lbs. 
25  500 
32  900 

Lbs. 
27  600 
35  8oo 

Lbs. 

4i  5°3 
40  708 

Mean  41 033  lbs. 


Result  of  Experiments  on  .875  Inch.  Wrought-iron 
Bolts.  (E.  Clark.) 

Lbs.  I Tons.  II  Lbs.  I Tons. 

Single  shear 54096  24.15  Double  shear  of  two  .625-inch  plates 

Double  “ 46904  I 22.1  II  riveted  together  (one  section) 45  696  | 20.4 

Tensile  strength 50 176  lbs. 

Riveted  Joints. 

Experiments  on  strength  of  riveted  joints  showed  that  while  the  plates 
were  destroyed  with  a stress  of  43  546  lbs.,  the  rivets  were  strained  by  a 
stress  of  39  088  lbs. 

Cast  Iron. 

Resistance  to  shearing  is  very  nearly  equal  to  its  tensile  strength.  An 
average  of  English  being  24  000  lbs.  per  sq.  inch. 

Steel. 

Shearing  strength  of  steel  of  all  kinds  (including  Fagersta)  is  about  72  per 
cent,  of  its  tensile  strength. 

Treenails. 

Oak  treenails,  1 to  1.75  ins.  in  diameter,  have  an  average  shearing  strength 
of  1.8  tons  per  sq.  inch,  and  in  order  to  fully  develop  their  strength,  the  planks 
into  wrhich  they  are  driven  should  be  3 times  their  diameter. 

Woods. 

When  a beam  or  any  piece  of  wrood  is  let  in  (not  mortised)  at  an  inclina- 
tion to  another  piece,  so  that  thrust  will  bear  in  direction  of  fibres  of  beam 
that  is  cut,  depth  of  cut  at  right  angles  to  fibres  should  not  be  more  than  .2 
of  length  of  piece,  fibres  of  which,  by  their  cohesion,  resist  thrust. 

Ash 650  lbs.  I Deal 625  lbs.  I Pine 650  lbs. 

Chestnut 600  “ | Oak 2300  “ | Spruce 625  “ 


STRENGTH  OF  MATERIALS. TENSILE. 


TENSILE  STRENGTH. 

Tensile  Strength  is  resistance  of  the  fibres  or  particles  of  a body  to 
separation.  It  is  therefore  proportional  to  their  number,  or  to  area  of 
its  transverse  section,  and  in  metals  it  varies  with  their  temperature, 
generally  decreasing  as  temperature  is  increased.  In  silver,  tenacity 
decreases  more  rapidly  than  temperature ; and  in  copper,  gold,  and  plat- 
inum less  rapidly. 

Cast  Iron- 

Experiments  on  Cast-iron  bars  give  a tensile  strength  of  from  4000  to 
5000  lbs.  per  sq.  inch  of  its  section,  as  just  sufficient  to  balance  elasticity  of 
the  metal;  and  as  a bar  of  it  is  extended  the  12300th  part  of  its  length  for 
every  1000  lbs.  of  direct  strain,  or  one  sixteenth  of  an  inch  in  64.06  feet  per 
sq.  inch  of  its  section,  it  is  deduced  that  its  elasticity  is  fully  excited  when 
it  is  extended  less  than  the  2400th  part  of  its  length,  and  extension  of  it  at 
its  limit  of  elasticity,  which  is  about  .5  of  its  destructive  weight,  is  esti- 
mated at  1500th  part  of  its  length. 

Average  ultimate  extension  is  500th  part  of  its  length. 

A bar  will  contract  or  expand  .000006173  inch,  or  the  162000th  of  its 
length,  for  each  degree  of  heat ; and  assuming  extreme  of  moderate  range 
of  temperature  in  this  country  140°  (—  20°  + 120°),  it  will  contract  or  ex- 
pand with  this  change  .0008642  inch,  or  the  1157th  part  of  its  length. 

It  follows,  then,  that  as  1000  lbs.  will  extend  a bar  the  12300th  part  of 
its  length,  contraction  or  extension  for  1157th  part  will  be  equivalent  to  a 
force  of  10648  lbs.  (4.75  tons)  per  sq.  inch  of  section.  It  shrinks  in  cooling 
from  one  eighty-fifth  to  one  ninety-eighth  of  its  length. 

Mean  tensile  strength  of  American,  as  determined  by  Maj.  Wade  for  U.  S. 
Ordnance  Corps,  is  31  829  lbs.  (14.21  tons)  per  sq.  inch  of  section;  mean  of 
English,  as  determined  by  Mr.  E.  Hodgkinson  for  Commission  on  Applica- 
tion of  Iron  to  Railway  Structures,  1849,  is  19484  lbs.  (8.7  tons)  ; and  by  Col, 
Wilmot,  at  Woolwich,  in  1858,  for  gun-metal,  is  23257  lbs.  (10.35  tons), 
varying  from  12320  lbs.  (5.5  tons)  to  25  520  lbs.  (10.5  tons). 

Mean  ultimate  extension  of  four  descriptions  of  English,  as  determined 
for  Commission  above  referred  to,  was,  for  lengths  of  10  feet,  .1997  inch, 
being  600th  part  of  its  length ; and  this  weight  would  compress  a bar  the 
775th  part  of  its  length. 

Tensile  strength  of  strongest  piece  ever  tested — 45  97°  lbs.  (2°*52  tons). 
This  was  a mixture  of  grades  1,  2,  and  3 from  furnace  of  Robert  P.  Parrott 
at  Greenwood,  N.  Y.,  and  at  3d  fusion. 

At  2.5  tons  per  sq.  inch  it  will  extend  same  as  wrought  iron  at  5.6  tons. 

From  experiments  of  Maj.  Wade  he  deduced  the  following  mean  results  : 

Density.  I Tensile.  I Transverse.  I Torsion.  1 Crushing.  I Hardness. 

7.225  |1  31829  1 8182  I 8614  I 144916  I 22.34 

Tensile  per  sq.  inch  of  section ; Transverse  per  sq.  inch,  one  end  fixed, 

load  applied  at  other  end  at  a distance  of  i foot ; and  Torsion  per  sq.  inch, 
stress  applied  at  end  of  a lever  i foot  in  length. 

Green  sand  castings  are  6 per  cent,  stronger  than  dry,  and  30  per  cent, 
stronger  than  chilled ; but  when  castings  are  chilled  and  annealed,  a gain  of 
1 15  per  cent,  is  attained  over  those  made  in  green  sand. 

Resistance  to  crushing  and  tensile  stress  is  for  American  as  4.55  to  1,  and 
for  English  as  5.6  to  7 to  1.  Strength  increasing  with  density. 


STRENGTH  OF  MATERIALS. TENSILE. 


785 


Remelting. — Strength,  as  well  as  density,  are  increased  by  repeated  re- 
meltings. The  increase  is  the  result  of  the  gradual  abstraction  of  the  con- 
stituent carbon  of  the  iron,  and  the  consequent  approximation  of  the  metal 
to  wrought  iron. 

Result  of  the  4th  melting  of  pig  iron,  as  determined  by  Major  Wade,  was 
to  increase  its  strength  from  12880  lbs.  (5.75  tons)  to  27888  lbs.  (12.45 
tons),  and  its  specific  gravity  from  6.9  to  7.4. 

Three  successive  meltings  of  Greenwood  iron,  N.  Y.,  gave  tensile  strength 
of  21 300,  30 100,  and  35  700  lbs. 

Result  of  5th  melting  by  Mr.  Bramwell  was  to  increase  strength  of  Acadian 
iron  from  16800  lbs.  (7.5  tons)  to  41  440  lbs.  (18.5  tons). 

Remelting  increases  its  resistance  to  a crushing  stress  from  70  to  80  tons 
(14  per  cent.)  per  sq.  inch  of  section. 

Hot  and.  Cold  Blast. 

Mr.  Hodgkinson  deduced  from  experiments  that  relative  strength  of  1.2 
and  3 ins.  square  was  as  100,  80,  and  77,  and  that  hot  blast  had  less  tensile 
strength  than  cold  blast,  but  greater  resistance  to  a crushing  stress. 

Captain  James  ascertained  that  tensile  strength  of  .75  inch  bars,  cut  out 
of  2 and  3 inch  bars,  had  only  half  strength  of  a bar  cast  1 inch  square. 

Mr.  Robert  Stephenson  concluded,  from  experiments  of  recent  date,  that 
average  strength  of  hot  blast  was  not  much  less  than  that  of  cold  blast ; but 
that  cold  blast,  or  mixtures  of  cold  blast,  were  more  regular,  and  that  mixt- 
ures of  cold  blast  and  hot  blast  were  better  than  either  separate. 

Stirling’s  IVLixed  or  Toughened  Iron. 

By  mixture  of  a portion  of  malleable  iron  with  cast  iron,  carefully  fused 
in  a crucible,  a tensile  strain  of  25  764  lbs.  has  been  attained.  This  mixt- 
ure, when  judiciously  managed  and  duly  proportioned,  increases  resistance 
of  cast  iron  about  one  third ; greatest  effect  being  obtained  with  a propor- 
tion of  about  30  per  cent,  of  malleable  iron. 

NXallea'ble  Cast  Iron. 

Tensile  strength  of  annealed  malleable  is  guaranteed  by  some  Manufact- 
urers of  it  at  56000  lbs. ; it  is  capable  of  sustaining  22400  lbs.  without  per- 
manent set. 

AWr 011  glit  Iron. 

Experiments  on  English  bars  gave  a tensile  strength  of  from  22  000  lbs. 
to  26400  lbs.  per  sq.  inch  of  its  section,  as  just  sufficient  to  balance  elasticity 
of  the  metal ; and  as  a bar  of  it  is  extended  the  28  oooth  part  of  its  length 
for  every  1000  lbs.  of  direct  strain,  or  one  sixteenth  of  an  inch  m 116.66  teet 
per  sq.  inch  of  its  section,  it  is  deduced  that  its  elasticity  is  fully  excited 
when  it  is  extended  the  1000th  part  of  its  length,  and  extension  of  it  at  its 
limit  of  elasticity,  which  is  from  .45  to  .5  of  its  destructive  weight,  is  esti- 
mated at  1520th  part  of  its  length. 

A bar  will  expand  or  contract  .000  006  614  inch,  or  151  200  part  of  its  length 
for  each  degree  of  heat ; and  assuming,  as  before  stated  for  cast  iron,  that 
extreme  range  of  temperature  in  air  in  this  country  is  140°,  it  will  contract 
or  expand  with  this  change  .000926,  or  1080th  of  its  length,  which  is  equn  a- 
lent  to  a force  of  20  740  lbs.  (9.25  tons)  per  sq.  inch  of  section. 

Mean  tensile  strength  of  American  bars  and  plates  (45000  to  76000), 
60  500  lbs.  (27  tons)  per  sq.  inch  of  section ; as  determined  by  Prof.  Johnson 
in  1836,  is  55900  lbs. ; and  mean  of  English,  as  determined  ly  Capt.  Brown, 
Barlow,  Brunei,  and  Fairbairn,  is  53  900  lbs. ; and  by  Mr.  Kirkaldy,  bars  and 
plates  (47040  to  55910)  51 475  lbs.  (22.97  tons). 

- U* 


786  STRENGTH  OF  MATERIALS. TENSILE. 

Greatest  strength  observed  73449  lbs.  (32.79  tons). 

Ultimate  strength,  as  given  by  Mr.  D.  K.  Clark,  59  732  lbs.  (26.66  tons). 

Average  ultimate  extension  is  600th  part  of  its  length. 

Strength  of  plates,  as  determined  by  Sir  William  Fairbairn,  is  fully  9 per 
cent,  greater  with  fibre  than  across  it. 

Resistance  of  wrought  iron  to  crushing  and  tensile  strains  is,  as  a mean, 
as  1.5  to  1 for  American;  and  for  English  1.2  to  1. 

Reheating. — Experiments  to  determine  results  from  repeated  heating  and 
laminating,  furnished  following : 

From  1 to  6 reheatings  and  rollings,  tensile  stress  increased  from  43  904 
lbs.  to  61  824  lbs.,  and  from  6 to  12  it  was  reduced  to  43904  again. 

Effect  of  Temperature . — Tensile  strength  at  different  temperatures  is  as 
follows:  6o°,  1;  1140,  i-Hi  2I2°,  1.2;  250°,  1.32;  270°,  1.35;  3250,  1.41 ; 
435°)  1-4- 

Experiments  of  Franklin  Institute  gave  at 

8o° 56000  lbs.  I 7200. 55  000  lbs.  I 12400 22  000  lbs. 

5700 66500  44  I 10500 32000  u I 13170 9000  “ 

Annealing. — Tensile  strength  is  reduced  fully  1 ton  per  sq.  inch  by  an- 
nealing. 

Cold  Rolling. — Bars  are  materially  stronger  than  when  hot  rolled,  strength 
being  increased  from  one  fifth  to  one  half,  and  elongation  reduced  from  21 
to  8 per  cent. 

Hammering  increases  strength  in  some  cases  to  one  fifth. 

Welding. — Strength  is  reduced  from  a range  of  3 to  44  per  cent.  20  per 
cent.,  or  one  fifth,  is  held  to  be  a fair  mean. 

Temperature. — From  o°  to  400°  strength  is  not  essentially  affected,  but  at 
high  temperature  it  is  reduced.  When  heated  to  redness  its  strength  is  re- 
duced fully  25  per  cent. 

Tensile  strength  at  230  was  found  to  be  .024  per  cent,  less  than  at  64°. 

Cutting  Screw  Threads  reduces  strength  from  n to  33  per  cent. 

Hardening  in  water,  oil,  etc.,  reduces  elongation,  but  does  not  essentially 
increase  the  strength. 

Case  Hardening  reduces  strength  fully  10  per  cent. 

Galvanizing  does  not  affect  strength  of  plates. 

Angled  Bars , etc. — Their  strength  is  fully  10  per  cent,  less  than  for  bolts 
and  plates. 


Elements  connected,  with.  Tensile  Resistance  of  various 
Substances. 


Substances. 

Stress  per  Sq. 
Inch  for  limit 
of  Elasticity. 

Ratio  of  Stress 
to  that  causing 
Rupture. 

Substances. 

Stress  per  Sq. 
Inch  for  limit 
of  Elasticity. 

Ratio  of  Stress 
to  that  causing 
Rupture. 

Lbs. 

Lbs. 

Beech 

3 355 

4 000 

. 'I 

Wrought  iron,  ordinary 

“ “ Swedish.... 

17  600 
24  400 
(18850 
(22  4OO 

15  OOO 

.3 

Cast  iron,  English 

. 22 

•34 

•35 

•35 

.26 

“ “ American 

Oak 

5000 
2 856 
52  000 

.2 

•23 

.62 

44  44  English 

Steel  plates,  .5  inch 

“ “ American. . . 

“ wire . 

757°° 

• 5 

44  wire,  No.  9,  unannealed 

47  532 

.46 

Yellow  pine 

3 332 

•23 

44  “ 44  annealed.. 

36300 

•45 

Turning. — Removing  outer  surface  does  not  reduce  the  strength  of  bolts. 


STRENGTH  OF  MATERIALS. TENSILE. 


787 


TIE-RODS. 

Results  of  Experiments  on  Tensile  Strength  ofWrought- 
iron  Tie-rods. 


Common  English  Iron,  1.1875  Ins.  in  Diameter. 


Description  of  Connection. 

1 Breaking  Weight. 

Semicircular  hook  fitted  to  a circular  and  welded  eye 

Lbs. 

Two  semicircular  hooks  hooked  together 

16220 
29  120 
48  160 
56  000 

Right-angled  hook  or  goose-neck  fitted  into  a cylindrical  eye  '■ 
Two  links  or  welded  eyes  connected  together 

Straight  rod  without  any  connective  articulation 

Ratio  of  Ductility  and  Malleability  of  Metals. 


In  order  of 
Wire-drawing 
Ductility. 

In  order  of 
Laminable 
Ductility. 

In  order  of 
Wire-drawing 
Ductility. 

In  order  of 
Laminable 
Ductility. 

In  order  of 
Wire-drawing 
Ductility. 

In  order  of 
Laminable 
Ductility. 

Gold. 

Silver. 

Platinum. 

Iron. 

Copper. 

Zinc. 

Tin. 

Lead. 

Nickel. 

Gold. 

Silver. 

Copper. 

Tin. 

Platinum. 

Lead. 

Zinc. 

Iron. 

Nickel. 

Relative  resistance  of  Wrought  Iron  and  Copper  to  tension  and  compres- 
sion is  as  100  to  54.5. 

Steel. 


Experiments  of  Mr.  Kirkaldy,  1858-61,  give  an  average  tensile  strength 
for  bars  of  134400  lbs.  (60  tons)  per  sq.  inch  for  tool-steel,  and  62720  lbs. 
(28  tons)  for  puddled.  Greatest  observed  strength  being  148  288  lbs.  (66.2 
tons).  Plates,  mean,  86800  lbs.  (32  to  45.5  tons)  with  fibre,  and  81  760  lbs. 
(36.5  tons)  across  it. 

Its  resistance  to  crushing  compared  to  tension  is  as  2.1  to  1. 

Hardening.— Its  strength  is  very  materially  increased  by  being  cooled  in 
oil,  ranging  from  12  to  55  per  cent. 

Crucible.— Experiments  by  the  Steel  Committee  of  Society  of  Civil 
Engineers,  England,  1868-70,  give  a tensile  strength  of  91  571  lbs.  per  sq. 
inch  (40.88  tons),  with  an  elongation  of  .163  per  cent.,  or  1 part  in  613,  and 
an  elastic  extension  of  .000  034  7th  part  for  every  1000  lbs.  per  sq.  inch,  or 
1 part  in  28818. 

Bessemer.— Experiments  by  same  Committee  give  a tensile  strength 
of  76653  lbs.  per  sq.  inch  (34.22  tons)  with  an  elongation  of  .144  per  cent., 
or  1 part  in  695,  and  an  elastic  extension  of  .oooo34  82d  part  for  every  1000 
lbs.  per  sq.  inch,  or  1 part  in  28  719. 

Result  of  Experiments  by  Committe  of  Society  of  Civil 
Engineers  of  England,  1868-70,  and  Mr.  Daniel  Kir- 
kaldy, 1875. 

Per  Sq.  Inch. 


Steel. 

Elastic  Strength. 

Elastic  E 
in  Parts 
of 

Length. 

ixtension 

per 

iooo^  Lbs. 

Ratio  of 
Elastic 
to 

Ultimate 

Strength. 

Destructive 

Weight. 

Crucible 

Lbs. 
49  840 
44  800 
48  608 
39  200 
32  080 
28784 

Tons. 

Per  Cent. 
.225 
.204 

In  Length. 
.0005 
.00045 

Per  Cent. 
58.2 
59 

59* 2 
5i-5 

46.4 

44.4 

Lbs. 
86  464 
75  757 
78 176 
7 2 576 
69888 
64512 

Tons. 

38-6 

Bessemer 

20 

Fagersta,  unannealed  . 

“ annealed 

Siemens,  unannealed. . 
“ annealed.... 

21.7 

17-5 

14.56 

12.85 

33- 82 

34- 9 
32-4 
31-2 
28.8 

STRENGTH  OF  MATERIALS. TENSILE, 


Average  Tensile  Elasticity-  of  Steel  Bars  and  3?lates. 
(Com.  of  Civil  Engineers , 1870.) 


Description. 


lasticity  per 
Sq. Inch. 

Elastic  Exten- 
sion in  Parts  of 
Length. 

Ratio  of  Elas- 
tic to  Destruc- 
tive Strength. 

Lbs. 

Parts. 

Per  Cent. 

50  557 

1 in  485 

58.2 

43  814 

i in  675 

55 

56  56o 

— 

64.8 

34048 

— 

55-6 

55  574 

— 

64-7 

40  858 

— 

54 

30710 

1 in  980 

59-2 

26940 

1 in  1020 

56-5 

32  5 00 

— 

46.4 

28780 

— 

44.4 

40174 

— 

58.8 

42  112 

1 in  185 

— 

Bars. 

Crucible,  hammered  and  rolled 

Bessemer,  “ “ ...  .. 

Fagersta,  rolled  

“ unannealed 

u hammered  and  rolled 

“ “ “ annealed., 

“ plates,  unannealed. 

“ “ annealed 

unannealed... 

annealed , 

“ tires 

Krupp’s  shaft 


Siemens, 


Tensile  strength  of  steel  increases  by  reheating  and  rolling  up  to  second 
operation,  but  decreases  after  that. 

Tensile  Strength,  of  Various  Materials,  deduced  from 
Experiments  of  TJ.  S.  Ordnance  Department,  Eair- 
hairn,  Hodgkinson,  KLirkaldy-,  and  "by-  the  Author. 
Power  or  Weight  required  tq,  tear  asunder  One  Sq.  Inch , in  Lbs. 

Metals.  Lbs. 

Steel,  Pittsburgh,  mean 94450 

Bessemer,  rolled - - - f 76  650 

“ hammered 


Metals.  Lbs. 

Antimony,  cast 1 053 

Bismuth,  cast 3248 

Cast  Iron,  Greenwood 45  97° 

mean,  Major  Wade. . . 31  829 

gun-metal,  mean 37  232 

malleable,  annealed. . 56000 

Eng.,  strong 29000 

“ weak 13400 

, , ( 1 c 600 

“ averages { 2J28o 

u gun-metal 23257 

“ mean* 19484 

11  Low  Moor,  No.  2 14076 
“ Clyde,  No.  1.. . . 16125 

“ “ No.  3....  23468 

u Stirling,  mean..  25764 

Copper,  wrought 34000 

rolled 36  000 

cast 24350 

bolt 36800 

wire 61  200 

Gold 20384 

Lead,  cast 1 800 

“ pipe.....:. ....... ........  2240 

“ “ encased 3 759 

“ rolled  sheet 3320 

Platinum  wire 53000 

Silver,  cast 40000 

Steel,  cast,  maximum 142000 

“ mean 88560 

puddled,  maximum 173817 

Amer.  Tool  Co 179980 

„.ivo  i 2IOOOO 

( 3OOOOO 

plates,  lengthwise 96  300 

u crosswise ....  93  700 

Chrome  bar 180000 


125  000 
152900 

“ Eng.,  cast 134000 

“ u “ plates,  mean 93500 

“ “ plates 86800 

“ “ puddled  plates 62720 

u u crubible 91 570 

“ “ homogeneous 96280 

“ “ blistered,  bars 104000 

“ “ Fagersta  bars 89600 

“ “ “ plates 98560 

“ “ Whitworth’s { 89600 

( 152000 

“ “ Siemens’s  plates. . j ^ |g° 

“ “ Krupp’s  shaft 92243 

Tin,  cast. .... f; 5000 

u Banca.. 2100 

Wire  rope,  per  lb.  w’t  per  fathom  4480 
“ “ galvanized  steel,  u 6720 

Wrought  Iron,  boiler  plates. . . j J^ooo 

“■  rivets....... 65000 

“ bolts,  mean 

“ “ inferior 30000 

“ hammered 54000 

“ shaft 44  750 

u wire 73600 

tc  u No.  9 100000 

“ “ No.  20 120000 

“ “ diam.  .0069  inch  301 168 

“ “ galv’ized  .058  “ 64960 

“ Eng.,  heavy  forging.  33600 

“ “ plates,  lengthw’e  53800 

“ “ “ crosswise  48800 


* By  Coram’fl  on  application  of  Iron  to  Railway  Structure. 


STRENGTH  OF  MATERIALS. TENSILE. 


789 


Metals.  Lbs. 


Wrought  Iron,  Eng.,  mean 51000 

Eng.,  Low  Moor 57600 

“ Lancashire 48800 

“ Thames 65921 

“ armor-plates  ....  40000 

“ bar...... (31300 

( 50000 

“ “ charcoal 63000 

“ rivet,  scrap 51  760 

Russian,  bar,  best 59500 

“ “ 49000 

Swedish,  “ best 72000 

“ “ 48900 

Zinc 3 500 


sheet. 


Alloys  or  Compositions. 


Alloy,  Cop. 60,  Iron  2,  Zinc  35, Tin  2.  85  120 

“ Tin  10,  Antimony  1 noco 

Aluminium,  Cop.  90 71  6co 

“ maximum 96320 

Bell-metal 3670 

Brass,  cast 18000 

“ wire 49000 

Bronze,  Phosphor.,  extreme 50915 

“ mean 34464 

“ ordinary . 23500 

“ Cop.  10,  Tin  1 33000 

“ “ 9,  “ 1 38080 

“ “ 8,  “ 1 36000 

“ “ 2,  Zinc  1 29000 

Gun-metal,  ordinary 18000 

mean , . 33600 

“ bars 42040 

Speculum  metal 7000 

Yellow  metal 48  700 

Woods. 


Ash,  white 14000 

“ American 9500 

“ English 16000 

Bamboo 6 300 

Bay 14000 

Beech,  English 11500 

Birch 15000 

“ Amer.,  black 7000 

Box,  African 23000 

Bullet 19000 

Cedar,  Lebanon 11400 

West  Indian 7500 

“ American 600 

Chestnut 500 

“ horse 10000 

Cypress 6 000 

Deal,  Christiana 12400 

Ebony 27000 

Elm \ 6000 

l 13000 

Gum,  blue 18000 

“ Alabama 15860 

Hackmatack 12000 

Hickory 11000 

Holly .. . 16000 

Lance ( x7  35° 

( 23000 


Woods. 

Larch  1 ... .. 

Lignum  vitae 

Locust 

Mahogany,  Honduras 

“•  Spanish.. 

Oak,  Pa. , seasoned 

“ Va.,  “ ........ 

“ white 

‘ ‘ live,  Ala 

“ red 

u African 

“ English 

“ Dantzic 

Pear 

Pine,  Ya 

“ Riga 

“ yellow' 

“ white 

“ red 

Poon 

Poplar 

Redwood,  Cal.. 

Spruce,  white 

Sycamore. 

Teak,  India 

“ African 

Walnut,  Eng 

“ black 

“ Mich 

Willow 

Yewr 

Across  Fibre. 

Oak 

Pine 


Lbs. 

( 4 200 

i 9500 

..II  800 

I 16000 
( 20500 

. . 21  OOO 

I 8000 
( 12  000 
••  20333 
. . 25  222 

. . l6  500 

. . l6  380 

. . IO  250 

. . 9 500 
J 4 500 

l 7 571 
. • 4 200 

. . 9 860 

. . 19  200 

. . 14  OOO 
. . 13 OC  O 

..11 8co 
. . 13  000 
. . 13300 
. . 7 000 
..  10833 
| 10290 
\ 12400 
( 9 660 

l *3 

. . 15000 
. . 2 1 000 
. . 7 800 

. . 16633 

■ • 17  580 
. . 13  000 
. . 8 000 

. . 2 300 

• 550 


Miscellaneous. 

Basalt,  Scotch 

Beton,  N.  Y.  Stone  Con’g  Co. . . . j 


Blue  stone 
Brick,  extreme 

inferior. 


Cement,  Portland,  7 days. 


“ pure,  1 mo 

“ sand  2,  320  days 

“ U J u 

“ pure,  •“ 

“ sand  1,  in  water 
1 mo 

“ “ 1 “ 1 y’r 

u “ 3?  1 year. . 

“ “ 5,.  1 “ 

“ “ 7,  1 “ 

Hydraulic 

Rosedale, Ulst.  Co.,  7 days 
“ sand  i,  30 

“ 9 mos 


1469 

300 

500 

77 

75o 

100 

290 

400 

860 

393 

713 

948 
1 152 

201 

3X9 

310 

214 

163 

284 

104 

102 

560 

700 


79  o 


STRENGTH  OP  MATERIALS. TORSION. 


Miscellaneous. 

Cement,  Roman,  in  water  7 days 
“ “ 1 mo., 

“ “ 1 year 

“ sand  1, 42  days. 


Flax 

Glass,  crown  . . . . 

Glue 

Granite 

Gutta  Percha’. . . , 

Hemp  rope 

Ivory 

Leather  belting. . 

Limestone 


Marble,  statuary. , 
“ Italian.., 

Marble,  white 

“ Irish 


Lbs. 

90 

«5 

286 

284 

1 99 
160 
25  000 
2546 
4000 
578 
3 5oo 
12000 
16000 
1 000 
330 
670 

2 800 

3 200 
5 200 
9000 

17  600 


Miscellaneous.  Lbs. 

Mortar,  1 year j 

“ hydraulic j 

“ ordinary 35 

Oxhide 6 300 

Rope,  Manila 9000 

“ tarredhemp 15000 

Sandstone 150 

“ fine  green 1260 

“ Arbroath j i^3 

“ Caithness { 473 

l 1054 

“ Portland . { 857 

( 1 000 

“ Craigleth 453 

Silk  fibre  . . * . . 52  000 

Slate * {,;& 

Whalebone 7000 


TORSIONAL  STRENGTH. 

SHAFTS  AND  GUDGEONS. 

Shafts  are  divided  into  Shafts  and  Spindles , according  to  their  mag- 
nitude, and  are  subjected  to  Torsion  and  Lateral  Stress  combined,  or  to 
Lateral  Stress  alone. 

A Gudgeon  is  the  metal  journal  or  Arbor  upon  which  a wooden  shaft 
revolves. 


Lateral  Stiffness  and  Strength. — Shafts  of  equal  length  have  lateral  stiff- 
ness as  their  breadth  and  cube  of  their  depth,  and  have  lateral  strength  as 
their  breadth  and  square  of  their  depths. 

Shafts  of  different  lengths  have  lateral  stiffness  directly  as  tlieir  breadth 
and  cube  of  their  depth,  and  inversely  as  cube  of  their  length ; and  have 
lateral  strength  directly  as  their  breadth  and  as  square  of  their  depth,  and 
inversely  as  their  length. 

Hollow  Shafts  having  equal  lengths  and  equal  quantities  of  material  have 
lateral  stiffness  as  square  of  their  diameter,  and  have  lateral  strength  as  their 
diameters.  Hence,  in  hollow  shafts,  one  having  twice  the  diameter  of  an- 
other will  have  four  times  the  stiffness,  and  but  double  the  strength ; and 
when  having  equal  lengths,  by  an  increase  in  diameter  they  increase  in  stiff- 
ness in  a greater  proportion  than  in  strength. 

When  a solid  shaft  is  subjected  to  torsional  stress,  its  centre  is  a neutral 
axis,  about  which  both  intensity  and  leverage  of  resistance  increase  as  radius 
or  side ; and  the  two  in  combination,  or  moment  of  resistance  per  sq.  inch,  t, 
increase  as  square  of  radius  or  side. 

Round  Shaft. — Radius  of  ring  of  resistance  is  radius  of  gyration  of  sec-  f 
tion,  being  alike  to  that  of  a circular  plate  revolving  on  its  axis,  viz.,  .7071 
radius.  The  ultimate  moment  of  resistance  then  is  expressed  by  product 
of  sectional  area  of  shaft,  by  ultimate  shearing  resistance  per  sq.  inch  of 
material  by  radius,  and  by  .7071. 


Or,  .7854  d2  r S X -7071  = .278  d3  S = Rf.  (D.  K.  Clark.) 
d representing  diameter  of  shaft  and  r radius,  S ultimate  shearing  stress  of  mate- 
rial in  lbs.  per  sq.  inch,  R radius  through  which  stress  is  applied , in  ins. , and  W 
moment  of  load  or  destructive  stress,  in  lbs. 

.278  d3  S _ R W _ _ , /R  W 

— 5 — =W;  ^5  = S;  an,Jv/^_X  1,534  = - 


Hence, 


STRENGTH  OF  MATERIALS. — TORSION. 


791 


Round  Shaft. — Strength,  compared  to  a square  of  equal  sectional,  area, 
is  about  as  1 to  .85.  Diameter  of  a round  section,  compared  to  side  of 
square  section  of  equal  resistance,  is  as  1 to  .96. 

Square  Shaft. — Moment  of  torsional  resistance  of  a square  shaft  exceeds 
that  of  a round  of  same  sectional  area,  in  consequence  of  projection  of  cor- 
ners of  square ; but  inasmuch  as  material  is  less  disposed  to  resist  torsional 
stress,  the  resistance  of  a square  shaft,  compared  to  a round  one  of  like  area 
of  section,  is  as  1 to  1.18,  and  of  like  side  and  diameter,  as  1.08  to  1. 

Hence,  -^78  X n°8  S = w HoUwo  Round  Shafts.  '■*?■ S = W. 

When  Section  is  comparatively  Thin.  x' 57  ^ = W.  s representing  side , 
d and  c V external  and  internal  diameters , and  t thickness  of  metal  in  ins. 

Torsional  Angle  of  a bar,  etc.,  under  equal  stress,  will  vary  as  its  length. 
Hence,  torsional  strength  of  bars  of  like  diameters  is  inversely  as  their 
lengths. 

Stress  upon  a shaft  from  a weight  upon  it  is  proportional  to  product  of  the  parts 
of  shaft  multiplied  into  each  other.  Thus,  if  a shaft  is  10  feet  in  length,  and  a weight 
upon  centre  of  gravity  of  the  stress  is  at  a point  2 feet  from  one  end,  the  parts  2 
and  8,  multiplied  together,  are  equal  to  16;  but  if  weight  or  stress  were  applied  in 
middle  of  the  shaft,  parts  5 and  5,  multiplied  together,  would  produce  25. 

When  load  upon  a shaft  is  uniformly  distributed  over  any  part  of  it,  it  is  consid- 
ered as  united  in  middle  of  that  part;  and  if  load  is  not  uniformly  distributed,  it  is 
considered  as  united  at  its  centre  of  gravity. 

Deflection  of  a shaft  produced  by  a load  which  is  uniformly  distributed  over  its 
length  is  same  as  when  .625  of  load  is  applied  at  middle  of  its  length. 

Resistance  of  body  of  a shaft  to  lateral  stress  is  as  its  breadth  and  square 
of  its  depth;  hence  diameter  will  be  as  product  of  length  of  it , and  length 
of  it  on  one  side  of  a given  point , less  square  of  that  length. 

Illustration. — Length  of  a shaft  between  centres  of  its  journals  is  10  feet;  what 
should  be  relative  cubes  of  its  diameters  when  load  is  applied  at  1,  2,  and  5 feet 
from  one  end?  and  what  when  load  is  uniformly  distributed  over  length  of  it? 
lX  l1  — I3  = d3]  and  when  uniformly  distributed,  d3-r-2  — dx. 

10  X 1 = 10  — 1 2 — 9 — cube  of  diameter  at  1 foot ; 10X2  = 20  — 22  = 16  ==  cube 
of  diameter  at  2 feet ; 10  X 5 = 50  — 5 2 = 25  = cube  of  diameter  at  5 feet. 

When  a load  is  uniformly  distributed,  stress  is  greatest  at  middle  of  length,  and 
is  equal  to  half  of  it;  25  -f-  2 = 12. 5 — cube  of  diameter  at  5 feet. 

Torsional  Strength  of  any  square  bar  or  beam  is  as  cube  of  its  side,  and 
of  a cylinder  as  cube  of  its  diameter.  Hollow  cylinders  or  shafts  have  great- 
er torsional  strength  than  solid  ones  containing  same  volume  of  material. 

To  Compute  Diameter  of*  a Solid.  Shaft  of*  Cast  or 
Wrought  Iron  to  Tfcesist  Lateral  Stress  alone. 

When  Stress  is  in  or  near  Middle.  Rule. — Multiply  weight  by  length  of 
shaft  in  feet ; divide  product  by  500  for  cast  iron  and  560  for  wrought  iron, 
and  cube  root  of  quotient  will  give  diameter  in  ins. 

Example. — Weight  of  a water-wheel  upon  a cast-iron  shaft  is  50000  lbs.,  its  length 
30  feet,  and  centre  of  stress  of  wheel  7 feet  from  one  end ; what  should  be  diameter 
of  its  body  ? 

3J  ^5° 000  X 3°^  _ I4  42  ins  } weight  was  in  middle  of  its  length. 

Hence  diameter  at  7 feet  from  one  end  will  be,  as  by  preceding  Buie,  30X7  — 
72  — 161  = relative  cube  of  diameter  at  7 feet ; 30  X 15  — if2 — 22s  — relative  cube 
of  diameter  at  15  feet,  or  at  middle  of  its  length. 

Then,  as  ^225  : 14.42  ::  -^161  : 12.91  ins.,  diameter  of  shaft  at  7 feet  from  one  end. 


79  2 


STRENGTH  OF  MATERIALS. TORSION. 


For  Bronze,  420;  Cast  steel,  1000  to  1500;  and  Puddled  steel,  500. 

When  Stress  is  uniformly  laid  along  Length  of  Shaft.  Rule.  — Divide 
cube  root  of  product  of  weight  and  length  by  9.3  for  Cast  iron  and  10.6 
for  Wrought  iron,  and  quotient  will  give  diameter  in  ins. 


For  Bronze,  8.5  ; Cast  steel,  18.6  to  27.9 ; and  Puddled  steel,  9.3. 

When  Diameter  for  Stress  applied  in  Middle  is  given . Rule. — Take  cube 
root  of  .625  of  cube  of  diameter,  and  this  root  will  give  diameter  required. 

Example. — Diameter  of  a shaft  when  stress  is  uniformly  applied  along  its  length 
is  14.42  ins. ; what  should  be  its  diameter,  stress  being  applied  in  middle? 


To  Compute  Diameter  of*  a,  Solid.  Shaft  of  Cast  Iron  to 
Pfcesist  its  'Weight  alone. 

Rule. — Multiply  cube  of  its  length  by  .007,  and  square  root  of  product 
will  give  diameter  in  ins. 

Example. — Length  of  a shaft  is  30  feet;  what  should  be  its  diameter  in  body? 
V(3°3  X .007)  = ^189  — ,3.75  ins 


To  Compute  Diameter  of  a Hollow  Shaft  of  Cast  Iron 
to  Sustain  its  Load  in  Addition  to  its  Weight. 

When  Stress  is  in  or  near  Middle.  Rule. — Divide  continued  product  of 
.012  times  cube  of  length,  and  number  of  times  weight  of  shaft  in  lbs.,  by 
square  of  internal  diameter  added  to  1,  and  twice  square  root  of  quotient 
added  to  internal  diameter  will  give  whole  diameter  in  ins. 

Example. — Weight  of  a water-wheel  upon  a hollow  shaft  30  feet  in  length  is  2.5 
times  its  own  weight,  and  internal  diameter  is  9 ins. ; what  should  be  whole  diam- 
eter of  shaft? 


To  Compute  Diameter  of  a Ffcoxxnd  or  Square  Shaft  to 
Ptesist  Combined  Stress  of  Torsion  and  'Weight. 

Rule. — Multiply  extreme  of  pressure  upon  crank-pin,  or  at  pitch-line  of 
pinion,  or  at  centre  of  effect  upon  the  blades  of  a water-wheel,  etc.,  that  a 
shaft  may  at  any  time  be  subjected  to ; by  length  of  crank  or  radius  of 
wheel,  etc.,  in  feet ; divide  the  product  by  Coefficient  in  following  Table,  and 
cube  root  of  quotient  will  give  diameter  of  shaft  or  its  journal  in  ins. 


Example. — What  should  be  diameter  for  journal  of  a wrought-iron  water-wheel 
shaft,  extreme  pressure  upon  crank-pin  being  59  400  lbs.,  and  crank  5 feet  in  length  ? 


When  Two  Shafts  are  used , as  in  Steam-vessels , etc.,  with  One  Engine . 
Rule. — Divide  three  times  cube  of  diameter  for  one  shaft  by  four,  and 
cube  root  of  quotient  will  give  diameter  of  shaft  in  ins. 


Example. — Area  of  journal  of  a shaft  is  113  ins. ; what  should  be  diameter,  two 
shafts  being  used? 


Example. —Apply  rule  to  preceding  case. 


v 50  000  x 30 
9-3 


— = 12.31  ins. 


V.625  x 14.42  3 = -^.625  X 3000  — 12.33  ins. 


HOLLOW  SHAFTS. 


2 


A 


'.012  X 303  X 2.  5' 
V * + 92  > 


:)+9=:2\/1 


6.28  ms. , and  6. 28  -f-  9 = 15. 28  ins. 


C = 120. 


' 59  4oo  X 5 


12° 


2475,  and  V2475  = *3-53 


STRENGTH  OF  MATERIALS. TORSION.  793 


Torsional  Strength,  of  Various  NLetals. 

[Maj.  Wm.  Wade , U.  S.  Ordnance  Corps , 1851,  Steel  Committee  [ England , 1868],  and 
Stevens  Institute , N.  J.,  1878.) 

Reduced  to  a Uniform  Measure  of  One  Inch  in  Diameter  or  Side- 
Stress  applied  at  One  Foot  from  Axis  of  Body  and  at  Face  of  Axis. 

„ „ . t C d.a 


Destructive  Stress 

Bars  and  Metals. 

Tensile 

Strength. 

at 

25  Ins. 

Computed 

at 

12  Ins. 

Cast  Iron. 

Lbs. 

Lbs. 

Lbs. 

I 

Area  1 sq.  inch  ) 

45  000 

520 

1082 

Area  2. 97  sq.  ins.  ) 

ll 

3800 

7904 

*7^.  Diam.  1 Least . . . 
IP  1 =1.9  [Mean... 
ins.  ) Greatest. 

§111  Side  1 inch ) 

111  Area  1 sq.  inch  j 

9000 
31  829 
45000 
u 

1550 

2145 

2840 

350 

3664 

4462 

59°7 

728 

Wrought  Iron. 

Diam.  f Least  . . . 

Wm  =1.9 'Mean 

ins.  ( Greatest. 
Area  2. 83  sq.  ins. 

38027 
56  300 
74  592 

1250 

1375 

1500 

2600 

2860 

3120 

Bronze. 

u Diam.=  ( Least 

1.9  ins.  (Greatest. 
Area  2.83  sq.  ins. 

17  698 
56  786 

500 

650 

1040 

1352 

Cast  Steel. 

tt  Diam.=  (Least 

1.9  ins.  (Greatest. 
Area  2.83  sq.  ins. 

42  000 
128000 

2600 

7760 

5408 

16140 

Bessemer  Steel. 

u Diam.  = 1.382  ins. ) 
Area  1.5  sq.  ins.  j 

36  960 

1568 

3261 

492 


530 

650 

850 

728 


152 

197 


788 

2353 


1236 


245 


115 


235 


230 


80 


105 


To  Compute  Diameter  of  Shafts  of  Oak  and.  Pine. 

Multiply  diameter  ascertained  for  Cast  Iron  as  follows:  Oak  by  1.83, 
Yellow  Pine  by  1.716. 

Metals  and  Woods. 

Ultimate  Torsional  Strength. — Of  Cast  Iron  may  be  taken  as  equal  to  its 
transverse  strength  for  American  and  .9  for  English,  or  as  .26  of  its  tensile 
strength  for  American  and  .23  for  English.  Of  Wrought  Iron,  as  .7  to  .8  of 
its  transverse  strength  for  American  and  .7  to  1 for  English,  and  of  Steel,  as 
.72  of  its  tensile  strength. 

Elastic  Torsional  Strength. — Of  Cast  Iron  may  be  taken  as  equal  to  its 
transverse  strength,  of  Wrought  Iron  40  per  cent,  of  its  ultimate  torsional 
strength,  of  Steel  44  per  cent,  of  its  tensile  strength,  and  45  per  cent,  of  its 
ultimate  torsional  strength. 

# Bessemer  Steel. — Has  a torsional  strength  of  6670  lbs.  per  sq.  inch  at  a ra- 
dius of  one  foot,  being  somewhat  less  than  that  of  Cast  Iron,  Fagersta  has  50 
per  cent,  of  its  ultimate  transverse  strength,  and  Siemens  44.5  per  cent,  of 
its  ultimate  tensile.  _ 

3X 


794 


STRENGTH  OF  MATERIALS.— TORSION. 


Note.— Examples  here  given  are  deduced  from  instances  of  successful  practice* 
where  diameter  has  been  less,  fracture  has  almost  uuiversally  taken  place  stress 
being  increased  beyond  ordinary  limit.  * ’ 

2.— When  shafts  of  less  diameter  than  12  ins.  are  required,  Coefficients  here  given 
may  be  slightly  reduced  or  increased,  according  to  quality  of  the  metal  and  diame- 
ter of  shaft;  but  when  they  exceed  this  diameter,  Coefficients  may  not  be  increased 
as  strength  of  a shaft  decreases  very  materially  as  its  diameter  increases. 

Order  of  shafts,  with  reference  to  degree  of  torsional  stress  to  which  they 
may  be  subjected,  is  as  follows : J 

1.  Fly-wheel.  | 2.  Water-wheel.  | 3.  Secondary  shaft.  J 4.  Tertiary,  etc. 

Hence,  diameters  of  their  journals  may  be  reduced  in  this  order. 

To  Compute  Diameter  of  a Wrought-iron  Centre  Shaft 
for  connecting  Two  Engines  at  a Right  Angle. 

Conditions  of  such  a shaft  are  as  follows : 

Greatest  stress  that  it  is  subjected  to  is  when  leading  engine  is  at  ,75  of 
its  stroke,  and  following  engine  .25  of  its  stroke ; hence,  position  of  each 
crank  is  as  sin.  220  30  x 2 = .70?1  of  length  of  crank  or  radius  of  power. 

Consequently,  ■ = d.  P representing  extreme  pressure  on  piston. 

Note.— In  computing  P it  is  necessary  to  take  very  extreme  pressure  that  piston 
may  be  subjected  to,  however  short  the  period  of  time.  Average  pressure  does  not 
meet  requirement  of  case. 

Illustration. — Extreme  pressure  upon  each  piston  of  two  engines  connected  at 
a right  angle  was  hi  592  lbs.,  and  stroke  of  pistons  10  feet;  what  should  have  been 
diameter  of  centre  shaft  ? and  what  of  each  wheel  or  driving  shaft? 

3 / fill  592  X 2 x .707  „ /788q55  „ _ . 

V \ ^ — 7 = y — 125  =18.48  ms.  centre  shaft. 

For  ordinary  mill  purposes,  driving  shafts  should  be  as  cube  roots  of  2^  of  ? 
times  cube  of  centre  shaft.  , 0 Oow  J 

Thus  ^11^  = ,6.79  »s. 

To  Compute  Torsional  Strength  of  Hollow  Shafts  and 
Cylinders. 

Rule.— From  fourth  power  of  exterior  diameter  subtract  fourth  power  of 
interior  diameter,  and  multiply  remainder  by  Coefficient  of  material ; divide 
this  product  by  product  of  exterior  diameter  and  length  or  distance  from  axis 
at  which  stress  is  applied  in  feet,  and  quotient  will  give  resistance  in  lbs. 


d±—d'*C  n 
0r’  - =R- 

Example. —What  torsional  stress  may  be  borne  by  a hollow  cast-iron  shaft,  hav- 
ing diameters  of  3 and  2 ins.,  power  being  applied  at  one  foot  from  its  axis? 

C = 130.  34  — 2*  X 130  = 8450,  which -f- 3 X 1 = = 2816.6  lbs. 

To  Compute  Torsional  Strength  of  Round  and  Square 
Shafts. 


Rule.— Multiply  Coefficient  in  preceding  Table  by  cube  of  side  -or  of 
diameter  of  shaft,  etc.,  and  divide  product  by  distance  from  axis  at  which 
stress  is  applied  in  feet ; quotient  will  give  resistance  in  lbs. 

Illustration; — What  torsional  stress  may  be  borne  by  a cast-iron  shaft  of  best 
material,  2 ins.  in  diameter,  power  applied  at  2 feet  from  its  axis. 


C from  table  = 130. 


130  X ?3  1040 


= 520  lbs. 


For  steamers,  when  from  heeling  of  vessel  or  roughness  of  sea  the  stress  may  be 
confined  to  one  wheel  alone,  diameter  of  journal  of  its  shaft  should  be  equal  to 
that  of  centre  shaft. 


STRENGTH  OF  MATERIALS. — TORSION. 


795 


GUDGEONS. 


'Po  Compute  Diameter  of  a Single  Gudgeon  of  Cast 
Iron,  to  Support  a given  Weiglit  or  Stress. 

Rule.— Divide  square  root  of  weight  in  lbs.  by  25  for  Cast  iron,  and  26 
for  Wrought  iron,  and  quotient  will  give  diameter  in  ins. 

Example.— Weight  upon  a gudgeon  of  a cast-iron  water-wheel  shaft  is  62  500  lbs. ; 
what  should  be  its  diameter  ? 


To  Compute  Diameter  of  Two  Gudgeons  of  Cast  Iron, 
to  Support  a given  Stress  or  "W eight. 

Rule.— Proceed  as  for  two  shafts,  page  792. 

To  Compute  XJltimate  Torsional  Strength,  of  Hound,  and 
Square  Shafts.  (D.  K.  Clark.) 


S representing  ultimate  shearing  strength , and  W moment  of  load,  both  in  lbs.,  s side 
of  square  shaft,  and  R radius  of  stress,  both  in  ins. 

Illustration.  — What  is  ultimate  torsional  strength  of  a round  cast-iron  shaft 
4 ins.  in  diameter,  stress  applied  at  5 feet  from  its  axis  ? 


By  experiments  of  Major  Wade,  ordinary  foundry  iron  has  a torsional  strength 
of  7725  lbs.,  or  644  lbs.  per  sq.  inch  at  radius  of  one  foot. 


When  Torsional  Strength  per  sq.  inch  for  radius  of  1 inch  is  ascertained, 
substitute  C for  .278,  .4,  .2224,  or  .32.  n . v 

Stress  which  will  give  a bar  a permanent  set  of  .5°  is  about  .7  of  that 
which  will  break  it,  and  this  proportion  is  quite  uniform,  even  when  strength 
of  material  mav  vary  essentially. 

Wrought  Iron,  compared  with  Cast  Iron,  has  equal  strength  under  a stress 
which  does  not  produce  a permanent  set,  but  this  set  commences  under  a less 
force  in  wrought  iron  than  cast,  and  progresses  more  rapidly  thereafter. 
Strongest  bar  of  wrought  iron  acquired  a permanent  set  under  a less  strain 
than  a cast-iron  bar  of  lowest  grade. 

Strongest  bars  give  longest  fractures. 

.2  d3  S When  S is  not  known,  substitute  for 

Steel.  Round.  ^ — w.  g 72  s=z  y2  per  cent.  of  tensile  strength. 

Torsional  Strength  of  Cast  Steel  is  from  2 to  3 times  that  of  Cast  Iron. 

Following  rules  are  purposed  to  apply  in  all  instances  to  diameters  of 
journals  of°shafts,  or  to  diameter  or  side  of  bearings  of  beams,  etc.,  where 
length  of  journal  or  distance  upon  which  strain  bears  does  not  greatly  ex- 
ceed diameter  of  journal  or  side  of  beam,  etc. ; hence,  when  length  or  distance 
is  greatly  increased,  diameter  or  side  must  be  correspondingly  increased. 

Coefficients  for  torsional  breaking  stress  of  Iron,  Bronze,  and  Steel,  as  de- 
termined by  Major  Wade,  are : Wrought  Iron,  640 ; Cast  Iron,  560 ; Bronze, 
460 ; Cast  Steel,  1 120  to  1680.  Puddled  Steel  does  not  differ  essentially  from 
that  of  cast  iron. 


^62  500 250 

25  — 25 


— “ 10  ins. 


Cast  Iron.  Round. 
Square.  — = W,  and  1. 


Hollow. 


.278  ( d 4 — d'*)  S 


R d 


Assume  S = 20000  lbs. 


Thus,  take  preceding  illustration.  Then  ?723  x 4_  __  g24Q 
v 5 x 12 


•30  S 3 s 

= W.  Square.  — = W. 


STRENGTH  OF  MATERIALS. TORSION. 


F jrmulas  for  Minimum  and  Maximum  Diam.  of  Wrouglit-iron  Shafts. 
(A.  E.  Seaton , London , 1883,  and  Board  of  Trade,  Eng.) 

Compound  Engines . \J^ 


1 d2  ] 


S — diameter.  D and  d representing  diam- 
eter of  low  and  high  pressure  cylinders , and  S half  stroke,  all  in  ins.,  p pressure  of 
steam  m boiler , in  lbs.  per  so.  inch,  and  C a rnpifirieni.  ns  • 


Angle 

of 

Crank. 

Shai 

Crank. 

fts. 

Pro- 

peller. 

Angle 

of 

Crank. 

Shafts.  I 

Crank.  I Pro- 
peller. 

Angle 

Crank. 

Shai 

Crank. 

?t8.  I 

Pro- 
peller. 

Angle 

of 

Crank. 

Shai 

Crank. 

fts. 

Pro- 

peller. 

O 

O 

OS 

(2468 

(4000 

2880 

5400 

IOO° 

| 2279  2659 

( 4000  1 5400  | 

IlO° 

i 2I3I, 
1 4000 

2487 
5400  1 

120° 

| 2016 
(4000 

2352 

5400 

/iH? 

y — C = diameter.  A.  E.  Seaton,  London,  1883. 

Side-wheel  Engines,  Sea  Service.—  One  cylinder  crank  journal,  C = 8o;  outboard 
100;  Two  cylinder  crank  journal  50;  outboard  65 ; and  centre  shaft  58. 

Propeller  Engines.— One  cylinder  crank  journal  150:  Tunnel  130:  Two  cylinder 
compound  crank  130;  Tunnel  no;  Two  cranks,  crank  100;  Tunnel  85;  Three  cranks, 
crank  90;  and  Tunnel  78.  ’ 

River  Service.—  C may  be  reduced  one  fifth. 

Illustration. — With  a compound  propeller  engine,  steam  cylinders  20  and  40 
ins.  in  diameter,  by  40  ins.  stroke,  operating  under  a pressure  of  80  lbs.  steam 
(mercurial  gauge),  what  should  be  the  diameter  of  the  shafts  of  wrought  iron? 

a /202  X 80 —J—  40 2 X 15  . . „ A6000  _ 

V ^ X 40  = y X 40  = 8.24  ms.  crank  shaft ; 


4000 

and 


, /56000 

V ^5400"  X 4°  = 7- 46  ms.  propeller  shaft. 


«T ournals  of  Shafts,  etc. 

Journals  or  bearings  of  shafts  should  be  proportioned  with  reference  to 
pressure  or  load  to  be  sustained  by  the  journal.  Simplest  measure  of  bear- 
ing capacity  of  a journal  is  product  of  its  length  by  its  diameter,  in  sq.  ins.  • 
and  axial  area  or  section  thus  obtained,  multiplied  by  a coefficient  of  pressure 
per  sq.  inch,  will  give  bearing  capacity. 

Sir  William  Fairbairn  and  Mr.  Box  give  instances  of  weights  on  bearings  of 
shafts  etc.  from  which  following  deductions  are  made,  showing  pressure  per  sq 
inch  of  axial  section  of  journal : . F H 

Crank  pins,  687  to  1150  lbs.  per  sq.  inch. 

Link  bearings,  456  to  690  lbs.  per  sq.  inch. 
axTa? Sea6  ^ bearings’  as  a &eneral  ™le,  should  not  exceed  750  lbs.  per  sq.  inch  of 

Length  of  Journals  should  be  1.12  to  1.5  times  diameter. 

Journals  of  Locomotives  or  Like  Axles  are  usually  made  twice  diameter,  and  to 
sustain  a pressure  of  300  lbs.  per  sq.  inch  of  axial  area,  or  10  sq.  ins.  per  ton  of  load. 

Solid.  Cylindrical  Couplings  or  Sleeves. 

*?’  3 7^7“  hi  = ^ ’ .25  d-}-.i2  — 1c.  d representing  diameter , 

ZnhnYjffX  °f  l!e^  °S  top  °r  scarf  of  shaft,  k breadth  of  key,  its  depth  be- 
ing half  its  bieadth,  and  D diameter  of  coupling  or  sleeve , all  in  ins. 

Flanged  Couplings. 

d+V3;5  d=D;  3 d-}-  1 = F;  .3df-.4z=l;  d- fi=L;  Z-h4  — s.  D repre- 
sentingdiameter  of  body  of  coupling  F diameter  of  flanges,  l thickness  of  both  flanges, 
L length  of  each  coupling , s projection  of  end  of  one  shaft  and  retrocession  of  other 
from  centre  of  coupling,  and  d diameter  of  shaft,  all  in  ins. 

Supports  for  Shafts.  ( Molesworth .) 

- L-  L representing  distance  of  supports  apart,  in  feet. 


STRENGTH  OF  MATERIALS. TORSION, 


797 


To  Resist  Lateral  Stress.  = D-  W representing  weight  or  pressure 

at  centre  of  length  in  lbs.,  and  D diameter  or  side , if  square,  in  ins. 

Value  of  C. Wrought  Iron,  560;  Cast  Iron,  500;  Cast  Steel,  1000  to  1500;  Bronze, 

420;  and  Wood,  40.  When  Weight  is  distributed  put  2 C. 

Values  of  C for  Shafting  of  Various  Metals , as  observed  by  different 

16  W r 


Authorities , and  deduced  from  Formulas  of  Navier. 

Ultimate  Resistance. 


- = C. 


Metal. 

! c 

Metal. 

c 

Metal. 

C 

Wrought  Iron. 

Cast  Iron. 

Steel. 

American,  Pembe,  Me. 

61673 

( 

36846 

American,  Conn. . 

82  926 

Ulster 

61 815 

American,  mean  { 

3830° 

“ Spindle 

102  131 

“ mean 

66  436 

l 

42  821 

“ Nash.  I.  Co. 

95213 

English,  refined 

Swedish 

49!48 

“ 18  trials 

44  957 

English,  Shear 

hi  191 

54  585 
61  909 

English,  mean. . | 

22  132 
38217 

Bessemer j 

73060 

79662 

16  W R 

7T  <23 


HVIill  and.  Factory  Shafts.  ( J . B.  Francis.) 
Cylindrical.  Square. 

3V2WR 


= T. 


- d 3 T 
16  R 


= W. 


/s3  T \ 

=T-  hr^3)^2 


_W. 


Mean  value  of  T. 


Cast  Iron 


{22000 
65000 

mean 35000 

“ Eng.  30000 


Wrought  Iron { 49  000 

° ( 94000 

“ mean 50000 

“ “ Eng.  45000 


Steel 


\ 76000 
( in  000 

mean 86000 

“ Bessemer  78000 


Illustration.— What  is  the  ultimate  or  destructive  weights  that  may  be  borne 
by  a Round  Cast-iron  shaft  2 ins.  in  diameter,  and  by  a Square  shaft  1.75  ins.  side, 
stress  applied  at  25  ins.  from  axis?  Assume  T = 36000. 

Round.  Square. 

3.1416X23X36000  /1.753  X 36000  . 

16x25  y:>  V 25 

Their  lengths  should  be  reduced,  and  diameter  increased,  in  following  cases : 
1st.  At  high  velocities,  to  admit  of  increased  diameter  of  journals,  thereby 
rendering  them  less  liable  to  heating.  2d.  As  they  approach  extremity  of  a 
line  of  shafting.  3d.  Attachment  of  intermediate  pulleys  or  gearing. 


■4-3^  4- ^2  = 1837.8  lbs. 


Prime  Movers  of  Power. 


Wrought 


Iron, 


Cast 

Iron, 


ght  3 j* 

;.  V 


> JIP 


Transmitters  of  Power. 

- d,  and  .02  nd3  = IEP. 
= d,  and  .032  nd*  = IIP. 


= d,  and  .01  wd3  = IIP. 

Steel.  ^/6“-„IIP  = <*,  and  .016  n d»  = IBP. 

3 /167IIP  _ , and  oo6nd3  = up.  , /83.5  IEP  = d and  0,2  n d*  = jip 
. V n V n 

IIP  representing  horse-power  transmitted , n number  of  revolutions,  and  d diameter 
of  shaft  in  ins. 

Illustration  1.— What  should  be  diameter  of  a wrought-irom  shaft,  to  simply 
transmit  128  IP  at  100  revolutions  per  minute? 

3 /50XJ28  _ 3 /6400  _ {ns 

V 100  100 

2.— What  BP  will  a steel  shaft  of  4 ins.  diameter  transmit  at  100  revolutions  per 
minute? 

.032  X 100  X 4 3 = 204. 8 horses , 


79$  STRENGTH  OF  MATERIALS. — TRANSVERSE. 


TRANSVERSE  STRENGTH. 

Transverse  or  Lateral  Strength  of  any  Bar , Beam , Rod,  etc.,  is  in  propor- 
tion to  product  of  its  breadth  and  square  of  its  depth;  in  like-sided  bars, 
beams,  etc.,  it  is  as  cube  of  side,  and  in  cylinders  as  cube  of  diameter  of 
section. 

When  One  End  is  Fixed  and  the  Other  Projecting , strength  is  inversely  as 
distance  of  weight  from  section  acted  upon  ; and  stress  upon  any  section  is 
directly  as  distance  of  weight  from  that  section. 

When  Both  Ends  are  Supported  only,  strength  is  4 times  greater  for  an 
equal  length,  when  weight  is  applied  in  middle  between  supports,  than  if  one 
end  only  is  fixed. 

When  Both  Ends  are  Fixed , strength  is  6 times  greater  for  an  equal  length, 
when  weight  is  applied  in  middle,  than  if  one  end  only  is  fixed. 

When  Ends  Rest  merely  upon  Two  Supports,  compared  to  one  When  Ends  are 
Fixed , strength  of  any  bar,  beam,  etc.,  to  support  a weight  in  centre  of  it,  is 
as  2 to  3. 

When  Weight  or  Stress  is  Uniformly  Distributed,  weight  or  stress  that  can 
be  supported,  compared  with  that  when  weight  or  stress  is  applied  at  one  end 
or  in  middle  between  supports,  is  as  2 to  1. 

HVTetals. 

In  Metals,  less  dimension  of  side  of  a beam,  etc.,  or  diameter  of  a cylinder, 
greater  its  proportionate  transverse  strength,  in  consequence  of  their  having 
a greater  proportion  of  chilled  or  hammered  surface,  compared  to  their  ele- 
ments of  strength,  resulting  from  dimensions  alone. 

Strength  of  a Cylinder , compared  to  a Square  of  like  diameter  or  sides,  is 
as  5.5  to  8.  Strength  of  a Hollow  Cylinder  to  that  of  a Solid  Cylinder,  of 
same  area  of  section,  is  about  as  1.65  to  1,  depending  essentially  upon  the 
proportionate  thickness  of  metal  compared  to  diameter. 

Strength  of  an  Equilateral  Triangle,  Fixed  at  One  End  and  Loaded  at  the 
Other,  having  an  edge  up,  compared  to  a Square  of  the  same  area,  is  as  22  to 
27 ; and  strength  of  one,  having  an  edge  down,  compared  to  one  with  an  edge 
up,  is  as  10  to  7. 

Note.— In  Barlow  and  other  authors  the  comparison  in  this  case  is  made  when 
the  beam,  etc.,  rested  upon  supports.  Hence  the  stress  is  contrariwise. 

Strongest  rectangular  bar  or  beam  that  can  be  cut  out  of  a cylinder  is  one 
of  which  the  squares  of  breadth  and  depth  of  it,  and  diameter  of  the  cylinder, 
are  as  1,  2,  and  3 respectively. 

, Oast  Iron. 

Mean  transverse  strength  of  American,  as  determined  by  Major  Wade,  is 
681  lbs.  per.  sq.  inch,  suspended  from  a bar  fixed  at  one  end  and  loaded  at 
the  other ; and  mean  of  English,  as  determined  by  Eairbairn,  Barlow,  and 
others,  is  500  lbs. 

Experiments  upon  bars  of  cast  iron,  1,  2,  and  3 ins.  square,  give  a result 
of  transverse  strength  of  447,  348,  and  338  lbs.  respectively ; being  in  the 
ratio  of  1,  .78,  and  .756. 

Woods. 

Beams  of  wood,  when  laid  with  their  annular  layers  vertical,  are  stronger 
than  when  they  are  laid  horizontal,  in  the  proportion  of  8 to  7. 

Relative  Stiffness  of  ^Materials  to  Resist  a Transverse 

Stress. 

Ash 089  I Cast  Iron 1 I Oak 095  I Wrought  iron  1.3 

Beech 073  | Elm 073  | White  pine. ..  .1  | Yellow  pine..  .087 


STRENGTH  OF  MATERIALS. — TRANSVERSE.  799 


Strength  of  a Rectangular  Beam  in  an  Inclined  position,  to  resist  a vertical 
stress,  is  to  its  strength  in  a horizontal  position,  as  square  of  radius  to  square 
of  cosine  of  elevation ; that  is,  as  square  of  length  of  beam  to  square  of  dis- 
tance between  its  points  of  support,  measured  upon  a horizontal  plane. 


Transverse  Strength,  of  Various  [Materials. 

[U  S.  Ordnance  Department , Hodglcinson , Fairbairn , KirJcaldy,  and  by  the  Author.) 

Power  reduced  to  uniform,  Measure  of  One  Inch  Square , and  One  Foot  in  Length ; 
Weight  suspended  from  one  End. 


Metals. 

Brass 

Cast  Iron,  mean  of  4 grades 

“ “ (Maj.  Wade) 

“ ordinary 

u extreme,  West  P’t  F’dry 

“ gun-metal,*  “ “ 

“ Eng.,  Low  Moor, cold  blast. 

“ u Ponkey,  “ 

“ “ Ystalyfera  \l 

“ “ mean,  65  kinds 

“ “ “ 15  kinds, coldblast 

“ u planed  bar 

“ “ rough  bar 

Copper 

Steel,  hammered,  mean 

“ cast,  soft 

“ “ hard 

“ hematite,  hammered 

“ Krupp’s  shaft 

“ Fagersta,  hammered 

Wrought  Iron,  mean 

“ u English 

“ 11  Swedish  t 


260 

660 

681 


575 


740 

472 


581 


770 

500 

641 

5*8 

534 

244 

1500 

1540 

4200 

1620 

2096 

1200 

606 


475 


665 


Woods. 

Ash 

“ English 

“ Canada 

Balsam,  Canada 

Beech 

“ white 

Birch 

Cedar,  white 

f‘  Cuba 

Chestnut 

Elm 

“ Canada,  red 

Fir,  Baltic,  mean 

“ Canada,  yellow 

“ red 

“ Norway 

“ Dantzic 

Riga 

“ Memel 

“ “ red  

Greenheart,  Guiana 

Gum,  blue 

Hackmatack 

Hemlock 


168 

160 

120 

87 

130 

112 

160 

115 

160 

63 

105 

160 

125 

170 

i53 


58 


117 

120 

123 

163 

112 

161 


75 

160 

136 

102 

100 


Woods. 

Hickory. 

Iron  wood,  Burmah 

Larch,  Russian 

Lignumvitse 

Locust 

Mahogany 

Mangrove 

Maple ...  

Oak,  white 

“ live 

“ red,  black 

“ African 

“ English 

“ French 

“ Dantzic 

“ Canada 

“ Sardinia 

“ Spanish 

Pine,  white 

“ pitch 

“ yellow 

“ Georgia 

Poon 

Poplar 

Spruce,  Canada 

“ black 

Sycamore 

Tamarack 

Teak V. . . . 

Walnut 

Willow 

White  wood 


170 

250 

240 

118 

162 

295 

112 

162 

202 

150 

160 

135 

207 

105 


88 

146 

142 

105 

125 

!37 

130 

200 

184 

112 

125 

87 

125 

ICO 

165 
1 12 
87 

Il6 


Stones,  Bricks,  etc. 

Brick,  common,  mean 

“ pressed,  “ . ..* 

u English,  stock 

“ “ fine  

Brick  arch 

Cement,  mean 

“ “ Portland. j 

“ 11  Sheppey 

“ “ hydraulic,  Portland. 

11  “ Roman 

u Puzzuolana 

“ Portland,  1 year 

“ Roman,  “ 

Concrete,  Eng.,  fire-brick  beam,  ) 
cement j 


20 
40 
11. 8 

14 

15 
15 

10.2 

37-5 

5 

5 

2 


2.5 

3-i 


* This  was  with  a tensile  strength  of  27000  lbs. 

t With  840  lbs.  the  deflection  was  1 inch,  and  the  elasticity  of  the  metal  destroyed. 


800  STRENGTH  OF  MATERIALS. TRANSVERSE. 


Stones,  Bricks,  etc. 

Concrete,  Eng. , fire-brick,  sand  3, 1 

lime  1 j '7 

“ Eng. , clay  and  chalk 5.4 

Flagging,  blue,  New  York 31*25 

Freestone,  Conn 13 

“ Dorchester 10.8 

New  Jersey,  mean 19 

“ New  York 24 

“ Eng.,  Craigleth 10.7 

“ “ Darby,  Victoria. . . 1.3 

11  “ Park  Spring 4.3 

Glass,  flooring 42.5 

Granite,  blue,  coarse 18 

“ Quincy 26 

“ mean 25 

“ Eng.,  Cornish 22 

Limestone 

“ English 11 

Marble 


Stones,  Bricks,  etc. 

Marble,  Adelaide  

“ Italian... 

Mortar,  lime,  60  days 

“ 1 lime,  1 sand 


Oolite,  English,  Portland 

Paving,  Scotch,  Caithness 

“ Ireland,  Valentia 

“ Welsh 

“ English,  Yorkshire,  blue. . 

“ “ Arbroath 

Slate 

“ Bangor 

“ English,  Llangollen 

Stones,  English,  Bath 

“ “ Kentish,  Rag 

“ “ Yorkshire,  landing 

“ Caen 


4*5 

2*5 

2 

*•75 

1*25 

21.2 

68 

68.5 

57 

10.4 
17 
81 
90 
43 

5*2 

35*8 

22.5 

12.5 


Elastic  Transverse  Strength  of  Woods,  compared  with  their  Breaking  Weight , 
is  as  follows : 


Ash . 


Elm... 

Larch.. 


Per  Cent. 
..  29 

Norway  Spruce.. 
Oak,  Dantzic  . . . . 
‘k  English  . . . . 

Per  Cent. 
..  30 

..  36 

Red  Pine.. . . , 

Riga  Fir 

Teak 

..  38 

Pitch  Pine 

Yellow  Pine, 

Increase  in.  Strength,  of  several  Woods  "by  Seasoning. 
Per  Cent. 

Ash 44.7  ] Beech ..61.9  | Elm 12.3  | Oak 26.1  | White  pine. ..  .9 


Concretes,  Cements,  etc. 


Materials. 

Breaking 

Weight. 

Materials. 

Breaking 

Weight. 

concretes  (English). 
Fire-brick  beam,  Portl’d  cement 

Lbs. 
3. 1 

bricks  (English). 

Best  stock 

Lbs. 
11. 8 

I A 

“ sand  3 parts,  lime  1 part 
cements  (English). 

Blue  clay  and  chalk 

. 7 

Fire-brick 

New  brick 

IO.  7 

5*4 
37*5 
10. 2 
5 

Old  brick 

AW.  / 

0. 1 

Portland J 

Sheppey 

Stock-brick,  well  burned 

5*  8 

“ inferior,  burned. . . 

2*5 

Transverse  Strength  of  Various  Figures  of*  Cast  Iron. 
Reduced  to  Uniform  Measure  of  Sectional  Area  of  One  Inch  Square  and  One  Foot  in 
Length.  Fixed  at  one  End ; Weight  suspended  from  the  other. 


Form  of  Bar  or  Beam. 


Breaking 

Weight. 


Form  of  Bar  or  Beam. 


Breaking 

Weight. 


Lbs. 


Lbs. 


Square 


Square,  diagonal  vertical. . . 


Cylinder. 


673 


568 


573 


Rectangular  prism. 

2 X .5  ins.  in  depth. . . 

3 X .33  “ in  depth. . . 
4X.25  “ in  depth... 


Equilateral  triangle,  an 

edge  up 

Equilateral  triangle,  an 
edge  down 


1456 

2392 

2652 

560 

958 


O Hollow  cylinder;  greater 
diameter  twice  that  of 
lesser 


794 


T2  ins.  in  depth  X 2 X ) 
.268  inch  in  width. . . j 

A 2 ins.  in  depth  X 2 X ) 
.268  inch  in  width  . . ) 


2068 

555 


STRENGTH  OF  MATERIALS. — TRANSVERSE.  801 


Solid,  and.  Hollow  Cylinders  of*  various  Materials. 
One  Foot  in  Length.  Fixed  at  one  End  ; Weight  suspended  from  the  other. 


Materials. 

[ External 
Diam. 

Internal 

Diam. 

Breaking 

Weight. 

Materials. 

External 

Diam. 

; Internal 
| Diam. 

Breaking 

Weight. 

WOODS. 

Ash 

Ins, 

2 

Inch. 

Lbs. 

685 

604 

772 

75 

610 

METAL. 

Cast  iron,  cold) 

Ins. 

Ins. 

Lbs. 

2 

1 

blast ) 

3 

— 

12  COO 

Fir* 

2 

STONEWARE. 

White  pine. . 

u a 

1 

2 

1 1 

Rolled  pipe  of  ) 
fine  clay ) 

2.87 

1.928 

190 

* An  inch-square  batten,  from  same  plank  as  this  specimen,  broke  at  139  lbs. 

Formulas  for  Transverse  Stress  of  Rectangular  Bars, 
Beams,  Cylinders,  etc. 


Fixed  at  One  End.  Loaded  at  the  Other. 


Bars,  Beams,  etc. 


IW 

Vd2 


= S; 


Sbd2 

l 


= W 
IW 


Sbd 2 


IW 

SO2 


jiw 

/lTa“ 


Bars,  Beams,  etc. 


n ~\y 

and  Cylinder  3/  — b and  d. 

Fixed  at  Both  Ends.  Loaded  in  Middle . 
IW  • „ . 6 S b d*  _ 6 Sbd2 


6 bd2 


— S; 


zW: 


W 


— l: 


\ Sb 


IW 

6 Sd2~~ 


/-  = d ; and  Cylinder  3 = b and  d. 

V 6 S b ’ J V 6 S 

Fixed  at  Both  Ends.  Loaded  at  any  Other  Point  than  in  Middle . 


Bars,  Beams,  etc. 


2 m n W 


2,1b  d 
mnW 


■S' 


3lbd2S 


2 mnW 


d2~  ’ V 

}2mnW  , .......  „ /2mnW  , , 

V7sTF  = d;  and  Cylinder  3^^- = 6 and  d 


'2  mn  W 


’ 3$b 

2 m n W 


3S  Id- 


Bars,  Beams,  etc. 


3 s i 0 ' * v 3SI 

Supported  at  Both  Ends.  Loaded  in  Middle, 
t W _o  4S6  4S&rf2 

ffd2~S]  Z 


= W; 

'IW 


w 


\ 


/ l ^-z=d:  and  Cylinder  3 b and  d. 

/ 4 S b ’ V 4s 


Z W 
4 S d2 


Supported  at  Both  Ends.  Loaded  at  any  Other  Point  than  in  Middle . 


™ . mnW 

Bars,  Beams,  etc.  ^ — = S ; 

Imn  W 


Slbd2 


= W; 


mnW m n W 

S b d2  ~ 1 Sld2~ 


VmnW  , , _ , . , „ A 

-gIF  = d;  and  Cylinder 


'm  w W 


S « 


- — 6 and  d. 


In  Square  Beams,  etc.,  for  b and  d put  = d.  In  Cylinders,  for 

b d2  put  d3  as  above. 


When  weight  is  uniformly  distributed , same  formulas  will  apply , W repre- 
senting only  half  required  or  given  weight. 

S representing  stress  in  a Bar , fleam,  or  Cylinder , one  foot  in  length,  and  one  inch 
square,  side , or  in  diameter;  and  W weight , m Z&s. ; 6 breadth,  and  d depth , in  ins.; 
‘ length,  m distance  of  weight  from  one  end,  and  nfrom  the  other,  all  in  feet. 

B rick-work. 

A brick  arch,  having  a rise  of  2 feet,  and  a span  of  15  feet  9 ins.,  and  2 
feet  in  width,  with  a depth  at  its  crown  of  4 ins.,  bore  358400  lbs.  laid.  along 
its  centre.  0 


802  STRENGTH  OF  MATERIALS. TRANSVERSE. 


Coefficient  or  Factor  of  Safety. 


Coefficient  or  factor  of  safety  of  different  materials  must  be  taken  in  view 
of  importance  of  structure,  or  instrument,  probable  or  required  period  of  du- 
ration of  it,  and  if  it  is  to  bear  a quiescent,  vibratory,  gradual,  or  percussive 
stress,  and  to  meet  these  varied  conditions,  it  will  range  from  .125  to  .3  of 
the  maximum  or  ultimate  strength  here  given  or  ascertained. 

To  Compute  Transverse  Strength,  of  a Rectangular  Bar 
or  Beam. 

When  a Bar  or  Beam,  is  Fixed  at  One  End , and  Loaded  at  the  Other. 
Rule. — Multiply  Coefficient  of  material  in  preceding  Tables,,  or,  as  may  be 
ascertained,  by  breadth  and  square  of  depth  in  ins.,  and  divide  product  by 
length  in  feet. 

Note.  —When  a beam,  etc.,  is  loaded  uniformly  throughout  its  length,  result  must 
be  doubled. 


Example.— What  weight  will  a cast-iron  bar,  2 ins.  square  and  projecting  30  ins. 
in  length,  bear  without  permanent  injury? 

Assume  strength  of  material  at  660,  and  its  elasticity  at  one  fifth  or  .2  of  its 
strength. 

Then  660  X - 2 X = lbs . 

2.5  2.5 


If  Dimensions  of  a Beam  or  Bar  are  Required  to  Support  a Given  Weight 
at  its  End.  Rule. — Divide  product  of  weight  and  length  in  feet  by  Coeffi- 
cient of  material,  and  quotient  will  give  product  of  breadth  and  square  of 
depth. 

Example. — What  is  the  depth  of  a wrought- iron  beam,  2 ins.  broad,  necessary  to 
support  576  lbs.  suspended  at  30  ins.  from  fixed  end  ? 

Assume  strength  of  iron  at  150. 

Then  = 9.6,  and  /— = 2.19  ins.  depth. 

150  V 2 


When  a Beam  or  Bar  is  Fixed  at  Both  Ends,  and  Loaded  in  the  Middle. 
Rule. — Multiply  Coefficient  of  material  by  6 times  breadth  and  square  of 
depth  in  ins.,  and  divide  product  by  length  in  feet. 

Note.— When  beam  is  loaded  uniformly  throughout  its  length,  result  must  be 
doubled. 

Example.— What  weight  will  a bar  of  cast  iron,  2 ins.  square  and  5 feet  in  length, 
support  in  middle,  without  permanent  injury? 

Assume  strength  of  material  as  in  a previous  case  at  .2  of  660. 


Then 


660  X-2X2X6X22  6336 


= ~ --  — 1267.2  lbs. 


If  Dimensions  of  a Beam  or  Bar  are  Required  to  Support  a Given  Weight 
in  Middle,  between  Fixed  Ends.  Rule.— Divide  product  of  weight  and 
length  in  feet  by  6 times  Coefficient  of  material,  and  quotient  will  give  prod- 
uct of  breadth  and  square  of  depth. 

Example.— What  dimensions  will  a square  cast-iron  bar,  5 feet  in  length,  require 
to  support  without  permanent  injury  a stress  of  2160  lbs.  ? 

Assume  strength  of  material  at  .2  of  660  or  132,  as  preceding. 

Then  2I^°— — =. 10  — 13.64,  which , divided  by  2 for  assumed  breadth  — 6.82, 

132  X 6 792 

and  -\J 6. 82  = 2.61  ins.  depth. 

When  Breadth  or  Depth  is  Required.  Rule.— Divide  product  obtained 
by  preceding  rules  by  square  of  depth,  and  quotient  is  breadth ; or  by 
breadth,  and  square  root  of  quotient  is  depth. 

Example. — If  128  is  the  product,  and  depth  is  8;  then  128  -r-  82  = 2,  breadth. 
Also,  128  2 = 64,  and  V64  = 8?  depth. 


STRENGTH  OF  MATERIALS. TRANSVERSE.  803 

When  Weight  is  not  in  Middle  between  Ends.  Rule:.— -Multiply  Coefficient 
of  material  by  3 times  length  in  feet,  and  breadth  and  square  of  depth  in 
ins.,  and  divide  product  by  twice  product  of  distances  of  weight,  or  stress 
from  either  end. 

Example.— What  weight  will  a cast-iron  bar,  fixed  at  both  ends,  2 ins.  square  and 
5 feet  in  length,  bear  without  permanent  injury,  2 feet  Irom  one  end? 

Assume  strength  of  material  at  .2  of  660  or  132,  as  preceding. 


When  a Beam  or  Bar  is  Supported  at  Both  Ends , and  Loaded  in  Middle . 
Rule:.— Multiply  Coefficient  of  material  by  4 times  breadth  and  square  of 
depth  in  ins.,  and  divide  product  by  length  in  feet. 

Note. When  beam  is  loaded  uniformly  throughout  its  length,  result  must  be 

doubled. 

Example.— What  weight  will  a cast-iron  bar,  5 feet  between  the  supports,  and  2 
ins.  square,  bear  in  middle,  without  permanent  injury? 

Assume  strength  of  iron  at  132,  as  preceding. 


If  Dimensions  are  Required  to  Support  a Given  Weight.  Rule.— Divide 
product  of  weight  and  length  in  feet  by  4 times  Coefficient  of  material,  and 
quotient  will  give  product  of  breadth,  and  square  of  depth. 

When  Weight  is  not  in  Middle  between  Supports.  Rule.— Multiply  Coef- 
ficient of  material  by  length  in  feet,  and  breadth  and  square  of  depth  in  ins., 
and  divide  product  by  product  of  distances  of  weight,  or  stress  from  either 
support. 

Example.— What  weight  will  a cast-iron  bar,  2 ins.  square  and  5 feet  in  length, 
support  without  permanent  injury,  at  a distance  of  2 feet  from  one  end,  or  support  ? 

Assume  strength  of  iron  at  132,  as  preceding. 


To  Compute  Pressure  upon  Ends  or  upon  Supports. 

Rule  i. — Divide  product  of  weight  and  its  distance  from  nearest  end  or 
support,  by  whole  length,  and  quotient  will  give  pressure  upon  end  or  sup- 
port farthest  from  weight. 

2.— Divide  product  of  weight  and  its  distance  from  farthest  end,  or  sup- 
port, by  whole  length,  and  quotient  will  give  pressure  upon  end  or  support 
nearest  weight. 

Example.— What  is  pressure  upon  supports  in  case  of  preceding  example? 

880  X 2 — 352  lbs.  upon  support  farthest  from  the  weight ; 880  X - = 528  lbs.  upon 

5 _ 5 

support  nearest  to  weight. 

When  a Bar  or  Beam , Fixed  or  Supported  at  Both  Ends , bears  Two 
Weights  at  Unequal  Distances  from  Ends. 

m W l w A T . n w . V W , „r  , 

1 = pressure  at  w end , and  — — | — - — — pressure  at  W end . 

L L L L 

m and  n representing  distances  of  greatest  and  least  weights  from  their  nearest 
end , W and  w greatest  and  least  weights , L whole  length , l distance  from  least  weight 
to  farthest  end , and  V distance  of  greatest  weight  from,  farthest  end. 

Illustration. — A beam  10  feet  in  length,  having  both  ends  fixed  in  a wall,  bears 
two  weights— viz.,  one  of  1000  lbs.,  at  4 feet  from  one  of  its  ends,  and  the  other  of 
2000  lbs.,  at  4 feet  from  the  other  end;  what  is  pressure  upon  each  end? 


Then  132  X 2 X 4 X 2 2 = 4224  -r-  5 — 844. 8 lbs. 


Then 


132  x 5 X 2 X 22 
2 X (5  — 2) 


5^  = 880  lbs. 
6 


...  4 X 1000  . 6 X 2000  _ 

1400  lbs.  at  w ; - = 1600  lbs.  at  W. 

^ 10  10 


804  STRENGTH  OF  MATERIALS. TRANSVERSE. 


When  Plane  of  Bar  or  Beam  Projects  Obliquely  Upward  or  Downward. 
When  Fixed  at  One  End  and  Loaded  at  the  Other.  Rule. — Multiply  Co- 
efficient  of  material  by  breadth  and  square  of  depth  in  ins.,  and  divide  product 
by  product  of  length  in  feet  and  cosine  of  angle  of  elevation  or  depression. 


Note.— When  beam  is  loaded  uniformly  along  its  length,  result  must  be  doubled. 
Example. — What  is  weight  an  ash  beam,  5 feet  in  length,  3 ins.  square,  and  pro- 
jecting upward  at  an  angle  of  70  15',  will  bear  without  permanent  injury? 


Assume  breaking  weight  of  ash  at  160,  and  its  elasticity  at  .25  of  its  strength,  and 
cosine  of  70  15'  = .992. 


Then 


160  X -25  X 3 X 32 
5 X -992 


= 1080  ■=  217.74  lbs. 


To  Compute  Transverse  Strength,  of  an  Eqnilateral  Tri- 
angle or  T Beam. 

Rule. — Proceed  as  for  a rectangular  beam,  taking  following  proportions 
of  Coefficient  of  material : 


Fixed  at  One  or 
Both  Ends. 

Supported  at 
Both  Ends. 


Equilateral  triangle,  edge  up. . . 
Equilateral  triangle,  edge  down 

X beam,  flange  up 

Equilateral  triangle,  edge  up. . . 
Equilateral  triangle,  edge  down 
X beam,  flange  up 


bxd2X.  2 C 
b X d2  X -34  “ 
b X d2  X .42  “ 
b X d2  X .34  “ 
bXd2X-  2 “ 
b X d2X  .42  “ 


To  Compute  Transverse  Strength  of  a Solid.  Cylinder. 

Rule. — Proceed  as  for  a rectangular  beam,  and  take  .6  of  Coefficient  or 
of  product. 

A mean  of  18  results  with  cold  blast  gun-metal,  gave  a coefficient  for  740  lbs. 


When  Fixed  at  One  End , and  Loaded  at  the  Other.  Rule. — Multiply 
weight  to  be  supported  in  lbs.  by  length  of  cylinder  in  feet ; divide  product 
by  .6  of  Coefficient  of  material,  and  cube  root  of  quotient  will  give  diameter. 

Note.— When  cylinder  is  loaded  uniformly  throughout  its  length,  cube  root  of 
half  quotient  will  give  diameter. 

Example.— What  should  be  diameter  of  a cast-iron  cylindrical  beam  of  gun-metal,  i 
8 ins.  in  length,  to  break  at  15  000  lbs.  ? t 


15  000  X t 


> „ /10000 

- = 3 / = 2. 

V 444 


61  ins. 


.6  X 74° 

When  Fixed  at  Both  Ends , and  Loaded  in  Middle.  Rule.-— Multiply 
weight  to  be  supported  in  lbs.  by  length  of  cylinder  between  supports  in 
feet;  divide  product  by  .6  of  Coefficient  of  material,  and  cube  root  of  one 
sixth  of  quotient  will  give  diameter. 

Note.— When  cylinder  is  loaded  uniformly  along  its  length,  cube  root  of  half  the 
quotient  will  give  diameter. 

Example. — What  is  the  diameter  of  a cast-iron  cylinder  of  gun-metal,  2 feet  be- 
tween supports,  that  will  break  at  35  964  lbs.  ? 

3j^64>C2  = i6  3/^  = 3 ins. 

.6x74°  V 6 

Mean  results  of  cylinder  and  square  bars  gave  444  and  740  lbs.  Hence,  strength 
of  a cylinder  compared  to  a square  is  as  444  to  740  or  .6  to  1. 


Then 


4 X 33  X 444 


= 47  952 


lbs. 


To  Compute  Diameter  of*  a.  Solid.  Cylinder  to  Support 
a given  Weight. 

When  Supported  at  Both  Ends , and  Loaded  in  Middle.  Rule. — Multiply 
weight  to  be  supported  in  lbs.  by  length  of  cylinder  between  supports  in 
feet;  divide  product  by  .6  of  Coefficient  of  material,  and  cube  root  of  one 
fourth  of  quotient  will  give  diameter. 


STRENGTH  OF  MATERIALS. TRANSVERSE.  805 


Note.— When  cylinder  is  loaded  uniformly  along  its  length,  cube  root  of  half  the 
quotient  will  give  diameter. 

Example.— What  is  diameter  of  a cast-iron  gun-metal  cylinder,  1 foot  between  its 
supports,  that  will  break  at  48000  lbs.  ? 

48  000  X 1 0.0  A 08  ,,  . 

2 = 108,  and  3 / — — 3.61  ms. 

.6X740  V 4 

Ttecta.iagu.lar,  ( D . K.  Clark.) 

(1)  Loaded  at  Middle,  —j—  — W.  (2)  Loaded  at  One  End.  —j—  — W. 
Cylindrical. 

(3)  Loaded  at  Middle.  5'5  ^ ■ — = W.  (4)  Loaded  at  One  End.  I’375^---  =:  W. 

W representing  ultimate  stress  in  tons. 

Above  Coefficients  are  for  iron  of  a tensile  strength  of  7 tons  per  sq.  inch. 


(1) 

(2) 

(3) 

(4) 

(1) 

(2) 

(3) 

(4) 

9.2 

2-3 

6.3 

1.6 

For  12  tons  put 

13  “ 

13-8 

3-4 

9.4 

2.4 

10.4 

2.6 

71 

1.8 

14-5 

3-6 

10.2 

2.6 

ii-5 

2.9 

7-9 

2 

14 

16 

4 

11 

2.8 

12.7 

3-2 

8.6 

2.2 

15  “ 

17.2 

4-3 

11. 8 

3 

To  Compute  Destructive  W eiglit,  or  Loads  tliat  may  "be 
borne  "by  Wrought-iron  Rolled.  Beams  and  Grirders, 
or  Riveted  Tubes  of*  various  Figures  and  Sections. 

Supported  at  Both  Ends.  Load  applied  in  Middle. 

When  Section  of  Beam  or  Girder  is  that  of  any  of  the  Figures  in  follow- 
ing Table.  Rule. — Divide  product  of  area  of  section,  depth,  and  Coefficient 
for  girder,  etc.,  from  following  Table,  by  length  between  supports  in  feet, 
and  quotient  will  give  destructive  weight  in  lbs. 

Note. — The  Coefficients  given  are  based  upon  experiments  with  English  iron. 

Solid  Beams. 

Illustration. — What  load  will  destroy  a wro  ight-iron  grooved  beam  of  following 
dimensions,  10  feet  in  length  between  supports,  and  loaded  in  its  middle? 

Flanges,  5.7  X .64  inch;  Web,  .6  inch;  Depth,  n.75  ins. ; Area,  13.34  sq.  ins. 

Assume  Coefficient  4638  as  for  like  case  (12)  in  following  table,  page  806. 

13.34X11.75X4638  726821 

= = 72  682.  i lbs. 

10  10 

Ultimate  stress  for  such  a beam  by  experiment  was  estimated  at  97997  lbs. 

Formulas  of  Various  Authors  give  following  Results: 


D.  K.  Clark. 


d (4  a + 1. 1555  a') 
.6  l 


= W.  a representing  area  of  section  of  lower 

flange , a area  of  section  of  web,  less  one  flange,  d depth  of  beam.,  less  average  depth 
of  one  flange,  all  in  ins.,  I length  in  feet , and  W ultimate  destructive  weight  in  tons. 
This  formula  is  based  upon  the  assumption  that  the  beam  has  lateral  support. 


1T-75  — -6  (4  X 5-7  X .6 -f-  1. 155  X ix.75  — .6  X .6)  _ 238.69 


.6  X 10 


: 39.78,  which  X 2240 


= 88 107  lbs. 

Molesworth.  - - b--  _ 


W.  C = 7616  lbs. , and  for  b d2  put  b d'  — 2 b' d'2. 


b and  d representing  exterior  and  b'  and  d'  interior  dimensions,  and  l length  in  ins. 

’)]  = 786.6  — 558. 

- = 57  805.4  lbs. 


5.7  X ii-752  — [5-7  — *6  X 11.75  — (.64  X 2 2)]  = 786.6  — 558.9  = 227.7. 
Then  4 X 7616  x 227.7  ^693665  _ 


Fairbairn’s  formula  would  give  a result  less  than  half  of  the  first,  and  Hodgkin- 
son’s  alike  to  that  of  Molesworth. 

3 v 


806  STRENGTH  OF  MATERIALS. TRANSVERSE. 


WROUGHT  IRON. 

Transverse  Strength,  of  Wrought-iron  Rolled  Beams 
and  Girders.  {Barlow,  Fairbairn , Hughes , KirTcaldy , dc.) 

Reduced  to  Uniform  Measure  of  One  Foot  in  Length. 

Supported  at  Both  Ends  ; Stress  or  Weight  applied  in  Middle. 


Section. 


Area 

(A). 

Destructive  Weieht. 

l w 

Flanges. 

Web. 

Depth  {d).  Distance. 

For 

Length  of 

Distance. 

One  Foot(/). 

, A d u 

Ins. 

Ins. 

Ins. 

Feet.  Ins. 

Sq.  Ins. 

Lbs. 

Lbs.  (W). 

— 

1 

1 

1 

1 

2 500 

2500 

2 500 

2 

2 

2 

9 

4 

6 600 

18  150 

2 266 

— 

i-5 

3 

2 

9 

4-5 

10080 

27  720 

2053 

— 

1 

3 

3 

3 

7050 

21  150 

2 350 

— . 

1 

1 

5 

•78 

474 

2370 

2370 

3-5  X .6 

.8 

3-5 

2 

7 

5-65 

20  160 

52480 

2654 

25  Xi 
4 X .38 

} -325 

7 

j'v  > f i - > . : 

2 

9 

5-9 

- : 

44000 

121  000 

2930 

2.6  X1.25 

.85 

5 

4 

6 

7-44 

19000 

85  500 

2 298 

3 X -49 

•5 

7-°7 

10 

5.87 

24  200 

242  000 

5830 

4.6  X .8 

•5 

9-85 

20 

n-5 

38  080 

761  600 

6724 

5-7  X .64 

.6 

ii-75 

10 

x3-34 

72  688 

726  880 

4 638 

2.85X  .38 

•3i 

2.5 

4 

x-75 

3150 

12  600 

2 880 

7 X-5 
4 X-5 

} -38 

16.5 

22 

6 

18.9 

49  280 

1 108  800 

3 556 

4-5X-375 

2X2X.3125 

} .38 

14.25 

1 6 

5 

10.5 

47  000 

775  500 

5 183 

4- 5X28 
4-5X3 

} -25 

7 

7 

6-35 

24380 

170660 

3840 

3-9 

3-9 

}■* 

6 

7 

6 

2.62 

9976 

74  820 

4766 

i5-5 

I5-5 

} -53 

24 

30 

41.4 

128  885 

3 866  550 

3 896  ' 

24 

24 

•75 

•75 

} 35-75 

45 

87.38 

257  080 

11  568600 

3 703  * 

- 

.131 

{12.4 

(12.138 

■h 

5-05 

17885 

178850 

2 856  j 

- 

•x43 

I15 

X 9-75 

I10 

5-56 

26  250 

262  500 

3 x47  $ 

J 

•75 

5-2 

5 

7.72 

102  480 

512400 

12  760. 

Steel. 


These  results  are  very  conclusive  of  the  correctness  of  above  formula,  as 
will  be  seen  in  cases  given,  and  they  are  deduced  from  beams  and  girders 
varying  from  i to  45  feet  in  length ; hence,  when  length  qf  a beam  or  girder 
of  any  of  the  sections  given  is  less,  relative  breaking  weight  may  be  in- 
creased, in  consequence  of  increased  stability  of  beam  or  girder. 

For  full  experiments  on  Tubes  and  Tubular  Girders,  etc.,  see  Rep.  of  Commas  on 
Railway  Structures , London , 1849. 

Tensile  strength  of  iron  assumed  at  45  000  lbs.  per  sq.  inch. 


STRENGTH  OF  MATERIALS. TRANSVERSE. 


807 


Elements  of  Rolled  Wrought  - iron  Beams  and 
Channel  Bars. 

With  Safe  Load  Uniformly  Distributed.  F or  Length  of  One  Foot. 
Tlie  ISTew  Jersey  Steel  and  Iroix  Co.,  Trenton,  IN'.  J. 

( Beams  Supported  Sidewise.) 

Width. 


Designation. 


inch. 


X Beams. 
Extra  Light. 
Light. 
Heavy. 
Light. 
Heavy. 
Light. 
Heavy. 
go  lbs. 

120  “ 

55  “ 
Light. 
Heavy. 
Light. 
Heavy. 
Extra  Heavy. 
Extra  Light. 
Light. 
Heavy. 
Light. 
Heavy. 
Light. 
Heavy. 

Deck  Beams. 


Channels. 

Extra  Light. 


Light. 
Heavy. 
Extra  Light. 
Light. 

Extra  Light. 

Light. 

% “ 
Heavy. 
Light. 

Heavy. 

Light. 

Heavy. 


Weight 


Web. 

Flange. 

Web. 

Flange. 

Total. 

Foot. 

Inch. 

Ins. 

Sq.  Ins. 

Sq.  Ins. 

Sq.  Ins. 

Lbs. 

•1875 

2 

•75 

1.02 

1.77 

6 

•25 

2-75 

1 

1.91 

2.91 

10 

•3125 

3 

1.25 

2.41 

3.66 

12.3 

•25 

2-75 

1.2 

1.79 

2.99 

10 

•3125 

3 

1.56 

2.34 

3-9 

13-3 

•25 

3 

i-5 

2.51 

4.01 

13-3 

•3 

3-5 

1.8 

3-n 

4.91 

16.7 

•5 

5 

3 

5-7 

8.7 

30 

.625 

5-25 

3-75 

8.09 

11.84 

40 

•3 

3-75 

2. 1 

3-4 

5-5 

18.3 

• 3 

4 

2.4 

3-97 

6-37 

21.7 

•375 

4-5 

2.96 

5-07 

8.03 

26.7 

.3 

4 

2.7 

4-3 

7 

23- 3 

•375 

4-5 

3-38 

5.12 

8-5 

28.3 

•57 

4-5 

5-i3 

7.2 

12.33 

41.7 

•3125 

4-5 

3.28 

5.62 

8.9 

30 

•375 

4-5 

3-94 

6.5 

10.44 

35 

•47 

5 

4-93 

8-43 

13-36 

45 

•47 

4.8 

5-75 

6.58 

12.33 

41.7 

.6, 

5-5 

7-39 

9-38 

16.77 

56.7 

• 5 

5 

7-59 

7-45 

15.04 

50 

.6 

5-75 

9.07 

10.95 

20.02 

66.7 

•31 

4-5 

2.17 

3.18 

5-35 

18.3 

•38 

4-5 

3-04 

3-25 

6.29 

21.7 

.2 

1.5 

.6 

.85 

i-45 

5 

.2 

i-5 

.8 

.85 

1.65 

5-5 

.2 

1.625 

1 

.92 

1.92 

6-3 

.18 

1-875 

1.08 

1.17 

2.25 

7-5 

.28 

2.25 

1.68 

1.52 

3-2 

11 

•4 

2-5 

2.4 

1.92 

4-32 

15 

.2 

2 

1.4 

1. 14 

2-54 

8-5 

•25 

2.5  « 

i-75 

1.85 

3-6 

12 

.2 

2.2 

1.6 

i-7 

3-3 

11 

.26 

2-5 

2.08 

2.4 

4.48 

i5 

•33 

2-5 

2.97 

2. 11 

5.08 

16.7 

•43 

3-125 

3-87 

3-i5 

7.02 

23-3 

•375 

2-75 

3-94 

2.06 

6 

20 

•33 

3 

4.04 

2.96 

7 

23-3 

.68 

4 

8-33 

5-77. 

14. 1 

46.7 

•5 

4 

7-5 

4-5 

12 

40 

•75 

1 4-75 

1125 

7.6 

18.85 

63-3 

Load. 


18  000 
30  100 
36  800 
38  700 
49  100 
62  600 
7 6 800 
132000 
172  000 
1 01  000 
135000 
168  000 
167  000 
199  000 
268  000 
250000 
286  000 
360000 

377  000 
51 1 000 
551000 
748  000 


63  500 
91  800 


10  500 

15700 

22  800 
33  680 
45  7 00 

583°° 

39  5°o 
62  000 
65  800 
88950 
104  000 
146  000 
134  750 
200  100 
381 000 
401  000 
625000 


The  loads  given  in  the  table  are  such  as  will  effect  a maximum  strain 
upon  the  metal  of  12000  lbs.  per  sq.  inch.  For  permanent  stress,  absolutely 
free  from  vibration,  a greater  strain  would  be  allowable,  and,  contrariwise, 
if  the  stress  is  mainly  that  of  a live  load  the  loads  here  given  should  be  re- 

A difference  of  25  per  cent,  in  either  direction  should  be  made,  according 
to  the  character  of  the  load  to  be  supported  or  stress  to  be  borne. 

Steel  beams  have  greater  estimated  strength  than  iron,  but  their  stiffness 
is  not  materially  greater. 


808  STRENGTH  OF  MATERIALS. TRANSVERSE. 


Elastic  Transverse  Strength  of  W rough t-iron  Bars  is  about  45  per  cent,  of 
their  transverse  strength,  and  of  Plates  55  per  cent.,  or  48  per  cent,  of  their 
tensile  strength ; of  solid  rolled  beams,  50  per  cent. ; and  of  double-headed 
rails,  46  per  cent,  of  their  transverse  strength ; of  Fagersta  Steel,  56  per  cent, 
of  its  transverse  strength  ; of  double-headed  Steel  rails,  47  per  cent. ; of  Bes- 
semer Steel,  37.5  to  48  per  cent. ; of  Steel  flanged,  68  per  cent. ; and  of 
Wrought-iron  Steel  flanged,  62  per  cent,  of  its  transverse  strength. 

Transverse  strength  of  Solid  Cast-iron  Beams  or  Girders  is  about  50  per  cent, 
of  ultimate  strength;  of  double-headed  or  flanged  rails,  46  per  cent. ; and  of  single- 
flanged  rails,  62  per  cent,  of  its  tensile  strength. 

Note. — The  actual  breaking  weight  of  a 10.5  ins.  beam  of  New  Jersey  Steel  and 
Iron  Co.,  weight  35  lbs.  per  foot,  for  a length  of  span  of  20  feet,  is  60000  lbs. 

Channel  and  Deck  13 earns  and  Strut  Bars. 


With  Safe  Load  Uniformly  Distributed  for  Length  of  One  Foot. 
( Beam  supported  Sidewise.) 


Depth. 

Designation. 

Wi 

Web. 

dth. 

Flange. 

Area. 

Section. 

Weight 
per  Foot. 

Load. 

Strength 

Sidewise. 

as  Strut. 
Edgewise. 

Ins. 

Channel. 

Inch. 

Ins. 

Sq.  Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

3 

Extra  Light 

u tt 

.2 

i-5 

i-45 

5 

10500 

51 

341 

4 

.2 

i-5 

1.65 

5-5 

15  700 

49 

597 

5 

it  It 

.2 

1.625 

1.92 

6-33 

22  800 

57 

930 

6 

u u 

.18 

i-875 

2.25 

7-5 

33680 

77 

1403 

6 

Light 

.28 

2.25 

3-2 

11 

45700 

IOI 

1343 

6 

Heavy 

•4 

2.5 

4-32 

i5 

583°° 

123 

1257 

7 

Extra  Light 

li  it 

.2 

2 

2.54 

8-33 

39  5oo 

82 

1700 

7 

•25 

2-5 

3-6 

12 

62  000 

136 

1883 

8 

it  it 

.2 

2.2 

3-3 

11 

65  800 

109 

2493 

8 

Light 

U 

.26 

2-5 

4.48 

15 

88  950 

142 

2480 

9 

•33 

2-5 

5.08 

16-33 

104  000 

124 

2892 

9 

Heavy 

•43 

3-125 

7.02 

23-33 

146000 

190 

2925 

10.5 

Light 

•375 

2-75 

6 

20 

134  750 

160 

3685 

12.25 

.42 

3 

8.62 

28.33 

238  000 

172 

5275 

12.25 

Heavy 

.68 

4 

14. 1 

46.66 

381  000 

3i7 

5170 

15 

Light 

•5 

4 

12 

40 

401 000 

301 

7833 

15 

Heavy 

•75 

4-75 

18.85 

63-33 

625  000 

428 

7762 

7 

) Deck  ( 

•3i 

4-5 

5-35 

21.66 

63  500 

35i 

36 

8 

j Beams.  ( 
Strut  Bars. 

.38 

4-5 

6.29 

18.38 

91  800 

547 

37 

5 

Light,  Single 
Heavy,  u 

— 

— 

i-55 

5-33 

9 100 

44 

457 

5 

— 

— 

2.15 

7-33 

11  900 

48 

433 

Operation!  of  Table. 

To  Compute  Depth  of  a Beam,  to  Support  a Uniformly 
Distri.bn.ted.  Load. 

Rule. — Multiply  load  in  lbs.  by  length  of  span  in  feet,  and  take  from 
table  the  beam,  the  load  of  which  is  nearest  to  and  in  excess  of  the  product 
thus  obtained. 

Example.— What  should  be  depth  of  a beam  to  sustain  with  safety  a uniformly 
distributed  load  of  30000  lbs.,  over  a span  of  15  feet? 

30000  X 15  = 450000,  which  is  load  for  a heavy  beam  12.25  ins.  in  depth. 
Weight  of  beam  should  be  added  to  load. 

Inversely.— If  the  load  is  required,  divide  load  in  table  by  span  of  beam  in  feet, 
and  subtract  weight  of  beam. 

To  Compute  Deflection  of  Like  Beams. 

Rule. — Divide  square  of  span  in  feet  by  70  times  depth  of  beam  in  ins. 
Example.— Assume  beam  as  preceding. 

152  225  , 


STRENGTH  OF  MATERIALS. TRANSVERSE.  8O9 

Comparative  Strength,  and  Deflection  of  Cast-iron 


Description  of  Beam. 


Beam  of  equal  flanges 

“ with  only  bottom  flange. 
“ with  flanges  as  i to  2. . . 
with  flanges  as  1 to  4. . . 


Comp. 

Strength. 


•58 

.72 

.63 

•73 


Description  of  Beam. 

Comp. 

Strength. 

Beam  with  flanges  as  1 to  4. 5 . . . 

.78 

“ with  flanges  as  1 105.5... 

.82 

“ with  flanges  as  1 to  6 

1 

with  flanges  as  1 to  6.73. . 

.92 

Dimensions  and  Proportions  of  Wronght-iron  Flanged 
13 earns.  (D.  K.  Clark.) 


Depth. 


25 


Ins 

3 

3 
3' 

4 

4 

4- 75 

5 

5 

5- 5 

6 

6.25 

6.25 

6.25 

7 

7 

7 

7 

8 
8 
8 
8 
8 

9- 25 
9-5 
10 
10 


14 

14 

16 


Breadth 
of  Flanges. 


Thickness. 

Web.  Flanges. 


3 

1.625 


3 

4-5 


2.25 
3-25 
2.25 
2.25 
3- 625 

3.625 
2-375 

2- 5 

4 

5 

5-125 

3- 75 

4- 5 
4-5 
4-75 

4- 75 

5 

6 

5- 5 
6 

5.625 


Inch. 

1875 

5 

875 

5 

5 

5 

3i25 

j7S 

375 

4375 

3125 

3i25 

3i25 

281 

3i25 

3i25 

3i25 

3i25 

375 

375 

375 

4375 

4375 

375 

4375 

4375 

75 

5625 

5625 

5625 

5625 

75 


Weight  per 
Lineal  Foot. 


Inch. 

,2187 

25 

187 

125 

75 

3I25 

4375 

4375 

5625 

4375 

375 

(.062 

375 

4375 

4375 

5 

4375 

375 

5 

5625 

5625 

6875 

5625 

5625 

625 

8i75 

9375 

875 

8x75 

8175 


Lbs. 

5-5 


5-5 

8 


11 

18 
12.5 

14 

14 

19 
*9 

15 

15 

21 

29 

29 

24 

30 
32 
32 
36 
42 
56 

60 

60 

62 


Ultimate 

Strength. 

Loaded 
in  Middle. 


Lbs. 

2 800 
5600 
2 490 

5 49° 
8510 

6 940 
13  440 
19  270 
11  880 

23  830 
13  440 
13000 
17470 
14790 
17  020 
23300 
25980 
20830 
21  280 
34  5oo 
44  800 
47  040 
41  560 
59  36o 
56  000 
58  240 
76  160 
100  800 
136  640 
150020 
1 52  260 
188  160 


Safe  Stress 

Uniformly 

Distributed. 


Lbs. 

910 
I 860 
830 

1 830 

2 830 
2 310 
4480 
642O 
3960 
794O 
4 440 

4 330 

5 820 
4 930 
5670 
7760 
8660 
6940 
7 090 

11  500 

14  930 

15  680 
13850 
19  750 
18660 
19  410 

25390 

33  600 
45  530 
50000 
50  750 
62  720 


■Wrought-iron  Rectangular  Girders  or  Tubes.  (Sit'd.) 

Supported  at  Both  Ends.  Loaded  in  Middle. 

AdG  — w.  A representing  area  of  section  in  sq.  ins.,  d depth  in  ins.,  I length  be- 
tween supports  in  feet,  and  W destructive  weight  in  lbs. 

Ti  lustration  —What  is  the  destructive  weight  of  a rectangular  girder,  35.75  ins^ 
in  depth  by  24  in  breadth,  metal  .75  inch  thick,  and  length  between  supports  45  feet? 

Assume  C or  coefficient  = 37  coo,  as  per  case  (r7)  in  preceding  table,  page  806, 
and  area  = 87. 375  ins.. 

Then  87-375  X 35-375  X 3?oo  _ 1 1 557  523  _ ^ 8 

45  45 

W l , 

By  experiment  it  was  257  080  lbs.  By  Inversion  — A,  and  A c — 

, result  of  259373  lbs.,  and  Molesworth’s 


Hodgkinson’s  formula  would  give 
303  907  lbs. 


8lO  STRENGTH  OF  MATERIALS. TRANSVERSE. 


I2  W 


Unequally  Loaded  Beams,  etc. 


m n — w-  l representing  length  between  supports , and  m and  n distances  from 

points  of  support,  all  in  like  denomination , and  W and  w destructive  and  safe  weights 
also  in  like  denomination. 

To  Compute  Destructive  Weight  and.  Area  of  Bottom 
Blate. 


AdC 


= W; 


W l 


: = A;  and 


W m n 


l “ " ’ c d ~~  25  c d l = A‘  A rePreseniin9  Mea  of  plate  in  sq. 

ins.,  d and  l depth  and  length,  m and  n distances  of  load  at  other  points  than  in 
middle , all  in  feet,  and  W weight  in  lbs. 

Note. — Sufficient  metal  should  be  provided  in  sides  to  resist  transverse  and 
shearing  stress,  and  in  upper  flange  to  resist  crushing. 

Illustration.— What  area  of  wrought  iron  is  necessary  in  bottom  plate  of  a rec- 
tangular tubular  girder,  3 feet  in  depth,  supported  at  both  ends,  and  loaded  in  middle 
with  130000  lbs.  ? 

C,  ascertained  by  experiment  for  destructive  stress,  180000  lbs.,  and  area  7.  x sq.ms. 
130000  X 30 

—5 3 — — 7. 22  sq.  ins. 

180000  X 3 

Wronght-iron  Cylindrical  Beams  or  Tubes. 

— W.  Illustration. — What  is  destructive  weight  of  a cylindrical  tube, 

12.4  ins.  in  diameter,  .131  inch  in  thickness,  and  10  feet  between  its  supports? 

Area  of  metal  = 5.05  sq.  ins.,  and  C = 2856,  as  in  the  19th  case  of  table,  page  806. 


Then  5-°5Xl2-4X  2856  = i7 


= 17  884.2  lbs. 

D.  K.  Clark.  ^ ^ = W.  d representing  diameter , t thickness  of  metal,  and 

l length,  all  in  ins.,  S tensile  strength  of  metal  per  sq.  inch,  and  W weight,  both  in  lbs. 
3.14  X 12. 42  X .131  X 45000 2 846  250 


S = 45  000  lbs. 


- = 23  718.7  lbs. 


10  X 12 

Molesworth’s  formula  gives  a result  of  23286.1  lbs. 

"Wr onght-ir on  Elliptical  Beams  or  Tribes. 
A d'C 
l 


- = W.  Illustration. — Assume  diameter  of  tube  9.75  and  15  ins.,  metal 
.143  inch  n thickness,  and  distance  between  supports  10  feet. 

A = 5.56  sq.  ins.  C = 3147,  as  per  case  (20)  in  preceding  table,  page  806. 


Then 


5. 56  X 15  X 3M7  _ 262459.8 


= 26  245.9  lbs. 


D.  K.  Clark. 


1. 57  (&2  + d2)<S 


: W.  b and  d representing  conjugate  and  trans- 


verse diameter , l length  between  supports,  t thickness  of  metal,  all  in  ins.,  Si  tensile 
strength  of  metal  per  sq.  inch,  and  W destructive  weight,  both  in  lbs. 

a J-57  (9*75 2 -f-152)  x .143  x 44000  3161840  - 0 ^ 77_ 

S = 44  000  lbs. = = 26  348. 6 lbs. 

10  X 12  120 

Note.— B.  Baker,  in  liis  work  on  Strength  of  Beams,  etc.,  London,  1870,  page  26, 
shows  that  ordinary  method  of  computing  transverse  strength  of  a hollow  shaft  by 
difference  of  diameter  alone  is  erroneous,  in  consequence  of  loss  of  resistance  to 
flexure  in  a hollow  beam. 

Grirders  and  Beams  of  Unsymmetrical  Section. 

4 S d 

— - — — W.  S representing  tensile  resistance  of  metal,  and  W destructive  weight, 

both  in  lbs. , d distance  between  centres  of  compression  and  extension,  or  crushing  and 
tensile  resistances,  in  ins.,  and  l length  between  supports , in  feet. 

Note. — To  ascertain  d,  see  Rule,  page  819, 


STRENGTH  OF  MATERIALS. TRANSVERSE.  8 I I 


Illustration.— Dimensions  of  a rolled  wrought-iron  girder,  n feet  in  length  be- 
tween  its  supports,  is  as  follows  : 


Top  flange 2.5X1  inch. 

Web “ 


I Bottom  flange 4 X .38  inch. 

I Depth 7 ins. 


L W 


What  is  its  destructive  weight? 

d = 5.22  ins.  S assumed  at  45000  lbs.  Then  4 * ^ ^ = 7118,18  ^s' 

Strength  of  Riveted  Beams  or  Girders,  compared  with  Solid,  is  less,  and  deflec- 
tion is  greater. 

■Wronglit-irorL  Inclined.  Beams,  etc. 

— w.  L and  l representing  lengths  or  inclination , and  horizontal  line , in  like 

denominations , and  W and  w destructive  and  safe  weights  on  horizontal  line  and  in- 
clination, also  in  like  denominations. 

[Plate  Grirders. 

W.  A representing  section  in  sq.  ins.,  d depth  in  ins.,  and  l length  be- 

b 

tween  supports  in  feet. 

Illustration.— What  load  will  destroy  a wrought-iron  plate  girder  or  beam  of 
following  dimensions,  10  feet  in  length  between  its  supports? 

Top  flange 4.5  X -375  inch.  I Width  of  web 375  inch. 


AdC 


Depth  of  web 13.5  ins. 

Depth  of  beam 14-25 


Bottom  flange 4-5  X -375 

Angle  pieces 2 X -3I25  ‘ 

Area  of  Section  = 13  sq.  ins. 

Assume  coefficient  of  5180  as  per  case  (14)  in  preceding  Table,  page  806. 

Then  »3X.4-»iX5.te  = 96oi54  = 9601S.4  lbs. 

IO  IO 

Molesworth.  ^ = W.  L representing  load  equally  distributed,  and  W destruc- 
tive weight,  both  in  tons , and  d effective  depth  of  girder  in  feet. 

By  actual  experiment  L = 48  tons  for  16.5  feet  between  supports;  hence, 

10:  i6.5:;48:7g.2  tons=.  39.6  when  supported  in  middle,  and  14.25  ins.  = 1.1875  feet. 


Then 


D.  K.  Clark. 


39-6  X 1 
8X1.1875  9*5 

d (4^4- 1. 155  a') 

.61 


__  39^  __  68,  which  X 2240  = 93  363.2  lbs. 


— W.  d representing  depth  of  girder  or  beam 

less  depth  of  lower  flange  in  ins.,  a and  a ' areas  of  sections  of  bottom  flange  and  of 
web,  at  its  reputed  depth,  both  in  sq.  ins.,  and  l length  between  supports  in  feet, 
d — 14.25  — .375  = 13.875  ins.  a = 3,  and  a'  = 5 sq.  ins. 


Then 


13.875  (4  X 3+1.155  X 5)  _ 246.63 


= 41.105,  which  X 2240  = 92075.2  lbs. 


.6  X 10 

Mr.  Clark  assumes,  however,  that  for  girders  of  like  construction  the  destructive 
stress  should  be  taken  at  two  thirds  of  that  deduced  by  the  formula. 

Girders  or  Beams  without  Upper  and  Lower  Flanges. 

Illustration.  — Assume  angles  2.125  X .28  above,  2.125  X -3  below,  web 
.25,  depth  7 ins.,  and  length  between  supports  7 feet. 

Area  of  section  = 6.35  sq.  ins.,  and  C = 3840,  as  per  case  (15)  in  preceding  Table, 
page  806. 

Then  fi-35X  7X3840  = £Z£688  = ^ ^ 

7 7 


Approximate. 


--f  .25  a'X  5 d 


W.  a representing  area  of  sections  of  upper 


and  lower  angles , a ' area  of  section  of  web  for  total  depth , both  in  sq.  ins.,  d depth  of 
girder  in  ins. , and  W load  or  stress  in  lbs. 


8 12  STRENGTH  OF  MATERIALS. TRANSVERSE, 


a = 4.6  sq.  ins. , and  w = 7 x .25  = 1.75  sg.  ins. 


4-6  , 1.75 
T + “X5X7 


Then 


90.81  . . , 

= — - — = 12.973,  which  X 2240  = 29059.5  lbs. 


IRON  AND  STEEL  RAILS. 
Symmetrical  Section. 
To  Compute  Transverse  Strength.. 


S (4  a + 1. 155  1 d 2) 


:W,  and 


W l 


(4aj  + I-l5sid2) 


(D.  K.  Clark.) 

= S.  S representing  ten- 


sile strength  in  lbs.  or  tons  per  sq.  inch , a area  of  one  head  or  flange  exclusive  of  cen- 
tral portion  composing  web , in  sq.  ins.,  d'  depth  or  distance  between  centres  of  heads, 
d depth  of  rail , t thickness  of  web,  l distance  between  supports , all  in  ins. , and  W 
weight  in  lbs.  or  tons,  alike  to  S. 

Illustration  i.— What  is  destructive  weight  of  a wrought-iron  double-headed 
rail,  5.4  ins.  deep,  having  a web  of  .8  ins.,  an  area  of  head  of  1.9  sq.  ins.,  distance 
between  centres  of  its  heads  4.2  ins.,  and  between  its  supports  5 feet? 

S assumed  at  50000  lbs. 


Then 


50000  ^4  X 1.9  X + 1. 155  X .8  x 5-42^ 


5 X 12 


_ 50000  X (25.23  -f-  26.93) 
60 


43  466.6  lbs. 

2. — What  is  destructive  weight  of  a Bessemer  steel  double-headed  rail,  5.4  ins. 
deep,  having  a web  of  .75  inch,  an  area  of  head  of  2 sq.  ins.,  and  distance  between 
heads  4.2  ins.  ? 

S assumed  at  80000  lbs. 


Then 


80000  ^4  X 2 X + z*  J55  X .75  X 5-42^ 


5 X 12 


80000  X 5*-39 

60 


: 68  520  lbs. 


Note.— Transverse  strength  of  Bessemer  Rails  increases  very  generally,  in  direct  proportion  with 
the  proportion  of  Carbon  in  it. 

TTn  symmetrical  Section. 

W.  d"  representing  vertical  distance  between  centres  of  tension 


6. 92  S d"  A 
Ih  " 


and  compression , h height  of  neutral  axis  above  base  of  section , and  l length  between 
supports,  all  in  ins.,  and  A sum  of  products,  obtained  by  multiplying  areas  of  strips 
of  reduced  section  under  tensile  stress,  by  their  mean  distances,  respectively,  that  is, 
the  distances  of  their  centres  of  gravity,  from  the  neutral  axis,  in  ins. 

Bowstring  Griixler. 

To  Compute  Diameter  of  a Wrought-iron  Tie-rod.  of  an 
Arched  or  Bowstring  Grirder  of  Cast  Iron. 

/ W l 

. / -~d.  W representing  weight  distributed  over  beam  in  lbs.,  I length 

V 45°°  X h 

between  piers  or  supports  in  feet,  and  li  height  between  centre  of  area  of  section  of 
girder  and  centre  of  rod  in  ins. 

Illustration. — Required  diameter  of  tie-rod  for  an  arched  girder,  25  feet  be- 
tween its  piers,  and  30  ins.  between  centres  of  its  area  and  of  rod,  to  safely  support 
a uniformly  distributed  load  of  25  000  lbs.  ? 

V25  000X25  /625000  . , 

— . / = y/ 4*62  = 2. 1 5 ms. 

4500X3°  V 135°°° 

If  two  rods  are  used.  Then  = 1.52  ins.  = diameter  of  each  rod. 


STRENGTH  OF  MATERIALS. TRANSVERSE.  8 I 3 


CAST  IRON. 

Transverse  Strength  of  Grirders  and  Beams. 
(.Deduced  from  Experiments  of  Barlow , Hodgkinson , Hughes , Bramah , Cubitt, 
' Tredgold , and  others. ) 

Reduced  to  a Uniform  Measure  of  One  Foot  in  Length. 

Supported  at  Both  Ends.  Stress  or  Weight  applied  in  Middle. 


Destructive  Weight. 

1 w 

Section. 

Flanges. 

Web. 

Depth. 

Distance. 

Area. 

For  Dis- 
tance. 

Length 
of  One  Foot. 

_=c. 

Ad 

Ins. 

Ins. 

Feet.  Ins. 

Sq.  Ins. 

Lbs. 

Lbs. 

1 

1 

1 

1 

2 240 

2 240 

2240 

1 

X 

4 

6 

1 

500 

2250 

2250 

IS]  | 

_ 

3 

3 

13 

6 

9 

5080 

68  580 

2540 

p ( 



1 

3 

4 

6 

3 

5 100 

22  950 

2550 

1 1 

- 

1 

4 

4 

6 

4 

10300 

46350 

2896 

+ 

4 X 2 

2 

4 

5 

12 

6 720 

33  600 

700 

n 

1.52  X -78 

1.56 

4.07 

4 

6 

2.35 

6666 

30  000 

3136 

JL 

1.5  x .5 

•5 

3 

3 

1 

2 

5 208 

16145 

2676 

T 

15  X -5 

•5 

3 

3 

1 

2 

4 536 

14062 

2331 

X 

1.5  x .5 

•5 

4 

3 

1 

1 

7104 

22420 

5475 

T 

i- s'  x .5 

•5 

4 

3 

1 

1 

3 312 

10  267 

2553 

X 

i-53X  1 

•5 

2.04 

4 

2.6 

4004 

16  016 

3019 

H 

2 X .51 

1 

2.02 

4 

2-59 

2569 

10276 

1963 

# 

- 

- 

2.52 

5 

4.98 

4i43 

20715 

1650 

- 

- 

2.83 

5 

4 

2 988 

14940 

1320 

JL  1 

2.28  x -53 

f *3 
l -425 

} 5i3 

4 

6 

2.28 

9 503 

42  763 

3656 

X \ 

23.9  X 3-i2 

3-29 

36.1 

20 

i83-5 

403  312 

8 066  240 

1220 

I 

1.76  X .4 

.29 

5-i3 

4 

6 

2.82 

6678 

30512 

2077 

■f 

1.74  X .26 
1.78  X .55 

5- 13 

4 

6 

2.87 

7368 

33  200 

2250 

jj 

1.07  X .3 
2.1  X .57 

} -32 

5-i3 

4 

6 

3.02 

8 270 

37  215 

2402 

1 

i-54  X .32 
6.5  X .51 

} -34 

5-i3 

4 

6 

5-41 

21 009 

94  54° 

3406 

<( 

2.5  X 1.5 
1 3-75  x 1.4 

j.-25 

8.18 

11 

i5 

35  620* 

391  853 

3i93 

* Stirling  iron. 

Hence,  — - — = W.  A representing  area  of  section,  d depth  in  ins.,  I length  in  feet, 
and  W destructive  weight  in  lbs. 

Note. — When  lengths  are  less  than  those  instanced,  breaking  weight  will  be  in* 
creased,  in  consequence  of  increased  stability  of  girder. 


8 14  STRENGTH  OF  MATERIALS. — TRANSVERSE. 


To  Compute  Transverse  Strength,  or  Destructive  Stress 

of  Cast-iron  Beams  or  Girders,  of  various  Figures. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 

When  Section  of  Beam  or  Girder  is  alike  to  any  of  Examples  given  in 
preceding  Table.  Rule  i.*— Divide  product  of  area  of  section  and  depth 
in  ins.,  and  Coefficient  for  girder,  etc.,  from  preceding  Table,  by  length  be- 
tween supports  in  feet,  and  quotient  will  give  breaking  weight  in  lbs. 

Example. — Dimensions  of  a beam,  having  top  and  bottom  flanges  in  proportion 
of  i to  6,  give  an  area  of  section  of  25.6  sq.  ins.,  a depth  of  15.5  ins.,  and  a length 
between  its  supports  of  18  feet;  what  is  its  destructive  weight? 

Note.— In  consequence  of  increased  area  of  metal  over  case  No.  21  in  Table,  Coef- 
ficient of  3402  is  reduced  to  3300. 

Dimensions. — Top  flange,  3 x .75  ins-  bottom,  18  X .75  a = 13.5  sq.  ins.  ; web, 
15-  5 X • 7 a'  = io- 8 sq.  ins. ; and  d'  = 15. 5 — . 75  = 14. 75  ins. 


Then  X 15.5  X 3300  = i32|44£  = ^ m 

10  10 

D.  K.  Clark.  — - — 5 ^ ^ = W.  a representing  area  of  bottom  flange , a' 

of  web  at  depth  d'  of  beam,  less  depth  of  bottom  flange  in  sq.  ins.,  I length  between 
supports  in  feet,  and  W destructive  weight  in  tons. 


Then  5*75  ft  X x3-5 + s 

3 X 18 


54 


: 31.71,  which  a 2240  = 


Hodgkinson’s  formula  would  give  a result  of  53491.2  lbs.,  and  Molesworth’s 
54248.3  lbs. 


Rule  2. — From  product  of  breadth  and  square  of  depth  in  ins.  of  rec- 
tangular solid,  the  dimensions  of  which  are  the  depth  and  greatest  breadth  of 
beam  in  its  centre,  subtract  product  of  breadths  and  square  of  depths  of 
that  part  of  the  beam  which  is  required  to  make  it  a rectangular  solid,  and 
then  determine  its  resistance  by  rule  for  the  particular  case  as  to  its  being  • 
supported  or  fixed,  etc. 

This  rule  is  applicable  only  in  case  referred  to,  viz.,  when  area  of  section  is  great 
compared  with  area  of  extreme  dimensions. 

Mr.  Baker,  in  case  of  a hollow  cylindrical  shaft,  where  thickness  of  metal  is  but 
one  eighth  of  extreme  diameter,  computes  result  at  but  .4  of  that  of  a solid  beam. 

This  is  in  consequence  of  resistance  to  flexure  in  hollow  beam  being  more  than 
proportionally  greater  than  in  solid. 

Example. — Take  7th  case  from  preceding  Table,  page  813,  for  length  of  one  foot. 

Coefficient  for  cold-blast  iron  = 500. 


Then  1.52  X 4.072  — 1.52  X 2.51 2 X 4 X 500  = 25.17  — 9.58  X 2000  = 31 180  lbs. 

Result  as  by  experiment,  30000  lbs. 

Note  l— These  rules  are  applicable  to  all  cases  where  flange  of  beam  is  as  shown 
in  Table,  and  beam  rests  upon  tw  o supports,  or  contrariwise,  as  to  position  of  flange, 
when  beam  is  fixed  at  one  end  only. 

2. — When  case  under  consideration  is  alike  in  its  general  character  to  one  in 
Table,  but  differs  in  some  one  or  more  points,  an  increase  or  decrease  of  metal  is  ob 
tained  by  an  increase  or  reduction  of  the  Coefficient,  according  as  the  differences  may  1 
affect  resistance  of  beam. 

3. — The  Coefficients  here  given  are  based  altogether  upon  experiments  with  Eng- 
lish iron. 


* Utility  of  these  rules  in  preference  to  those  of  Hodgkinson,  Fairbairn,  Tredgold,  Hughes,  and 
Barlow  is  manifest,  as  in  one  case  the  Coefficient  of  the  metal  is  considered,  and  in  the  other  cases  the 
metal  is  assumed  to  be  of  a uniform  value  or  strength. 

Only  variable  element  not  embraced  in  this  rule  is  that  consequent  upon  any  peculiarity  of  form  of 
section  ; as,  for  instance,  in  that  of  a Hodgkinson,  or  like  beam,  where  area  of  one  flange  greatly  ex- 
ceeds the  rest  of  section,  and  this  flange  is  other  than  below,  when  beam  rests  upon  two  supports  or  is 
fixed  at  both  ends,  or  than  above,  when  beam  is  fixed  at  one  or  both  ends. 

This  deficiency  is  met  to  some  extent  by  the  three  cases  iu  table,  where  proportion  of  flanges  are  1 to 
2,  i to  3,  and  1 to  6.5. 

t For  thick  castings  put  7,  and  put  Coefficient  same  as  tensile  strength  of  metal  in  tons  per  sq.  inch. 


STRENGTH  OF  MATERIALS. TRANSVERSE.  8 I 5 

in  a,  n"  eel  Hollow  or  Annular  Beams  of*  Symmetrical 
Sections.  (B.  K.  Clark.) 

When  Depth  is  Great  Compared  with  ThicJcness  of  Flanges. — Figs,  i,  2,  and  3. 
!.  2.  3.  X S (4  a- f- 1.155  afj  _ w a representing  area  of  one 

flange , a'  area  of  web  or  ribs , both  in  sq.  ins.,  d depth  of 
beam , less  depth  of  one  flange,  and  l distance  between  sup- 
ports, both  in  ins.,  S tensile  strength  of  metal,  and  W 
weight  between  supports,  both  in  lbs. 

When  Depth  of  Flanges  is  Great  Compared  with  Depth  of  Beam—  Figs. 

4 and  5. 

d'2 


S (40— +-  1. 155  td2) 

. _ = W.  a representing  area  of  one  flange  less 

thickness  of  web,  in  sq.  ins. , t thickness  of  web,  d!  reputed l depth  or 
distance  between  centres  of  flanges,  and  d depth  of  beam,  all  in  ins. 

When  Section  of  Circular  or  Elliptic  Beam  is  Small  Compared  with  Diam- 
eter.— Figs.  6,  7,  and  8. 

6. 


3.14  d2  t S 
l 


= W. 


b and  d representing  mean  breadth  and  depth. 

Illustration  1.— Assume  Figs.  1,  2,  and  3,  20  ins.  in  depth,  width  of  flanges  on 
top  and  bottom  ribs  5 ins.,  thickness  of  flanges  and  webs  1 inch,  and  of  sides  of 
Fig.  3 .5  inch;  length  between  supports  10  feet,  and  S 20000  lbs. ; what  would  be 
breaking  weight  of  each? 

.X200°0(4X5  + i~i55Xi8)  = 380000(20  + 20-79)  = I2g  j6g  + m 


Then- 


10  X 12 

2. -Assume  Figs.  4 and  5, 6 ins.  in  depth,  area  of  flanges  3 ins.,  widths  of  webs  1 
inch,  and  length  and  S as  in  preceding  case. 


Then 


20000  ^4  X 3 X — g b I I55  X ) 


X 6 


10  X 12 


= 20  000  X 9^5^  — x 5 263. 3 lbs. 


Assume  Fig.  6 10  ins.  in  diameter,  Fig.  7, 7.5  ins- in  depth  and  12  ins.  in  width, 

and  Fig.  8, 12  ins.  in  depth  and  7.5  ins.  in  width,  and  thickness  of  ail  metal  1 inch. 

Then  Fie  6 3- 14  X 103  X tX  20000 _ 6280000  = ^ tts.,  which  is  .4  of 

men,  mg.  ° 10X12  120 

that  of  solid  cylinder. 

, 1.57  X (-22  + 7- 52)x  1X20000  _ 6287850  _ 52  39g_75  lbs 
IO  x 12  !20 


Figs.  7 and  { 


Note For  all  ordinary  purposes,  operation  of  computing  their  strength,  by  first 

computing  that  if  [heir  circumscribing  figure,  and  then  deducting  from  it  strength 
due  to  difference  between  it  and  section  of  beam  under  computation,  will  be  su 
ficiently  accurate.  See  Illustration,  page  814. 

If  greater  accuracy  is  required,  see  page  810,  or  D.  K.  ClarFs  Manual , pp.  5I3~I7* 
Note. -To  compute  location  of  neutral  axis  of  beams  of  unsymmetrical  section, 
see  also  D.  K.  Clark,  pp.  514-15-  - 

even*™  toa  thickness  of  metal  of  one  eighth  of  diameter.  He  assigns  their  strength  so  low  as  .4  of 
that  of  solid  cylinder,  in  consequence  of  loss  of  resistance  to  flexure. 


8l6  STRENGTH  OF  MATERIALS. TRANSVERSE. 


General  Formulas  for  Destructive  Weight  of  Solid. 
Beams  of  Symmetrical  Section. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 

Line  of  Neutral  Axis  runs  through  centre  of  gravity  of  section. 

2 a f — = W,  and  1 ^ = S.  In  square  beams  for  a d put  d3.  a and  d rep- 

l ’ 2 a d r * 

resenting  area  and  depth  of  section , r radius  of  gyration  (half  depth  of  beam  = i), 
l length  of  beam  between  its  supports  in  ins.,  W destructive  weight  in  tons  or  lbs., 
and  S tensile  strength  of  material  in  like  tons  or  lbs.  per  sq.  inch. 


Illustration. — Assume  dimensions  of  cast-iron  beams,  Figs,  i,  2,  3,  4,  and  5,  as 
follows,  viz. : 1 and  2,  5 X 5 ins. ; 3,  2.5  x 10;  4,  5.64  diameter;  and  5,  7.25  x 4.39, 
or  equal  areas;  distance  between  supports  60  ins.,  and  tensile  strength  of  iron  = 
20000  lbs. 


Areas  of  each  25  sq.  ins.  Radius  of  gyration,  No.  1,  .5775;  2,  .4083;  3,  .5775;  -< 
.5;  and  5,  1.43. 


2 X 25  X 10  X -5775  X 26 000 


60 


= 125  125  lbs. 


2 X 25  X 7-°7*  X .4083  X 26000 


60 


= 62  545  lbs. 


3.  2 ^ 53 1 X • 5775  X 26 poo _ _6^  ^ 


60 


4.  For  formula  for  square  beams  substitute  _ w 


25X5-64X26000  .7854  6 d2S  _ 

Then  4.  — = 57  766  lbs.;  and  for  5.  - ^ — W. 


■ 7854  X 4-39  X 7«252  X 26000 


60 


- = 78  532  lbs. 


These  formulas  give  a result  equal  to  a transverse  strength  for  Cast  iron  of  550  for 
a tensile  strength  of  26 000  lbs.,  and  of  Wrought  iron  of  600  lbs.  for  a like  strength 
of  50  000  lbs.  (as  per  table,  page  788). 


4 C b d2  _ _ 

= W.  C representing  coefficient  of  strength  of  metal  in  lbs.,  b and  a 


breadth  and  depth  in  ins.,  I length  in  feet,  and  W destructive  weight  in  tons. 


R4:  7*4 

6.  — p — 4 7 — 6 d'z.  R and  r representing  external  and  internal  radius. 


b d3  — 5' d'3 

= b d2.  b and  d/  representing  interior  breadth  and  depth. 


8.  .38  R 3 — bd2. 


b d2, 

9.  — W.  d representing  depth  or  height. 

4 


10.  bd2  -j-  2 b' d'2  — W.  b and  d representing  breadth  and  depth  of  centre  and 
vertical  rib,  and  b'  and  d'  breadth  and  depth  of  horizontal  rib , external  to  central  rib. 

Values  of  C 550  for  a tensile  strength  of  Cast  Iron  of  26000  lbs.  per  sq.  inch,  and 
of  600  for  a like  strength  of  Wrought  Iron  of  50000  lbs.,  and  pro  rata. 


* Diagonal  of  aquare. 


f In  square  beams  d*  = ay.4. 


STRENGTH  OF  MATERIALS. TRANSVERSE.  8l7 

Flanged  Beams  of  Unsymmetrioal  Section.  (D.  K.  Clark.) 

X L i T 

i!i  = w s representing  total  tensile  strength  of  section  in  lbs.  per  sq.  inch,  d 
vertical  distance  between  centres  of  tension  and  compression  in  ins.,  I length  in  ins., 

and  weight  in  Ws.  sectional  area  of  a beam  0f  cast  iron  is  5.9  sq.  ins.,  the 

“e 

4x^.9x30000x5.6  _ 3964800  _ 6oo?2  7 m 
lnen  5-5X12  66 

STEEL. 

rpQ  Compute  Transverse  Strength  of  Steel  Bars. 
Supported  at  Both  Ends.  Weight  applied  in  Middle. 

1. 155  S hd2  w g r epresenting  tensile  strength  in  lbs .,  I length  between  supports 

in  ins.  and  W weight  in  lbs. 

Illustration What  is  ultimate  destructive  stress  of  a bar  of  Crucible  steel, 

2 ins.  square,  and  2 feet  between  supports  ? s = 9°  000  lbs- 

Then  155  X 9°°°°  X s3  _ gjigoo  _ fien  as 
2X12  24 

To  Compute  Section  of  Lower  Flange  of  a Girder  or 
Cylindrical  Shaft  of  Cast  Iron  to  Sustain  a Sate  Load 
in  its  ^Middle.  {Baker.) 

1 d w _ M 1 representing  distance  between  supports  in  feet,  d depth  of  girder,  etc. , 
in  ins.,  W weight  in  tons,  C coefficient,  and  M moment  of  weight  around  support. 

Illustration. -What  should  be  section  of  a girder  12  ins  deep,  to  sustain  a safe 
load  of  10  tons  in  its  middle,  between  supports  16  feet  apart  t 


Stress  assumed  2 tons  per  sq.  inch,  and  Factor  of  safety  4. 
M 


16  X 12  X 10 


1:480—  M. 


An(}  ^ — a.  S representing  stress  assumed  in  tons,  and  a area  of  section  of 

d X S 
s in  sq.  ins. 


Then  ■ 48  — = zosq.ins. 
12  X 2 


For  Rectangular,  Diagonal,  or  Circular  Beam  or  Shaft. 


d2  b 


— M. 


d 3 


= M. 


General  Formulas  for  Computation  of  Destructive 
Weight  of  a Beam  or  CGirder  of  any  form  of  Cross 
Section  and  of  any  Material.  ( B . Baker.) 

Load  applied  at  Middle. 

S M (1  -j-  Q ) _ g representing  tensile  strength  of  material  per  sq.  inch  in  tons, 

M moment  of  resistance  of  section  = product  of  effective  depth  of  girder  orteam,  and 
effective  areu  of  flange  portion  of  section,  in  sq.  ins.,  Q resistance  due  to  flexure,  l dis- 
tance between  supports  in  feet,  and  Q'  = QX  thickness  of  web  of  section,  both  in  ms. 

Average  Values  of  S for  Various  Materials . 

Tons  I Tons.  I Tons. 

7 Steel 40  to  50  Oak 2.5  to  4.5 

..2I  | “ plates 35  1 Pine 2 3-5 

3Z 


Cast  Iron 

Wrought  Iron 


8 1 8 


STRENGTH  OF  MATERIALS. TRANSVERSE. 


Substituting  Values  of  S and  Q in  a General  Equation. 


Section. 

Cast  Iron. 

W rought 
Iron. 

Steel. 

Oak. 

Pine. 

• 

W=.S75^ 

d3 

W=.  5625  ~ 

d2b 

=I‘75T 

d3 

~lV5T 

d3 

= I.!25T 

d2b 

=3  t05  — 

=2.625  to  4.25  — 

* d3 

=2  to  3. 25  — ' • 

+ d2b 

=.14  to. 2 5 — 

d3 

=.1  to  . 16  —j- 

0*  d3 

=.08  to.  14  — 

. d2  b 

= . II  tO. 2 -y- 
o+  d 3 

=.08  to.  14  — 
d 3 

=.06  to  . II  — 

d representing  depth  of  a rectangular  bar , side  of  a square , or  diameter  of  a round 
b breadth  of  a vertical  bar , all  in  ins.,  and  l distance  between  supports  in  feet.  ’ 


Moment  of  Resistance. 


Moment  of  Resistance  of  a cross  section  is  the  static  force  resisting  an  ex- 
ternal force  of  tension  or  compression,  and  it  is  equal  to  moment  of  Inertia, 
divided  by  distance  of  centre  of  effect  of  the  area  of  fibres  which  are  respec- 
tively the  most  extended  or  compressed  from  the  neutral  axis  of  the  section. 

I To  Compute  Moment  of  Resistance. 

— = M.  I representing  moment  of  inertia,  and  d distance  of  centre  oj  effect  of 
area  of  fibres  of  extension  or  compression. 

Worls:  of  Resistance. 

Under  a Quiescent  Load. — Intensity  of  Elastic  resistance  increases  uni- 
formly with  total  space  through  which  action  of  stress  operates ; hence,  it 
may  be  defined  by  a triangular  section. 

Consequently,  . 5 s L ==  R.  5 representing  space  passed  through,  L load,  and  R re- 
sistance. 

To  Compute  Moment  of  Resistance. 

6CI  3 MI  M 0 

— and  = R.  C a coefficient  = one  sixth  of  destructive  weight , I moment 

of  inertia,  h height  of  neutral  axis  from  base  of  section,  R moment  of  resistance,  and 
M modulus  of  rupture. 

Note.— Neutral  axis,  for  all  practical  purposes,  is  at  centre  of  gravity  of  any 
section. 

For  Radius  of  Gyration,  see  Centre  of  Gyration,  page  609. 

For  other  rule  for  computation  of  Moment  of  Resistance,  see  Strength  of  Beams. 
B.  Baker,  London,  1870. 


Moment  of  Inertia. 

Moment  of  Inertia  is  resistance  of  a beam  to  bending,  and  moment  of  any 
transverse  section  is  equal  to  sum  of  products  of  each  particle  of  its  area  into 
square  of  their  distance  from  neutral  axis  of  section. 


Illustration.— If  transverse  section  of  a beam,  A B C D,  Fig.  1,  is 
8 X 20  ins.,  its  neutral  axis  will  be  at  middle  of  its  depth,  0 r;  divide 
A B,  o r,  into  any  number  of  equal  spaces,  as  shown,  then  each  space 
will  be  2X2  = 4 sq.  ins.,  and  the  distances  of  the  centre  of  each 
square  from  neutral  axis  will  be  as  follows  : 


1,  1.  2X2X4X  i2=  16 

2,  2.  2 X 2 X 4 X 32  = 144 
3,3.  2 X 2 X 4 X 52  = 4oo 


4,  4.  2 X 2 X 4 X 72=  784 
5,5.  2 X 2 X 4 X 92— 1296 

2640  x 2 for  low- 
er half  = 5280  ==  moment. 


Note.— If  the  area  of  the  figure  in  illustration  had  been  more  minutely  divided, 
the  result  would  have  approximated  more  nearly  to  the  above  result. 


For  Moment  of  Inertia  of  a Revolving  Body , see  Centre  of  Gyration,  page  609. 


i 

i 


1 


STRENGTH  OF  MATERIALS. TRANSVERSE.  819 


To  Compute  .Moment  of  Inertia  of  a Solid  Beam.— Fig.  2. 

Illustration.—’ Take  elements  of  preceding  case. 

Then  = = 5333.33  moment. 

12  12 

Or  a <3  n3  6 = M t representing  breadth  of  vertical  divisions  n number  of  hori- 
zontal divisions  from  plane  of  neutral  axis , b breadth , and  d depto  0/  beam. 
Illustration.— Take  elements  of  preceding  case. 

t = 2,  n==  5,  and  6 = 8. 

Then  .3X2SX53X8  = 2400  X 2 for  lower  half=:  4800  = moment 

3.  4.  5.  Beams  of  Various  Figures. — Figs.  3,  4,  5. 

fed3  — &'d'3  & d3  — 2 &'  d'3 

fc.  3-  — ,4  and  5.- -- -M. 

5'  and  d'  representing  respectively  breadth  less 
thickness  of  web,  and  depth  less  thicJcness  of  flanges. 


.7854  r 2 = M. 


.7854  c t3  — M. 


.3  *><*3 


— L-  = M. 


Q *7854  (r*  — r'*)'  = M.  ^ - = M-  “f!  *11  r2  = M. 

t representing  rcidius , £ transverse  and  c conjugate  diameters,  and  s side. 

To  Compute  Common  Centre  of  Gravity  and  Vertical 
Distance  between  Centres  of  Crushing  and  Tensile 
Stress  of  a.  Grirder  or  Beam. 

Rule.  — Multiply  surface  of  section  of  each  part,  or  figure  composing 
whole,  by  distance  of  its  centre  from  centre  of  one  of  the  two  extreme  parts 
or  figures,  as  . ; divide  sum  of  their  products  by  sum  of  surfaces  of  sec- 
tion, and  result  will  give  distance  of  common  centre  of  gravity  from  centres 
of  each  extreme  part  or  figure. 

Example.— Take  annexed  figure. 

C 2.5  X 1 X o =2.5  Xo  = .0 


Above 


.325  X 


.325x3.31=1.076 

.38  X4X  (^  + 5-62  + 1)  = 1.52  X 6.31  = 9-591 


4.345  10.667 

Dividing  10.667  by  4.345  = 2. 455  = distance  of  common  centre  from  centre  of  upper 
part. 

1.52  Xo  1=1.52  Xo  = .0 


Below 


•325  X 5*62  X (5|?  + v)  = 1-826  X 3 =5-478 

2.5  X (|  + 5-62  + '-f)=2-5  X6.3i=25- 


v 5.846  21.253 

Dividing  21.225  by  5.846  = 3-631  = distance  of  common  centre  from  centre  of  lower 
part. 

Hence,  3.631  + — = 3.821  = distance  of  common  centre  from  bottom , and  3.631  + 
2.652  = 6.283  =s  distance  between  centres  of  gravity. 


820  STRENGTH  OF  MATERIALS. TRANSVERSE. 

To  Compute  Neutral  Axis  of  a Beam  of  Unsymmetrical 
Section.— Figs.  3,  4,  5,  6,  7,  8,  and  9.  (D.  K.  Clark.) 

Operation.— Divide  section  as  reduced  into  its  simple  elements  and 
assume  a datum-line  from  which  moments  of  elements  are  to  be  computed. 
Multiply  area  of  each  element  by  distance  of  its  own  centre  of  gravity  from 
datum-line,  to  ascertain  its  moment.  Divide  sum  of  these  moments  bv  to- 
tal reduced  area ; and  quotient  is  distance  of  centre  of  gravity  of  reduced 
section,  or  of  neutral  axis  of  whole  section,  from  datum-line. 

Illustration.— Fig.  8 annexed  is  i2  ins.  deep,  i2  ins.  wide,  and  i inch  thick. 

Extend  web,  c d,  to  the  lower  surface  at  d'  and  d",  leaving  5.5  ins! 
of  web,  a d'  and  d"  b , on  each  side.  Reduce  this  width  in  the  ratio 
of  1.73  to  1,  or  to  (5.5-1-  1. 73=)  3. 2 ins.,  and  set  off  d'  a'  and  d"  b' 
each  equal  to  3. 2 ms.  Then  reduced  flange,  a'  b',  is  (3. 2 x 2 = 6. 4 -f 
1 =)  7-4  ins-  wide,  and  reduced  section  consists  of  two  rectangles 
a ' b'  and  c d.  Assume  any  datum-line,  as  ef  at  upper  end  of  sec- 
tion, and  bisect  depths  of  rectangles,  or  take  intersections  of  their 
diagonals  at  g and  o,  for  their  centres  of  gravity.  Distances  of  these 
from  datum-line  are  5.5  and  11.5  ins.  respectively,  and  areas  of  the 
1 rectangles  are  nXi^nsq.  ins.,  and  7.4  x 1 = 7.4  sq.  ins. 

Then,  cd—n  x 5.5=60.5 
a'  ~ 7-4  X 11.5=  85.1 

18.4  145.6  = 7.91  ins. 

Showing  that  centre  of  gravity  of  reduced  section,  being  neutral  axis  of  whole 
section,  is  7.91  ins.  below  upper  edge,  in  line  it.  Centre  of  gravity  of  entire  section 
at  . , it  may  be  added,  is  8.65  ins.  below  upper  edge,  or  .74  inch  lower  than  that  of 
reduced  section. 

Neutral  axes  of  other  sections,  Figs.  3 to  7,  found  by  same  process,  are  marked  on 
the  figures.  Section  of  a flange  rail,  No.  7,  which  is  very  various  in  breadth,  may  be 
treated  in  two  ways:  either  by  preparatorily  averaging  projections  of  head  and 
flange  into  rectangular  forms;  or,  by  taking  it  as  it  is,  and  dividing  it  into  a con- 
siderable number  of  strips  parallel  to  base,  for  each  of  which  the  moment,  with  re- 

STlftPit  to  ASRIIITIPH  Hntnm.Hno  1C?  in  ooonrfoinA/1  F i ^ a • 


u-  e-' 

/ 

f 

-9 

d 

1 u 

-~rr 

TT~ f 

0 a' d' d"  b'  b 

paicwiei  tu  u u&u,  iui  eauu  ui  wiiicTi  ine  moment,  witn  re- 
spect  to  assumed  datum-line,  is  to  be  ascertained.  First  mode  of  treatment  is  ap- 
proximate; second  is  more  nearly  exact. 

To  Compute  Ultimate  Strength  of  Homogeneous  Beams 
of  Unsymmetrical  Section. 

Operation. — Resuming  section,  Fig.  9,  for  which  neutral  axis  has  been 
ascertained, 

To  Compute  Tensile  Resistance , 

Divide  portion  below  neutral  axis  i 1,  Fig.  9,  with  reduced  width  of 
flange,  a'  b\  into  parallel  strips,  say  .5  inch  deep,  as  shown, 
and  multiply  area  of  eacli  strip  by  its  mean  distance  from 
neutral  axis  for  proportional  quantity  of  resistance  at 
strip.  Divide  sum  of  products,  amounting  in  this  case 
to  31.3,  by  extreme  depth  below  neutral  axis  = 4.09  ins., 
and  multiply  quotient  by  1.73  S (ultimate  tensile  resist- 
rTF-"^  T~~1  ance  at  lower  surface).  ‘The  final  product  is  total  tensile 
of  v resistance  of  section ; or, 

31.3  X i • 73  S 

— — r — ==  13- 24  S total  tensile  resistance. 

4.°9 

S representing  ultimate  tensile  strength  of  material  per  sq.  inch. 

Again,  multiply  area  of  each  strip  by  square  of  its  mean  distance  from  neu- 
tral axis,  and  divide  sum  of  these  new  products,  amounting  to  104.64,  by 
sum  of  first  products.  The  quotient  is  distance  of  resultant  centre  of  tensile 
stress,  d',  from  neutral  axis.  Or,  resultant  centre  is, 

104  ^4  = 3. 34  ins.  below  neutral  axis. 

3i-3 

This  process  is  that  of  ascertaining  centre  of  gravity  of  all  the  tensile  resistances. 


— vr 

1 

y: 

x-t — 

— i 

X 

STRENGTH  OF  MATERIALS. TRANSVERSE.  821 

By  a similar  process  for  upper  portion  in  compression,  sum  of  first  products  is 

ascertained  to  be  same  as  for  lower  portion  = 3i-3- 

as 

,„SX  ^ = 3-34  8,  and  3£-3  X 3-34_g  ^ S total  compressive  resistance, 

wbich  is4same  as  total  tensile  resistance,  in  conformity  with  general  law  of  equal- 
ity of  tensile  and  compressive  stress  in  a section. 

Sum  of  products  of  areas  of  stress,  divided  by  squares  of  tlieir  distances  respec- 
tively from  neutral  axis,  is  164.9,  and  resultant  centre  c,  Fig.  9,  is  -——  = 5.27 
ins.  above  neutral  axis. 

onm  nf  distances  of  centres  of  stress  or  of  resistance  from  neutral  axis,  3.34  + 

, 27  = 8 6 ^‘“Ssmnceapart  of  these  centres  as  represented  by  central  line.c  d. 

Abbreviated  Computation. -As  upper  part  of  section  is  a rectangle,  its  resultant 
centre  =4  of  height,  or  7.91  X f = 5-  27  ins.  above  neutral  axis  Average  resist- 
ance  is  half  maximum  stress,  viz.,  that  at  upper  portion,  which  is  3.34  S per  sq. 

^nCk'  ! 7.9IX3-34S  O 

Area  of  rectangle  therefore  =17.91  X 1 = 7.91  sq.  ins.,  and  — p ~ — 13.21 

compressive  resistance , as  before  determined.  ^ ^ 4 S cZ 

Moment  of  tensile  resistance  = 13. 21  X 8.61  ins.  = 113. 76  S,  also  = — , or  —j~  — 

W.  S representing  total  resistance  of  section  in  lbs.,  d vertical  drawee  apart  of 
centres  of  tension  and  compression , and  l length  between  supports , all  m in  . 

Strength- of  Beam  Inverted.—  When  inverted,  maximum  tensional  resistance  of 
beam  at  its  lower  surface  c,  Fig.  8,  is  1.73  S. 

„ t „ ari  :nQ  nnfi  7 91  X t. 73  o g i0fai  tensile  re- 

Area  of  rectangle  nc  — 7.91  sq.  ins.,  ana  - . , , ^ — °-79  0 

sistance,  or  about  one  half  of  beam  in  its  normal  position. 

Note.— For  other  rule  for  computation  of  centre  of  gravity,  see  Strengt 
Loudon, 1870. 


Metals. 


Density. 


( Least 

Cast  Iron....  'Greatest. 

( Mean 

Wrought  Iron  { Neatest! 

Cast  Steel. .. . j Greatest. 

Bronze (Greatest. 


6.9 

7-4 

7.225 

7- 7°4 
7.858 
7.729 

8- 953 
7.978 
8-953 


Compres-  TenBile. 
sion.  1 


Sq.  Ins. 
84529 
174  120 
144916 
40  OOO 
127  720 
198944 
391  985 


Sq.  Tns. 
9000 

45  97° 
31  829 
38027 

74  592 

128  OOO 
17  698 
56786 


Torsion. 


Trans- 

verse. 


Sq.  Ins. 

8 614 
2913 
3 643 


28  280  1916 

1 852  — 

2656  | — 


Sq.  Ins. 
416 
958 
680 
542 


mis,  etc.  B.  Baker , 

Major  Wade. 

Tensile 

Hard- 

to Com- 

ness. 

pression. 

I to  9.4 

4-57 

1 “ 3.8 

33-51 

1 “ 4.6 

22.34 

I “ I 

10.45 

I “ 1.7 

12. 14 

1 to  3.1 

— 

; — 

4-57 

— 

5-94 

IT  actors  of  Safety. 

Girders , Beams,  etc.,  of  cast  iron  should  not  be  subjected  to  a greater  stress 
than  one  sixth  of  their  destructive  weight,  and  they  should  not  be  subjected 
to  an  impulsive  stress  greater  than  one  eighth. 

The  following  are  submitted  by  English  Board  of  Trade,  Commission- 
ers,  etc. 


Structure. 


Cast  Iron. 

Girders 

Columns 

Tanks 

Machinery 


I Stress. 

Factor. 

j Structure. 

Stress. 

Factor. 

- — 

1 

Dead 

3 to  6 

I Wrought  Iron. 
Girders 

Dead 

3 

Live 

6 

• 

0 

Bridges 

Mixed 

4 

Live 
. ! Shock 

4 

8 

10 

Steel, 

S Bridges 

Mixed 

4 

822  STRENGTH  OP  MATERIALS. TRANSVERSE. 


Girders,  Beams,  Lintels,  etc. 

Transverse  or  Lateral  Strength  of  any  Girder,  Beam,  Breast-summer 
Lintel,  etc  is  in  proportion  to  product  of  its  breadth  and  square  of  its 
depth,  and  area  of  its  cross-section. 

Best  form  of  section  for  Cast-iron  girders  or  beams,  etc.,  is  deduced 
from  experiments  of  Mr.  E.  Hodgkinson,  and  such  as  have  this  form  of 
section  mT_  are  known  as  Hodgkinson’s. 

7 R^7e  deduc.ed  fr07m  his  experiments  directs,  that  area,  of  bottom  flanqe 
should  be  6 times  that  of  top  flange— flanges  connected  by  a thin  ver- 
tical web,  sufficiently  rigid,  however,  to  give  the  requisite  lateral  stiff- 
ness tapering  both  upward  and  downward  from  the  neutral  axis  - and 
in  order  to  set  aside  risk  of  an  imperfect  casting,  bv  any  great  dispro- 
portmn  between  web  and  flanges,  it  should  be  tapered  so  as  to  connect 
with  them,  with  a thickness  corresponding  to  that  of  flange. 

As  both  Cast  and  Wrought  iron  resist  compression  or  crushing  with  a 
grea  er  force  than  extension,  it  follows  that  the  flange  of  a girder'or  beam 
of  +1  2 • °,f  thes?  meta.ls*  which  is  subjected  to  a crushing  strain,  according 
as  the  girder  or  beam  is  supported  at  both  ends , or  fixed  at  one  end , should  be 
of  less  area  than  the  other  flange,  which  is  subjected  to  extension  or  a ten- 
siiB  stress. 

When  girders  are  subjected  to  impulses,  and  sustain  vibrating  loads,  as  in 
budges,  etc.,  best  proportion  between  top  and  bottom  flange  is  as  i to  4 • as 
a general  rule,  they  should  be  as  narrow  and  deep  as  practicable,  and  should 
never  be  deflected  to  more  than  .002  of  their  length. 

^ub,lc  Halls,  Churches,  and  Buildings  where  weight  of  people  alone 
t ptr“vided  .f,or>  an  estimate  of  175  lbs.  per  sq.  foot  of  floor  surface 

is  sufficient  to  provide  for  weight  of  flooring  and  load  upon  it.  In  comput- 
ing other  weight  to  be  provided  for  it  should  be  that  which  may  at  any  time 
bear  upon  any  portion  of  their  floors;  usual  allowance,  however,  is  for  a 
weight  of  280  lbs.  per  sq.  foot  of  floor  surface  for  stores  and  factories. 

noJedato  3X!!’-h  as, in  baildinS?  and  bridges,  where  the  structure 'is  ex- 
posed  to  sudden  impulses  the  load  or  stress  to  be  sustained  should  not  ex- 
ceed horn  .2  to  .16  of  breaking  weight  of  material  employed ; but  when  load 
weight*™  °r  Str6SS  C1U,eScent:  11  may  be  increased  to  .3  and  .25  of  breaking 

An  open-web  girder  or  beam,  etc.,  is  to  be  estimated  in  its  resistance  on 
he  mad1®  PnnclPle  as  lf  had  a solid  web.  In  cast  metals,  allowance  is  to 
and  flanges  °f  Strength  due  to  uneclual  contraction  in  cooling  of  web 

In  Cast  Iron,  the  mean  resistances  to  Crushing  and  Extension  are,  for 
American  as  4.55  to  1,  and  for  English  as  5.6  to  7 to  1 ; and  in  Wrought  Iron 
't;  American  as  1.5  to  1 and  for  English  as  ,.2  to  1 ; hence  the  inass  of 
metal  below  neutral  axis  will  be  greatest  in  these  proportions  when  stress  is 
intei  mediate  between  ends  or  supports  of  girders,  etc. 

„r]V™td ™.Girder*  or  Beams,  w hen  sawed  in  two  or  more  pieces,  and  slips 
® Ijctw®en  them,  and  whole  bolted  together,  are  made  stifler  by  the 
operation,  and  are  rendered  less  liable  to  decay. 

nr^orrteionCofTWrnh  ‘A®?  are  stronSer  tba"  "'hen  cast  on  a side,  in  the 
flange  up  f * aad  1 iey  are  stronSest  also  when  cast  with  bottom 

Most  economical  construction  of  a Girder  or  Beam,  with  reference  to  at- 
taining greatest  strength  with  least  material,  is  as  follows : The  outline  of 


STRENGTH  OF  MATERIALS. TRANSVERSE.  823 


top  bottom,  and  sides  should  be  a curve  of  various  forms,  according  as 
breadth  or  deptli  throughout  is  equal,  and  as  girder  or  beam  is  loaded  only 
at  one  end,  or  in  middle,  or  uniformly  throughout. 

Breaking  Weights  of  Similar  Beams  are  to  each  other  as  Squares  of  their 
like  Linear  Dimensions. 

By  Board  of  Trade  regulations  in  England,  iron  may  be  strained  to  5 tons 
per  sq.  inch  in  tension  and  compression,  and  by  regulation  of  the  Ponts  et 
Chaussees,  France,  3.81  tons. 

Rivets  .75  and  1 inch  in  diameter,  and  set  3 ins.  from  centre  in  top  of 
girder,  and  4 ins.  at  bottom. 

Character  of  fracture,  as  to  whether  it  is  crystalline  or  fibrous,  depends 
upon  character  of  blows ; thus,  sharp  blows  will  render  it  crystalline,  and 
slow  will  not  disturb  its  fibrous  structure. 

For  spans  exceeding  40  feet,  wrought  iron  is  held  to  be  preferable  to 
cast  iron. 

Riveting,  when  well  executed,  is  not  liable  to  be  affected  by  impact  or 
velocity  of  load. 

A Coupled  Girder  or  Beam  is  one  composed  of  two,  fastened  together,  and 
set  one  over  the  other. 

Trussed  Beams  or  Grirclers. 

Wrought  and  Cast  Iron  possess  different  powers  of  resistance  to  tension  and  com- 
pression- and  when  a beam  is  so  constructed  that  these  two  materials  act  in  uni- 
son with  each  other  at  stress  due  to  load  required  to  be  borne , their  combination  will 
effect  an  essential  economy  of  material.  In  consequence  of  the  difficulty  of  adjust- 
ing a tension-rod  to  the  stress  required  to  be  borne,  it  is  held  to  be  impracticable  to 
construct  a perfect  truss  beam. 

Fairbairn  declares  that  it  is  better  for  tension  of  truss-rod  to  be  low  than  high, 
which  position  is  fully  supported  by  following  elements  of  the  two  metals  : 

Wrought  Iron  has  great  tensile  strength,  and,  having  great  ductility,  it  undergoes 
much  elongation  when  acted  upon  by  a tensile  force.  On  the  contrary,  Cast  Iron 
has  great  crushing  strength,  and,  having  but  little  ductility,  it  undergoes  hut  little 
elongation  when  acted  upon  by  a tensile  stress-,  and,  when  these  metals  are  re- 
leased from  the  action  of  a high  tensile  stress,  the  set  of  one  differs  widely  from 
that  of  the  other,  that  of  wrought  iron  being  the  greatest. 

Under  same  increase  of  temperature,  expansion  of  wrought  is  considerably  great- 
er than  that  of  cast  iron;  1.81*  tons  per  sq.  inch  is  required  to  produce  in  wrought 
iron  same  extension  as  in  cast  iron  by  1 ton. 

Fairbairn,  in  his  experiments  upon  English  metals,  deduced  that  within  limits 
of  stress  of  13440  lbs.  per  sq.  inch  for  cast  iron,  and  30240  lbs.  per  sq.  inch  for 
wrought  iron,  tensile  force  applied  to  wrought  iron  must  be  2.25  times  tensile  force 
applied  to  cast  iron,  to  produce  equal  elongations. 

Relative  tensile  strengths  of  cast  and  wrought  iron  being  as  1 to  1.35,  and  their 
resistance  to  extension  as  1 to  2.25,  therefore,  where  no  initial  tension  is  applied  to 
a truss-rod,  cast  iron  must  be  ruptured  before  wrought  iron  is  sensibly  extended. 

Resistance  of  cast  iron  in  a trussed  beam  or  girder  is  not  wholly  that  of  tensile 
strength,  but  it  is  a combination  of  both  tensile  and  crushing  strengths,  or  a trans- 
verse strength;  hence,  in  estimating  resistance  of  a trussed  beam  or  girder,  trans- 
verse strength  of  it  is  to  be  used  in  connection  with  tensile  strength  of  truss. 

Mean  transverse  strength  of  a cast-iron  bar,  one  inch  square  and  one  foot  in 
length,  supported  at  both  ends,  stress  applied  in  the  middle,  without  set , is  about 
900  lbs. ; and  as  mean  tensile  strength  of  wrought  iron,  also  without  set , is  about 
20000  lbs.  per  sq.  inch,  ratio  between  sections  of  beams  and  of  truss  should  be  in 
ratio  of  transverse  strength  per  sq.  inch  of  beam  and  of  tensile  strength  of  truss. 

Girders  under  consideration  are  those  alone  in  which  truss  is  attached  to  beam 
at  its  lower  flange,  in  which  case  it  presents  following  conditions: 


* Elongation  of  cast  and  wrought  iron  being  5500  and  10000,  hence  10  000  -r-  5500  = 1.81. 


824  STRENGTH  OF  MATERIALS. TRANSVERSE. 


1.  When  truss  runs  parallel  to  lower  flange.  2.  When  truv ? nt  ™ ,• 

to  lower  flange,  being  depressed  below  its  centre.  3.  When  beam  is  archeTunwnvd 
and  truss  runs  as  a chord  to  curve.  arched  upward , 

Consequently  in  all  these  cases  section  of  beam  is  that  of  an  onen  one  with  a 
cast-iron  upper  flange  and  web,  and  a wrought-iron  lower  flange  increased  fnTt^  re 
sistance  over  a wholly  cast-iron  beam  in  proportion  to  the  increased  tensile  strength 
of  wrought  iron  over  cast  iron  for  equal  sections  of  metals.  strength 

From  various  experiments  made  upon  trussed  beams,  it  is  shown  : 

1.  That  their  rigidity  far  exceeds  that  of  simple  beams:  in  some  cases  it 
?nl°h8rnreshgreater”  2'-  Thatuwhen  truss  resilts  ruptur^, uppe“  fiSS  of  beam T 
en- by  comPr®sslon) there  is  a great  gain  in  strength.  3.  ThaUheir  strength 
s gieatly  increased  by  upper  flange  being  made  larger  than  lower  one  , Th  , t 

arei  of  mefah  'S  Sreater  tha“  tUat  ofa  wrought-iron  tubular  beam  containing'  same 

Comparative  Value  of  Wrought-iron  Bars,  Hollow 
Grirders,  or  Tubes  ol  'Various  Figures  {English). 


Circular  tubes,  riveted 

Flanged  beams x 2 

Elliptic  tubes,  riveted *.!!.!.  n 

Rectangular  tubes,  riveted * * * * * x 5 


Circular,  uniform  thickness 

Plate  beams * 

Elliptic,  uniform  thickness!.*.’.*..*.**”  jg 
Rectangular,  uniform  thickness 


General  Deductions  from  Experiments  of  Stephenson , Fairbaim  Cubitt 
Hughes , etc. 

in“irn  ®h.ows  in  his  experiments  that  with  a stress  of  about  12  320  lbs  ner  sn 

ibs- 0,1  wrought  iron’ the  sets  and  asara 

A cast-iron  beam  may  be  bent  to  .3  of  its  breaking  weight  if  load  is  laid  on 

ufiy  ’ 1? Dd  'l6  i-f1laid  on  at  once’  wi]1  Pr°duce  same  effect,  if  weight  of  beam 
compared  with  weight  laid  on.  Hence,  beams  of  cast  ironXuld  be  S 
capable  of  bearing  more  than  6 times  greatest  weight  which  will  be  laid  upon  them 
^ams  of  cast  or  wrought  iron,  if  fixed  or  supported  at  both  ends  flanffp«? 
should  be  m proportion  to  relative  resistances  of  material  to  crushing  or  extension 
Breaking  weights  in  similar  beams  are  to  each  other  as  squares  of  their  like  linear 
dimensions;  that  is,  breaking  weights  of  beams  are  computed  by  multiplying  to- 
gether area  of  their  section,  depth,  and  a Constant , determined  from  experiments  on 
beawee^si?pports.'CU  ar  f°r“  UDder  illvestigation>  and  dividing  product  by  distance 

2 4C41‘and  wrought'iron  learns,  having  similar  resistances,  have  weights  nearly  as 

ri.^0<Xflbeam/,r  girder'  eeestructed  of  plates  of  wrought-iron,  compared  to  a single 
rib  and  flanged  beam  X,  of  equal  weights,  has  a resistance  as  100  to  93.  g 

Resistance  of  beams  or  girders,  where  depth  is  greater  than  their  breadth  when 
supported  at  top,  is  much  increased.  In  some  cases  the  difference  is  fully  one  third. 
When  a beam  is  of  equal  thickness  throughout  its  length,  its  curve  of  equilibrium 

shmdd  hP  u!  • SUPP°h  Py  Unif°rm  StreSS  with  ecP,al  ^(stance  in  every  part! 
should  be  an  Ellipse,  and  if  beam  is  an  open  one,  its  curve  of  equilibrium  for  a uni! 
foi  m load  should  be  that  of  a Parabola.  Hence,  when  middle  portion  is  not  wholly 
removed,  its  curve  should  be  a compound  of  an  ellipse  and  a parabola  approaching 
nearer  to  the  latter  as  the  middle  part  is  decreased.  ? 1 P S 

iron  °f  CaSt  ir0D’  UP  t0  a SI>an  °f  40  feet’  involve  a less  cost  than  of  wrought 

Cast-iron  beams  and  girders  should  not  be  loaded  to  exceed  .2,  or  subjected  to  a 
greater  stress  than  166  of  their  destructive  weight ; and  when  the  stress  is  attended 
with  concussion  and  vibration,  this  proportion  must  be  increased. 

WnJ^Ll^t',I!?n’girl®r8*n?y  be  ™ade  50  feet  in  length>  and  best  f°rm  is  that  of 

a fixed  ioadj  flanges  sbouid  be  as  1 to  6’ whei 

Forms  of  girders  for  spaces  exceeding  limit  of  those  of  simple  cast  iron  are  vari- 
inc-ipa  0nB!' adopted  are  those  of  straight  or  arched  cast-iron  girders  in 
and  TubidarCeS’  and  b°  ted  together— Trussed>  Bowstring,  and  wroughtdron  Box 


STRENGTH  OF  MATERIALS. — TRANSVERSE.  825 


Straight  or  Arched  Girder , formed  of  separate  castings,  is  entirely  dependent 
upon  bolts  of  connection  for  its  strength. 

Trussed  or  Bowstring  Girder  is  made  of  one  or  more  castings  to  a single  piece, 
and  its  strength  depends,  other  than  upon  the  depth  or  area  of  it,  upon  the  proper 
abstinent  of  the  tension,  or  the  initial  strain,  upon  the  wrought-iron  truss. 

Box  or  Tubular  Girder  is  made  of  wrought  iron,  and  is  best  constructed  with 
cast-iron  tops,  in  order  to  resist  compression:  this  form  of  girder  is  best  adapted  to 
afford  lateral  stiffness. 

When  a girder  has  four  or  more  supports,  its  condition  as  regards  a stress 
upon  its  middle  is  essentially  that  of  a beam  fixed  at  both  ends. 

The  following  results  of  the  resistances  of  materials  will  show  how  they 
should  be  distributed  in  order  to  obtain  maximum  of  strength  with  minimum 
of  dimensions : 


To  Tension,  j 

Cast  iron 

t 21  OOO 

{ 32  OOO 

“ English.. 

( 13000 
) 23  OOO 

Granite 

578 

Limestone 

f 670 
\ 2 800 

To  Tension. 


Oak,  white,  mean. 
“ English  “ . 

Wrought  iron 


“ English 
Yellow  pine 


11  000 
6 500 
S 45  000 
1 59  000 
( 31  000 
i 53  000 
16000 


To  Crush’g. 


7500 
3 100 
47  000 
83  000 
40  000 
65000 
4000 


The  best  iron  has  greatest  tensile  strength,  and  least  compressive  or  crushing. 

Conditions  of  Forms  and  Dimensions  of  a Symmetrical 
Beam  or  Grirder. 


When  Fixed  at  One  End,  and  Loaded  at  the  Other . 

1.  When  Depth  is  uniform  throughout  entire  Length , section  at  every  point 
must  be  in  proportion  to  product  of  length,  breadth,  and  square  of  depth,  and 
as  square  of  depth  is  in  every  point  the  same,  breadth  must  vary  directly  as 
length ; consequently,  each  side  of  beam  must  be  a vertical  plane,  tapering 
gradually  to  end. 

2.  When  Breadth  is  uniform  throughout  entire  Length , depth  must  vary 
as  square  root  of  length ; hence  upper  or  lower  sides,  or  both,  must  be  deter- 
mined by  a parabolic  curve. 

3.  When  Section  at  every  point  is  similar , that  is , a Circle , an  Ellipse , a 
Square , or  a Rectangle , Sides  of  which  bear  a fixed  Proportion  to  each  other , 
the  section  at  every  point  being  a regular  figure,  for  a circle,  the  diameter 
at  every  point  must  be  as  cube  root  of  length ; and  for  an  ellipse  or  a rec- 
tangle, breadth  and  depth  must  vary  as  cube  root  of  length. 

Illustration.— A rectangular  beam  as  above,  6 ins.  wide  and  1 foot  in  depth  at 
its  extreme  end,  and  4 feet  in  length,  is  capable  of  bearing  6480  lbs. ; what  should 
be  its  dimension  at  3 feet  ? x 5g7?  and  .^/3  — 1. 442. 

Then  1.587  : 1.442  ::  1 : .9086,  and  6 and  12  X .9086  ==  5.452  and  10.9. 

TT  5.452  X 10.92  , . 6 X 122 

Hence  — 2i6,  and  = 216. 

3 4 

When  Fixed  at  One  End , and  Loaded  uniformly  throughout  its  Length. 

1.  When  Depth  is  uniform  throughout  its  entire  length , breadth  must  in- 
crease as  the  square  of  length. 

2.  When  Breadth  is  uniform  throughout  its  entire  length,  depth  will  vary 
directly  as  length. 

3.  When  Section  at  every  point  is  similar , as  a Circle , Ellipse , Square , and 
Rectangle , section  at  every  point  being  a regular  figure,  cube  of  depth  must 
be  in  ratio  of  square  of  length. 


826  STRENGTH  OF  MATERIALS. TRANSVERSE. 


Illustration.  —Take  preceding  case. 

Then  42  : 32  ::  I23  ; 9?2j  and  = ^ in  depth 

When  Supported  at  Both  Ends. 

1.  When  Loaded  in  the  Middle,  Coefficient  or  Factor  of  Safety  of  the  beam 
or  product  of  breadth  and  square  of  depth,  must  be  in  proportion  to  distance 
from  nearest  support ; consequently,  whether  the  lines  forming  the  beam  are 
straight  or  curved,  they  meet  in  the  centre,  and  of  course  the  two  halves  are 

2.  When  Depth  is  Uniform  throughout,  breadth  must  be  in  ratio  of  length, 
of' len^tlr  Breadih  is  Uniform  throughout,  depth  will  vary  as  square  root 

4.  When  Section  at  every  point  is  similar , as  a Circle,  Ellipse,  Square , and 
Rectangle,  section  at  every  point  being  a regular  figure,  cube  of  depth  will 
be  as  square  of  distance  from  supported  end.  1 

When  Supported  at  Both  Ends,  and  Loaded  uniformly  throughout  its 
Length. 

1.  When  Depth  is  Uniform,  breadth  will  be  as  product  of  length  of  beam 

and  length  of  it  on  one  side  of  given  point,  less  square  of  length  on  one  side 
of  given  point.  0 

2.  When  Breadth  is  Uniform,  depth  will  be  as  square  root  of  product  of 

length  of  beam  and  length  of  it  on  one  side  of  given  point,  less  square  of 
length  on  one  side  of  given  point.  4 

3.  When  Section  at  every  point  is  similar , as  a Circle,  Ellipse,  Square , and 

Rectangle,  section  at  every  point  being  a regular  figure,  cube  of  depth  will 
be  as  product  of  length  of  beam  and  length  of  it  on  one  side  of  given  point 
less  square  of  length  on  one  side  of  given  point.  ’ 

DRIlliptical-sided.  Beams. 

To  Determine  Side  or  Curve  of  an  Elliptical-sided  Beam 
J LI 

L rePresentm9  load  in  lbs.,  I length  in  feet , C coefficient,  and  b 

breadth  in  ins. 

Illustration.— What  should  be  depth  in  centre  of  abeam  of  white  pine  10  feet 
in  length  between  its  supports,  and  5 ins.  in  breadth,  to  support  a load  of  ioioo  lbs.? 

Assume  C = 100.  Then  7™°°°*™^  A^oooo  _ 

V 2 X 100  x 5 V 1000 

Hence,  outline  of  beam  is  that  of  a semi-ellipse,  having  10  feet  for  its  transverse 
diameter,  and  9 ins.  for  its  semi-conjugate. 

Note.— Weight  of  Girder,  Beam,  etc.,  should  in  all  cases  be  added  to  stress  or  load. 

Miscellaneous  Illustrations. 

1.— What  should  be  side  of  a rectangular  white  oak  beam,  2 ins.  in  width  and  6 

feet  between  its  supports,  to  sustain  a load  of  360  lbs.  ? ’ 

Assume  stress  at  .2  of  breaking  weight  of  150  lbs.  = 30. 

/ 6 X 360  _ /2160 

v 4 X 2 X 30  \ 240  — * 


= 3 ins. 


v 4 X 2 X 3®  v 
2.— What  should  be  breadth  and  depth  of  such  a beam  if  square? 
3 /6  x 36°  ? /'2160 

V 4X30  ~v 


. 4 X 30  V 120 

3* — What  should  be  diameter  of  a cylinder? 
360  X 6 


= 2. 62  ins. 


.6  X 3° 


= 120,  and 


ins- 


STRENGTH  OF  MATERIALS. TRANSVERSE.  827 


STEEL. 

To  Compute  Transverse  Strength,  of  Steel  Bars. 

Supported  at  Both  Ends.  Weight  applied  in  Middle. 

x.155  S bd*  =w  g representing  tensile  strength  in  lbs. , l length  between  supports 
in  ins. , and  W weight  in  lbs. 

Illustration  —What  is  ultimate  destructive  stress  of  a bar  of  Crucible  steel, 
2 ins.  square,  and  2 feet  between  supports  ? s = 9°  000  lbs. 

1.155  X 90000  X 23  = 831600  6 J6s. 

2 X 12  24 

Elastic  Transverse  Strength  is  50  per  cent,  of  its  ultimate  strength. 

Hardening  in  oil  increases  its  strength  from  12  to  56  per  cent.  Thus, 

Soft  steel,  121  520  lbs. ; soft  steel,  cooled  in  water,  90  160  lbs. ; soft  steel, 
cooled  in  oil,  215  120  lbs. 

Knipps  is  about  .45  of  its  tensile  breaking  weight,  .24  of  its  compressive 
or  crushing  strength,  .38  of  its  transverse,  and  .39  of  its  torsional. 

Friction  of  a steel  shaft  compared  to  one  of  wrought  iron  is  as  .625  to  1. 

Capacity  of  steel  to  resist  a transverse  stress  is  much  less  than  to  resist 
torsion. 

Relative  diameters  of  steel  and  wrought-iron  shafts,  to  resist  equal  trans- 
verse stress,  are  as  .98  to  1,  and  weight  of  such  a proportion  of  steel  shaft 
compared  with  one  of  wrought  iron  will  be  about  4 per  cent,  less,  and  friction 
of  bearing  will  be  6 per  cent.  less. 


CYLINDERS,  FLUES,  AND  TUBES. 

Hollow  Cylinders.  Cast  Iron. 

To  Compute  Elements  of*  Hollow  Cylinders  ■vvithin 
Limits  of*  Elastic  Strength.  (D.  K.  Clark.) 

P P 

S X hyp.  log.  R = P.  r y— ^ = S.  ~ = hyp.  log.  R.  S representing 

**  * hyp.  log.  R o 

elastic  tensile  strength  of  metal  in  lbs.  per  sq.  inch , R ratio  of  external  diameter  to  in- 
ternal = — = — , and  P internal  pressure  in  lbs.  per  sq.  inch,  d and  d'  representing 
’ dr 

internal  and  external  diameter , and  r and  r'  internal  and  external  radii , all  in  ins. 

Note.— Hyperbolic  Logarithm  of  a number  is  equal  to  product  of  its  common  logarithm  and  2.3026. 
Illustration  i.  — Diameters  of  a hydrostatic  cylinder  5.3  by  13.125  ins. ; what 
pressure  within  its  elastic  strength  will  it  sustain  per  sq.  inch? 

Assume  S = 10000  lbs.  Hyp.  log.  R = -3‘ ---  X 2.3026  = log.  2.5  X 2.3026  = .92. 

5-3 

Then  10  000  X .92  = 9200  lbs.  per  sq.  inch. 

Note.— For  Bursting  Strength  take  maximum  strength  of  metal. 

2. —A  water-pipe  .75  inch  thick  has  an  internal  diameter  of  10  ins.,  what  is  its 
bursting  pressure  ? 

S = 30  000  lbs.  Hyp.  log. 10  — = . 1398. 


Then  30000  X *1398  = 4194  Ibd. 

3. — If  it  were  required  of  a hydrostatic  press  to  sustain  a pressure  of  589050  lbs. 
upon  a ram  of  5 ins.  in  diameter,  what  would  be  pressure  on  ram,  and  what  should 
be  thickness  of  metal,  assuming  it  equal  to  an  elastic  tensile  stress  of  15000  lbs. 
per  sq.  inch  ? 


Area  of  5 ins.  = 19.635. 


589  050  _ ooQ  —pressure  per  sq.  inch  on  ram. 
i9-635 


Then  30000  = 2,  which  = hyp.  log.  R — 7. 39,  and  7. 39  X 5 = 36-95  = external  di< 
15000 

ameter.  36.95  — 5 = 31.95,  which  -f-  2 = 15-975  ins.  thickness  of  metal. 


828  STRENGTH  OF  MATERIALS. TRANSVERSE. 


Wrought  Iron.  and.  Steel. 

R + hyp.  log.  •—  — 1 

d Q -p  2P  „ 2 P , 

2 S“P*  tT.  7,  7 d'  ~S'  ^+I  = (p+hypiog.R). 

Illustration  i.—  If  diameters  of  a wrought-iron  cylinder  are  5 and  u ins  and 
ingp^ssure^?  StmCtlVe  strengt^  of : metal  is  4°  000  lbs.  per  sq.  inch,  what  is5 its  break- 
~ = 3-  Hyp.  log.  3 = .477  12  X 2.3026  = 1.0986. 

tviqv.  3'f_I-°986  — 1 

men  X 40000  = 61  972  lbs.  per  sq.  inch  = 61  972  X 5 -r- 15  — 5 = 

30  986. 2 lbs.  per  sq.  inch  of  section  of  metal. 

steam-boiler  6 feet  in  internal  diameter,  of  wrought-iron  plates  .375  inch 
thick  and  double  riveted  longitudinally,  burst  at  a joint  bv  a pressure  of  mi  lbs  per 
sq.  inch ; what  was  resistance  of  joint  per  sq.  inch  of  its  section  ? J y 


72  + . 375  X 2 


— 1. 0104.  Hyp.  log.  1. 0104  = .010345. 


Then  - 


2 x 300 


_ 600 

1. 0104  -J-.  010  345  — 1 “ .020745 


= 29  405  Ms.  per  sq.  inch  of  section  of  joint. 


SHIP  AND  BOILER  PLATES. 

(- Seepages  751-757 /or  Boiler  Riveting.) 

Ultimate  Tensile  Strength,  of  Riveted.  and.  "Welded 
Joints  of  Wrought-iron  IPlates.  (D.  K.  Clark.) 

Entire  Plate  = 100. 


Joints. 

•5 

Plate. 

•375 

•4375 

Aver- 

age. 

Joints. 

•5 

Plate. 

•375 

•4375 

Aver- 

age. 

Scarf- welded 

Lap-welded 

Single  hand  riveted. 
“ “ snap-1 

headed f 

50 

40 

50 

102 

66 

60 

56 

106 

69 

50 

52 

104 

62 

50 

53 

Double  riv’d,  snap- 
headed  

“ “counter-] 

sunk  and  snap- 
headed   j 

I 

1 

59 

53 

72 

69 

70 

72 

67 

65 

“ “by  machine 
“ “ counter-) 

sunk  head  . . . J 

40 

44 

52 

S2 

54 

50 

1 

49 

49 

“ “with  single] 
welt,  counters’k 
and  snap-headed) 

1 

52 

65 

60 

59 

Strength  of  Riveted  Joints  per  Sq.  Inch  of  Single  Plate . ( Wm.  Fairbairn.) 

Single  Lapped—  Machine  riveted.  Pitch  3 times,  25 000  lbs. 

Hand  riveted.  Pitch  3 times,  24  000  lbs. 

Rivets  “staggered,”  and  equidistant  from  centres,  3050Q  lbs. 

Abut  Joints.— Hand  riveted.  Rivets  not  “ staggered,”  and  equidistant 
from  centres,  single  cover  or  strap,  30  000  lbs. 

Rivets  “square,”  single  cover  or  strap,  42 000  lbs. ; double  covers  or 
straps,  55  000  lbs. 

Comparative  Strength  of  Riveted  Joints. 


Entire  Plate  .375  ins.  thick  = 100. 


Double  riveted,  double  strap,  or  fish- ) Q 

plated  joint j 80 

Double  riveted  lap  joint 72 


Double  riveted,  single  strap,  or  fish- 1 , 

plated  joint j 

Single  riveted  lap  joint 60 


For  all  joints  of  plates  over  .5  inch,  other  than  double  welded,  these  proportions 
are  too  high. 

^A  closer  pitch  of  rivets  should  be  adopted  in  single  than  in  double  riveted  abuts, 


STRENGTH  OF  MATERIALS. TRANSVERSE, 


829 


Dimensions  of  R,ivets, 

Pitcli,  Lap,  etc, 

Plate. 

Thickness. 

Diam. 
of  Rivet. 

Length 
from  Head. 

Pitch. 

Single. 

**  a P- 
Double. 

Staggered. 

Inch. 
•25 
• 3I25 
•375 

•5625 

.625 

•75 

•875 

1 

Ins. 

•5 

.625 

•75 

.8125 

•9375 

1 

1. 125 
1.25 
i-5 

Ins. 

1. 125 

1-375 

1.625 

2.25 

2.75 

3 

3-25 

4 

' 4-5 

Ins. 

i-5 

1.625 

1- 75 
2.125 

2- 375 

2.625 
3 

3- 375 

4- 375 

Ins. 

1.5625 

2 

2.4375 

2.625 

3 

3-25 

3.625 

4 

4.875 

Ins. 

2.75 

3- 4375 
4.125 

4- 4375 

5.1875 

5- 5 

6.1875 
6.875 
8.25 

Ins. 

2- 4375 
3 

3.625 

3- 9375 

4- 5625 
4.8125 

5- 4375 

6.0625 
7-25 

1 1.5  1 4-5  4-375  4-»75  °*25  7-^3 

Straps.  — Single,  .125  thicker  than  the  plate;  Double,  each  .625  of  thickness  of 
plate. 


To  Compute  Diameter  of  Trivet. 

Ordinarily , T 1.25  + . 1875  — d.  T representing  thickness  of  plate , and  d diameter 
of  rivet. 

Fitcli  of  Rivets.  {Nelson  Foley.) 


Plates. 

Metal  between  the 
Holes. 

Diam. 
of  Rivets. 

1 Plates. 

Metal  between  the 
Holes. 

Diam. 
of  Rivets. 

Single 

Staggered. 

52  to  62  per  cent. 
68  to  75  “ “ 

1.4  to  2.3 

1.4  tO  2. 1 

I Square . . 
(Triple. . . 

70  to  78  per  cent. 
76  to  80  “ “ 

.99  to  1.7 
.77  to  1 

Proportions  of  Single  Rivet  Wrought-iron  Joints. 

{French. ) 


Thickness 
of  Plate. 

Diameter 
of  Rivets. 

Pitch  of 
Rivets. 

Width  of 
Lap. 

Mil’s 

Inch. 

Mil’s 

Inch. 

Mil’s 

Ins. 

Mil’s 

Ins. 

3 

.118 

8 

•315 

27 

1.06 

3° 

1. 18 

4 

.158 

10 

•394 

32 

1.26 

34 

i-34 

5 

.197 

12 

.472 

37 

1.46 

40 

1.58 

6 

.236 

*4 

•551 

43 

1.69 

44 

I-73 

7 

.276 

16 

•63 

48 

1.89 

50 

1.97 

8 

•3I5 

17 

.669 

5i 

2.01 

54 

2.13 

9 

•354 

19 

.748 

54 

2.13 

56 

2.2 

Thickness  1 
of  Plate. 

Diameter  1 
of  Rivets. 

Pitch  of 
Rivets. 

Width  of 
Lap. 

Mil’s 

Inch. 

Mil’s 

Ins. 

Mil’s 

Ins. 

Mil’s 

Ins. 

10 

-394 

20 

.787 

56 

2.2 

58 

2.28 

11 

•433 

21 

.827 

57 

2.  24 

60 

2.36 

12 

.472 

22 

.866 

58 

2.28 

60 

2.36 

13 

.512 

23 

.906 

60 

2.36 

62 

2.44 

14 

-551 

24 

-945 

62 

2.44 

64 

2.52 

15 

•591 

25 

.984 

63 

2.48 

66 

2.6 

16 

.63 

26 

1.024 

65 

2.56 

68 

2.68 

Result  of  Experiments  on  Dorible  Riveted,  and  Double 
Strapped  Rlate  Joints.  [Mr.  Brunei.) 

Plates  20  ins.  in  width,  5 inch  thick , Abut  jointed,  with  a Strap  or  Fish-plate  on 
each  side , 10  ins.  in  width.  Holes  Punched. 


For  Boiler  Riveting  see  pp.  755-57* 

4a 


830 


STRENGTH  OF  MATERIALS. TRANSVERSE. 


Hi.il Is  of  Vessels. 


Diameter  of  Rivets. 


Plate. 

U.  S.  and 
British 
Lloyds. 

Liverpool 

Regfy. 

Admiralty, 

Eng. 

Mill  wall, 
Eng. 

■■.’I 

Pitch 
of  Rivets. 

Length  ( 
Counter- 
sunk. 

>f  Ri  7eta. 
Snap- 
headed. 

Inch. 

Inch. 

Ins. 

Ins. 

Inch. 

Ins. 

Ins. 

Ins. 

•3125 

.625 

•5 

•5 

.625 

*•75 

1-125 

1.5 

•375 

.625 

.625 

.625 

.625 

2 

1.25 

1.625 

•4375 

.625 

.625 

•75 

.625 

2.125 

1-375 

1-75 

•5 

•75 

•75 

•75 

•75 

2.25 

i-5 

2 

•5625 

•75 

•75 

•875 

•75 

2-437 

1.6875 

2.1875 

.625 

•75 

.8125 

•875 

^875 

2.56 

1-9375 

2-375 

.6875 

•875 

•875 

•875 

•875 

2.812 

2.1875 

2.625 

•75 

•875 

•875 

1 

•875 

3- 125 

2-375 

2-75 

.8125 

•875 

•9375 

1 

.875 

3-375 

2-5 

2.875 

•875 

1 

1 

1. 125 

1 

3-625 

2.625 

3 

•9375 

1 

1.0625 

1.125 

i 

3-875 

' 2.75 

3-125 

1 

1 

1.125 

1. 125 

1 

4.125 

2.875 

3-25 

Lap  of  Joint  or  Course  should  be  .5  pitch  of  rivets  added  to  .3  diam.  of  rivet. 

Note.— Lloyd’s  requires  a spacing  of  4.5  diameter.  Liverpool  Registry,  4.  Ad- 
miralty, 4.5  to  5 in  edges  and  abuts  of  bottom  and  bulkhead  plates,  and  5 to  6 in 
other  water-tight  work.  Bureau  Veritas , 4 diameters  for  single  riveting,  and  4 5 
for  double. 


STEEL  PLATES. 

Steel  Plates,  according  to  M.  Barba,  .354  inch  thick  are  equal  to  wrought 
iron  .472  inch  thick,  or  as  3 to  4;  consequently,  when  iron  rivets  are  used, 
their  diameter  should  be  in  proportion  to  an  iron  plate. 

It  is  ascertained  also  that  they  are  best  united  by  iron  rivets. 

A steel  plate  .3125  inch  thick  requires  an  iron  rivet  .5625  inch  in  diam- 
eter, and  1.375  ins.  apart. 

Bridge  Blates  and  Rivets. 

Plates  .25  to  .5  inch  thick.  Rivets  .75  to  1 inch  diameter,  and  3 ins.  apart 
from  centres  in  upper  flange  or  girder,  and  4 ins.  in  lower 

Rivet  Heads. 


Ellipsoidal , Fig.  1.  — D diameter , R radius  of  head  = D,  r radius  of 
flange  = .4  D,  c depth  at  centre  p ,5  D. 

Segmental , Fig.  2. — D diameter , c depth  at  centre  = .625  /'ITX2' 

~P  Tiortrl nr - IV  n rl  I n /'i  /I  IV  \ y 1 


D,  R radius  of  head  — .75  D,  0 depth  below  head  = . 125  D, 


Countersunk. — Head  1.52  D,  angle  6o°.  Countersink  .45  diam.  of  plate. 
Cheesehead  or  heads,  section  of  which  is  a parallelogram.  Head  .45  D, 
diameter  1.5  D. 

Rivets. 


Shearing  strength  of  a Lowmoor  rivet  = 40  320  d2  or  18  d2  in  tons. 

d representing  diameter  of  rivet  in  ins. 


Memoranda. 


Punching  holes  for  riveting  weakens  plates,  varying  from  10  to  20  per  cent.,  ac- 
cording to  their  temper,  hardest  losing  most. 

Countersunk  riveting  does  not  impair  strength  of  joint,  as  compared  with  ex- 
ternal head. 

Diagonal  abut  joints  are  stronger  than  square. 

Shearing  strength  of  rivets  should  not  exceed  that  of  plates. 

Maximum  strength  of  joint  is  attained  at  90  to  100  per  cent,  of  net  section  of  plate. 

Shearing  strength  of  English  wrought  iron  is  taken  at  80  per  cent,  of  its  tensile 
strength. 


STRENGTH  OF  MATERIALS. — TRANSVERSE.  83  I 


LEAD  PIPE. 

Resistance  of  Lead  Bipe  to  Internal  Pressure. 
( Kirkaldy , Jar  dine,  and  Fairbairn.) 


Diam. 

Thick- 

ness. 

Weight 

Foot. 

Bursting 

Pressure. 

Diam. 

Thick- 

ness. 

Weight 

per 

Foot. 

Bursting 

Pressure, 

Diam. 

Thick- 

ness. 

Weight 

per 

Foot. 

Bursting 

Pressure. 

Inch. 

•5 

.625 

•75 

X 

Inch. 

.2 

.2 

.22 

.2 

Lbs. 

2- 3 
2.6 

3- 8 

4. 1 

Lbs. 

1579 
1349 
1191 
9t 1 

Ins. 

1.25 

1-5 

1-5 

i-5 

Inch. 
.21 
.24 
. 2 
• 2 

Lbs. 
5-3 
7- 1 

Lbs. 

6$3 

734 

528 

62$ 

Ins. 

2 

2 

3 

3 

Inch. 

.21 

.2 

•25 

•25 

Lbs. 

9.2 

Lbs. 

498 

448 

364 

374 

Tensile  strength  of  metal  = 2240  lbs.  per  sq.  inch. 

To  Compute  Thickness  of  a Lead.  Bipe  when  Diameter 
and  Pressure  in  Lbs.  per  Sq.  Inch  is  given. 

j>ULE Multiply  pressure  in  lbs.  per  sq.  inch  by  internal  diameter  of  pipe 

in  ins.,  and  divide  product  by  twice  tensile  resistance  of  metal  in  lbs.  per  sq. 
inch. 

Illustration  —Diameter  of  a lead  pipe  is  3 ins.,  and  pressure  to  which  it  is  to 
be  submitted  is  370  lbs.  per  sq.  inch;  what  should  be  thickness  of  metal? 

370  X 4 mo  . 

' • AL — tLA  = — . 248  ins. 

2240  X 2 4480 

Difference  in  Weight  between  Pipes  of  “Common,”  “Middling,”  and  “Strong” 
is  12  per  cent. 

To  Compute  Weiglit  of  Lead  Bipe. 

D 2 — dp  3. 86  ==  W.  D and  d representing  external  and  internal  diameters  in  ins., 
and  W weight  of  a lineal  foot  in  lbs. 


To  Compute  Maximum  or  Bursting  Pressure  tliat  may 
be  Lome  by  a Lead  Bipe. 

Rule.— Multiply  tensile  resistance  of  metal  in  lbs.  per  sq.  inch  by  twice 
thickness  of  pipe,  and  divide  product  by  internal  diameter,  both  in  ms. 
Illustration.— What  is  bursting  pressure  of  a lead  pipe  3 ins.  in  diameter  and 

.5  inch  thick?  

2^°  X -5XJ  _ ^12  ==  6 6 lbs 

3 3 

Assume  a column  of  water  34  feet  in  height  to  weigh  15  lbs.  per  sq.  inch;  what 
head  of  water  would  such  a pipe  sustain  at  point  of  rupture? 

15  • 34  ••  746.6  : 1692.3  feet. 

Resistance  of  Grlass  GHLoLes  and  Cylinders  to  Internal 
Pressure  and  Collapse.  ( Flint  Glass.) 

Bursting  Pressure. 


Diameter. 

GLOBES. 

Thickness. 

Per  Sq.  Inch. 

Diameter. 

cylin: 

Length. 

DER. 

Thickness. 

Per  Sq.  Inch. 

* 

Inch. 

Lbs. 

Ins. 

Ins. 

Inch. 

Lbs. 

4 

.024 

84 

4 

7 

.079 

282 

5 

.022 

90 

Elliptical  (Crown  Glass). 

6 

•059 

152 

Colla 

4- 1 1 7 1 

psing  Pressure. 

[ *019  1 

109 

5 

1 -OI4 

292 

II  3 

1 14 

I *OI4 

1 85 

4 

.025 

1000* 

4 

7 

•034 

202 

6 

1 -°59 

j 900* 

ll  4 

* Unbroken. 

1 *4 

1 .064 

1 297 

832  STRENGTH  OF  MATERIALS. — TRANSVERSE. 


Manganese  Bronze. 

Manganese  Bronze , No.  2,  has  a Tensile  strength  of  72  000  to  78  600  lbs. 
per  sq.  inch,  its  elastic  limit  is  from  35000  to  50000  lbs.,  its  ultimate  elon- 
gation 12  to  22  per  cent.,  and  its  hardness  alike  to  that  of  mild  steel. 

Transverse  Strength—  Destructive  stress  of  a bar  1 inch  square,  supported 
at  both  ends  at  a distance  of  1 foot  = 4200  lbs.,  bending  to  a right  angle  be- 
fore breaking,  and  requiring  1700  lbs.  to  give  it  a permanent  set. 

MEMORANDA. 

Cast  Iron. 

Beams  cast  horizontally  are  stronger  than  when  cast  vertically. 

Relative  strength  of  columns  of  like  material  and  of  equal  weights  is : 
Cylindrical,  100;  Square,  93;  Cruciform,  98;  Triangular,  no.  ( Hodgkinson .) 

If  strength  of  a cylindrical  column  is  100,  one  of  a square,  a side  of  which 
is  equal  to  diameter  of  the  cylinder,  is  as  150. 

Repetition  of  Stress . — A piece  submitted  to  transverse  stress  broke  at 
1956th  strain,  with  a stress  .75  of  that  of  its  original  ultimate  resistance. 

Resistance  to  Bursting  of  Thick  Cylinders. — Mean  resistance  to  bursting, 
of  chambers  of  cast-iron  guns  is  as  follows  ( Major  Rodman , U.S.A.)  : 

Thickness  of  metal  — 1 calibre,  length  = 3 calibres,  52  217  lbs.  per  sq.  inch. 

Thickness  of  metal  — .5  calibre,  length  ==  3 calibres,  49  100  lbs.  per  sq.  inch. 

The  tensile  strength  of  the  iron  being  18  820  lbs. 

Diam.  of  cylinder  2 ins.,  length  12  ins.,  metal  2 ins.,  80229  lbs-  Per  sq.  inch. 

Diam.  of  cylinder  3 ins.,  length  12  ins.,  metal  3 ins.,  93702  lbs.  per  sq.  inch. 

Tensile  strength  of  iron  being  26  866  lbs. 

Sudden  Applications  of  Stress.— 'Loss  of  strength  by  sudden  application 
of  load  was,  by  experiment,  18.6  per  cent,  in  excess  of  load  applied  gradually, 
and  its  elongation  20  per  cent,  greater. 

Low  Temperature. — Tensile  strength  at  230  under  sudden  application  of 
load,  was  reduced  3.6  per  cent.,  and  elongation  18  per  cent. 

"W i*  outfit  Iron. 

Increased  Hammering  gives  20  per  cent,  greater  strength  with  decreased 
elongation. 

Hardening. — Water  increases  strength  more  than  oil  or  tar.  A bar  .87 
inch  in  diameter,  forged  and  hardened  in  water,  attained  a tensile  strength 
of  73448  lbs.  {Mr.  Kirkaldy.) 

Case  Hardening. — Loss  of  tensile  strength  4950  lbs.  per  sq.  inch. 

Cold  Rolling  added  18.5  per  cent,  to  tensile  strength,  and  when  plates 
were  reduced  .33  in  thickness,  strength  was  nearly  doubled,  with  but  .1  per 
cent,  elongation.  Specific  gravity  was  reduced. 

Fibre. — Plates  are  about  12  per  cent,  stronger  with  fibre  than  across  it. 

Angles , Tees,  etc.,  have  from  2200  to  4500  lbs.  less  tensile  strength  than 
rectangular  bars. 

Galvanizing  does  not  perceptibly  affect  strength. 

W elding. — Strength  as  affected  by  welding  varies  by  experiment  from  2.6 
to  43.8  per  cent,  less,  average  being  19.4. 

Elastic  Strength  is  about  .45  of  its  tensile  breaking  weight,  .15  of  its  com- 
pressive or  crushing  strength,  and  .5  of  its  transverse  strength. 

Effect  of  Screw  Threads. — 1 inch  bolts  lose  by  dies  6.1 1 per  cent.,  and  by 
chasing  28  per  cent. 

Steel. 

Steel  can  be  hardened  in  water  at  a temperature  of  310°. 


STRENGTH  OF  MATERIALS. — TRANSVERSE.  833 


WOODS. 

To  Compute  Transverse  Strength,  of  Large  Timber. 
Destructive  Stress. 

.3S  b d2 

Fixed  at  One  End , and  Loaded  at  the  Othei . — ^ 

. x.8S^2 

Fixed  at  Both  Ends , and  Loaded  in  Middle.  ■ 


zW. 


4 Supported  at  Both  Ends , arcd  Loaded  in  Middle. 


1.2  S be?2 


= W. 


iVzed  aZ  j&rfA  Lrcds,  arad  Loaded  at  any  other  point  than  | -45  s 6 d2  — w, 
the  Middle.  3 l 

Supported  at  Both  Ends , cmd  Loaded  at  any  other  point ) • 3 B ^ 

the  Middle.  T w ; ^ m n 

i S. 


* Hence, 


W Z „ , Wi 


5 tZ,  and  l representing  breadth , tZepto,  and  length  to  or  between  supports  allin 
ins  S m^an  of  tensile  and  crushing  strengths  of  material  at  two  thirds  of  its  Value, 
as  determined:  by  experiments , W ultimate,  weight  or  stress  m lbs.,  and  m and  n dis- 
tances of  load  from  nearest  supports  in  ins. 

When  a beam  is  uniformly  loaded,  the  stress  is  twice  that  if  applied  in  its  middle 
or  at  one  end. 

Values  of  1.3  S. 

Hence,  for  other  coefficients,  as  .3,  1.8,  etc.,  the  values  will  be  proportional. 


Woods. 


Ash,  white 

“ ' Canadian 

“ English 

Beech 

Birch 

Cedar 

“ Cuban 

Chestnut 

Cypress 

Elm,  English.. 

“ Rock,  Canada. . 

Fir,  Dantzic  

Greenheart 

Gum,  blue 

Hackmatack 

Iron  wood 

Larch 


Woods. 


Locust 

Mahogany,  Honduras. 

Oak,  Pa 

“ Va 

u white 

“ English 

u Dantzic 

“ French 

Pine,  Va 

“ pitch 

“ white 

“ yellow 

“ Canada.. 

Redwood,  Cal 

Spruce 

Teak 

Walnut,  black 


Illustration  i. — What  is  destructive  stress  of  a beam  of  English  oak,  2 ins. 
square,  and  6 feet  between  its  supports? 

1.2  from  table  — 1.7,  and  S = .66  of  5700  (mean  of  tensile  and  crushing  strength) 
= 3762  lbs. 

1.7  X 2X^X3762  _ 5ii«3  = Jlo6  lbSi 
6 X 12  72 

By  experiment  of  Mr.  Laslett  it  was  688  lbs. 

2. — What  is  destructive  stress  of  a beam  of  yellow  pine,  3 ins.  by  12,  and  14  feet 
between  its  supports? 

1. 2 from  table  = 3. 87,  and  S = .66  of  10  200  (mean  of  tensile  and  crushing  strength) 
= 6732  lbs. 

3.87  X 3 X 122  X 6732  ___  11  654827 
14  X 12  168 


- = 69  374  lbs. 


If  the  beam  was  fixed  at  both  ends  then  3.87  would  be  5.8. 


Or, 


s 1.2  : 1.8  ::  3.87  : 5.8. 


Safe  Statical  Loads  for  Rectangular  Beams  of  Various  ^Materials,  One  Inch  in  Breadth,  and 

One  Boot  in  Length. 

Supported  at  Both  Ends  and  Loaded  in  Middle. 

A American,  Af  African,  B Baltic,  B’k  Black,  C Canadian,  D Dantzic,  HZ  English,  G Georgia,  M Memel,  P Pitch,  R Riga,  W White, 
Y Yellow. — Figures  at  Head  of  Columns  denote  Destructive  Weight  of  Material  in  Lbs. 


834 


STRENGTH  OF  MATERIALS. TRANSVERSE. 


Locust. 
1 180 

O -vf  »i-0  O O -vhvo  O VO  ^ -^O  0 vO 

a*  m Tj-  oi  c->.  O ON  vO  0 m 0 vo  00  00  10  O it 

CN  ON  hi  (V  Oi  V in  H m O m on  00  01  hi  rh 

h3  n ro  1000  h to  ov  moo  m ovvo  m 0 ' 

mmciNNCOCOtJ-  10  VO 

Hickory. 
Maple. 
Af  Oak. 
G Pine. 
800 

Lbs. 
160 
640 
1440 
2560 
4 000 
5760 
7840 
10240 
12  960 
16  000 
19360 
23040 
27040 
31  36° 
36  000 
40  960 

Ash. 

D Fir. 
Rock  Elm. 

680 

Lbs. 

136 

544 

1 224 

2 176 
3400 
4 896 
6 664 
8704 

11  016 
13  600 
16456 

19584 

22  984 
26  656 
31  600 
34816 

M FJr. 
Chestnut. 
E Ash 

640 

Lbs. 

128 

512 

1 152 

2 048 

3 200 
4608 
6 272 
8 192 

10  368 
12  800 
15488 
18432 
21  632 
25  088 
28  800 
32768 

W Oak. 
B Fir. 

600 

.0000000000000000 

n NOOOO  M 0 CICOCO  M 0 NOOOO  NON 

h vO  CNO  moo  vo  no  ion  m mo 
H H roviONON  -v  c-.  0 m n-  0 
H H M M d M m 

gie  S ° 
£0  g *r> 

^ n.  0 
PhOm 

0000000000  000000 

» h v 0 vo  mvo  on  h 0 >-i  -v-  onvo  mvo 
,c  m voNNNO\foo  ovo  moo  m m * 
•-I  h n m m c^oo  h m mco  m tj-oo 

Hi  H M H d Cl  01 

i-ii  m 
« so  « 
10 

momomomomomomomo 

n*  0 M ”^00  NOO  VN  O 0 O N -VOO  01  OO 

,0  m •<+■  ovo  'O  nh  mo  mo  m m mo  00 
h oi  m mo  00  0 n m no  mo 

H HI  H M 01  N 01 

Spruce. 

Sycamore. 

Elm. 

W Pine. 
500 

Lbs. 

100 

400 

900 

1 600 

2 500 
3600 
4 900 
6 400 
8 100 

10000 
12  100 
14400 
16  900 
19  600 
22  500 
25  600 

A . . 

« g-Sr  0 
$ 

Lbs. 

90 
360 
810 
1440 
2 250 
3240 
4410 
5760 
7 290 
9000 
10  890 
12  960 
15  210 
17  640 
20  250 
23.040 

Birch. 

Hack- 

matack. 

Hemlock. 

400 

Lbs. 

80 

320 

720 

1 280 

2 000 
2 880 
3920 

5 120 

6 480 
8 000 
9680 

11  520 
13520 
15  680 
1 8 000 
20480 

E N 0 
« 

-to  0^0  Tf-vo  0+0  'S-O  O 0 't- 
aJOmt-^NOOm  ovoo  0 >1  h v 0 00 

.0  oi  moo  m h 0 « Ti-r^oioo  m + m 

^ H h n m '+■  mo  n On  0 N -v-o 

4j » 0 

^0  0 

T3  °° 

&(2  ' 

.OOOOOOOOOOOOOOOO 
« O ■+  VO  OO  '+-'+0  OO  -cj-  -+-0  OO 
•g  oi  m On  m m ovoo  ooooiOHt^mm 

1-5  h 01  01  m tj-o  Choo  0 hi  m m 

•£.  « h n ro  mo  r^oo  o O h n m ij-  mo 

G IHMHHHHH 

« l 


2 

^ O 

a ^ 


*S 


5 *S 
^ % 


A,  o 

3 

o a, 

& 3 


s 

N 

& 


S v 

.w  > 

O <D 

3 

5 a 


o a> 

.5  3 

© g 


, © 
y 2 


ca  a 

m 

rS3  03 

"o  ^ 

S3  c3 
.3  n't 


a £ 

»c  £ £ 

33  Ctf  „ 


STRENGTH  OF  MATERIALS. TRANSVERSE.  835 


Following  Coefficients  or  Factors  of  Safety  are  for  .125  of  average  de- 
structive weight : 


Coefficients 

Ash 

“ English 

“ Canada 

Beech 

Chestnut 

Elm,  Canada 

“ English 

Fir,  Riga 

Hemlock 


for  Various  Woods. 


85 

80 

60 

58 


65 

80 

42 


Hickory 

Larch 

Locust 

Maple 

Oak,  white  . . , 
“ English 
“ Dantzic 
“ Adriatic, 


100 

130 

40 

150 

80 

60 

62 

55 


( Hatfield  and  others.) 

Ogk,  Canada 

“ French 

Pine,  pitch 

“ yellow 

“ red 

Georgia • 

o white......... 

“ Canada  red. . . . 


70 

85 


68 


65 

55 

100 

120 

62 

60 


Spruce 65 

Illustration. — What  safe  weight  will  a beam  of  white  pine  sustain,  4 ins.  in 
breadth.  12  in  depth,  and  15  feet  between  its  supports,  when  loaded  in  its  middle? 
and  what  when  uniformly  loaded?  Coefficient  as  above,  62. 


Then  4 X 12  X 62  _ 2^gag  ^ loaded  in  its  middle , and  2380.8  X 2 = 4761.6  lbs. 
if  uniformly  loaded. 

Floor  Beams  of  Wood. 

Condition  of  stress  borne  by  a Floor  beam  is  that  of  a beam  supported 
at  both  ends  and  uniformly  loaded ; but  from  irregularity  in  its  loading 
and  unloading,  and  from  necessity  of  its  possessing  great  rigidity,  it  is 
proper  to  estimate  its  capacity  as  a beam  loaded  at  middle  of  its  length. 


To  Compute  Capacity  of  Floor  Beams,  Grirdei-s,  etc. 
Supported  at  Both  Ends. 

Rule.— Divide  product  of  breadth  and  square  of  depth,  in  ins.,  and  Coef- 
cient  for  material,  by  length  in  feet,  and  result  will  give  weight  in  lbs. 

Or,  & = W.  Fixed  at  Both  Ends.  C = W. 

I t 

Example.— The  dimensions  of  a white-pine  floor  timber  are  4 by  12  ins.,  and  its 
length  between  supports  15  feet;  what  weight  will  it  sustain  in  its  centre? 

A X I22  X 62 

C as  per  preceding  table  ==  62.  Then  — = 2380.8  lbs. 

When  Uniformly  Loaded.  Multiply  the  results  by  2. 


To  Compute  Deptti  of  a Floo:1  Beam. 

Supported  at  Both  Ends. 

When  Length  between  Supports , Breadth  and  Distance  between  Supports , 
for  One  Foot , between  Centres  of  Beams  are  Given.  Rule.— Divide  product 
of  length  in  feet,  and  weight  to  be  borne  in  lbs.,  by  product  of  breadth  in 
ins.,  and  Coefficient  for  material,  and  square  root  of  quotient  will  give  depth 
in  ins.,  for  distance  between  centres  of  one  foot. 

Or,  A-77  = d.  Fixed  at  Both  Ends.  . / r— ; = d. 

’ V bG  V i-5  b c 

When  Uniformly  Loaded , W represents  but  half  required  or  given  weight. 

Example. — Take  elements  of  preceding  case,  distance  between  centres  of  beam 
15  ins.  0 = 62.  /i 5 X 2380. 8 /35712 

Then  / — ^ A — = 12  ms. 

V 4 X 62  V 248 

When  Distance  between  Centres  of  Beams  is  greater  or  less  than  one  Foot. 
Rule. — Divide  product  of  square  of  depth  for  a beam,  When  distance  between 
centres  is  one  foot , by  distance  given,  by  12,  and  square  root  of  quotient  will 
give  depth  of  beam. 


836  STRENGTH  OF  MATERIALS. — TRANSVERSE. 


Example.— Assume  beam  in  preceding  case  to  be  set  15  ins.  from  centres  of  ad 
joining  beams;  what  should  be  its  depth? 

2X  15 

— ' 3.42  ins. 


Then 


/122  X 15  /2160 

V“^— =v/^  = I3'4 


To  Compute  Breadth,  of'  a Floor  Beam  or  Grirder. 
/Supported  at  Both  Ends. 

When  Length  and  Depth  are  given.  Rule.— Divide  product  of  length  in 


feet,  and  weight  to  be  borne  in  -lbs.$  by  product  of  square  of  depth  in  i 
and  Coefficient  for  material,  and  quotient  will  give  breadth  in  ins. 

„ l W _.  ' 7 w 

- = &:  Fixed  at  Both  Ends.  — — b. 


’ d2C  ' 


1.5  d2  C ~ 

When  Uniformly  Loaded,  W represents  but  half  required  or  given  weight. 
Example.— Take  elements  of  preceding  cases. 

Then  *SXH*o-*±  35712  _ . 

i22X  62  8928  4 mS' 

When  Distance  between  Centres  of  Beams  is  greater  or  less  than  One  Foot. 
Rule.  Divide  product  of  breadth  for  a beam,  When  distance  between  centres 
is  one  foot,  and  distance  given,  by  12,  and  result  will  give  breadth. 

Example.— Assume  beam,  as  in  preceding  case,  to  be  set  15  ins.  from  centre  of 
adjoining  beams;  what  should  be  its  breadth? 

Then 

When  Weight  is  Suspended  or  Stress  borne  at  any  other  point  than  the 
Middle , See  Formulas,  page  801. 


Header  and  Trimmer  or  Carriage  Beams. 

Conditions  of  stress  borne  or  to  be  provided  for  by  them  are  as  follows : 

Header  supports  .5  of  weight  of  and  upon  tail  beams  inserted  into  or  at- 
tached to  it,  and  stress  upon  it  is  due  directlv  to  its  length  and  weight  of 
and  upon  tail  beams  it  supports,  alike  to  a girder  loaded  at  different  points. 

. Trimmer  or  Carriage  beams  support,  in  addition  to  that  borne  by  them 
directly  as  floor  beams,  each  .5  weight  on  headers. 


Note. — In  consequence,  of  effect  of  mortising  (when  bridles  or  stirrups  are  not 
used),  a reduction  of  tully  one  inch  should  be  made  iu  computing  the  capacity  of 
depth  of  headers  and  trimmers. 


To  Compute  Breadth  of  a Header  Beam. 

When  Uniformly  Loaded.  Rule.— Compute  weight  to  be  borne  in  lbs.  by 
tail  beams,  divide  it  by  two  (one  half  only  being  supported  by  header),  mul- 
tiply result  by  length  of  beam  in  feet,  and  divide  product  by  product  of 
twice  Coefficient  of  material  and  square  of  depth,  and  result  will  give  breadth 
in  ins.  ■ - 

. W-r-zl 

2 (j  ^2  = W representing  weight  per  sq.  foot. 


Example. — What  should  be  breadth  of  a Georgia  pine  header,  13  ins.  in  depth. 
10  feet  in  length,  supporting  tail  beams  12  feet  in  length,  bearing  200  lbs.  per  sq. 
foot  of  area  supported? 

C,  as  per  preceding  table,  100,  and  depth  = 13  — 1 = 12  ins. 


Then  12  X 10  X 2oo-f-2  x 10 


^toS'  = 4'I7<MS- 


2 X 100  X 12 


STRENGTH  OF  MATERIALS. TRANSVERSE.  837 


To  Compute  Depth  of  a Header  Beam. 

Rule. — See  rule  for  depth  of  a floor  beam,  page  835,  with  the  exception 
that  a header  is  assumed  to  be  always  uniformly  loaded. 


Or, 


- =<2. 


To  Compute  Breadth  of  a Trimmer  Beam. 


With  One  Header  and  One  Set  of  Tail  Beams. 


Rule. — Proceed  as  for 


computation  of  dimension  of  a beam  loaded  at  any  other  point  than  middle. 

m n \\  — m an(i  n representing  distances  of  the  weight  or  load  from  each 
Id2  C 
end  in  feet 

Illustration  —What  should  be  breadth  of  a trimmer  or  carriage  beam  of  Georgia 
nine  22  feet  in  length,  15  ins.  in  depth,  .sustaining  a header  10  feet  m length,  with 
tail  beams  19  feet,  and  designed  for  a load  of  540  lbs.  per  sq.  foot  ot  floor? 

1 and  n — 19  and  4 feet 


Assume  C = 


•5  X 


x>;  <2  = 15  — 1 = 14; 

19X4X  19  X 10  = 2 X 540 


• • 5 X 


450  800 


= 4. 32  ms. 


23  X 100  X 142 

Note  1.— Depth  of  trimmer  beams  is  usually  determined  by  depth  of  floor  beams; 
when  not,  proceed  to  determine  it  as  tor  a header. 

2 when  a trimmer  beam  is  mortised  to  receive  headers,  it  is  proper  to  deduct 

1 inch  from  its  depth,  as  in  preceding  illustrations.  When  bridle  or  stirrup  irons 
are  used  to  suspend  headers,  a deduction  of  the  thickness  of  the  iron  only  is  neces- 
sary, usually  .5  inch. 

With  Two  Headers  and  One  Set  of  Tail  Beams— Fig.  1. 

Operation. — Proceed  for  each  weight  or  load  as  for  a beam,  when  weights 
are  sustained  or  stress  borne  at  other  point  than  the  middle. 

a L-  — W and  w.  a representing  area  of  floor  in  sq.  feet , L load  per  sq.  foot , 
•5X5 

and  W and  w weights  or  loads  at  points  of  rest  on  trimmers. 

Note. —Hatfield  and  some  other  authors  give 
complex  and  extended  formulas,  to  deduce  the  di- 
mensions of  a Girder  or  Beam,  under  a like  stress. 

Upon  consideration,  however,  it  will  readily  be 
recognized  that  a beam  loaded  at  more  than  one 
point  is  simply  two  or  more  beams,  as  the  case 
may  be,  loaded  at  different  points,  and  connected 
together. 

Illustration. — What  should  be  breadth  of  a 
trimmer  beam  of  Yellow  or  Georgia  pine,  25  feet 
in  length,  12  ins.  in  depth,  sustaining  two  headers 
12  feet  in  length,  set  at  15  feet  from  one  wall  and  5 feet  from  the  other,  to  support 
with  safety  300  lbs.  per  sq.  foot  of  floor? 

I — 25,  m — 15,  n=  10,  5 = 5,  r = 20,  C=  100,  and  d=  12  — 1 = 11  for 

loss  by  mortising. 

..XsXjg  = I«2S2  = 4500  lbs.  at  W,  and  12X5X-^  = = 4500  lbs.  at  w. 

.5X5  -25  5X5  -25 

Then  X 10  X 4500  __  675000  _ g im  breadth  for  load  on  header  at  is  feet, 
25  X 112  X 100  302  500 

an(j  5 X 20  X 45go,  _ 45000g  _ 1-4g  inSt  breadth  for  load  on  header  at  ant* 

25  X 112  X 100  302500 

2.23  + 1.48  = 3.71  ins.  combined  breadth. 


<—m — * 

r--  ■ 

l 1 

W 

OD 


838 


STRENGTH  OF  MATERIALS. TRANSVERSE. 


With  Two  Headers  and  Two  Sets  of  Tail  Beams. — Fig.  2. 

Fig  2t  Operation.— Pi  oceed  as  directed  for  Fig.  1. 

Illustration.  — What  should  be  breadth  of  a 
trimmer  beam  of  yellow  pine  25  feet  in  length,  15 
ins.  in  depth,  sustaining  two  headers  12  feet’  in 
length,  set  at  15  feet  from  one  wall  and  5 feet  from 
the  other,  to  support  with  safety  300  lbs.  per  sq 
foot  of  floor?  ’ 

1 — 25,  m = 15,  n = 10,  5 = 5,  r — 20,  C = 100, 
and  d—  15  — 1 = 14  for  loss  by  mortising. 
12X15X300  54000 

— ■ - 13  500  lbs.  at  W,  and 


K— dr 4 

II — 72 — i 

J _ : 

1 

tl 71 

<:~4  1 

3 

1 — 

-5X-5 


= 4500  lbs.  at  w. 


12X5X  300  _ 18000 
•5X.5  _ -25 

Then  15  x IQ  X 13  5QQ _ 2025000: _ 

25  X 142  X 100  490000  "T 

ins. , and  4.14  .92  = 5.06  ins. . combined  oreadth. 


4-  »4  ins. , ami  A_X  20  X 45°o  = 45£°°o  = 
25  X 14  X 100  490000 


Fig.  3- 


With  Three  Headers  and  Two  Sets  oj  Tail  Beams.— Fig.  3. 


— 1 

HM 

1— T 

l — 

f— - 

M 



*| —rri— 

l-H> 

1 

1 1 

1 

15  X 7 — 3 x 200 


Then 


5X-5  .25 

7 X 13  X 5250  _ 47 7 750 


Operation.— Proceed  as  directed  for  Fig.  1. 

Illustration.  —What  should  be  breadth  of  a 
trimmer  beam  of  yellow  pine,  20  feet  in  length  13 
ins.  in  depth,  sustaining  3 headers  15  feet  in  length, 
set  at  3,  7j  and  13  feet  from  one  wall,  to  sustain  a 
load  of  200  lbs.  per  sq.  foot  of  floor? 

1 = 20,  m = 7,  n = i3,  s = 7,  0 = 3,  d = i3  — 1 
= 12  ins.,  and  C = 100. 

15X7  X 200 21 000 

.5X.5  “ -25 

15X7  — 3X20  , 

= 3000  Lbs.  at  w . 


- 5250  lbs.  at  W ; 


20  X 12 2 X 100  288000 

3 X 17  X 3000  __  i53°oo 
288  000 


= 3000  lbs.  at  w ; and 
:.66  ins. ; 


5 X .5 
7 X 13  X 3000 
20  X 122  X 100 " 


273000 


— .95  ins.; 


and 

20  X 122  X 100 
bined  breadth. 


= .53  ins. 


288  OOQ 

Hence,  1.66 + .95 + .53  = 3.,  14  ins.  com - 


Stirrmps  ox-  Bridles. 

Stirrups  are  resorted  to  in  flooring  designed  for  heavy  loads,  in  order  to 
avoid  the  weakening  of  the  trimmers  by  mortising. 

Average  wrought  iron  will  sustain  from  40000  to  50000  lbs.  per  sq.  inch. 
Hence  45  000  lbs.  as  a mean,  which  -4-  5 for  a factor  of  safety,  = 9000  lbs. 

A stirrup  supports  one  half  weight  of  header,  and  being  doubled  (looped), 
the  stress  on  it  is  but  .5^-2  = .25  of  load  on  header. 

To  Compute  Dimensions  of  Stirrups  or  Bridles. 


W-f-2 
2 X 9000 


= area.  Hence 


area 


thickness 


— = width. 


Illustration. — What  should  be  area  and  width  of  .75  inch  wrought-iron  stirrup 
irons  for  a weight  on  a header  beam  of  240000  lbs.  ? 


240  000  — 2 
2 X 9000 


120000  „ „ ' . 6.66 

= — ^ = 6.66  sq.  ms. , and = 8. 8 ins.  = width. 

10000  .75 


STRENGTH  OF  MATERIALS. TRANSVERSE.  839 


Condition  of  stress  borne  by  a Girder  is  that  of  a beam  fixed  or  supported 
at  both  ends,  as  the  case  may  be,  supporting  weight  borne  by  all  beams 
resting  thereon,  at  the  points  at  which  they  rest. 


Rule.— Multiply  length  in  feet  by  weight  to  be  borne  in  lbs.,  divide 
product  "by  twice*  the  Coefficient,  and  quotient  will  give  product  of  breadth 
and  square  of  depth  in  ins. 


Example.— It  is  required  to  determine  dimensions  of  a yellow-pine  girder,  15  feet 
between  its  supports,  to  sustain  ends  of  two  lengths  of  beams,  each  resting  upon  it 
and  adjoining  walls,  15  feet  in  length,  having  a superincumbent  weight,  including 
that  of  beams,  of  200  lbs.  per  sq.  foot. 

Condition  of  stress  upon  such  a girder  is  that  of  a number  of  beams,  30  feet  in 
length  (15X2),  supported  at  their  ends,  and  sustaining  a uniform  stress  along  their 
length,  of  200  lbs.  upon  every  superficial  foot  of  their  area. 

Coefficient  . 2 of  500  =*  100. 


To  Compute  Greatest  Load  upon  a,  Grirder,  and.  Dimen- 
sions thereof.— Fig.  1. 


Then,  for  weight  and  dimensions,  same  formulas  will  apply. 

Illustration — Assume  weight  of  8000  lbs.  at  3 feet  from  one  end  of  a white-pine 
beam  12  feet  in  length  between  its  bearings,  and  another  weight  of  3000  lbs.  at  5 
feet  from  other  end.  C . 2 of  500  — 100. 

8000  X 3 X 12  — 3 = 216  000  effect  of  weight  at  location  1,  and  3000  X 5 X 12  — 5 
— io5  000  effect  of  weight  at  location  2.  Hence  1,  being  greatest,  = W,  and  2 = w. 


Grirder. 


To  Compute  Dimensions  of  a Grirder. 


ins.  Or,  if  15  ins.,  then 


Then 


30  X 15  X 200  ~T 


1 5 X 200  -4-  2,  for  half  support  on  their  walls  = 45  000  lbs. 

— ==  3375  = b and  d2.  Assuming  b = 12  ins.,  then =16.77 

3.,  then = 15  ins. 


then 


ins. , then 


When  a Beam  is  Loaded  at  Two  Points. 


Fig.  1.  < 


-1 


—m— 


ni^  ==  effect  of  weight  at  1, 

^ = effect  of  weight  at  2, 
y ( W X n + w s)  t=  the  two  effects 


at  1,  and  j {tb  r -f  W m)  ==  two  effects  at  2. 


Then,  3 — X 8000  = 18  000  at  W,  and  x 8750  = 3750  at  w ; and 

’to  12 


12 


(8000  X 9 + 3000  X 5)  = 21  750  = total  effect  at  W,  and  ~ (3000  X 7 + 8000  X 3) 
= 18  750  = total  effect  at  tv. 


5 


Hence,. to  ascertain  dimensions  at  greatest  stress, 


Hence,. to  ascertain  dimensions  at  greatest  stress, 


12  x 300 


* For  being  uniformly  loaded. 


STRENGTH  OF  MATERIALS. TRANSVERSE. 


Verification. — Assume  a beam  as  above  loaded  with  21  750  lbs.  at  3 feet  from  end. 

Then,  by  formula  for  801,  — Xg2*  750  = **1^.  _ 8 ins 

12  X 102  X 100  120000  y 


Fig.  2.  < 


Equivalent  Weight  at  Middle, — Fig.  2. 

1 > w'  o 

;=A; 


in  y 

H*trt 


Z-f- 


W n 
1^2 


= B; 


= E ; 


and  ^ -4-  2*  = D = 


ic  $ 

equivalent  load  at  middle. 

Illustration.— What  should  bo 
breadth  of  a beam  of  Georgia  pine, 

font  in  1 rr!  it,7 


w i |d  l w 

A.  w-  E 

.......  i®  20  feet  in  length,  15  ins.  in  depth' 

uniformly  loaded  with  4000  lbs.,  and  sustaining  3 headers  or  concentrated  loads  of 
6000  lbs.,  at  respective  distances  of  4 and  9 feet  from  one  end  and  7000  lbs.  at  6 feet 
from  other  end  ? 


4,  r — 16,  m — 9,  n = 
6000  x 4 


C = 850  X .2  = 170. 
4000  X 20 


20  - 


= 2400; 


= 6,  ^^15  — 1 = 14,  L = 4000,  and 

6coo  X 9 7000  X 6 

■ = 54°°  5 : = 4200; 


and  ■ 


Hence, 


Then 

20  X 70000 


20-^  2 

2 = 2000.  2400  -j-  5400  -j-  42CO  -j-  2000  = 14  000  lbs. 

14000  X 10  X 10 


1 400  000 


- = 70  000  lbs .,  effect  at  middle. 
■=z  10. 5-f-  ins. 


' 4 X 142  X 170  133  280 

Operation  deduced  by  Graphic  Delineation  of  Greatest  Stress  without  uni- 
form Load. 

Fig.  3.  < 1 > Moments  of  weights  = 

A r 


W m n 


and 


, t , j 

19200,  29700,  and  29400,  and 
let  fall  perpendiculars  1,  2,  and  3 
proportionate  thereto. 

Connect  w',  W,  and  w with 
A B,  and  sum  of  distances  of  in- 
tersections of  these  lines  upon  perpendiculars,  from  1,  2,  and  3,  respectively,  will 
give  stress  upon  A B at  these  points. 

Whence,  greatest  stress  at  greatest  load  will  be  ascertained  to  be  61 800  lbs. 

When  Loaded  at  Three  Points,  m no 

as  in  Fig.  2.  J (W  n + w s)  + w T = Gnatest  Stms • 

Illustration. — Take  elements  of  above  case,  omitting  uniformly  distributed  load, 

— (6000  X 11  X 7000  X 6)  -f-  6000  11  ^ ^ = — X 108  000 -f-  13  200  =.  61 800  lbs. 


Deflection  of  G-irders  and  Beams. 
W13__d<  Cbd* 


' CM*-*"  ~7^W;  = aadl/—lr^  = l 1 represent- 

ing  length  in  feet , b and  d breadth  and  depth,  and  D defection  in  ins. 

Values  of  C for  Various  Woods.  {Hatfield.) 


Ash 4000 

Chestnut 2550 

Hemlock 2800 

Hickory 3850 


Larch 2093  Pine,  Georgia 5900 

Oak,  white 3100  “ pitch 2836 

“ English,  mean. . 2686  “ white 2900 

Spruce 3500  “ red 4259 

Illustration. — What  would  be  deflection  of  a floor  beam  of  white  pine,  10  feet 
in  length,  4 ins.  in  breadth,  and  8 in  depth,  with  4000  lbs.  loaded  in  its  middle? 

_ 4000  X 10 3 4000000  . . . 

C = 2900.  — -r  = = .674  inch. 

2900  X 4 X 8 3 5 939  200 


* Load  uniformly  distributed. 


STRENGTH  OF 


MATERIALS. TRANSVERSE.  84 1 


When  Weight  is  Uniformly  Distributed . 


A fair  allowance  for  deflection  of  floor  beams,  etc.,  is  .03  inch  per  foot  of  length; 
04  inch  may  be  safely  resorted  to. 

AW  eights  of  Floors  and.  of  Loads. 

Dwellings.— Weight  of  ordinary  floor  plank  of  white  pine  or  spruce,  3 lbs. 
per  sq.  foot,  and  of  Georgia  pine,  4.5  lbs. 

Plastering,  Lathing,  and  Furring  will  average  9 lbs.  per  sq.  foot. 

Clay  Blocks  ( Flat  Arch)  5.25  X 7-25  ins.  in  depth  and  1 foot  in  length, 
21  lbs.  = 80  lbs.  per  cube  foot  of  volume. 

Floors  of  dwellings  will  average  5 lbs.  per  sq.  foot  for  white  pine  or  spruce, 
and  on  iron  girders  will  average  from  17  to  20  lbs.  per  sq.  foot. 

Weight  of  men,  women,  and  children  over  5 years  of  age,  105.5  ^s.,  and 
one  third  of  each  will  occupy  an  average  area  of  12  X 16  ins.  = 192  sq.  ins. 
= 78.5  lbs.  per  sq.  foot. 

Of  men  alone  15  X 20  ins.  = 300  sq.  ins.  =48  m 100  sq.  ieet. 

Bridges , etc.— Weight  of  a body  of  men,  as  of  infantry  closely  packed,  = 
138  lbs.  each,  and  they  will  occupy  an  area  of  20  X 15  ins.  = 300  sq.  ms.— 
66.24  lbs.  per  sq.  foot  of  floor  of  bridge,  and  as  a live  or  walking  load,  80  lbs. 

per  sq.  foot.  „ „ „ . 

Weight  of  a dense  and  stationary  crowd  of  men,  120  lbs.  per  sq.  toot. 

Bridging  of  Floor  Beams  increases  their  resistance  to  deflection  in  a very 
essential  degree,  depending  upon  the  rigidity  and  frequency  of  the  bridges. 

Weight  on.  Floors,  etc.,  in  addition  to  Weight  ofStruct- 
ure,  per  Sq..  Foot. 


Relative  resistance  of  scarfs  in  Oak  and  Pine,  2 ins.  square,  and  4 feet  in 
length,  by  experiments  of  Col.  Beaufoy. 

Scarf  12  ins . in  Length  and  13  ins.  from  End , or  1 inch  from  Fulcrum . 

Vertical. — no  lbs.  gave  away  in  scarf. 

Horizontal , large  end  uppermost  and  towards  fulcrum.— 101  lbs.  fastenings 
drew  through  small  end  of  scarf;  small  end  uppermost , etc.,  87  lbs.  gave 
away  in  thick  part  of  scarf. 


Statical  or  Dead  Load  at  .2  of  destructive  stress,  but  for  ordinary  pur- 
poses it  may  be  increased  to  .25,  and  in  some  cases  with  good  materials  to  .3. 
Live  Load  at  .1  to  .125  of  destructive  stress. 

See  also  page  802. 


Ball  rooms, 


Brick  or  stone  walls 

Churches  and  Theatres. . . 


1x5  to  150  “ 
80  “ 


85  lbs. 


Roofs,  wind  and  snow 30  to  35  lbs. 

Slate  roofs 


Dwellings 
Factories. 
Grain 


200  to  400  “ 
100  “ 


4°  “ 


Snow,  per  inch 
Street  bridges. , 

Warehouses 

Wind 


250  to  500 

50 


Scarfs, 


Factors  of  Safety. 


842 


SUSPENSION  BRIDGE. 


SUSPENSION  BRIDGE. 
Compute  Elements. 


= stress  at  • . C representing  chord  or  span , a half  chord , and  v versed  sine  of 
chord  or  curve  of  deflection,  in  feet,  L distributed  load  inclusive  of  suspended  struct- 
ure, Q load  per  lineal  foot,  and  S stress  at  centre,  all  in  tons,  x distance  of  any  point 
from  centre  of  curve,  and  h height  of  chain  at  x above  centre  of  it,  both  in  feet,  5 
stress  on  chain  at  any  point,  as  x,  from  centre  of  span,  s stress  on  any  tension-rod, 
and  t stress  at  abutments,  all  in  tons,  n number  of  tension-rods,  o angle  of  tangent 
of  chain  with  horizon  at  any  point,,  as  x,  r angle  of  chain  with  vertical  at  abutments 
l length  of  chain,  infect,  and  z angle  of  direction  of  chain. 

Assume  C = 300  feet,  L ==  1000  tons,  v = 25  feet,  x = 100  feet,  n = 30,  r = 71°  34', 
and  0 = 120  32'. 


Then,  ^ 


300  X 1000 


8X25 


= 1500  tons  - 


1500  -J-  1 = 1536.56  tons  = s', 


25  X 1002 * * S * * * 
(.  5 X 300)2 

4 X hi 


2 X 100 


= 11. 11  feet  — h) 

— . 2222  = 12°  32* — 


tan.  0 ; 

4 X 25 

3°° 

1500 


2 \/(' 

= -3333  = 7l0  34'—  = cot.  angle  r; 


5 X 3°°) 2 “h  — 25  2 = 305. 5 feet  =.  I ; 


300  X 1000 
8X1 500 


: 25  = v ; 


yj  + 1 = 1428.6  tons  — t)  and 


30 
2 X 25 


1000 

= 34. 48  tons  — s ; 


- = .3162  = 180  26'. 


V(2  X 25) 2 (300  -4-  2) 2 

For  a deflection  of  .125  of  span,  horizontal  stress  is  equal  to  total  load. 

To  Construct  curve,  see  Geometry,  page  230. 

To  Compute  Batio  wliicli  Stress  01a  Oliains  or  Cables  at 
either  Boint  of  Suspension  Bears  to  whole  Suspended 
"Weight  of  Structure  and.  Load. 


==  R.  R representing  ratio. 


2 X sin.  z 

Illustration. — Assume  elements  of  preceding  case. 

-■  x * 6g  ==  i-  58  ratio.  By  a preceding  formula  it  would  be  1. 536. 

Stress  on  Back  Stays. — The  cables  being  led  over  rollers,  having  free  mo- 
tion, tension  upon  them  is  same,  whether  angle  z is  same  as  that  of  r or  not. 

Stress  on  Piers. — When  angles  r and  i are  alike,  stress  on  piers  will  be 
vertical,  but  when  angle  of  i is  greater  or  less  than  r,  stress  will  be  oblique. 

To  Compute  Horizontal  Stress  and  Vertical  Pressure 
on  Biers. 

S cos.  z = Si,  S cos.  n r=  S 0,  S sin.  « = and  S sin.  w = Po.  St  and  S 0 

representing  stress , and  P i and  P 0 pressure,  inward  and  outward. 

No-te  —Span  of  New  York  and  Brooklyn  Bridge  1505.5  feet,  deflection  128  feet, 

angle  of  deflection  at  piers  from  horizontal  150  10'. 


TRACTION. 


843 


TRACTION*. 

JR, e suits  of'  Experiments  on  Traction  of  Roads 
and.  Pavements.  ( M . Morin.) 

1 st.  Traction  is  directly  proportional  to  load,  and  inversely  proportional 
to  diameter  of  wheel. 

2d.  Upon  a paved  or  Macadamized  road  resistance  is  independent  of 
width  of  tire,  when  it  exceeds  from  3 to  4 ins. 

3d.  At  a walking  pace  traction  is  same,  under  same  circumstances,  for 
carriages  with  or  without  springs. 

4th.  Upon  hard  Macadamized,  and  upon  paved  roads,  traction  increases 
with  velocity:  increments  of  traction  being  directly  proportional  to  incre- 
ments of  velocity  above  velocity  qf  3.28  feet  per  second,  or  about  2.25  miles 
per  hour.  The  equal  increment  of  traction  thus  due  to  each  equal  increment 
of  velocity  is  less  as  road  is  more  smooth,  and  carriage  less  rigid  or  better 
hung. 

5th.  Upon  soft  roads  of  earth,  sand,  or  turf,  or  roads  thickly  gravelled, 
traction  is  independent  of  velocity. 

6th.  Upon  a well-made  and  compact  pavement  of  dressed  stones,  traction 
at  a walking  pace  is  not  more  than  .75  of  that  upon  best  Macadamized 
roads  under  similar  circumstances ; at  a trotting  pace  it  is  equal  to  it. 

7th.  Destruction  of  a road  is  in  all  cases  greater  as  diameters  of  wheels 
are  less,  and  it  is  greater  in  carriages  without  springs  than  with  them. 

Experiments  made  with  the  carriage  of  a siege  train  on  a solid  gravel 
road  and  on  a good  sand  road  gave  following  deductions  : 

1.  That  at  a walk  traction  on  a good  sand  road  is  less  than  that  on  a good 
firm  gravel  road. 

2.  That  at  high  speeds  traction  on  a good  sand  road  increases  very  rapidly 
with  velocity. 

Thus,  a vehicle  without  springs,  on  a good  sand  road,  gave  a traction  2.64! 
times  greater  than  with  a similar  vehicle  on  same  road  with  springs. 

Ttesxilts  -witli  a Dynamometer. 

Wagon  and  Load  2240  lbs.  * 


Roadway. 

Relat’e  num- 
ber of  horses 
for  like  effect. 

Roadway. 

Relat’e  num- 
ber of  horses 
for  like  effect. 

t>n  railway  R ]hs,  t . . , , 

Telford  road,  46  lbs 

C,7C 

On  best  stone  tracks,  12.5  lbs. 
Good  plank  road,  32  to  50  lbs. 
Stone  block  pavement,  32.5  “ 
Macadamized  road,  65  lbs. . . . 

1.56 

4 to  6.25 
4.06 
8. 12 

Broken  stone  or  con’te,  46  lbs. 
Gravel  or  earth,  140-147  lbs.  j 
Common  earth  road,  200  lbs. . 

3/3 

5-75 

17- 5 

18- 37 

25 

Note. — By  recent  experiments  of  M.  Dupuit,  he  deduced  that  traction  is  inversely 
proportional  to  square  root  of  diameter  of  wheel. 

Relation  of  force  or  draught  to  weight  of  vehicle  and  load  over  6 different  con- 
structions of  road,  gave  for  different  speeds  as  follows: 

Walk.  Trot.  Walk.  Trot. 

Stage  coach,  5 tons. . 1.3  1 | Carriage,  seats  only,  on  springs. . 1.29  1 

Resistance  to  Traction  on  Common  JRoacls. 

On  Macadamized  or  Uniform  Surfaces.  {M.  Dupuit.) 

1.  Resistance  is  directly  proportional  to  pressure. 

2.  It  is  independent  of  width  of  tire. 

3.  It  is  inversely  as  square  root  of  diameter  of  wheel. 

4.  It  is  independent  of  speed. 


* See  Treatise  on  Roads , Streets , and  Pavements , by  Brev.  Maj.-Gcn'l  Q.  A.  Gillmore,  U.  S.  A. 
t Telford  estimated  it  at  3.5. 


844 


TRACTION. 


On  Paved  and  Rough  Roads. 

Resistance  increases  with  speed,  and  is  diminished  by  an  enlargement  of 
tire  up  to  a moderate  limit. 

Traction  on  Various  Roads. — Traction  of  a wheeled  vehicle  is  to  its  weight 
upon  various  roads  as  follows : 

Per  Ton. 

Stone  track,  best  12.5  to  15 
“ “ ....  28  to  39 

“ pavement.  14 
Asphalted. ......  22 

Plank 22 

Block  stone  ( 
pavement. . . . J 32 


to  36 
to  28 
to  45 

to  35 


Per  100  lbs. 

Per  Ton. 

• 55  to  .58 

Telford  road 

46  to  78 

x.25  to  1.3 

Macadamized. . . 

46  to  90 

.5  to  1.5 

“ loose 

67  tO  I 12 

I to  1.25 

Gravel 

134  to  180 

.98  to  2 

Sandy 

140  to  3*3 

1.4  to  1.6 

Earth 

4 

200  to  29c 

Per  100  lbs. 
2. 1 to  3. 5 

2 to  4 

3 to  s 

6 to  8 

6.3  to  14 
9 to  13 


Hence,  a horse  that  can  draw  140  lbs.  at  a walk,  can  draw  upon  a gravel  road 
6 -f-  8 

140  X 100  = 2000  lbs. 

2 

Resistance  on  Common  Roads  or  Fields. 

( Bedford  Experiments , 1874.*) 


Gravelled  Road. 

(Hard  and  dry , rising  1 in  430.) 

Maxi- 

mum 

Draft. 

Average 

Draft. 

Average 

Speed 

per 

Hour. 

H>  de- 
veloped 
per 

Minute. 

Draft  per  Ton 
on  Level. 

Work 
per  IP 
per 
Horse. 

Lbs. 

Lbs. 

Miles. 

tP. 

Lbs. 

IP. 

2 horse  wagon  without  springs. 

320 

159 

2-5 

1.06 

43.5  or. 0192 

•53 

4 “ “ “ “ 

400 

251 

2.6 

1.74 

44-5  “ -02 

.87 

2 “ “ with  “ 

300 

133 

2.47 

.88 

34-7  “ *OI5 

•44 

1 “ cart  without  “ 

180 

49.4 

2.65 

•35 

28 

“ .Q125 

•35 

Arable  Field. 

(Hard  and  dry , rising  1 in  1000.) 
2 horse  wagon  without  springs. 

1000 

700 

2-35 

4.36 

210  or. 099 

2. 18 

4 !!  “ - u !! 

1200 

997 

2.52 

6.7 

194  “ .0^3 

3-35 

2 “ “ with  “ 

1000 

710 

^•35 

4-45 

210  “ .099 

1.22 

1 “ cart  without  “ 

400 

212 

2.61 

1.48 

140 

-0625 

1.48 

Fore  wheels  of  wagons  were  39  ins.,  and  hind  57  ins.  in  diam. ; tires  varying  from 
2.25  to  4 ins. ; and  wheels  of  cart  were  54  ins.  in  diam.,  and  tires  3.5  and  4 ins. 

Springs  reduced  resistance  on  road  20  per  cent.,  but  did  not  lessen  it  in  the  field. 

From  these  data  it  appears,  that  on  a hard  road,  resistance  is  only  from  .25  to  .16 
of  resistance  in  field.  Lowest  resistance  is  that  of  cart  on  road  = 28  lbs.  per  ton; 
due,  no  doubt,  to  absence  of  small  wheels  alike  to  those  of  the  wagons. 

Assuming  average  power  without  springs  to  be  .6  IP  on  road,  as  average  for  a 
day’s  work,  it  represents  .6  X 33000  = 19800  foot-lbs.  per  minute  for  power  of  a 
horse  on  such  a road. 

Resistance  of  a smooth  and  well-laid  granite  track  (tramway),  alike  to  those  in 
London  and  on  Commercial  Road,  is  from  12.5  to  13  lbs.  per  ton. 


Omnibus. f (Weight  5758  lbs.) 

Average  Speed  per  Hour.  Per  Ton.  Total. 

Granite  pavement  (courses  3 to  4 ins.) 2.87  miles.  17.41  lbs.  44.75  lbs. 

Asphalt  roadway 3.56  “ 27.14  “ 69-75  “ 

Wood  pavement. . 3.34  “ 41.6  “ 106.88  “ 

Macadam  road,  gravelly 3.45  “ 44-48  “ 114.32  “ 

“ “ granite,  new 3.51  101.09  “ 259.8  “ 

Note. — The  resistance  noted  for  an  asphalt  roadway  is  apparently  inconsistent 
with  that  for  a granite  pavement,  for  when  it  is  properly  constructed  it  is  least 
resistant  of  all  pavements. 


* See  report  in  Engineering , July  io,  1874,  page  23. 


f Report  Soc.  Arts , London , 1875. 


TRACTION, 


845 


Per  Ton. 

Total. 

31.2 

lbs. 

33  Jbs. 

44 

u 

46  “ 

62 

u 

65  “ 

140 

“ 

147  “ 

W agon.  ( Sir  John  Macneil. ) 

Weight  2342  lbs.  Speed  2.5  Miles  per  Hour. 

Well-made  stone  pavement 

Road  made  with  6 ins.  of  broken  hard  stone,  on  a foundation) 
of  stones  in  pavement,  or  upon  a bottom  of  concrete | [ 

Old  flint  road,  or  a road  made  with  a thick  coating  of  broken  | 

Road  made  with  a thick  coating  of  gravel,  on  earth 

Stage  Coach..  {Sir  John  Macneil) 

Weight  3192  lbs.  Gradients  1 to  20  to  600. 

Metalled  Road. 

At  6 mfie's  per  hour 62  lbs.  per  ton. 

::  ■*  ::  « W » « 

TJnnr  —It  was  found  that,  from  some  unexplained  cause,  the  net  frictional  resistance  at  equal  speeds 
. . prmRiderablv  according  to  gradient,  resistances  being  a maximum  for  steepest  gradient  , and  a 
varied  cons^erably , accormng . g , e less  than  x in  600.  Mode  of  action  of 

To  Compute  Resistance  to  Traction  on  Various  Roads. 
(Sir  John  Macneil.) 

ON  A LEVEL. 

Rule  —Divide  weight  of  vehicle  and  load  in  lbs.  by  its  unit  in  following 
table,  and  to  quotient  add  .025  of  load;  add  sum  to  product  of  velocity  of 
vehicle  in  feet  per  second,  and  Coefficient  in  following  table  for  the  particular 
road,  and  result  will  give  power  required  in  lbs. 

Or  W + w 1 w .025  + Cv  = T.  W and  w repi'esenting  weights  of  vehicle  and  load 
’ unit  ' 

Coefficients  for  Traction  of  Various  Vehicles. 

Stage  coach I 2 horse  wagon  without  springs 54 

TTpuvv  wiip-nn cn  2 “ u with  ‘ 42 


Heavy  wagon 93 

4 horse  wagon  without  springs 55  I 1 


cart  without 


4-3 


Coefficients  for  Roads  of  Various  Construction. 

Macadamized  road 

Gravel,  clean 13 

u muddy 32 

Stone  tramway • • 1.2 

....  12. 1 


Pavement 2 

Broken  stone,  dry  and  clean 5 

“ “ covered  with  dust. .. . 8 

“ “ muddy 10 

Sand  and  Gravel 

Illustration.  — What  is  the  traction  or  resistance  of  a stage  coach  weighing  2200 
lbs.,  with  a load  of  1600  lbs.,  when  driven  at  a velocity  of  9 feet  per  second  over  a 
dry  and  clean  broken  stone  road  ? 

2200  + 1600  i6qq  x -025  .j.  5 x 9 = 123  lbs. 

100 

To  Compute  Rower  necessary  to  Sustain  a "Vehicle  -upon 
aix  Inclined.  Road,  and  also  its  Rressure  thereon,  omit- 
ting Efi'ect  of  Friction. 

AT  AN  INCLINATION. 

W : A C : : 0 : B C,  and  W : A C : : p : A B. 

Or,  r e : e 0 ::  A B : B C;  W : c o ::  l : h:  whence, 

W j = eo- 

* A Assume  A B of  such  a length  that  vertical  rise, 


B 

BC  = i foot;  then, 
W _ W 
ac_Vab24-i 


= W sin.  A = 0.  and 


W A B 


WAB 


AG  VAB24-i 

B 


= W cos.  A — p. 


TRACTION. 


846 


W W V W ■ W Z'2 

Or,  — = P:  — — =P]  or,  - - ==  P,  and  — =p.  W representing 

1 1 VZ'2  + 1 Vz'2+i 

weights  of  vehicle  and  load  0,  and  ¥ power  or  force  necessary  to  sustain  load  on  road , 
p pressure  of  load  on  surface , all  in  lbs .,  h height  of  plane , Z inclined  length  of  road 
or  plane,  and  l'  horizontal  length,  all  in  feet. 

Illustration. — What  is  power  required  to  sustain  a carriage  and  its  load,  weigh- 
ing 3800  lbs.,  upon  a road,  inclination  of  which  is  1 in  35,  and  what  is  its  pressure 
upon  road  ? 

Sin.  A = .028  56.  Cos.  A = .99994.  Z = 35.014. 

Then  3800  X • 028  56  = 108. 53  lbs.  = power,  and  3800  X • 999  94  = 3799-  77  Z&s.  pressure. 

To  Compute  Resistance  of  a Load,  on  an  Inclined.  Road. 

Rule. — Ascertain  the  tractive  power  required,  and  add  to  it  the  power 
necessary  to  sustain  load  upon  inclination,  if  load  is  to  ascend,  and  subtract 
it  if  to  descend. 

Example  1.— In  preceding  example  tractive  power  required  is  123  lbs.,  and  sus- 
taining power  for  that  inclination  108.53;  hence  123-1-108.53  = 231.53  lbs. 

2. — If  this  load  was  to  be  drawn  down  a like  elevation. 

Then  123  — 108.53  = 14.47  ^ s • 

To  Compute  Rower  necessary  to  Move  and.  Sustain  a 
"Vehicle  either  Ascending  or  Descending  an  Elevation, 
and  at  a given  Velocity,  omitting  Effect  of  Friction. 

(W  -4-  w w\  — — 

— h 1-  — cos.  L + (W  -4-  w)  sin.  L.  4-  v c = R.  L.  representing  angle  of 

t 40/ 

elevation  for  a stage  wagon  and  a stage  coach,  and  t units  as  preceding  ; upper  sign 
taken  when  vehicle  descends  the  plane , and  lower  when  it  ascends. 

Illustration. — Assume  a stage  coach  to  weigh  2060  lbs.,  added  to  which  is  a 
load  of  1100  lbs.,  running  at  a speed  of  9 feet  per  second  over  a broken  stone  road 
covered  with  dust,  and  having  an  inclination  of  1 in  30;  what  is  power  necessary 
to  move  and  sustain  it  up  the  inclination,  and  what  down  it  ? 

v = g , c = 8,  sin.  of  L.  = sin.  of  i°  54'+ = .0333,  and  cos.  L- =.9995. 

/2060  + 1100  , noo\  . — : 7 . . 

Then  ( \-  - ) x .9995 + ( 2060+ 1100)  X -0333  + 8 X 9 = 59-°7  + 

\ 100  40  / 

105.23  + 72  = 236.3  lbs.  up  inclination. 

. , /2060  + 1 100  . 1 roo\  . - — — — - — : — ‘ . 

And  ^ — b J X .9995  + 8X9  — (2060  + 1100)  X .0333  = 59.07  + 72 

— 105. 23  = 25. 84  lbs.  down  inclination. 


Tractive  and  Statical  Resistance  of  Elevations.  (Gillmore.) 
T 

■■  - —g.  T representing  traction  in  lbs.  per  ton , W weight  of  load  in  lbs., 

VW2- T2 

and  g'  grade  of  road. 

Illustration.— Assume  traction  as  per  preceding  table,  page  844,  200,  and  weight 
of  vehicle  2 tons ; what  should  be  least  grade  of  road  ? 


200  X 2 


\J  44802  — 


- = .0897  = - + . 


200  X 2 


Showing  that,  for  a road  upon  which  traction  is  200  lbs.  per  ton,  the  grade  should 
not  exceed  one  in  height  to  one  eleventh  fall  of  base;  hence,  generally,  the  proper 
grade  of  any  description  of  road  will  be  equal  to  force  necessary  to  draw  load  upon 
like  road  when  level. 


Practically,  greatest  grade  of  a Telford  or  Macadamized  road  in  good  condition 
= .05,  and  a horse  can  attain  at  a walk  a required  height  upon  this  grade,  without 
more  fatigue  and  in  nearly  same  time  that  he  would  require  to  attain  a like  height 
over  a longer  road  with  a grade  of  .033,  that  he  could  ascend  at  a trot. 

For  passenger  traffic,  grades  should  not  exceed  .033. 


traction. 


847 


Resistance  of  Gravity  at  Different  Inclinations  of 
Grrade.  For  a Load  of  100  Los. 


1 in  5 
1 in  10 

1 in  15 

1 in  20 


Lbs. 

19.61 

9-95 

6.65 

4.99 


1 in  25 
1 in  30 
1 in  35 
in  40 


1 in  45 
1 in  50 
1 in  55 
1 in  60 


Lbs. 

2.22 


1 in  70 
1 in  80 
1 in  90 
1 in  100 


Inclination  of  Roads.- Power  of  draught  at  different  inclinations  and  velocities 
is  as  follows  ( Sir  John  Macneil) : 


Inclination. 

Angle. 

1 in  20 

2°  52' 

i in  26 

2°  X2' 

1 in  30 

1°  55' 

1 in  40 

1°  26' 

1 in  60 

57- S' 

Traction  at  Speeds  of  per 

Feet 

Hour  of 

per  Mile. 

6 Miles. 

8 Miles. 

10  Miles. 

264 

268 

296 

203.4 

213 

219 

225 

176 

165 

196 

200 

132 

160 

166 

172 

88 

in 

120 

128 

Frictional  Resistance  per 

Ton  at  Speeds  of  per  Hour  ol 

6 Miles. 

8 Miles. 

10  Miles. 

76 

96 

112 

63 

68 

41 

63 

66 

56 

61 

65 

72 

78 

81 

Grrade. 

Grade  of  a road  should  be  reduced  to  least  of  practicable  attainment,  and 
as  a general  rule  should  be  as  low  as  1 in  33,  and  steepest  grade  that  is  ad- 
missible on  a broken  stone  road  is  1 in  20. 

The  condition  of  traction  is  /+sin.  a L,  which  should  not  exceed  P,  and  sin.  a 
should  not  exceed  j-  —f  or/,  f representing  coefficient  of  friction , a angle  of  in- 
clination, L load , and  P power  in  lbs. 

Illustration.- In  case,  page  846,  weight  or  load  = 2060  + 1x00  = 3160  ’ 

efficient  of  friction  for  such  a road  = .042  per  100  lbs.,  and  sin.  1 54  - -°33 1 • 

Then  .042 + .033  16  X 3160  — 237.5  lbs. 

Traction  of  a Vehicle  compared  to  its  Weight  on  Different  Roads. 

(F.  Robertson , F.  R.  A.  S.) 

1 in  68  I Flint  foundation 1 hj  34 

1 “ 49  I Gravel  road 1 ‘ J5 

Sandy  road 1 in  7. 

Assuming  a horse  to  have  a tractive  force  of  140  lbs.  continuously  and  steadily  at 
a walk,  he  can  draw  at  a walk  on  a gravel  road  15  X 140  = 2100  lbs. 

Friction  of*  Ptoacls. 

Friction  of  Roads—  According  to  Babbage  and  others,  a wagon  and  load 
weighing  1000  lbs.  requires  a traction  as  follows . 

Of  Load.  of  Load* 

Macadamized °33 

Dry  high  road 025 

Well  paved  road 014 

( .0035 

Railroad | .0059 

033  of  load. 


Stone  pavement — 
Macadamized  road 


Gravelled  road 

in 

Gravel  road,  new 083 

Common  road,  bad  order. . .07 

Sand  road 063 

Broken  stone,  rutted 052 

“ “ fair  order...  .028 

“ “ perfect  order  .015 

Macadamized,  new 045 

“ 033 

“ gravelly 02 

Earth,  good  order 025 


Sled , hard  snow,  iron  shod.,,.  . 

Coefficients  of  Friction  in  Proportion  to  Load. 

Per  100  lbs.  Per  Ton. 


Per  100  lbs. 

Wood  pavement 019 

Asphalt  roadway 012 

Stone  pavement 015 

Granite  “ 008 

Stone  “ very  smooth  .006 

Plank  road 01 

1 -0035 

Railway j .0059 

Stone  track 05 


848 


TRACTION. 


To  Compute  Frictional  Resistance  to  Traction  of*  a 
Stage  Coach  on  a NXetalled  Road  in  Good  Condition. 

3°  + 4 v + V10  v — R-  v representing  speed  in  miles  per  hour , and  R frictional 
resistance  to  traction  per  ton. 

Note.— Formula  is  applicable  to  wagons  at  low  speeds. 

Canal,  Slackwater,  and.  River. 

On  a canal  and  water,  resistance  to  traction  varies  as  square  of  velocity 
from  that  of  2 feet  per  second  to  that  of  11.5  feet. 

When  velocity  is  less  than  .33  miles  per  hour,  resistance  varies  in  a less 

degree. 

In  towing,  velocity  is  ordinarily  1 to  2.5  miles  per  hour. 

Resistance  of  a boat  in  a canal  depends  very  much  upon  the  comparative 
areas  of  transverse  sections  of  it  and  boat,  it  being  reduced  as  difference 
increases. 

In  a mixed  navigation  of  canal  and  slack-water,  3 horses  or  strong  mules 
will  tow  a full-built,  rough-bottomed  canal  boat,  with  an  immersed  sectional 
area  of  94.5  sq.  feet,  and  a displacement  of  240  tons,  1.75  to  2 miles  per  hour 
for  periods  of  12  hours. 

With  a section  of  but  24.5  sq.  feet,  or  a displacement  of  65  tons,  an  aver- 
age speed  of  2.5  miles  is  attained  for  a like  period. 

By  the  observations  of  Mr.  J.  F.  Smith,  Engineer  of  the  Schuylkill  Navigation 
Co.,  a canal  boat,  with  an  immersed  section  alike  to  that  above  given,  can  be  towed 
for  10  hours  per  day  as  follows: 

Per  Hour. 


By  1 horse  or 
mule. 

I By  2 horses 'or 
mules. 

By  3 horses  or 
mules. 

By  4 horses  or 
mules. 

By  8 horses  or 
mules. 

1 mile. 

1.5  miles. 

1.75  miles. 

1.875  miles. 

2.5  miles. 

Assuming  then,  the  tractive  power  of  a horse  as  given  in  table,  page  437,  the  above 
elements  determine  results  as  follows: 


Horses. 

Miles. 

Tractive  Power 
divided  by  Load. 

in  Lbs.  per  Ton. 

rraction 

in  Lbs.  per  Sq.  Foot  of 
immersed  Section. 

1 

1 

2504-240 

1.04 

2.65 

2 

i-5 

165  X 24-240 

1.38 

3-49 

3 

I-75 

140  X 3 -S-  240 

i-75 

4.44 

3 

i-875 

132  X 3 4-  240 

1-65 

4.19 

3 

2 

125  x 34-240 

1.56 

3-98 

3 (hght) 

2-5 

100  X 3 -4-  65 

4.61  * 

| 12.24 

Upon  a canal  of  less  section  and  depth,  a displacement  of  105  tons,  with  an  im- 
mersed section  of  43  sq.  feet,  a speed  of  2 miles  with  2 horses  was  readily  obtained, 
which  would  give  a traction  of  2.38  lbs.  per  ton,  and  of  5.71  lbs.  per  sq.  foot  of  im- 
mersed section. 

Maximum  Power  of  a.  Horse  011  a Canal.  (Molesworth.) 


Miles  per  hour 2.5  3 3.5  45678  9 10 

Duration  of  work  in  ) 0 

hours } II-5  8 5-9  4-5  2.9  2 1.5  1.125  -9  -75 


Load  drawn  in  tons..  520  243  153  102  52  30  19  13  9 6.5 

Street  Railroads  or  Tramways.  ( GenH  Gillmore.*) 

Upon  a level  road,  and  at  a speed  of  5 miles  per  hour,  the  power  required  to  draw 
a car  and  its  load  is  from  to  of  total  weight,  varying  with  condition  of 
rails  and  dryness  or  moisture  of  their  surface. 


* Treatise  on  Roads,  Streets,  and  Pavements.  D.  Van  Nostrand,  1876,  N.  Y. 


TRACTION. WATER. 


849 


To  Compute  Resistance  of*  a Car. 

Ty6_f.  Txv  — c-  vl*L?L  — r ; and  /-f-c-f-r  = R.  T representing  weight 

A A u J ) } 400  _ 

in  tons,  f f notion  in  lbs.,  v speed  in  miles  per  hour , a area  of  front  or  section  of  car 
in  sq.  feet , c concussion , r resistance  of  atmosphere , and  R total  resistance , aii  in  tos. 

Illustration. — Assume  a car  and  load  of  8960  lbs.,  with  an  area  of  section  of  56 
sq.  feet,  and  a speed  of  5 miles  per  hour. 

Then 

52  X 56 
400 


896°  __  ^ £0ns  . ^ x 6 = 24  lbs.  friction  ; - = 6.66  Z&s.  concussion ; 

2240  3 

= 3. 5 Z&s.  resistance  of  air  ; and  24  -j-  6. 66  + 3. 5 = 34- 16 

In  average  condition  of  a road,  the  resistance  of  a car  may  be  taken  at  which, 
in  preceding  case,  would  be  74.66  lbs.  On  a descending  grade,  therefore,  of  1 in 
74.66,  the  application  of  a brake  would  not  be  required. 


WATER. 

Fresh  Water.  Constitution  of  it  by  weight  and  measure  is 

By  Weight.  By  Measure.  I By  Weight.  By  Measure. 

Oxygen...  88.9  1 I Hydrogen..  11.1  2 

Cube  inch  of  distilled  water  at  its  maximum  density  of  39.10,  barom- 
eter at  30  ins.,  weighs  252.879  grains,  and  it  is  772.708  times  heavier 
than  atmospheric  air. 

Cube  foot  (at  39.  i°)  weighs  998.8  ounces,  or  62.425  lbs. 

Noxe< For  facility  of  computation,  weight  of  a cube  foot  of  water  is 

usually  taken  at  1000  ounces  and  62.5  lbs. 

At  a temperature  of  320  it  weighs  62.418  lbs.,  at  62°  (standard  tem- 
perature) 62.355  lbs.,  and  at  2120  59-64  lbs.  Below  39.10  its  density 
decreases,  at  first  very  slow,  but  progressing  rapidly  to  point  of  conge- 
lation, weight  of  a cube  foot  of  ice  being  but  57.5  lbs. 

Its  weight  as  compared  with  sea-water  is  nearly  as  39  to  40. 

It  expands  .085  53  its  volume  in  freezing.  From  40°  to  120  it  ex- 
pands .00236  its  volume,  and  from  40°  to  2120  it  expands  .0467- 
times  = .000271  5 for  each  degree,  giving  an  increase  in  volume  of  1 
cube  foot  in  21.41  feet. 


"Volumes  of  Pure  Water. 


At  320  27.684  cube  ins.  = 1 cube  foot. 

“ 39-i°  27.68  u “ = 1 

“ 62°  27.712  “ “ = 1 

“ 212°  28.978  “ “ =1  U * 


At  62°  1 Ton  =35-923  cube  feet. 

“ “ 1 Lb.  =27.71  “ ins. 

“ 39.10  1 Tonneau  = 35.3156  “ feet. 

“ “ 1 Kilogr.  = .0353  “ “ 


Height  of  a Column  of  Water  at  62°  or  62.355  lbs. 

1 lb.  per  sq.  inch  = 2. 3093  feet,  and  at  pressure  of  atmosphere  = 33.947  feet  = 
10.347  meters. 

Ice  and.  Snow. 

Cube  foot  of  Ice  at  320  weighs  57.5  lbs.,  and  1 lb.  has  a volume  of 
30.067  cube  ins. 

Volume  of  water  at  320,  compared  with  ice  at  320,  is  as  1 to  1.085  53>  ex- 
pansion being  8.553  Per  cent* 

Cube  foot  of  new  fallen  snow  weighs  5.2  lbs.,  and  it  has  12  times  bulk  of 
water. 


850 


WATER. 


Rainfall. 

Annual  Fall  at  different  Places . 


Location. 


Alabama 

Albany 

Algiers  

Alleghany 

Antigua 

Archangel 

Auburn 

Bahamas 

Baltimore 

Barbadoes 

Bath,  Me 

Belfast 

Biskra 

Bombay 

Bordeaux 

Boston 

Brussels 

Buffalo 

Burlington,  Vt 

Calcutta 

Cape  St.  Franpois. . 

Cape  Town 

Charleston 

Cherbourg 

Cologne 

Copenhagen 

Cracow 

Demerara 


150 

23- 31 

54 
39-7 
24 

23 

*3-33 
91.2 
132.21 

37- 52 
30.87 
36.92 

38- 52 

24- 5 
v 1 29 

Fairfield ..........  | 32.93 


Dover  (Engl.)  . . 

Dublin 

Dumfries 

East  Hampton  . 

Edinburgh 


Ins. 


30-I7 

4i-35 

7-75 

46.66 

45 

14.52 

30.17 

42.19 

39-9 

55-87 

34-58 

39-46 


IIO 

29.7 

39-23 

29 

27.27 

32 


Location. 


Ft.  Crawford,  Wis.. 

Ft.  Gibson,  Ark 

Ft.  Snelling,  Iowa.. 
Fortr.  Monroe,  Va. 

Florence 

Frankfort,  Oder. . 
“ Main. . 

Geneva 

Gibraltar 


Glasgow 

Gordon  Castle,  Sc’d 
Granada 


Great  Britain  . 

Greenock 

Halifax 

Hanover 

Havana 

Hong-kong 

Hudson 


India. 


Jamaica 

Jerusalem 

Key  West 

Khassaya,  India. . . 

Lewiston 

Liverpool 

London 

Louisiana 


Madeira  . . . . 
Manchester . 


Marseilles. . 


Ins. 


29-54 

30.64 

30.32 

52-53 

35-9 

21.3 

16.4 
32.6 
47.29 
21.3 
3i 
29-3 

105 

126 


32 

61.8 


33 

22.4 

52 

8i.35 

39-32 

60 


130 

34-31 

65 

31.39 

610 

23.15 

34.12 

25.2 

51-85 


Location. 


22 


4l 

36.14 

43 

18.2 


Michigan 

Mississippi 

Mobile,  1842 

Naples 

Newburg 

New  York 

Ohio 

Palermo 

Paris 

Philadelphia 

Plymouth  (Engl.).. 

Port  Philip 

Poughkeepsie 

Providence 

Rochester 

Rome 

Santa  Fe 

Savannah 

Schenectady 

Siberia  . . . .' 

Sierra  Leone 

Sitka 

St.  Bernard 

St.  Domingo 

St.  Petersburg 

State  of  N.Y 

Sydney 

Tasmania 

Trieste 

Ultra  Mullay,  India 

Utica 

Venice 

Vera  Cruz 

Vienna 

Washington 

West  Point 


Ins. 


33-5 

45 

54-94 

41.8 

40.5 
36 
36 

22.8 
23.1 
49 
44 

29. 16 

32.06 
36.74 

29 

39 

74.8 
55 

47-77 

7-75 


85-79 

48 

120 

17.6 

33- 79 

49 
35 

46.4 

263.21 

39-3 

34- i 
62 

19.6 
34.62 

48.7 


Average  rainfaH  in  England  for  a number  of  years  was,  South  and  East,  34  ins  • 
W est  and  hilly,  43.02  to  50  ins.,  and  percolation  of  it  was  estimated  at  30  per  cent.  ’ 


Mean  volume  of  water  in  a cube  foot  of  air  in  England  is  3.789  grains. 

Globe,  mean  depth ,5  ins 

Cape  of  Good  Hope  in  1846 in  3 hours,  6.2  “ 

At  Khassaya,  in  6 rainy  months 550  ins. ; in  1 day,  25.5  “ 

Evaporation.—  Mean  daily  evaporation,  in  India  .22  inch;  greatest  .56*  in  Eng- 
land .08.  At  Dijon,  when  mean  depth  of  rainfall  was  26.9  ins.  in  7 years’  evapora- 
tion was  for  a like  period  26.1  ins.,  and  in  Lancashire,  Eng.,  when  fall  was  as  06 
ins.,  evaporation  was  25.65. 


Volume  of*  Rainfall. 

Rainfall,  depth  in  ins. , x 2 323  200  = cube  feet  per  sq.  mile. 


X 17-37874  = millions  of  gallons  per  sq.  mile. 
X 3630  = cube  feet  per  acre. 

X 27  154. 3 = gallons  per  acre. 


Mineral  Waters  are  divided  into  5 groups,  viz. : 


_.  Carbonated,  containing  pure  Carbonic  acid  — as,  Seltzer,  Germany;  Spa,  Bel- 
gium; Pyrmont,  Westphalia;  Seidlitz,  Bohemia;  and  Sweet  Springs,  Virginia.’ 


2.  Sulphurous,  containing  Sulphuretted  hydrogen— as,  Harrowgat’e  and  Chelten- 
ham, England;  Aix-la-Chapelle,  Prussia;  Blue  Lick,  Ky. ; Sulphur  Springs,  Va.,  etc. 


3.  Alkaline,  containing  Carbonate  of  soda— these  are  rare,  as,  Vichy,  Ems. 


WATER. 


851 


a.  Chalybeate,  containing  Carbonate  of  iron— as,  Hampstead,  Tunbridge,  Chelten- 
ham, and  Brighton,  England;  Spa,  Belgium;  Ballston  and  Saratoga,  N.  Y. ; and 
Bedford,  Penn. 

c.  Saline,  containing  salts— as,  Epsom,  Cheltenham,  and  Bath,  England;  Baden- 
Baden  and  Seltzer,  Germany;  Kissingen,  Bavaria;  Plombieres,  France;  Seidlitz, 
Bohemia  ; Lucca,  Italy  ; Yellow  Springs,  Ohio ; Warm  Springs,  N.  C. ; Congress 
Springs,  N.  Y. ; and  Grenville,  Ky. 

Brief  Rules  for  Qualitative  Analysis  of  Mineral  Waters. 

First  point  to  be  determined,  in  examination  of  a mineral  water,  is  to  which  of 
above  classes  does  water  in  question  belong. 

1.  If  water  reddens  blue  litmus  paper  before  boiling,  but  not  afterwards,  and  blue 
color  of  reddened  paper  is  restored  upon  warming,  it  is  Carbonated. 

2.  If  it  possesses  a nauseous  odor,  and  gives  a black  precipitate,  with  acetate  of 
lead,  it  is  Sulphurous. 

3.  If,  after  addition  of  a few  drops  of  hydrochloric  acid,  it  gives  a blue  precipitate, 
with  yellow  or  red  prussiate  of  potash,  water  is  a Chalybeate. 

4.  If  it  restores  blue  color  to  litmus  paper  after  boiling,  it  is  Alkaline. 

5.  If  it  possesses  neither  of  above  properties  in  a marked  degree,  and  leaves  a 
large  residue  upon  evaporation,  it  is  a Saline  water. 

River  or  canal  water  contains  .05  \ 0£  vojume  0f  gaseous  matter. 

Spring  or  well  water  .07) 

Tie-agents. 

When  water  is  pure  it  will  not  become  turbid,  or  produce  a precipitate 
with  any  of  following  Re-agents : 

Baryta  Water,  if  a precipitate  or  opaqueness  appear,  Carbonic  Acid  is  present. 

Chloride  of  Barium  indicates  Sulphates,  Nitrate  of  Silver,  Chlorides,  and  Oxalate 
of  Ammonia,  Lime  salts.  Sulphide  of  Hydrogen,  slightly  acid,  Antimony,  Arsenic, 
Tin,  Copper,  Gold,  Platinum,  Mercury,  Silver,  Lead,  Bismuth,  and  Cadmium;  Sul- 
phide of  Ammonium , solution  alkaloid  by  ammonia,  Nickel,  Cobalt,  Manganese, 
Iron,  Zinc,  Alumina,  and  Chromium.  Chloride  of  Mercury  or  Gold  and  Sulphate  of 
Zinc,  organic  matter. 

Filter  Beds. 

Fine  sand,  2 feet  6 ins. ; Coarse  sand,  6 ins. ; Clean  shells,  6 ins.,  and  Clean  gravel 
2 feet,  will  filter  700  gallons  water  in  24  hours  by  gravitation. 


Sea  Water.  Composition  of  it  per  volume: 


Chloride  of  Sodium  (common  salt).  .2.51 


Sulphuret  of  Magnesium 53 

Chloride  of  “ 33 


Carbonate  of  Lime 
“ of  Magnesia 

Sulphate  of  Lime 

Water 


.01 

96.6 


By  analysis  of  Dr.  Murray,  at  specific  gravity  of  1.029,  it  contains 

Muriate  of  Soda 220.01  I Muriate  of  Magnesia 42.08 

Sulphate  of  Soda 33. 16  | Muriate  of  Lime 7.84 


303.09 

Or,  1 part  contains  .030309  parts  of  salt  = 3V  Part  °f  its  weight. 

Mean  volume  of  solid  matter  in  solution  is  3.4  per  cent.,  .75  of  which  is 
common  salt. 

Boiling  Points  at  Different  Degrees  of  Saturation. 


Salt,  by  Weight, 

Boiling 

Salt,  by  Weight, 

Boiling 

Salt,  by  Weight, 

Boiling 

in  100  Parts. 

Point. 

in  100  Parts. 

Point. 

in  100  Parts. 

Point. 

3-°3  = it 

213. 2° 

IS-15  = WS 

217. 90 

27.28=,^ 

222. 5° 

6-°6  = Ts 

214. 40 

l8.l8  = -^ 

2190 

30- 31  = if 

223.7° 

9.09= A 

2iS-5° 

2I.22  = -373 

220.2° 

33-34  = ii 

224.9° 

I2-12  — -h 

216. 70 

24-25  = A 

221 .4° 

*36.37 =i| 

226° 

* Saturated. 


WATER. WAVES  OF  THE  SEA. 


852 


Deposits  at  Different  Degrees  of  Saturation  and.  Tem- 
perature. 

When  1000  Parts  are  reduced  by  Evaporation. 


Volume  of  Water. 

Boiling  Point. 

Salt  in  100  Parts. 

Nature  of  Deposit. 

1000 

2140 

3 

None. 

299 

2170 

10 

Sulphate  of  Lime. 

102 

228° 

29*5 

Common  Salt. 

It  contains  from  4 to  5.3  ounces  of  salt  in  a gallon  of  water. 


Saline  Contents  of  Water  from  several  Localities. 


Baltic 6.6  I 

Black  Sea 21.6 

Arctic 28.3  I 


British  Channel. .. . 35.5 

Mediterranean 39.4 

Equator 39-42 


South  Atlantic. .....  41.2 

North  Atlantic 42.6 

Dead  Sea 385 


There  are  62  volumes  of  carbonic  acid  in  1000  of  sea- water. 


Cube  foot  at  62°  weighs  64  lbs.  Its  weight  compared  with  fresh  water 
being  very  nearly  as  40  to  39. 

Height  of  a Column  of  Water  at  6o°  or  64.3125  lbs. 

At  62°,  1 Ton  = 35  cube  feet.  1 Lb.  per  sq.  inch  = 2.239  feet>  and  at  pressure  of 
atmosphere  = 32.966  feet  = 10.048  meters. 

'Weights. 

A ton  of  fresh  water  is  taken  at  36,  and  one  of  salt  at  35  cube  feet. 


WAVES  OF  THE  SEA. 

Arnott  estimated  extreme  height  of  the  waves  of  an  ocean,  at  a distance 
from  land  sufficiently  great  to  be  freed  from  any  influence  of  it  upon  their 
culmination,  to  be  20  feet. 

French  Exploring  Expedition  computed  waves  of  the  Pacific  to  be  22  feet 
in  height. 

By  observations  of  Mr.  Douglass  in  1853,  he  deduced  that  when  waves  had 
heights  of 

8 feet,  there  were  35  in  number  in  one  mile,  and  8 per  minute. 

15  “ “ 5 and  6 “ “ 5 “ 

20  “ “ 3 “ “ 4 “ 

J.  Scott  Russell  divides  waves  into  2 classes — viz. : 

Waves  of  Translation,  or  of  1st  order ; of  Oscillation,  or  of  2d  order. 

Waves  of*  the  Dirst  Order. 

1.  Velocity  not  affected  by  intensity  of  the  generating  impulse. 

2.  Motion  of  the  particles  always  forward  in  same  direction  as  wave,  and 
same  at  bottom  as  at  surface. 

3.  Character  of  wave,  a prolate  cycloid,  in  long  waves,  approaching  a true 
cycloid.  When  height  is  more  than  one  third  of  length,  the  wave  breaks. 

Waves  of*  the  Second  Order. 

1.  Ordinary  sea  waves  are  waves  of  second  order,  but  become  waves  of  the 
first  order  as  they  enter  shallow  water. 

2.  Power  of  destruction  directly  proportional  to  height  of  wave,  and  great- 
est when  crest  breaks. 

3.  A wave  of  10  feet  in  height  and  32  feet  in  length  would  only  agitate 
the  water  6 ins.  at  10  feet  below  surface;  a wave  of  like  height  and  too  feet 
in  length  would  only  disturb  the  water  18  ins.  at  same  depth. 

Average  force  of  waves  of  Atlantic  Ocean  during  summer  months,  as  de- 
termined by  Thomas  Stevenson , was  61 1 lbs.  per  sq.  foot;  and  for  winter 
months  2086  lbs.  During  a heavy  gale  a force  of  6983  lbs.  was  observed. 


WAVES  OF  THE  SEA. 


853 


J.  Scott  Russell  deduced  that  a wave  30  feet  in  height  exerts  a force  of  1 
ton  per  sq.  foot,  and  that,  in  an  exposed  position  in  deep  water,  1.75  tons 
may  be  exerted  upon  a vertical  surface. 

At  Cassis,  France,  when  the  water  is  deep  outside,  blocks  of  15  cube  me- 
ters were  found  insufficient  to  resist  the  action  of  waves. 

Breakwaters  with  vertical  walls,  or  faces  of  an  angle  less  than  1 to  1,  will 
reflect  waves  without  breaking  them.  Waves  of  oscillation  have  no  effect 
on  small  stones  at  22  feet  below  the  surface,  or  on  stones  from  1.5  to  2 feet, 
12  feet  below  surface. 

A roller  20  feet  high  will  exert  a force  of  about  1 ton  per  sq.  foot. 

Greatest  force  observed  at  Skerry vore,  3 tons  per  sq.  foot ; at  Bell  Rock, 
1.5  tons  per  sq.  foot. 

Waves  of  the  second  order,  when  reflected,  will  produce  no  effect  at  a depth 
of  12  feet  below  surface. 

Action  of  waves  is  most  destructive  at  low-water  line. 

Waves  of  first  order  are  nearly  as  powerful  at  a great  depth  as  at  surface. 

To  Compute  'Velocity'. 

When  l is  less  than  d.  .55  yjl  or  1.818  y/l  — V. 

When  l exceeds  1000  d.  V 32.17  d = V,  and  When  Height  of  Wave  becomes  a sen- 


sible Proportion  to  Depth , ^32.  *7  y + 3 = V. 

To  Compute  Height  of  Waves  in  Reservoirs,  etc. 

x.  5^/l  + (2. 5 — VL)  = height  in  feet.  L representing  length  of  Reservoir,  Pond, 
etc.,  exposed  to  direction  of  wind,  in  miles. 

Tidal  Waves. 

Wave  produced  by  action  of  sun  and  moon  is  termed  Free  Tide  Wave.  K 

Semi-diurnal  tide  wave  is  this,  and  has  a period  of  12  hours  24-f-  minutes. 

Professor  A iry  declared  that  when  length  of  a wave  was  not  greater  than 
depth  of  the  water,  its  velocity  depended  only  upon  its  length,  and  was  pro- 
portionate to  square  root  of  its  length. 

When  length  of  a wave  is  not  less  than  1000  times  depth  of  water,  velocity  of  It 
depends  only  upon  depth,  and  is  proportionate  to  square  root  of  it;  velocity  being 
same  that  a body  falling  free  would  acquire  by  falling  through  a height  equal  to  half 
depth  of  water. 

For  intermediate  proportions,  velocity  can  be  obtained  by  a general  equation. 

Under  no  circumstances  does  an  unbroken  wave  exceed  30  or  40  feet  in  height. 

A wave  breaks  when  its  height  above  general  level  of  water  is  equal  to  general 
depth  of  it. 

Diurnal  and  other  tidal  waves,  so  far  as  they  arc  free,  may  be  all  considered  as 
running  with  the  same  velocity,  but  the  column  of  the  length  of  wave  must  be 
doubled  for  diurnal  wave. 


Length  of  Wave. 


Depth  of  Water. 

Feet. 

1 j 

Feet. 

10  I 

Feet.  I Feet. 

100  1 1000  1 

Velocity  per  Second. 

Feet. 

10  000  1 

Feet. 

; IOO  OOO 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

1 

2.26 

5-34 

5-67 

— 

— 

— 

10 

2.26 

7-i5 

16.88 

17.92 

17-93 

— 

100 

— 

7-I5 

22.62 

53-19 

56.67 

56.71 

1 000 

— 

22.62 

7i-54 

168.83 

179.21 

10000 

— 

— 

— 

7i-54 

226.24 

533-9 

4c 


WHEEL  GEARING. 


854 


WHEEL  GEARING. 

Pitch  Line  of  a wheel  is  circle  upon  which  pitch  is  measured,  and  it 
is  circumference  by  which  diameter,  or  velocity  of  wheel,  is  measured. 

Pitch  is  arc  of  circle  of  pitch  line,  is  determined  by  number  of  teeth 
in  wheel,  and  necessarily  an  aliquot  part  of  pitch  line. 

True  or  Chordial  Pitch , or  that  by  which  dimensions  of  tooth  of  a 
wheel  are  alone  determined,  is  a straight  line  drawn  from  centres  of 
two  contiguous  teeth  upon  pitch  line. 

Line  of  Centres  is  line  between  centres  of  two  wheels. 

Radius  of  a wheel  is  semi  - diameter  bounded  by  periphery  of  the 
teeth.  Pitch  Radius  is  semi-diameter  bounded  by  pitch  line. 

Length  of  a Tooth  is  distance  from  its  base  to  its  extremity. 

Breadth  of  a Tooth  is  length  of  face  of  wheel. 

Depth  of  a Tooth  is  thickness  from  face  to  face  at  pitch  line. 

Pace  of  a Tooth , or  Addendum , is  that  part  of  its  side  which  extends 
from  its  pitch  line  to  its  top  or  Addendum  line. 

Flank  of  a Tooth  is  that  part  of  its  side  which  extends  from  pitch 
line  to  line  of  space  at  base  of  and  between  adjacent  teeth ; its  length, 
as  well  as  that  of  face  of  tooth,  is  measured  in  direction  of  radius  of 
wheel,  and  is  a little  greater  than  face  of  tooth,  to  admit  of  clearance 
between  end  of  tooth  and  periphery  of  rim  of  wheel  or  rack. 

Cog  Wheel  is  general  term  for  a wheel  having  a number  of  cogs  or  teeth  set  in  or 
upon,  or  radiating  from,  its  circumference. 

Mortice  Wheel  is  a wheel  constructed  for  reception  of  teeth  or  cogs,  which  are 
fitted  into  recesses  or  sockets  upon  face  of  the  wheel. 

Plate  Wheels  are  wheels  without  arms. 

Rack  is  a series  of  teeth  set  in  a plane. 

Sector  is  a wheel  which  reciprocates  without  forming  a full  revolution. 

Spur  Wheel  is  a wheel  having  its  teeth  perpendicular  to  its  axis. 

Bevel  Wheel  is  a wheel  having  its  teeth  at  an  angle  with  its  axis. 

Crown  Wheel  is  a wheel  having  its  teeth  at  a right  angle  with  its  axis. 

Mitre  Wheel  is  a wheel  having  its  teeth  at  an  angle  of  450  with  its  axis. 

Face  Wheel  is  a wheel  having  its  teeth  set  upon  one  of  its  sides. 

Annular  or  Internal  Wheel  is  a wheel  having  its  teeth  convergent  to  its  centre. 

Spur  Gear. — Wheels  which  act  upon  each  other  in  same  plane. 

Bevel  Gear  —Wheels  which  act  upon  each  other  at  an  angle. 

Inside  Gear  or  Pin  Gearing.—  Form  of  acting  surfaces  of  teeth  for  a pitch-circle 
m inside  gearing  is  exactly  same  with  those  suited  for  same  pitch-circle  in  outside 
gearing,  but  relative  position  of  teeth,  spaces,  and  flanks  are  reversed,  and  adden- 
dum-circle is  of  less  radius  than  pitch-circle. 

A Train  is  a series  of  wheels  in  connection  with  each  other,  and  consists  of  a 
series  of  axles,  each  having  on  it  two  wheels,  one  is  driven  by  a wheel  on  a preced- 
ing axis  and  other  drives  a wheel  on  following  axis. 

Idle  Wheel. — A wheel  revolving  upon  an  axis,  which  receives  motion  from  a pre- 
ceding wheel  and  gives  motion  to  a following  ivheel,  used  only  to  affect  direction  of 
motion. 

Trundle.  Lantern,  or  Wallower  is  when  teeth  of  a pinion  are  constructed  of  round 
bars  or  solid  cylinders  set  into  two  disks.  Trundle  with  less  than  eight  staves  can- 
not be  operated  uniformly  by  a wheel  with  any  number  of  teeth. 

Spur,  Driver,  or  Leader  is  term  for  a wheel  that  impels  another:  one  impelled  is 
Pinion , Driven , or  Follower. 


WHEEL  GEARING. 


855 


Teeth  of  wheels  should  be  as  small  and  numerous  as  is  consistent  with 
strength. 

When  a Pinion  is  driven  by  a wheel , number  of  teeth  in  pinion  should  not 
be  less  than  8. 

When  a Wheel  is  driven  by  a pinion , number  of  teeth  in  pinion  should  not 
be  less  than  jo. 

When  2 wheels  act  upon  one  another,  greater  is  termed  Wheel  and  lesser  Pinion. 
When  the  tooth  of  a wheel  is  made  of  a material  different  from  that  of  wheel  it  is 
termed  a Cog  ; in  a pinion  it  is  termed  a Leap  in  a trundle  a Stave,  and  on  a disk 
a Pin. 

Material  of  which  cogs  are  made  is  about  one  fourth  strength  of  cast  iron. 
Hence,  product  of  their  b d 2 should  be  4 times  that  of  iron  teeth. 

Number  of  teeth  in  a wheel’ should  always  be  prime  to  number  of  pmion  ; 
that  is,  number  of  teeth  in  wheel  should  not  be  divisible  by  number  of  teeth 
in  pinion  without  a remainder.  This  is  in  order  to  prevent  the  same  teeth 
coming  together  so  often  and  uniformly  as  to  cause  an  irregular  wear  ot  their 
faces.  An  odd  tooth  introduced  into  a wheel  is  termed  a Hunting  tooth  or  Cog. 

The  least  number  of  teeth  that  it  is  practicable  to  give  to  a wheel  is  regu- 
lated bv  necessity  of  having  at  least  one  pair  always  in  action,  in  order  to 
provide  for  the  contingency  of  a tooth  breaking;  and  least  number  that  can 
be  employed  in  pinions  having  teeth  of  following  classes  is : Involute,  25 , 
Epicycloidal,  12 ; Staves  or  Pins,  6. 

Velocity  Ratio  in  a Train  of  Wheels. — lo  attain  it  with  least  number  of 
teeth,  it  should,  in  each  elementary  combination,  approximate  as  near  as 
practicable  to  3.59.  A convenient  practical  rule  is  a range  from  3 to  6. 

Illustration.  i 6 36  216  1296  velocity  ratio. 

j 2 3 4 elementary  combination. 

To  increase  or  diminish  velocity  in  a given  proportion,  and  with  least 
quantity  of  wheel-work,  number  of  teeth  in  each  pinion  should  be  to  number 
of  teeth  in  its  wheel  as  1 : 3-59-  Even  to  save  space  and  expense,  ratio 
should  never  exceed  1 : 6.  (Buchanan.) 

To  Compute  IPitcli. 

Rule  —Divide  circumference  at  pitch-line  by  number  of  teeth. 

Example.— A wheel  40  ins.  in  diameter  requires  75  teeth;  what  is  its  pitch? 
3.1416  X 40-^75  — 1.6755  ins. 

To  Compute  True  or  CLordial  iPitcli. 
rule  —Divide  180°  by  number  of  teeth,  ascertain  sine  of  quotient,  and 
multiply  it  by  diameter  of  wheel. 

Example. — Number  of  teeth  is  75,  and  diameter  40  ins. ; what  is  true  pitch? 
l8o  -5-  75  = 20  24',  and  sin.  of  20  24'  = .041  88,  which  X 40  = 1-6752  ins. 

To  Compute  Diameter. 

Rule.— Multiply  number  of  teeth  by  pitch,  and  divide  product  by  3.1416. 
Example.— Number  of  teeth  in  a wheel  is  75,  and  pitch  1.6755  ins. ; what  is  di- 
ameter of  it  ? 75  x 1.6755  -5-  3. 1416  = 40  ins. 

When  the  True  Pitch  is  given.  Rule.— Multiply  number  of  teeth  in  wheel 
by  true  pitch,  and  again  by  .3184. 

" Example.  — Take  elements  of  preceding  case. 

75  x 1.6752  X .3184  = 40  ins • 

Or,  Divide  180°  by  number  of  teeth,  and  multiply  cosecant  of  quotient  by 
pitch. 

l8o^-  75  = 2o  24/,  and  cos.  20  24'  = 23.88,  which  X 1.6752  = 40  ins. 


856 


WHEEL  GEARING. 


To  Compute  INTurribei'  of*  Teeth. 
Rule. — Divide  circumference  by  pitch. 


To  Compute  INAxrxiber  of*  Teeth  in  a Finion  or  Follower 
to  have  a given  Velocity. 

Rule.— Multiply  velocity  of  driver  by  its  number  of  teeth,  and  divide 
product  by  velocity  of  driven. 

Example  i.— Velocity  of  a driver  is  16  revolutions,  number  of  its  teeth  and 
velocity  of  pinion  is  48;  what  is  number  of  its  teeth? 

16  X 54-4-48  = 18  teeth. 

2.— A vvheel  having  75  teeth  is  making  16  revolutions  per  minute:  what  is  num- 
ber of  teeth  required  in  pinion  to  make  24  revolutions  in  same  time? 

16  X 75  -7-  24  = 50  teeth. 


To  Compute  Proportional  Ptadins  of*  a Wheel  or  Finion. 

Rule.— Multiply  length  of  line  of  centres  by  number  of  teeth  in  wheel 
for  vvheel,  and  in  pinion,  for  pinion,  and  divide  by  number  of  teeth  in  both 
vvheel  and  pinion. 

Example.— Line  of  centres  of  a wheel  and  pinion  is  36  ins.,  and  number  of  teeth 
in  wheel  is  60,  and  in  pinion  18;  what  are  their  radii? 


33 * * 6  X 60  _ 7 36  X 18  0 . 

- = 27.69  ins.  wheel.  ^ ^ = 8.3  ms.  pinion. 


60 -f- 18 


To  Compute  Diameter  of*  a Fin  ion. 

. When  Diameter  of  Wheel  and  Number  of  Teeth  in  Wheel  and  Pinion  are 
gwen.  Rule.  Multiply  diameter  of  vvheel  by  number  of  teeth  in  pinion, 
and  divide  product  by  number  of  teeth  in  wheel. 

Example.— Diameter  of  a wheel  is  25  ins.,  number  of  its  teeth  210,  and  number 
of  teeth  in  pinion  30;  what  is  diameter  of  pinion? 

25  X 30-f- 210  = 3. 57  iws. 


To  Compute  Number  of*  Teeth  required  in  a Train  of 
Wheels  to  produce  a given  Velocity. 


Rule.  Multiply  number  of  teeth  in  driver  by  its  number  of  revolutions 
and  divide  product  by  number  of  revolutions  of  each  pinion,  for  each  driver 
and  pinion. 

inH^iMPLEi~I,f  a?river  in. a train  of  three  wheels  has  90  teeth,  and  makes  2 revo- 
of^thertwo  J elocitles  re(luired  are  2,  10,  and  18,  what  are  number  of  teeth  in  each 

10  : 90  : : 2 : 18  = teeth  in  2 d wheel.  18  : 90  : : 2 : 10  = teeth  in  3d  wheel. 


To  Compute  Velocity  of*  a Pinion. 

Rule.— Divide  diameter,  circumference,  or  number  of  teeth  in  driver,  as 
case  may  be,  by  diameter,  etc.,  of  pinion. 

When  there  are  a Series  or  Train  of  Wheels  and  Pinions.  Rule. — Divide 
continued  product  of  diameter,  circumference,  or  number  of  teeth  in  wheels 
by  continued  product  of  diameter,  etc.,  of  pinions. 

Example  1.— Tf  a vvheel  of  32  teeth  drives  a pinion  of  10,  upon  axis  of  which  there 
is  one  01  30  teeth,  driving  a pinion  of  8,  what  are  revolutions  of  last? 

32  ^ 30  960 

— X -y  — = 12  revolutions. 

10  8 80 

2.  Diameters  of  a train  of  wheels  are  6,  9,  9.  10,  and  12  ins. ; of  pinions,  6,  6,  6,  6, 
and  6 ins. ; and  number  of  revolutions  of  driving  shaft  or  prime  mover  is  10  • what 
are  revolutions  of  last  pinion  ? 

6 X 9 X 9 X 10  X 12  X 10  583  200  , . 

~6X  6X6  X6  X 6 = ~777T  = 75  reMms- 


WHEEL  GEARING.  857 

To  Compute  Proportion  that  Velocities  of  "Wheels  in 
a Train  should,  "bear  to  one  another. 

Rule. — Subtract  less  velocity  from  greater,  and  divide  remainder  by  one 
less  than  number  of  wheels  in  train  ; quotient  is  number,  rising  in  arithmet- 
ical progression  from  less  to  greater  velocity. 

Example. — What  should  be  velocities  of  3 wheels  to  produce  18  revolutions,  the 
driver  making  3? 

1 ^ 3 — — 7.5  = number  to  be  added  to  velocity  of  driver  — 7. 5 -f-  3 = 10. 5,  and 

3 — 1—2 

10.5  _j_  7.5  = !8  revolutions.  Hence  3,  10.5,  and  18  are  velocities  of  three  wheels. 

Pitcli  of*  "Wlieels. 


To  Compute  Diameter  of  a "Wheel  for  a given  Pitch, 
or  Pitch  for  a given  Diameter. 

From  8 to  192  Teeth. 


No.  of 
Teeth. 

Diame- 

ter. 

No.  of 
Teeth. 

Diame- 

ter. 

No.  of 
Teeth. 

Diame- 

ter. 

No.  of 
Teeth. 

Diame- 

ter. 

No.  of 
Teeth. 

Diame- 

ter. 

8 

2.6l 

45 

14.33 

82 

26.II 

1 19 

37.88 

156 

49.66 

9 

2-93 

46 

14.65 

83 

26.43 

120 

38.2 

157 

49.98 

10 

3-24 

47 

14.97 

84 

26.74 

121 

38.52 

158 

50.3 

11 

3*55 

48 

15.29 

85 

27.06 

122 

38.84 

159 

50.61 

12 

3.86 

49 

15.61 

86 

27.38 

123 

39.16 

160 

50.93 

13 

4.18 

50 

15-93 

87 

27.7 

I24 

39-47 

l6l 

51.25 

14 

4.49 

5i 

16.24 

88 

28.02 

125 

39*79 

162 

51-57 

15 

4.81 

52 

16.56 

89 

28.33 

126 

40.II 

163 

51.89 

16 

5.12 

53 

16.88 

90 

28.65 

127 

40-43 

164 

52.21 

1 7 1 

5-44 

54 

17.2 

91 

28.97 

128 

40.75 

165 

52.52 

18 

5-76 

55 

17.52 

92 

29.29 

I29 

4I.O7 

166 

52.84 

19 

6.07 

56 

17.8 

93 

29.61 

130 

41.38 

167 

53-16 

20 

6-39 

57 

18.15 

94 

29.93 

131 

4I.7 

168 

53-48 

21 

6.71 

58 

18.47 

95 

30.24 

132 

42.02 

169 

53-8 

22 

7-03 

59 

18.79 

96 

30.56 

133 

42.34 

170 

54-12 

23 

7-34 

60 

i9.II 

97 

30.88 

134 

42.66 

171 

54-43 

24 

7.66 

61 

I9.42 

98 

31*2 

135 

42.98 

172 

54-75 

25 

7.98 

62 

19.74 

99 

3!‘52 

I36 

43  29 

173 

55-07 

26 

8.3 

63 

20.06 

! 100 

31.84 

137 

43.61 

174 

55-39 

27 

8.61 

64 

20.38 

101 

32.15 

138 

4393 

J75 

55-71 

28 

8-93 

65 

2O.7 

102 

3247 

139 

44-25 

176 

56.02 

29 

9-25 

66 

21.02 

103 

32.79 

140 

44-57 

177 

56.34 

30 

9 57 

67 

21-33 

104 

33.11 

141 

44.88 

178 

56.66 

31 

9.88 

68 

21.65 

105 

33.43 

I42 

45-2 

179 

56.98 

32 

10.2 

69 

2I.97 

106 

33.74 

143 

45-52 

180 

57  23 

33 

10.52 

70 

22.29 

107 

34.06 

144 

45-84 

181 

57.62 

34 

10.84 

7i 

22.6l 

108 

34.38 

145 

46.16 

182 

57  93 

35 

11. 16 

72 

22.92 

109 

34-7 

I46 

46.48 

183 

58-25 

36 

n.47 

73 

23.24 

no 

35*02 

147 

46.79 

184 

58-57 

37 

n.79 

74 

23.56 

III 

35-34 

I48 

47.11 

185 

58.89 

38 

12. 11 

75 

23.88 

1 12 

35.65 

149 

47-43 

186 

59.21 

39 

12.43 

76 

24.2 

113 

35*97 

150 

47*75 

187 

59*53 

40 

12.74 

77 

24.52 

114 

36.29 

151 

48.07 

188 

59*84 

4i 

13.06 

78 

24.83 

“5 

36.61 

152 

4839 

189 

60.16 

42 

13.38 

79 

25.15 

Il6 

36.93 

153 

48.7 

190 

60.48 

43 

13-7 

80 

2547 

117 

37-25 

154 

49.02 

191 

60.81 

44 

14.02 

81 

25.79 

Il8 

37*56 

155 

49-34 

192 

61.13 

Pitch  in  this  table  is  true  pitch,  as  before  described. 

To  Compute  Circumference  of  a "Wheel. 
Rule. — Multiply  number  of  teeth  by  their  pitch, 


858 


WHEEL  GEARING. 


To  Compute  Tfcevolntions  of  a Wheel  or  IPinion. 

Rule. — Multiply  diameter  or  circumference  of  wheel  or  number  of  its 
teeth  in  ins.,  as  case  may  be,  by  number  of  its  revolutions,  and  divide  prod- 
uct by  diameter,  circumference,  or  number  of  teeth  in  pinion. 

Example.— A pinion  io  ins.  in  diameter  is  driven  by  a wheel  2 feet  in  diameter, 
making  46  revolutions  per  minute;  what  is  number  of  revolutions  of  pinion  ? 

2 X 12  X 46 -f- 10  = 110.4  revolutions. 


To  Compute  Number  of  Teeth,  of  a Wheel  for  a given. 
Diameter  and.  3?itch. 

Rule. — Divide  diameter  by  pitch,  and  opposite  to  quotient  in  preceding 
table  is  given  number  of  teeth. 


Example. — Diam.  of  wheel  is  40  ins.,  and  pitch  1.675;  what  is  number  of  its  teeth? 
40  H- 1.675  = 23.88,  and  opposite  thereto  in  table  is  75  = number  of  teeth. 


To  Compute  Diameter  of  a Wheel  for  a given  JPitch.  and 
Number  of  Teeth. 

Rule. — Multiply  diameter  in  preceding  table  for  number  of  teeth  by 
pitch,  and  product  will  give  diameter  at  pitch  circle. 

Example. — What  is  diameter  of  a wheel  to  contain  48  teeth  of  2.5  ins.  pitch? 

1 5. 29  X 2. 5 = 38. 225  ins. 

To  Compute  flitch  of  a Wheel  for  a given  Diameter  and 
IV n m her  of  Teeth. 


Rule. — Divide  diameter  of  wheel  by  diameter  in  table  for  number  of 
teeth,  and  quotient  will  give  pitch. 

Example. — What  is  pitch  of  a wheel  when  diameter  of  it  is  50.94  ins.,  and  num- 
50.94  -T-  25.47  — 2 ins- 


ber  of  its  teeth  80? 


G-eneral  Illustrations. 

1. — A wheel  96  ins.  in  diameter,  making  42  revolutions  per  minute,  is  to  drive  a 
shaft  75  revolutions  per  minute;  what  should  be  diameter  of  pinion? 

96  X 42  “J“  75  = 53-  76  ins- 

2. — If  a pinion  is  to  make  20  revolutions  per  minute,  required  diameter  of  an- 
other to  make  58  -revolutions  in  same  time. 

58  -4-  20  = 2.9  = ratio  of  their  diameters.  Hence,  if  one  to  make  20  revolutions  is 
given  a diameter  of  30  ins.,  other  will  be  30 -f-  2.9  = 10.345  ins. 

3.  — Required  diameter  of  a pinion  to  make  12.5  revolutions  in  same  time  as  one 
of  32  ins.  diameter  making  26. 

32  X 26  -4-  12. 5 = 66. 56  ins. 

4. — A shaft,  having  22  revolutions  per  minute,  is  to  drive  another  shaft  at  rate 
of  15,  distance  between  two  shafts  upon  line  of  centres  is  45  ins. ; what  should  be 
diameter  of  wheels? 

Then,  1 st.  22  -f-  15  : 22  : : 45  : 26.75  = ins.  in  radius  of  pinion. 

2d.  22  -j—  1 5 : 15  : : 45  : 18. 24  = ins.  in  radius  of  spur. 

5. — A driving  shaft,  having  16  revolutions  per  minute,  is  to  drive  a shaft  81  revo- 
lutions per  minute,  motion  to  be  communicated  by  two  geared  wheels  and  two  pul- 
leys, with  an  intermediate  shaft;  driving  wheel  is  to  contain  54  teeth,  and  driving 
pulley  upon  driven  shaft  is  to  be  25  ins.  in  diameter;  required  number  of  teeth  in 
driven  wheel,  and  diameter  of  driven  pulley. 

Let  driven  wheel  have  a velocity  of  V16  X 81  = 36,  a mean  proportional  between 
extreme  velocities  16  and  81. 

Then,  1st.  36  : 16  ::  54  : 24  — teeth  in  driven  wheel. 

2d.  81  : 36  ::  25  : 11. 11  =in5.  diameter  of  driven  pulley. 

6- — If,  as  in  preceding  case,  whole  number  of  revolutions  of  driving  shaft,  num- 
ber of  teeth  in  its  wheel,  and  diameters  of  pulleys  are  given,  what  are  revolutions 
of  shafts? 


Then,  1st.  18  : 16  : : 54  : 48  = revolutions  of  intermediate  shaft. 
2d.  15  : 48  ::  25  : 80  = revolutions  of  driven  shaft. 


WHEEL  GEARING. TEETH  OE  WHEELS.  859 


TeetPL  of  Wheels. 

Tr,r>ic'v'cloidal. — In  order  that  teeth  of  wheels  and  pinions  should  work 
evenly  and  without  unnecessary  rubbing  friction,  the  face  (from  pitch  line 
to  top)  of  the  outline  should  be  determined  by  an  epicycloidal  curve  (see 
page  228),  and  that  of  the  flank  (from pitch  line  to  base ) by  an  hypocycloidal 
(see  also  page  228). 

When  generating  circle  is  equal  to  half  diameter  of  pitch  circle,  hypocy- 
cloidal  described  by  it  is  a straight  diametrical  line,  and  consequently  out- 
line of  a flank  is  a right  line,  and  radial  to  centre  of  wheel. 

If  a like  generating  circle  is  used  to  describe  face  of  a tootli  of  other  wheel 
or  pinion  respectively,  the  wheel  and  pinion  will  operate  evenly. 

Illustration. — Determine  all  elements  of  wheel 
—viz. , Pitch  circle,  Number  of  teeth,  Pitch,  Length, 
Face,  and  Flank. 

Cut  a template  A to  pitch  circle  c c of  wheel,  and 
secure  it  temporarily  to  a board. 

Having  determined  depth  of  tooth,  set  it  off  on 
pitch  line,  as  a 0,  Fig.  1,  and  above  it  apply  a sec- 

, ond  template,  a;  radius  of  wheel  is  equal  to  half 

radius  of  ninion-  insert  into,  or  attach  exactly  at  its  edge,  a tracer  . roll  template 
a alon"  a!  and  tracer  will  describe  an  epicycloidal  curve,  a r,  and  by  m\erting  a 
describe  0 and  faces  of  a tooth  are  delineated. 

To  describe  flanks,  define  pitch  line  c c,  Fig.  2,  and  arc  n w, 
drawn  at  base  of  teeth  or  board  A (as  in  Fig.  1),  secure  a strip 
of  wood,  w,  equal  in  length  to  radius  of  wheel,  and  locate 
centre  of  it,  *,  draw  radii  x a and  x 0,  and  they  will  define 
flanks,  which  should  be  filleted,  as  shown  at  s s.  Define  arc 
^ ^ zz,  and  length  of  tooth  is  determined. 

\ ! ; 1 Proceed  in  like  manner  conversely  for  teeth  of  pinion,  and 

• ! ! wheel  and  pinion  thus  constructed  will  operate  truly. 

:l.  I In  construction  of  the  teeth  of  a wheel  or  pinion  in 

" iLjil  the  pattern-shop,  it  is  customary  to  construct  the  wheel 

or  pinion  complete,  out  to  face  of  wheel  at  base  of  teeth, 
and  then  to  insert  the  teeth  in  rough,  approximately 
shaped  blocks,  by  a dovetail  at  their  base,  fitting  into  face  of  wheel,  and  then 
the  outline  of  a tooth  is  described  thereon ; the  block  is  then  remo,\  ed,  fin- 
ished as  a tooth,  replaced,  fastened,  and  filleted. 

Involute. 

Teeth  of  two  wheels  will  work  truly  together  when  their  face  is  that  of  an 
involute  (see  page  229),  and  that  two  such  wheels  should  work  truly,  the 
circles  from  which  the  involute  lines  for  each  wheel  are  generated  must  be 
concentric  with  the  wheels,  with  diameters  in  same  ratio  as  those  of  the  wheels. 

Assume  Ac,  Be,  Fig.  3.  pitch  radii  of  two  wheels  designed 
to  work  together,  through  c,  draw  a right  line,  e i,  and  with 
perpendiculars  c c,  i c,  describe  arcs  n 0,  r.  s,  and  involutes 
n c o and  res  define  a face  of  each  of  the  teeth. 

To  describe  teeth  of  a pair  of  B 4* 

wheels  of  which  A c,  B c,  Fig.  4, 
are  pitch  radii,  draw  c i,  c e,  per- 
pendicular to  radials  B i and  A c, 
and  they  are  to  be  taken  as  the 
radials  of  the  involute  arcs  from 
which  the  faces  of  the  teeth  are 
to  be  defined;  then  fillet  flanks  at  ' ^ 

base,  as  before  described,  Fig.  2.  \J 

Involute  teeth  will  work  with  truth,  even  at  varying  A ‘ 

distances  apart  of  the  centres  of  the  wheels,  and  any  wheels  of  a like  pitch  will  woik 
in  union,  however  varied  their  diameters. 


WHEEL  GEARING. — TEETH  OF  WHEELS. 


„ 


Circular  teeth  are  defined  as  follows : 

Assume  A A,  Fig.  5,  pitch-line,  b b line  of  base 
of  teeth,  and  t i face-line.  Set  off  on  pitch-line 
divisions  both  of  pitch  and  depth  of  teeth  then 
1-^  . Wltb  a radius  of  1.25  pitch  describe  arcs  as  o s 
4 p!tch  line  for  faces  of  teeth'  then  draw  ra- 

" dial  hnes  ov,  ru,  to  centre  of  wheel  for  flanks 

strike  arc  1 1 to  define  length  of  tooth,  and  fillet 
flanks  at  base  as  before  described. 


Proportions  of  Teeth. 

In  computing  dimensions  of  a tooth,  it  is  to 
be  considered  as  a beam  fixed  at  one  end 
, ..  . . , weight  suspended  from  other,  or  face  of  beam  • 

and  it  is  essential  to  consider  the  element  of  velocity,  as  its  stress  in  opera- 
tion, at  high  velocity  with  irregular  action,  is  increased  therebv. 

Dimensions  of  a tooth  should  be  much  greater  than  is  necessary  to  resist 
direct  stress  upon  it,  as  but  one  tooth  is  proportioned  to  bear  whole  stress 
upon  wheel,  although  two  or  more  are  actually  in  contact  at  all  times  • but 
this  requirement  is  in  consequence  of  the  great  wear  to  which  a tooth  is’ sub- 
jected, shocks  it  is  liable  to  from  lost  motion,  when  so  worn  as  to  reduce  its 
depth  and  uniformity  of  bearing,  and  risk  of  the  loss  of  a tooth  from  a defect. 

. A t00th  running  at  a low  velocity  may  be  materially  reduced  in  its  dimen- 
sions,  compared  with  one  running  at  a high  velocity  and  with  a like  stress. 

Result  of  operations  with  toothed  wheels,  for  a long  period  of  time  has 
determined  that  a cast-iron  (Eng.)  tooth  with  a pitch  of  3 ins.  and  a breadth 
ot  7.5  ms.  will  transmit,  at  a velocity  of  6.66  feet  per  second,  power  of  59  16 


To  Compute  Dimensions  of*  a Tooth,  to  Desist  a given 
Stress. 

Rule.— Multiply  extreme  pressure  at  pitch-line  of  wheel  by  length  of 
00  1 in  decimal  of  a foot,  divide  product  by  Coefficient  of  material  of  tooth, 
and  quotient  will  give  product  of  breadth  and  square  of  depth. 


S l 


0r’  ~ b d2-  s representing  stress  in  lbs.  \ and  l length  in  feet 


The  Coefficient  of  cast  iron  for  this  or  like  purposes  may  be  taken  at  from  50  to  70. 

Pitch  A B = 1.  Depth  rs  = . 45. 

Length  co  = . 75.  Space  5 v = . 55. 

Working  length  ce  = . 7.  Play  sv—rs£.i 
[--  Clearance  e to  0 — .05.  Face  B c = .35. 


Note.  — It  is  necessary  first  to  determine  pitch,  in 
order  to  obtain  either  length  or  depth  of  a tooth. 


of  tooth,  pitch  being  3 ins.  ? 

3 X .75  = 2.25  length  of  tooth , which  H- 
Coefficient  of  material  is  taken  at  60. 
4886  x -1875 


Example  — Pressure  at  pitch-line  of  a cast- 
iron  wheel  (at  a velocity  of  6.66  feet  per  sec- 
ond) is  4886  lbs. ; what  'should  be  dimensions 


1875  = length  in  decimal  of  a foot 


60 


— 15.27.  If  length  = 2.25,  pitch  ==,  3,  and  depth  = 1.35  ins. 


Pitches  of  Equivalent  Strength  for  Cast  Iron  and  Wood Iron  1.  Hard  wood  *.  26. 


Then  = 8.39  ins.  breadth. 

I-352 

When  Product  of  b d2  is  obtained , and  it  is  required  to  ascertain  either 
dimension,  ff  - depth , and  ~ = breadth. 


■ 


WHEEL  GEARING. TEETH  OF  WHEELS.  86 1 


To  Compute  Depth  of  a Tooth. 

1.  When  Stress  is  given.  Rule.— Extract  square  root  of  stress,  and  mul- 
tiply it  by  .02  for  cast  iron,  and  .027  for  hard  wood. 

2.  When  IP  is  given.  Rule.— Extract  square  root  of  quotient  of  EP  di- 
vided by  velocity  in  feet  per  second,  and  multiply  it  by  .466  for  cast  iron, 
and  .637  for  hard  wood. 

Example.— IP  to  be  transmitted  by  a tooth  of  cast  iron  is  60,  and  velocity  of  it 
at  its  pitch  line  is  6.66  feet  per  second;  what  should  be  depth  of  tooth? 

\/6^X-466  = ,'398  ,'"S- 
To  Compute  LP  of  a Tooth. 

Rule.— Multiply  pressure  at  pitch-line  by  its  velocity  in  feet  per  minute, 
and  divide  product  by  33  000. 

Example. —What  is  IP  of  a tooth  of  dimensions  and  at  velocity  given  in  preced- 
ing example. 

4886  X 6. 66  X 60"  -r-  33  000  = 59. 16  horses. 


To  Com  pate  Stress  that  may  be  borne  by  a Tooth. 

Rule— Multiply  Coefficient  of  material  of  tooth  to  resist  a transverse 
strain,  as  estimated  for  this  character  of  stress,  by  breadth  and  square  of  its 
depth,  and  divide  product  by  extreme  length  of  it  in  decimal  of  a foot. 

Example. — Dimensions  of  a cast-iron  tooth  in  a wheel  are  1.38  ins.  in  depth,  2.1 
ins.  in  length,  and  7.5  ins.  in  breadth;  what  is  the  stress  it  will  bear? 

Coefficient  assumed  at  60.  * 7 — - — = 4886  lbs. 

2. 1 — 12 

Following  deductions  by  the  rules  of  different  authors  for  like  elements  are  sub- 
mitted for  a cast-iron  tooth: 


Pitch 3 ins.  1 Depth 1.38  ins.  | Breadth...  7.5  ins.  | Length 2.1  ins. 


Actual  Power  in  Stress  Exerted  Depth  of  | Actual  Power  in  Stress  Exerted  Depth  of 
at  a velocity  of  400  feet  per  min.,  4886  lbs.  Tooth,  at  a velocity  of  400  feet  per  min.,  4886  lbs.  Tooth. 


By  Above  Rule  ^ X -446-  ■ 
“ Fairbairn  .025  y/W 


Imperial  Journal  ^ - 


W 

576  ' 


Ins. 

1.398* 

*■75 

1.76 


/ V 

By  Ranlcine  , / — 

V 1500 

“ TredgoU  A . 

“ Buchanan  • 


Ins. 

1.8 


2.25 


2.24 


H representing  horsepower  (60),  W stress  in  lbs.,  and  v velocity  in  feet  per  second. 


Depth,  Ditch,  and  Breadth.  (M.  Morin.) 

Cast  iron j. ^ 028  y/W  — d.  .057  y/W  = p- 

Hard  wrood 038  -fW  = d.  .079  y/W  = P. 

W representing  weight  or  stress  upon  tooth  in  lbs. , d depth  of  tooth , and  P pitch 
in  ins. 

When  velocity  of  pitch-circle  does  not  exceed  5 feet  per  second  5 = 4 c?, 
when  it  exceeds  5 feet  b = 5 d,  and  if  wheels  are  exposed  to  wet  6 = 6 d. 

b representing  breadth. 

Illustration.— Assume  pressure  at  pitch-line  of  a cast-iron  wheel  upon  a tooth 
equal  6000  lbs.,  and  velocity  5 feet  per  second. 

Then  .028  -y/6ooo=:  2. 17  ins.  Depth , and  .057  y/6000  — 4. 46  ins.  Fitch. 

Note. —For  further  Illustrations  of  Formation  of  Teeth,  Bevel  Gearing,  Willis’s  Odontograph, 
Staves,  Trundles,  etc.,  see  Mosely’s  Engineering,  Shelton’s  Mechanic’s  Guide,  Fairbairn’s  Mechanism 
and  Machinery  of  Construction,  etc. 


* This  depth,  with  a breadth  of  7.5  ins.,  is  .1  of  ultimate  strength  of  average  strength  of  American 
Cast  Iron. 


862  TEETH  OF  WHEELS. — WINDING  ENGINES. 


PROPORTIONS  OF  WHEELS. 

With  six  flat  A rms  and  Ribs  upon  one  side  of  them , as  ; or  a Web 

in  centre , as 

Rim. — Depth,  measured  from  base  of  teeth,  .45  to  .5  of  pitch  of  teeth,  hav- 
ing a web  upon  its  inner  surface  .4  of  pitch  in  depth  and  .25  to  .3  of  it  in 
width. 

Note.— When  face  of  wheel  is  mortised,  depth  of  rim  should  be  1.5  times  pitch, 
and  breadth  of  it  1.5  times  breadth  of  tooth  or  cog. 

Hub. — When  eye  is  proportionate  to  stress  upon  wheel,  hub  should  be 
twice  diameter  of  eye.  In  other  cases  depth  around  eye  should  be  .75  to  .8 
of  pitch. 

Arm. — Depth  .4  to  .45  of  pitch.  Breadth  at  rim  1.5  times  pitch,  increas- 
ing .5  inch  per  foot  of  length  toward  hub. 

Rib  upon  one  edge  of  arm,  or  Web  in  its  centre,  should  be  from  .25  to  .3 
pitch  in  width,  and  .4  to  .45  of  it  in  depth. 

When  section  of  an  arm  differs  from  those  above  given,  as  with  one  with 
a plane  section,  as  msmm , or  with  a double  rib,  as  |sgaBsj , its  dimensions 
should  be  proportioned  to  form  of  section. 

In  a wheel  of  greater  relative  diameter,  length  of  hub  and  breadth  of  arms, 
or  of  the  rib  or  web,  according  as  plane  of  arm  is  in  that  of  wheel,  or  con- 
trariwise, should  be  made  to  exceed  breadth  of  face  of  wheel  (at  the  hub) 
in  order  to  give  it  resistance  to  lateral  strain. 

Number  of  arms  in  wheels  should  be  as  follows*. 

1.5  to  3.25  feet  in  diameter 4 I 5 to  8.5  feet  in  diameter 6 

3 25  “ 5 “ “ 5 I 8.5  “ 16  “ “ 8 

16  to  24  feet  in  diameter 10 

With  light  wheels,  number  of  arms  should  be  increased,  in  order  better  to 
sustain  rigidity  of  rim. 

Mortise  Wheels. — Their  rim  or  face  should  be  .9  pitch  of  tooth,  and  twice 
depth  of  rim  of  a solid  wheel. 


WINDING  ENGINES. 

With  Winding  Engines,  for  drawing  coals,  etc.,  out  of  a Pit,  where  it 
is  required  to  give  a certain  number  of  revolutions,  it  is  necessary  to 
have  given  diameter  of  Drum  and  thickness  of  rope,  which  is  flat  made, 
and  contrariwise. 


To  Compute  Diameter  of  -a  Drum. 

Where  Flat  Ropes  are  used,  and  are  wound  one  part  over  the  other.  Rule. 
— Divide  depth  of  pit  in  ins.  by  product  of  number  of  revolutions  and  3.1416, 
and  from  quotient  subtract  product  of  thickness  of  rope  and  number  of  rev- 
olutions ; remainder  is  diameter  in  ins. 

Example. — If  an  engine  makes  20  revolutions,  depth  of  pit  being  600  feet,  and 
rope  1 inch,  what  should  be  diameter  of  drum  ? 


600  X 12 
20  X 3-I4l6 


— — 7200 

- ! X 2°  = - 20  = 94-59  MU- 


To  Compute  Diameter  of  Roll. 

Rule. — To  area  of  drum  add  area  or  edge  surface  of  rope ; then  ascertain 
by  inspection  in  table  of  areas,  or  by  calculation,  diameter  that  gives  this 
area,  and  it  is  the  diameter  of  Roll. 


WINDING  ENGINES.' — WINDMILLS. 


863 


Example.— What  is  diameter  of  roll  in  preceding  example? 

Area  of  94.59  = 7027.2  + (area  of  7200  X i)  = 720O:=:l4  227.2,  and  V^22?-2" 
.7854  = 134. 59  ins. 

Or,  Radius  of  drum  is  increased  number  of  revolutions  multiplied  by  thickness 
of  rope;  as  |2oXi=  67.295  ins. 

To  Compute  TSTuniDer  of*  Bevolntions. 

Rule.— To  area  of  drum  add  area  of  edge  surface  of  rope ; from  diameter 
of  the"  circle  having  that  area  subtract  diameter  of  drum,  and  divide  re- 
mainder by  twice  thickness  of  rope ; quotient  will  give  number  of  revolutions. 
Example. Length  of  a rope  is  2600  ins.,  its  thickness  1 inch,  and  diametei  of 

drum  20  ins. ; what  is  number  of  revolutions? 

Area  of  2o-|-  area  of  rope  = 314. 16-1-2600  = 2914. 16,  diameter  of  which  is  60.91, 

and  60.91— 2q_2  5 revolutions. 

1X2 

Or,  subtract  diameter  of  drum  from  diameter  of  roll,  and  divide  remainder  by 
twice  thickness  of  rope;  as  60.91  — 20  = 40.91,  and  40.91  -r- 1 X 2 = 20.45  revolutions. 

To  Compute  Boint  of  Meeting  of  Ascending  and  De- 
scending Buckets  when  two  or  more  are  used. 

To  Compute  Point  of  Meeting  of  Buckets.  Rule. — Divide  sum  of  length 
of  turns  of  rope  by  2,  and  to  quotient  add  length  of  last  turn ; divide  sum 
bv  2,  multiply  quotient  by  half  number  of  revolutions,  and  product  will 
give  distance  from  centre  of  drum  at  which  buckets  will  meet. 

Note  1.— Meetings  will  always  be  below  half  depth  of  pit. 

2.— At  half  number  of  revolutions  buckets  will  meet. 

Example  —Diameter  of  a drum  is  9 feet,  thickness  of  rope  1 inch,  and  revolu- 
tions 20;  what  is  depth  of  pit,  and  at  what  distance  from  top  will  buckets  meet? 

28.54  + 384?^  8 8^2  x.gg_ZL22  — 35.995  x 10  = 359-95  fa*- 

2 2 2 

To  Compute  this  Depth.  Rule.— To  diameter  of  drum  add  thickness  of 
rope  in  feet,  and  ascertain  its  circumference ; to  diameter  of  drum  add  quo- 
tient of  product  of  twice  thickness  of  rope  and  number  of  revolutions  less  1, 
divided  by  12  for  a diameter,  and  circumference  of  this  diameter  is  length 
of  last  turn,  also  in  feet ; add  these  two  lengths  together,  multiply  their  sum 
by  half  number  of  revolutions,  and  product  will  give  depth  of  pit. 

9 -f  thickness  of  rope  = 9 + jj  of  * = 9-  083,  which  X 3-  J4l6  = 28. 54  feet  = length 

of first  turn.  9.0833  + I><2><26~1  x 3.  I4i6  _ 38.48  feet  = length  of  last  turn. 

Then  28.54  + 38.48  X ^ = 67.02  X 10  = 670.2  feet,  depth  of  pit. 


WINDMILLS. 

Driving  Shaft  of  a vertical  windmill  should  be  set  at  an  elevating  angle 
with  horizon  when  set  upon  low  ground,  and  at  a depressing  angle  when  set 
upon  elevated  ground.  Range  of  these  angles  is  from  30  to  15V  A velocity 
of  wind  of  10  feet  per  second  is  not  generally  sufficient  to  drive  a loaded 
mill,  and  if  velocity  exceeds  35  feet  per  second  the  force  is  generally  too 
great  for  ordinary  structures. 

Angle  of  Sails  should  be  from  18°  to  30°  at  their  least  radius,  and  from 
70  to  1 70  at  their  greatest  radius,  mean  angle  being  from  150  to  170  to  plane 
of  motion  of  sails.  Length  of  a whip  (arm)  is  divided  into  7 parts,  sails  ex- 
tending over  6 parts. 


864 


WIND-MILLS. 


™!p^ul\0(  i(f  lellffth : B.readth  -°33,  at  top  .016 ; Depth  .025,  at  ton 
;°I23>  Width  of  sail  .33,  at  axis  .2.  Distance  of  sail  from  axis5.oi4  of 
length  of  whip,  ancl  cross-bars  16  to  18  ins.  from  centres. 

To  Compute  Angles  of  Sails. 

18  d2 

23  r 2 angle  of  sail  with  plane  of  its  motion  at  any  part  of  it.  d repre- 

senting distance  of part  of  sail  from  its  axis , and  r extreme  radius  of  sail,  both  in  feet. 
Illustration.— Assume  r = i4,  and^ength  of  sail  I2  feet,  d = .5  of  12  or  three 


Then  230  — - 


= 23 


— 5.88°=  17. 12° 
?°  — 72.88°. 


sixths  of  sail  — 5 X 12  -j-  (14  — 12)  = 2 = 8 feet. 

18X8 

Hence,  angle  of  sail  with  axis  = 90°  — 17. 12*  _ # 

be  asrtollowsf  Sa'1S  'S  divided  into  6 equal  Parts>  an«les  at  each  of  these  parts  will 

Distance  from  Axis. 

Angle  of  sail  with  axis 6*s*  6*o  3o  *o  7Q5, 

with  plane  of  motion 22.5°  210  18.5°  150 


85° 

5° 


-~n 


3.16  v 
r'  sin.  x 
IP  X 1 080  000 
v3 


To  Compute  Elements  of  Windmills. 

5”  A v3 


= A: 


y 


.1047  iiz=av ; 


'R2  + r2_ 


1 080  000 


v representing  velocity  of  wind  per  sec- 

°/centre  of  percussion  of  sails,  and  R and  r outer  and  inner  radii  of 
sails,  all  in  feet , x mean  angle  of  sail  to  plane  of  motion  n number  of  revolution?  of 
arms  per  minute,  a v angular  velocity,  A area  of  sails  in  sq.feet,  and  IP  horse-power. 

iLLUSTRATmN.-If  a windmill  has  4 arms  of  28  feet,  with  a mean  angle  (x)  of  160 
area  °f  sai  of  sq.  feet  each,  having  an  inner  radius  of  4 feet  and  is  on’ 
eiated  by  wind  at  a velocity  of  40  feet  per  second;  what  are  its  elements?  P 


Then 


11. 5 x 40 


4 X 150  X 6 74°°o__  ^ 
1 080  000 


= 23 


: 35-55J 


/28s  4-  42 

; yJ—.2  -20 feet; 

35-55  X 1 080000 
64  000 


3. 16  X 40 


20  X.  275  64 
= A = 599_9  sq.feet. 


— n~  22.95; 


Deductions  from  Velocities  varying  from  4 to  9 Feet  per 
Second.  {Mr.  Smeaton.) 

. /•  Vel,oc!.t/  of  windmill  sails,  so  as  to  produce  a maximum  effect,  is  near- 
ly  as  velocity  of  wind,  their  shape  and  position  being  same. 

2.  Load  at  maximum  is  nearly,  but  somewhat  less  than,  as  square  of  ve- 
locity of  wind,  shape  and  position  of  sails  being  same. 

3.  Effects  of  same  sails,  at  a maximum,  are  nearly,  but  somewhat  less 
than,  as  cubes  of  velocity  of  wind. 

4.  Load  of  same  sails,  at  maximum,  is  nearly  as  squares,  and  their  effect 
as  cubes  of  their  number  of  turns  in  a given  time. 

5.  In  sails  where  figure  and  position  are  similar,  and  velocity  of  wind  the 
lTi^gth^sail1,  °f  revolutions  in  a given  time  will  be  reciprocally  as  radius  or 

6.  Load,  at  a maximum,  which  sails  of  a similar  figure  and  position  will 
overcome  at  a given  distance  from  centre  of  motion,  will  be  as  cube  of  radius. 

7.  Effects  of  sails  of  similar  figure  and  position  are  as  square  of  radius. 
Velocity  of  extremities  of  Dutch  sails,  as  well  as  of  enlarged  sails,  in 

all  their  usual  positions  when  unloaded,  or  even  loaded  to  a maximum,  is 
considerably  greater  than  that  of  wind. 


WINDMILLS. WOOD  AND  TIMBER. 


865 

Results  of  Experiments  on  Effect  of  Windmill  Sails. 

When  a vertical  windmill  is  employed  to  grind  corn,  the  millstone  usu- 
ally makes  5 revolutions  to  1 of  the  sail. 

1.  When  velocity  of  wind  is  19  feet  per  second,  sails  make  from  n to  12 
revolutions  in  a minute,  and  a mill  will  grind  from  880  to  990  lbs.  in  an 
hour,  or  about  22  440  lbs.  in  24  hours. 

2.  When  velocity  of  wind  is  30  feet  per  second,  a mill  will  carry  all  sail, 
and  make  22  revolutions  in  a minute,  grinding  1984  lbs.  of  flour  in  an  hour, 
or  47  616  lbs.  in  24  hours. 

Reaxxlts  of  Operation  of  Windmills.  {A.  R.  Woolf  M.  E.) 
Velocity  of  Wind  15  to  20  Miles  per  Hour. 

Revolutions  of  Wheel  and  Gallons  of  Water  raised  per  Minute. 


Desig- 

Revolutions 

Water  raised  to  an  Elevation  of 

Power 

Cost  per  Hour. 

nation 
of  Mill. 

of 

Wheel. 

25  Feet. 

50  Feet. 

100  Feet. 

200  Feet. 

developed. 

Actual.* 

Per  HP. 

Feet. 

No. 

Gallons. 

Gallons. 

Gallons. 

Gallons. 

HP 

Cents. 

Cents. 

8.5 

70  to  75 

6.16 

3-C2 

— 

— 

.04 

.60 

15 

IO 

60  to  65 

19.18 

9-56 

4-75 

— 

.12 

.70 

5-8 

14 

50  to  55 

45-14 

22.57 

11.25 

5 

.28 

1.63 

5-8 

18 

40  to  45 

97.68 

52.16 

24.42 

12.21 

.61 

2.83 

4.6 

20 

35  to  40 

124.95 

63-75 

3i-25 

15-94 

.78 

3-56 

4-5 

25 

30  to  35 

212.38 

106.96 

49-73 

26.74 

i-34 

4.26 

3-2 

* Including  interest  at  5 per  cent,  per  annum. 


WOOD  AND  TIMBER. 

Selection  of  Standing  Trees. — Wood  grown  in  a moist  soil  is  lighter, 
and  decays  sooner,  than  that  grown  in  dry,  sandy  soil. 

Best  Timber  is  that  grown  in  a dark  soil,  intermixed  with  gravel. 
Poplar,  Cypress,  Willow,  and  all  others  which  grow  best  in  a wet  soil, 
are  exceptions. 

Hardest  and  densest  woods,  and  least  subject  to  decay,  grow  in  warm 
climates ; but  they  are  more  liable  to  split  and  warp  in  seasoning. 

Trees  grown  upon  plains  or  in  centre  of  forests  are  less  dense  than 
those  from  edge  of  a forest,  from  side  of  a hill,  or  from  open  ground. 

Trees  (in  U.  S.)  should  be  selected  in  latter  part  of  July  or  first  part 
of  August ; for  at  this  season  leaves  of  sound,  healthy  trees  are  fresh 
and  green,  while  those  of  unsound  are  beginning  to  turn  yellow.  A 
sound,  healthy  tree  is  recognized  by  its  top  branches  being  well  leaved, 
bark  even  and  of  a uniform  color.  A rounded  top,  few  leaves,  some  of 
them  turned  yellow,  a rougher  bark  than  common,  covered  with  parasitic 
plants,  and  with  streaks  or  spots  upon  it,  indicate  a tree  upon  the  de- 
cline. Decay  of  branches,  and  separation  of  bark  from  the  wood,  are 
infallible  indications  that  the  wood  is  impaired. 

Green  timber  contains  37  to  48  per  cent,  of  liquids.  By  exposure  to 
air  in  seasoning  one  year,  it  loses  from  17  to  25  per  cent.,  and  when 
seasoned  it  retains  from  10  to  15  per  cent. 

According  to  M.  Leplay,  green  wood  contains  about  45  per  cent,  of  its 
weight  of  moisture.  In  Central  Europe,  wood  cut  in  winter  holds,  at  end  of 
following  summer,  fully  40  per  cent,  of  water,  and  when  kept  dry  for  sev- 
eral years  retains  from  15  to  20  per  cent,  of  water. 

Felling  Timber  — Most  suitable  time  for  felling  timber  is  in  midwinter  and 
in  midsummer.  Recent  experiments  indicate  latter  season  and  month  of  J uly. 

1 4 D 


866 


WOOD  AND  TIMBER. 


A tree  should  be  allowed  to  attain  full  maturity  before  being  felled.  Oak 
matures  at  75  to  100  years  and  upward,  according  to  circumstances ; Ash 
Larch,  and  Elm  at  75 ; and  Spruce  and  Fir  at  80.  Age  and  rate  of  growth 
of  a tree  are  indicated  by  number  and  width  of  the  rings  of  annual  increase 
which  are  exhibited  in  a cross-section  of  its  body. 

A tree  should  be  cut  as  near  to  the  ground  as  practicable,  as  the  lower 
part  furnishes  best  timber. 

Dressing  Timber— As  soon  as  a tree  is  felled,  it  should  be  stripped  of  its 
bark,  raised  from  the  ground,  reduced  to  its  required  dimensions,  and  its 
sap-wood  removed. 

Inspection  of  Timber—  Quality  of  wood  is  in  some  degree  indicated  by  its 
color,  which  should  be  nearly  uniform,  and  a little  deeper  towards  its  cen- 
tre, and  free  from  sudden  transitions  of  color.  White  spots  indicate  decay. 
Sap-wood  is  known  by  its  white  color ; it  is  next  to  the  bark,  and  soon  rots. 

Defects  of*  Timber. 

Wind-shakes  are  serious  defects,  being  circular  cracks  separating  the  con- 
centric layers  of  wood  from  each  other. 

Splits , Checks , and  Cracks , extending  toward  centre,  if  deep  and  strongly 
marked,  render  timber  unfit  for  use,  unless  purpose  for  which  it  is  intended 
will  admit  of  its  being  split  through  them. 

Brash  is  when  wood  is  porous,  of  a reddish  color,  and  breaks  short,  with- 
out splinters.  It  is  generally  consequent  upon  decline  of  tree  from  age. 

Belted  is  that  which  has  been  killed  before  being  felled,  or  which  has  died 
from  other  causes.  It  is  objectionable. 

Knotty  is  that  containing  many  knots,  though  sound ; usually  of  stinted 
growth. 

Twisted  is  when  grain  of  it  winds  spirally ; it  is  unfit  for  long  pieces. 

Dry-rot  is  indicated  by  yellow  stains.  Elm  and  Beech  are  soon  affected, 
if  left  with  the  bark  on. 

Large  or  decayed  knots  injuriously  affect  strength  of  timber. 

Heart-shake—  Split  or  cleft  in  centre  of  tree,  dividing  it  into  segments. 

Star-shake. — Several  splits  radiating  from  centre  of  timber. 

Cup-shake. — Curved  splits  separating  the  rings  wholly  or  in  part. 

Rind-gall.— Curved  swelling,  usually  caused  by  growth  of  layers  over  spot 
where  a branch  has  been  removed. 

Upset. — Fibres  injured  by  crushing. 

Foxiness. — Yellow  or  red  tinge,  indicating  incipient  decay. 

Doatiness. — A speckled  stain. 

Seasoning  and  Preserving  Timber. 

Seasoning  is  extraction  or  dissipation  of  the  vegetable  juices  and  moisture 
or  solidification  of  the  albumen.  When  wood  is  exposed  to  currents  of  air 
at  a high  temperature,  the  moisture  evaporates  too  rapidly,  and  it  cracks ; / 

and  when  temperature  is  high  and  sap  remains,  it  ferments,  and  drv-rot 
ensues. 

Wood  requires  time  in  which  to  season,  very  much  in  proportion  to  density 
of  its  fibres. 

Water  Seasoning  is  total  immersion  of  timber  in  water,  for  purpose  of 
dissolving  the  sap,  and  when  thus  seasoned  it  is  less  liable  to  warp  and  crack, 
but  is  rendered  more  brittle. 


WOOD  AND  TIMBER. 


867 


For  nurDose  of  seasoning,  it  should  be  piled  under  shelter  and  kept  dry , 

ctrlim  nl  iced  between  each  layer,  one  near  each  end  of  pile,  and  others  at 
ho£  distances  in  order  to  keep  the  timber  from  winding.  These  strips 

should  be  one  over  the  other,  and  in  large  piles  she ^ ^ ttob» 

tVnVlr  T io’ht  timber  may  be  piled  in  upper  portion  ot  shelter,  neavj  timuer 
upon'°Tound  floor.  Each  pile  should  contain  but  one  description  o±  timber, 
and  they  should  be  at  least  2.5  feet  apart.  . . . 

It  should  be  repiled  at  interval*  and  all  pieces  indicating  decay  should  be 
removed,  to  prevent  their  affecting  those  which  are  still  sound. 

It  reouires  from  2 to  8 years  to  be  seasoned  thoroughly,  according  to  its 
dimensions,  and  it  should  be  worked  as  soon  as  it  is  thoroughly  dry,  for 
deteriorates  after  that  time. 

Gradual  seasoning  is  most  favorable  to  strength  and  durability  of  timber. 
Various  methods  have  been  proposed  for  hastening  the  process,  as  Steaming, 
whicMias  been  applied  with  success;  and  results  of  experiments  of  various 
processes  of  saturating  it  with  a solution  of  Corrosive  sublimate  and  An 
\er)tic  fluids  are  very  satisfactory.  Such  process  hardens  and  seasons  wood, 
Se  same  tTme  thlt  it  securest  from  dry-rot  and  from  attacks  of  worms. 

Woods  are  densest  and  strongest  at  the  roots  and  at  their  centres.  Their 
strength  decreasing  with  the  decrease  of  their  density. 

Oak  timber  loses  one  fifth  of  its  might  in  seasoning,  and  about  one  third 
in  becoming  perfectly  dry. 

Pitch  pine,  from  the  presence  of  pitch,  requires  time  in  excess  ot  that  due 
to  the  density  of  its  fibre. 

Mahogany  should  he  seasoned  slowly,  Pine  quickly.  Whitewood  should 
not  be  dried  artificially,  as  the  effect  of  heat  is  to  twist  it. 

Salt  water  renders  wood  harder,  heavier,  and  more  durable  than  fresh. 
Condition  of  timber,  as  to  its  soundness  or  decay,  is  readily  recognized 
when  struck  with  a quick  blow. 

Timber  that  has  been  for  a long  time  immersed  in  water,  when  brought 
into  the  air  and  dried,  becomes  brashy  and  useless. 

When  trees  are  barked  in  the  spring,  they  should  not  be  felled  until  the 
foliage  is  dead. 

Timber  cannot  be  seasoned  by  either  smoking  or  charring  ; but  when  it 
is  exposed  to  worms  or  to  the  production  of  fungi , it  is  proper  to  smoke  or 
char  it,  and  it  may  be  partially  seasoned  by  being  boiled  or  steamed. 

Timber  houses  are  best  provided  with  blinds  which  keep  out  rain  and 
snow  but  which  can  be  turned  to  admit  air  in  fine  weather,  and  the  houses 
should  be  kept  entirely  free  from  any  pieces  of  decayed  wood. 

Kiln-drying  is  suited  only  for  boards  and  pieces  of  small  dimensions,  as  it 
is  apt  to  cause  cracks  and  to  impair  the  strength,  unless  performed  very 
slowly. 

Charring , Painting , or  covering  the  surface  is  highly  injurious  to  any  but 
seasoned  wood , as  it  effectually  prevents  drying  of  the  inner  part  of  the 
wood,  in  consequence  of  which  fermentation  and  decay  soon  take  place. 

Timber  is  subject  to  Common  or  Dry-rot , former  occasioned  by  alternate 
exposure  to  moisture  and  dryness,  and  as  progress  of  it  is  from  the  exterior, 
covering  of  the  surface,  if  seasoned,  with  paint,  tar,  etc.,  is  a preservative. 


868 


WOOD  AND  TIMBER. 


Common-rot  is  the  consequence  of  its  being  piled  in  badly-ventilated  slieds. 
Outward  indications  are  yellow  spots  upon  ends  of  pieces,  and  a yellowish 
dust  in  the  checks  and  cracks,  particularly  where  the  pieces  rest  upon  pil- 
i’  g-strips. 

Dry  or  Sap-rot  is  inherent  in  timber,  and  it  is  the  putrefaction  of  the  veg- 
etable albumen.  Sap  wood  contains  a large  proportion  of  fermentable  ele- 
ments. 

Insects  attack  wood  for  the  sugar  or  gum  contained  in  it,  and  f ungi  subsist 
upon  the  albumen  of  wood ; hence,  to  arrest  dry-rot,  the  albumen  must  be 
either  extracted  or  solidified. 

Most  effective  method  of  preserving  timber  is  that  of  expelling  or  ex- 
hausting its  fluids,  solidifying  its  albumen,  and  introducing  an  antiseptic 
liquid. 

Strength  of  impregnated  timber  is  not  reduced,  and  its  resilience  is  improved. 
In  desiccating  timber  by  expelling  its  fluids  by  heat  and  air,  its  strength 
is  increased  fully  15  per  cent. 

The  saturation  of  wood  with  creosote,  tar,  antiseptics,  etc.,  preserves  it 
from  the  attack  of  worms.  Jarrow  wood,  from  Australia,  is  not  subjected 
to  their  attack. 

In  a perfectly  dry  atmosphere  durability  of  woods  is  almost  unlimited. 
Rafters  of  roofs  are  known  to  have  existed  1000  years,  and  piles  submerged 
in  fresh  water  have  been  found  perfectly  sound"  800  years  from  period  of 
their  being  driven. 

Resistance  of  woods  to  extension  is  greater  than  that  of  compression. 

Impregnation  of  Wood. 

Several  of  the  successful  processes  are  as  follows : 

# Kyan , 1832.— Saturated  with  corrosive  sublimate.  Solution  1 lb.  of  chlo- 
ride of  mercury  to  4 gallons  of  water. 

Burnett  ( Sir  Wm .),  1838.  — Impregnation  with  chloride  of  zinc  by  sub- 
mitting the  wood  endwise  to  a pressure  of  150  lbs.  per  sq.  inch.  Solution, 
1 lb.  of  the  chloride  to  4 gallons  of  water. 

Boucheri. — Impregnation  by  submitting  the  wood  endwise  to  a pressure 
of  about  15  lbs.  per  sq.  inch.  Solution,  1 lb.  of  sulphate  of  copper  to  12.5 
gallons  of  water. 

Bethel. — Impregnation  by  submitting  the  wood  endwise  to  a pressure  of 
150  to  200  lbs.  per  sq.  inch,  with  oil  of  creosote  mixed  with  bituminous 
matter. 

Robbins , 1865. — Aqueous  vapor  dissipated  by  the  wood  being  heated  in  a 
chamber,  the  albumen  solidified,  then  submitted  to  vapor  of  coal  tar,  resin, 
or  bituminous  oils,  which,  being  at  a temperature  not  less  than  3250,  readily 
takes  the  place  of  the  vapor  expelled  by  a temperature  of  2120. 

Hay  ford,  187-. — Aqueous  vapor  dissipated  by  the  wood  being  heated  in  a 
chamber  to  a temperature  of  from  250°  to  270°,  the  albumen  solidified,  then 
air  introduced  to  assist  the  splitting  of  the  outer  surfaces.  When  vapor  is 
dissipated,  dead  oils  are  introduced  under  a pressure  of  75  lbs.  per  sq.  inch. 

Planks , Deals , and  Battens. — When  cut  from  Northern  pine  (Pinus  Sylve- 
stris)  are  termed  yellow  or  red  deal,  and  when  cut  from  spruce  (Abies,  alba , 
etc.)  they  are  termed  white  deal. 

Desiccated  wood,  when  exposed  to  air  under  ordinary  circumstances,  ab- 
sorbs 5 per  cent,  of  water  in  the  first  three  days ; and  will  continue  to  absorb 
it  until  it  reaches  from  14  to  16  per  cent.,  the  amount  varying  according 
to  condition  of  the  atmosphere. 


WOOD  AND  TIMBER. 


869 


Durability  of  Various  Woods. 

Pieces  2 feet  in  Length ,.  1.5  ins.  Square,  driven  28.5  ins.  into  the  Earth. 


After  2.5  Years. 


Acacia 

Ash,  Amer 

Cedar,  Va 

“ Lebanon.. 

Elm,  Eng 

44  Can 

Fir 

Larch 

Oak,  Can 

“ Memel 

“ Dantzic  . . . 

“ Chestnut.. 
Pine,  pitch 


Teak  . 


yellow  . 
white . . 


Good 

Much  decayed 

Very  good 

Good 

Much  decayed 

“ attacked 

Surface  only  attacked. , 
Very  much  decayed.., 


Very  good 

Surface  only  attacked. , 

Attacked 

Very  much  decayed. . 
Very  good 


After  5 Years, 


( Externally  decayed,  rest  per- 
( fectly  sound. 

Decayed. 

Sound  as  when  driven. 
Tolerable. 

Entirely  decayed. 

Decayed. 

Much  decayed. 

( Attacked  in  part  only,  rest  fair 
( condition. 

Very  rotten. 


( Some  moderately,  most  very 
{ much,  decayed. 

( Attacked  in  part  only,  rest  fair 
( condition. 

Much  decayed. 

Very  rotten. 

Somewhat  soft,  but  good. 


Effect  of  Creosoting. 

Results  of  Experiments  with  Various  Woods  {E.  R.  Andrews). 


Spruce 
Oak . . . 


Water 

absorbed. 


Per  cent. 
•2543 
.0261 


Hard  pine. . . . 


( dried 

\ creosoted. 

Gum, black..  ]*™soted. 

Birch,  white.  j^‘“‘0Ve'd; 


Per  cent. 
. 16 


( dried 

| creosoted . 

f dried 

"j  creosoted . 

Cotton- wood  | cre0soted. 

Sesquoia  Gigantea  of  California,  dried,  .4722 ; creosoted,  .0. 

Fluids  will  pass  with  the  grain  of  w^ood  with  great  facility,  but  will  not 
enter  it  except  to  a very  limited  extent  when  applied  externally. 
Absorption  of  Preserving  Solution  by  different  Woods 
for  a Period  of  7 Pays.  Average  Lbs.  per  Cube  Foot. 


Black  Oak 3.6 

Chestnut 3 


Hemlock. 
Red  Oak. 


I Rock  Oak 3.9 

White  Oak 3.1 


Proportion  of  W a ter  in  various  Woods. 


Alder  (Betula  alnus) 416 

Ash  (Fraxinus  excelsior) 28.7 

Beech  (Fagus  sylvatica) 33 

Birch  ( Betula  alba) 30.8 

Elm  ( Ulmus  campestris) 44-5 

Horse-chestnut  ( AEsculus hippocast. ) 38. 2 

Larch  ( Pinus  larix) 48.6 

Mountain  Ash  (Sorbus  aucuparia). . 28.3 
Oak  ( Quercus  robur) 34*7 


Pine  (Pinus  Sylvestris  L.) 39.7 

Red  Beech  (Fagus  sylvatica) 39.7 

Red  Pine  (Pinus  picea  dur) 45.2 

Spruce  (Abies,  alba , nigra,  rubra , ' 

excelsa) 

Sycamore  (Acer  pseudo -platanus) . . 

White  Oak  (Quercus  alba) 

White  Pine  (Pinus  abies  dur) 

White  Poplar  (Populus  alba) 


35 


Willow  (Salis  caprea) 26 

Pecrease  in  Pimensions  of  Timber  by  Seasoning 
Ins.  Ins. 

13-25 


Woods. 

Cedar,  Canada 14  to 

Elm 11  to  10.75 

Oak,  English 12  to  11.625 

Pitch  Pine,  North. . . xoXioto  9.75X9-75 
Weight  of  a beam  of  English  oak,  when  wet,  was  reduced  by  seasoning 
from  972.25  to  630.5  lbs. 


Woods.  Ins.  Ins. 

Pitch  Pine,  South 18.375  to  18.25 

Spruce 8.5  to  8.375 

White  Pine,  American..  12  to  11.875 
Yellow  Pine,  North 18  to  17.875 


870 


WOOD  AND  TIMBER. 


Weight  of  a Cube  Foot  of  Oalt  and  Yellow  Fine. 


White  Oak,  Va. 

Yellow  Pine.  Va. 

Age. 

Round. 

Square. 

Round. 

Square. 

Live  Oak. 

Green 

64.7 

67.7 

47.8 

39-2 

78.7 

1 Year 

53-6 

53-5 

39- 8 

34-2 

2 Years 

46 

49.9 

34-3 

33-5 

66.7 

In  England,  Timber  sawed  into  boards  is  classed  as  follows : 

6.5  to  7 ins.  in  width,  Battens ; 8.5  to  10  ins.,  Deals ; and  11  to  12  ins., 
Planks.  (/See  also  page  62.) 


Distillation. — From  a single  cord  of  pitch  pine  distilled  by  chemical  ap- 
paratus, following  substances  and  in  quantities  stated  have  been  obtained : 


Charcoal 50  bushels. 

Illuminating  Gas about  1000  cu.  feet. 

Illuminating  Oil  and  Tar. . . 50  gallons. 
Pitch  or  Besin 1.5  barrels. 


Pyroligneous  Acid 100  gallons. 

Spirits  of  Turpentine 20  “ 

Tar . 1 barrel. 

Wood  Spirit 5 gallons. 


Strength  of  Timber. 

Results  of  experiments  have  satisfactorily  proved:  That  deflection  was 
sensibly  proportional  to  load ; That  extension  and  compression  were  nearly 
the  same,  though  former  being  the  greater ; That,  to  produce  equal  deflection, 
load,  when  placed  in  the  centre,  was  to  a load  uniformly  distributed,  as  .638 
to  1 ; That  deflection  under  equal  loads  is  inversely  as  breadths  and  cubes 
of  the  depths,  and  directly  as  cubes  of  the  spans.  (M.  Morin.) 

It  has  also  been  shown,  that  density  of  wood  varies  very  little  with  its  age. 
That  coefficient  of  elasticity  diminishes  after  a certain  age,  and  that  it  de- 
pends also  on  the  dryness  and  the  exposure  of  the  ground  where  the  wood 
is  grown.  Woods  from  a northerly  exposure,  on  dry  ground,  have  a high 
coefficient,  while  those  from  swamps  or  low  moist  ground  have  a low  one. 
That  tensile  strength  is  influenced  by  age  and  exposure.  The  coefficient 
of  elasticity  of  a tree  cut  down  in  full  vigor,  or  before  it  arrives  at  this 
condition,  does  not  present  any  sensible  difference.  That  there  is  no  limit 
of  elasticity  in  wood,  there  being  a permanent  set  for  every  extension. 


Average  Result  of  Experiments  on  Tensile  Strength  of  Wood  in  Various 
Positions  per  Sq.  Inch.  {MM.  Chevandier  and  Werlheim.) 

With  the  fibre,  6900  lbs.  Radially,  683  lbs.,  and  Tangentially,  723  lbs. 


To  Compute  Volume  of  an  Irregular  Body'. 


By  “ Simpson's  Rule." 

Operation. —Take  a right  line  in  the  figure  for  a base  line,  as  A B,  divide  the  fig- 
ure into  any  number  of  equal  parts,  and  compute  the  areas  of  their  plane  sections 
as  1,  2,  3,  etc.,  at  the  points  of  division , by  rules  applicable  to  area  of  a plane.  Then, 
operate  these  areas  as  if  they  were  the  ordinates  of  a plane  curve  or  figure  of  same 
length  as  the  figure,  and  result  will  give  volume  required. 

Illustration. — Assume  a figure  having  areas  as  follows,  and  A B = 24  feet. 


Sections,  1 

c 4 3 2 1 2 


Areas,  3 feet 
5 “ 


Multiplier,  1 
4 
2 
4 


Products,  3 
20 

14 

36 

iz 


84 


and  84  x 24  -r-  4 -4-  3 = 168  cube  feet. 


MISCELLANEOUS  MIXTURES. 


871 


MISCELLANEOUS  MIXTURES. 

Cements. 

Much  depends  upon  manner  in  which  a cement  is  applied  as  upon  the 
cement  itself,  as  best  cement  will  prove  worthless  if  improperly  applied. 
Following  rules  must  be  rigorously  adhered  to  to  attain  success : 

1.  Bring  cement  into  intimate  contact  with  surfaces  to  be  united.  This  is  best 
done  by  heating  pieces  to  be  joined  in  cases  where  cement  is  melted  by  heat,  as 
with  resin,  shellac,  marine  glue,  etc.  Where  solutions  are  used,  cement  must  be 
well  rubbed  into  surfaces,  either  with  a brush  (as  in  case  of  porcelain  or  glass), 
or  by  rubbing  the  two  surfaces  together  (as  in  making  a glue  joint  between  pieces 
of  wood). 

2.  As  little  cement  as  practicable  should  be  allowed  to  remain  between  the  united 
surfaces.  To  secure  this,  cement  should  be  as  liquid  as  practicable  (thoroughly 
melted  if  used  with  heat),  and  surfaces  should  be  pressed  closely  into  contact  until 
cement  has  hardened. 

3.  Time  should  be  allowed  for  cement  to  dry  or  harden,  and  this  is  particularly 
the  case  in  oil  cements,  such  as  copal  varnish,  boiled  oil,  white  lead,  etc.  When 
two  surfaces,  each  .5  inch  across,  are  joined  by  means  of  a layer  of  white  lead 
placed  between  them,  6 months  may  elapse  before  cement  in  middle  of  joint  be- 
comes hard.  At  the  end  of  a month  the  joint  will  be  weak  and  easily  separated;  at 
end  of  2 or  3 years  it  may  be  so  firm  that  the  material  will  part  anywhere  else  than 
at  joint.  Hence,  when  article  is  to  be  used  immediately,  the  only  safe  cements 
are  those  which  are  liquefied  by  heat  and  which  become  hard  when  cold.  A joint 
made  with  marine  glue  is  firm  an  hour  after  it  has  been  made.  Next  to  cements 
that  are  liquefied  by  heat  are  those  which  consist  of  substances  dissolved  in  water 
or  alcohol.  A glue  joint  sets  firmly  in  24  hours;  a joint  made  with  shellac  varnish 
becomes  dry  in  2 or  3 days.  Oil  cements,  which  do  not  dry  by  evaporation,  but 
harden  by  oxidation  (boiled  oil,  white  lead,  red  lead,  etc.)  are  slowest  of  all. 

Stone.—  Resin,  Yellow  Wax,  and  Venetian  Red,  each  1 oz. ; melt  and  mix. 

Aquarium. 

Litharge,  fine  white  dry  Sand,  and  Plaster  of  Paris,  each  1 gill;  finely  pulverized 
Resin,  .33  gill. 

Mix  thoroughly  and  make  into  a paste  with  boiled  linseed  oil  to  which  drier  has  been  added.  Beat 
well,  and  let  stand  4 or  5 hours  before  using  it.  After  it  has  stood  for  15  hours,  however,  it  loses  its 
strength.  Glass  cemented  into  a frame  with  this  cement  will  resist  percolation  for  either  salt  or  fresh 
water. 

Adhesive  for  Fractures  of  all  Kinds. 

White  Lead  ground  with  Linseed-oil  Varnish,  and  kept  from  contact  with  the  air. 

Requires  a few  weeks  to  harden. 

Stone  or  Iron. 

Compound  equal  parts  of  Sulphur  and  Pitch. 

Brass  to  Glass. 

Electrical. — Resin,  5 ozs. ; Beeswax,  1 oz. ; Red  Ochre  or  Venetian  Red,  in  pow- 
der, 1 oz.  Dry  earth  thoroughly  on  a stove  at  above  212°  Melt  Wax  and  Resin 
together  and  stir  in  powder  by  degrees.  Stir  until  cold  lest  earthy  matter  settle 
to  bottom. 

Used  for  fastening  brass-work  to  glass  tubes,  flasks,  etc. 

Chinese  Waterproof*. 

Schio-liao. — To  3 parts  of  Fresh  Beaten  Blood  add  4 parts  of  Slaked  Lime  and  a 
little  Alum;  a thin,  pasty  mass  is  produced,  which  can  be  used  immediately. 

Materials  which,  are  to  be  made  specially  waterproof  are  painted  twice,  or  at  most  three  times. 
Mr  ooden  public  buildings  of  China  are  painted  with  schio-liao , vtfhich  gives  them  an  unpleasant  red- 
dish appearance,  but  adds  to  their  durability.  Pasteboard  treated  with  it  receives  anpearanee  and 
strength  of  wood. 

China. 

Curd  of  milk,  dried  and  powdered,  10 ozs. ; Quicklime,  1 oz. ; Camphor,  2 drachms. 

Mix,  and  keep  air-tight.  When  used,  a portion  is  to  be  mixed  with  a little  water  into  a paste. 

Cisterns  and  Water-casks. 

Melted  Glue,  8 parts;  Linseed  oil,  boiled  into  a varnish  with  Litharge,  4 parts. 

Tbi3  cement  hardens  in  about  48  hours,  and  renders  the  joints  of  wooden  cisterns  and  casks  air  and 


872 


MISCELLANEOUS  MIXTURES. 


Clotli  or  Leather. 

Shellac,  1 part;  Pitch,  2 parts;  India  Rubber,  4 parts;  and  Gutta  Percha,  10 
parts;  cut  small;  Linseed  oil,  2 parts;  melted  together  and  mixed. 


Earthen  and.  Grlass  Ware. 

Heat  article  to  be  mended  a little  above  2120,  then  apply  a thin  coating  of  gum 
Shellac  upon  both  surfaces  of  broken  vessel. 

Or,  dissolve  gum  Shellac  in  alcohol,  apply  solution,  and  bind  the  parts  firmly  to- 
gether until  cement  is  dry. 

Or,  dilute  white  of  egg  with  its  bulk  of  water  and  beat  up  thoroughly.  Mix  to 
consistence  of  thin  paste  with  powdered  Quicklime. 

Use  immediately. 

Entomologists’. 

Thick  Mastic  Varnish  and  Isinglass  size,  equal  parts. 

Grixtta  IPercha. 

Melt  together,  in  an  iron  pan,  2 parts  Common  Pitch  and  1 part  Gutta  Percha. 

Stir  well  together  until  thoroughly  incorporated,  and  then  pour  liquid  into  cold  water.  When  cold 
it  is  black,  solid,  and  elastic  ; but  it  softens  with  heat,  and  at  ioo°  is  a thin  fluid.  It  may  be  used  as  a 
soft  paste,  or  in  liquid  state,  and  answers  an  excellent  purpose  in  cementing  metal,  glass,  porcelain 
ivory,  etc.  It  may  be  used  instead  of  putty  for  glazing.  ’ 

Grlass. 

Sorer s.—  Mix  commercial  Zinc  White  with  half  its  bulk  of  fine  Sand,  add  a solu- 
tion of  Chloride  of  Zinc  of  1.26  spec,  grav.,  and  mix  thoroughly  in  a mortar. 

Apply  immediately,  as  it  hardens  very  quickly. 


Holes  iix  Castings. 

Sulphur  in  powder,  1 part;  Sal-ammoniac,  2 parts;  powdered  Iron  turnings,  80 
parts.  Make  into  a thick  paste. 


Make  only  as  required  for  immediate  use. 


ILydranlic  JPaint. 

Hydraulic  cement  mixed  with  oil  forms  an  incombustible  and  waterproof  paint 
for  roofs  of  buildings,  outhouses,  walls,  etc. 


Iron  Ware. 

Sulphur.  2 parts;  fine  Black-lead,  1 part.  Heat  sulphur  in  an  iron  pan  until 
it  melts,  then  add  the  lead;  stir  well,  and  remove.  When  cool,  break  into  pieces 
as  required.  Place  upon  opening  of  the  ware  to  be  mended,  and  solder  with  an 
iron. 

Kerosene  Lamps,  etc. 

Resin,  3 parts;  Caustic  Soda,  1;  Water,  5,  mixed  with  half  its  weight  of  Plaster 
of  Paris. 


It  sets  firmly  in  about  three  quarters  of  an  honr.  Is  of  great  adhesive  power,  not  permeable  to  kero- 
sene, a low  conductor  of  heat,  and  but  superficially  attacked  by  hot  water. 


Ijeatlier  to  Iron,  Steel,  or  Grlass. 

1.  — Glue,  1 quart,  dissolved  in  Cider  Vinegar;  Venice  Turpentine,  1 oz. ; boil  very 
gently  or  simmer  for  12  hours. 

Or,  Glue  and  Isinglass  equal  parts,  soak  in  water  10  hours,  boil  and  add  tannin 
until  mixture  becomes  “ropy;”  apply  warm. 

Remove  surface  of  leather  where  it  is  to  be  applied. 

2. — Steep  leather  in  an  infusion  of  Nutgall,  spread  a layer  of  hot  Glue  on  sur- 
face of  metal,  and  apply  flesh  side  of  leather  under  pressure. 


Leather  Eelting. 

Common  Glue  and  Isinglass,  equal  parts,  soaked  for  10  hours  in  enough  water  to 
cover  them.  Bring  gradually  to  a boiling  heat  and  add  pure  Tannin  until  whole  be- 
comes ropy  or  appears  alike  to  white  of  eggs. 

Clean  and  rub  surfaces  to  be  joined,  apply  warm,  and  clamp  firmly. 


^Molding  and  Temporary  Adhesion. 

Soft. — Melt  Yellow  Beeswax  with  its  weight  of  Turpentine,  and  color  with  finely 
powdered  Venetian  red. 

When  cold  it  has  the  hardness  of  soap,  but  is  easily  softened  and  molded  with  the  fingers. 


MISCELLANEOUS  MIXTURES. 


873 


Alai tli a,  or  Grreeli  Alastic. 

Lime  and  Sand  mixed  in  manner  of  mortar,  and  made  into  a proper  consistency 
with  milk  or  size  without  water. 

ALarUle. 

Plaster  of  Paris,  in  a saturated  solution  of  Alum,  baked  in  an  oven,  and  reduced 
to  powder.  Mixed  with  water,  and  color  if  required. 

Aletal  to  Glass. 

Copal  Varnish,  15  parts;  Drying  Oil,  5;  Turpentine,  3.  Melt  in  a water  bath  and 
add  10  of  Slaked  Lime. 

Alending  Shells,  etc. 

Gum  Arabic,  5 parts;  Rock  Candy,  2;  and  White  Lead,  enough  to  color. 

Large  01>j ects. 

Wollaston's  White.—  Beeswax,  1 oz. ; Resin,  4 ozs. ; powdered  Plaster  of  Paris,  5 
oz.  Melt  together. 

Warm  the  edges  of  the  object  and  apply  warm. 

By  means  of  this  cement  a piece  of  wood  may  be  fastened  to  a chuck,  which  will  hold  when  cool  ; and 
when  work  is  finished  it  may  be  removed  by  a smart  stroke  with  tool.  Any  traces  of  cement  may  be 
removed  by  Benzine. 

ALarble  Workers  and.  Coppersmiths. 

White  of  egg,  mixed  with  finely-sifted  Quicklime,  will  unite  objects  which  are 
not  submitted  to  moisture. 

Porcelain. 

Add  Plaster  of  Paris  to  a strong  solution  of  Alum  till  mixture  is  of  consistency 
of  cream. 

It  sets  readily,  and  is  suited  for  cases  in  which  large  rather  than  small  surfaces  are  to  be  united. 

Rust  Joint. 

{Quick  Setting.)  — Sal-ammoniac  in  powder,  1 lb. ; Flour  of  Sulphur,  2 lbs. ; Iron 
borings,  80  lbs.  Made  to  a paste  with  water. 

(Slow  Setting.)— Sal-ammoniac,  2 lbs. ; Sulphur,  1 lb. ; Iron  borings,  200  lbs. 

The  latter  cement  is  best  if  joint  is  not  required  for  immediate  use. 

Steam  Boilers,  Steam-pipes,  etc. 

Finely  powdered  Litharge,  2 parts;  very  fine  Sand,  1;  and  Quicklime  slaked  by 
exposure  to  air,  1. 

This  mixture  may  be  kept  for  any  length  of  time  without  injuring.  In  using  it,  a portion  is  mixed 
into  paste  with  linseed  oil,  boiled  or  crude.  Apply  quickly,  as  it  soon  becomes  hard. 

Soft. — Red  or  White  Lead  in  oil,  4 parts;  Iron  borings,  2 to  3 parts. 

Hard. — Iron  borings  and  salt  water,  and  a small  quantity  of  Sal-ammoniac  with 
fresh  water. 

Transparent— Glass. 

India-rubber,  1 part  in  64  of  chloroform;  gum  Mastic  in  powder,  16  to  24  parts. 
Digest  for  two  days,  with  frequent  shaking. 

Or,  pulverized  Glass,  10  parts;  powdered  Fluor-spar,  20;  soluble  Silicate  of  Soda, 
60.  Both  glass  and  fluor-spar  must  be  in  finest  practicable  condition,  which  is  best 
done  by  shaking  each  in  fine  powder,  with  water,  allowing  coarser  particles  to  de- 
posit, and  then  by  pouring  off  remainder,  which  holds  finest  particles  in  suspension. 

The  mixture  must  be  made  very  rapidly,  by  quick  stirring,  and  applied  immediately. 

Uniting  Leather  and.  ATetal. 

Wash  metal  with  hot  Gelatine;  steep  leather  in  an  infusion  of  Nutgalls,  hot, 
and  bring  the  two  together. 

Waterproof  Alastic. 

Red  Lead,  1 part ; ground  Lime,  4 parts;  sharp  Sand  and  boiled  Oil,  5 parts. 

Or,  Red  Lead,  1 part;  Whiting,  5;  and  sharp  Sand  and  boiled  Oil,  10. 

"Wood  to  Iron. 

Litharge  and  Glycerine.—  Finely  powdered  Oxide  of  Lead  (litharge)  and  Concen- 
trated Glycerine. 

The  composition  is  insoluble  in  most  acids,  is  unaffected  by  action  of  moderate  heat,  sets  rapidly, 
and  acquires  an  extraordinary  hardness. 

Turner's. — Melt  1 lb.  of  Resin,  and  add  .25  lb.  of  Pitch. 

While  boiling  add  Brick  dust  to  give  required  consistency.  In  winter  it  may  be 
necessary  to  add  a little  Tallow. 


874 


MISCELLANEOUS  MIXTURES, 


GLUES. 

Marine. 

Dissolve  India  Rubber,  4 parts,  in  34  parts  of  Coal-tar  Naphtha;  add  powdered 
Shellac,  64  parts. 

While  mixture  is  hot  pour  it  upon  metal  plates  in  sheets.  When  required  for 
use,  heat  it,  and  apply  with  a brush. 

Or,  India  Rubber,  1 part;  Coal  Tar,  12  parts;  heat  gently,  mix,  and  add  powdered 
Shellac,  20  parts.  Cool.  When  used,  heat  to  about  2500 

Or,  Glue,  12  parts;  Water,  sufficient  to  dissolve;  add  Yellow  Resin,  3 parts;  and, 
when  melted,  add  Turpentine,  4 parts. 

Strong  Glue. — Add  Powdered  Chalk  to  common  Glue. 

Mix  thoroughly. 

HVHucilage. 

Curd  of  Skim  Milk  (carefully  freed  from  Cream  or  Oil),  washed  thoroughly,  and 
dissolved  to  saturation  in  a cold  concentrated  solution  of  Borax. 

This  mucilage  keeps  well,  and,  as  regards  adhesive  power,  far  surpasses  gum  Arabic. 

Or,  Oxide  of  Lead,  4 lbs. ; Lamp-black,  2 lbs. ; Sulphur,  5 ozs. ; and  India  Rubber 
dissolved  in  Turpentine,  10  lbs. 

Boil  together  until  they  are  thoroughly  combined. 

Preservation  of  Mucilage. — A small  quantity  of  Oil  of  Cloves  poured  into  a bottle 
containing  Gum  Mucilage  prevents  it  from  becoming  sour. 

To  Resist  IVIoistnre. 

Glue,  5 parts;  Resin,  4 parts;  Red  Ochre,  2 parts;  mixed  with  least  practicable 
quantity  of  water. 

Or,  Glue,  4 parts;  Boiled  Oil,  1 part,  by  weight.  Oxide  of  Iron,  1 part. 

Or,  Glue,  1 lb.,  melted  in  2 quarts  of  skimmed  Milk. 

Rarclim  en  t. 

Parchment  Shavings,  1 lb. ; Water,  6 quarts. 

Boil  until  dissolved,  then  strain  and  evaporate  slowly  to  proper  consistence. 

Rice,  or  Japanese. 

Rice  Flour;  Water,  sufficient  quantity. 

Mix  together  cold,  then  boil,  stirring  it  during  the  time. 

Liquid. 

Glue,  Water,  and  Vinegar,  each  2 parts.  Dissolve  in  a water-bath,  then  add  Al- 
cohol, 1 part. 

Or,  Cologne  or  strong  Glue,  2.2  lbs. ; Water,  1 quart;  dissolve  over  a gentle  heat; 
add  Nitric  Acid  36°,  7 ozs.,  in  small  quantities. 

Remove  from  over  fire,  and  cool. 

Or,  White  Glue,  16  ozs.  f White  Lead,  dry,  4 ozs. ; Rain  Water,  2 pints.  Adc[  Al- 
cohol, 4 ozs.,  and  continue  heat  for  a few  minutes. 

Elastic  and.  Sweet.— Stamps  or  Rolls. 

Elastic. — Dissolve  good  Glue  in  water  by  a water-bath.  Evaporate  to  a thick  con- 
sistence, and  add  equal  weight  of  Glycerine  to  Glue^  submit  to  heat  until  all  water 
is  evaporated,  and  pour  into  molds  or  on  plates. 

Sweet—  Substitute Sugar  for  the  Glycerine. 

To  Adhere  Engravings  or  Litliograplis  -upon  Wood. 

Sandarach,  250  parts;  Mastic  in  tears,  64  parts;  Resin,  125  parts;  Venice  Tur 
pentine,  250  parts;  and  Alcohol,  1000  parts  by  measure. 

BROWNING,  OR  BRONZING,  LIQUID. 

Sulphate  of  Copper,  1 oz. ; Sweet  Spirit  of  Nitre,  1 oz. ; Water,  1 pint. 

Mix.  Let  stand  a few  days  before  use. 


MISCELLANEOUS  MIXTURES. 


875 


Grim  Barrels. 

Tincture  of  Muriate  of  Iron,  i oz. ; Nitric  Ether,  i oz. ; Sulphate  of  Copper,  4 
scruples;  rain  water,  1 pint.  If  the  process  is  to  be  hurried,  add  2 or  3 grains  of 
Oxymuriate  of  Mercury. 

When  barrel  is  finished,  let  it  remain  a short  time  in  lime-water,  to  neutralize  any  acid  which  may 
have  penetrated , then  rub  it  well  with  an  iron  wire  scratch-brush. 

After  Browning.  — Shellac,  1 oz. ; Dragon’s-blood,  .25  oz. ; rectified  Spirit,  1 qt. 
Dissolve  and  filter.  . , , , . . . _T  , 

Or  Nitric  Acid  spec.  grav.  1.2;  Nitric  Ether,  Alcohol,  and  Muriate  of  Iron,  each  1 
part.’  Mix,  then  add  Sulphate  of  Copper  2 parts,  dissolved  in  Water  10  parts. 

LACQUERS. 

Small  Ax* iris,  or  Waterproof  Paper. 

Beeswax,  13  lbs.;  Spirits  Turpentine,  13  gallons;  Boiled  Linseed  Oil,  1 gallon. 

All  ingredients  should  be  pure  and  of  best  quality.  Heat  them  together  in  a copper  or  earthen  ves- 
sel over  a gentle  fire,  in  a water-bath,  until  they  are  well  mixed. 

Bright  Iron.  Work. 

Linseed  Oil,  boiled,  80.5  parts;  Litharge,  5.5  parts;  White  Lead,  in  oil,  n.25  parts; 
Resin,  pulverized,  2.75  parts. 

Add  litharge  to  oil ; simmer  over  a slow  fire  3 hours ; strain,  and  add  resin  and  white  lea<|  -,  keep  it 
gently  warmed,  and  stir  until  resin  is  dissolved. 

Or,  Amber,  6 parts;  Turpentine,  6 parts;  Resin,  1 part;  Asphaltum,  1 part;  and 
Drying  Oil,  3 parts;  heat  and  mix  well. 

Or,  Shellac,  1 lb. ; Asphaltum,  6 lbs. ; and  Turpentine,  1 gallon. 

Iron  and.  Steel. 

Clear  Mastic,  10  parts;  Camphor,  5 parts;  Sandarac,  15  parts;  and  Elimi  Gum, 
5 parts.  Dissolve  in  Alcohol,  filter,  and  apply  cold. 

Brass. 

Shellac,  8 ozs. ; Sandarac,  2 ozs. ; Annatto,  2 ozs. ; and  Dragon’s-blood  Resin,  .25 
oz. ; and  Alcohol,  1 gallon. 

Or,  Shellac,  8 ozs. ; and  Alcohol,  1 gallon.  Heat  article  slightly,  and  apply  lacquer 
with  a soft  brush. 

Wood,  Iron,  or  Walls,  and  rendering  Cloth,  Paper,  etc., 
W aterproof. 

Heat  120  lbs.  Oil  Varnish  in  one  vessel,  33  lbs.  Quicklime  in  22  lbs.  water  in  an- 
other. Soon  as  lime  effervesces,  add  55  lbs.  melted  India  Rubber.  Stir  mixture, 
and  pour  into  vessel  of  hot  Varnish.  Stir,  strain,  and  cool. 

When  used,  thin  with  Varnish  and  apply,  preferably  hot. 

To  Clean  Soiled  Engravings. 

Ozone  Bleach,  1 part;  Water,  10;  well  mixed. 

INKS. 

Indelible,  for  Marking  Linen,  etc. 

1.— Juice  of  Sloes,  1 pint;  Gum,  .5  oz. 

This  requires  no  “ preparation  ” or  mordant,  and  is  very  durable. 

2 — Nitrate  of  Silver,  1 part  , Water,  6 parts,  Gum,  1 part;  Dissolve. 

3.— Lunar  Caustic,  2 parts;  Sap  Green  and  Gum  Arabic,  each  1 part;  dissolve  with 
distilled  water. 

“Preparation.”— Soda,  1 oz. ; Water,  1 pint;  Sap  Green,  .5  drachm.  Dissolve, 
and  wet  article  to  be  marked,  then  dry  and  apply  the  ink. 

Perpetual,  for  Tomb-stones , Marble,  etc.  — Pitch,  n parts;  Lamp-black,  1 part; 
Turpentine  sufficient.  Warm  and  mix. 

Copying  Ink. — Add  1 oz.  Sugar  to  a pint  of  ordinary  Ink. 

SOLDERING. 

Base  for  Soldering. 

Strips  of  Zinc  in  diluted  Muriatic,  Nitric,  or  Sulphuric  Acid,  until  as  much  is  de- 
composed as  acid  will  effect.  Add  Mercury,  let  it  stand  for  a day;  pour  off  the 
Water,  and  bottle  the  Mercury. 

When  required,  rub  surface  to  be  soldered  with  a cloth  dipped  in  the  Mercury. 


8/6 


MISCELLANEOUS  MIXTURES. 


VARNISHES. 

W aterproof. 

combined*  Sulphur’  1 lb' ’ Linseed  0il>  1 Sal1- ; boil  ^em  until  they  are  thoroughly 
Good  for  waterproof  textile  fabrics. 

Harness. 

India  Rubber,  .5  lb. ; Spirits  of  Turpentine,  1 gall. ; dissolve  into  a jelly ; then  mix 
hot  Linseed  Oil,  equal  parts  with  the  mass,  and  incorporate  them  well  over  a slow  fire. 

Fastening  Reatker  on  Top  Rollers. 

Gum  Arabic,  2.75  ozs.,  and  a like  volume  of  Isinglass,  dissolved  in  Water. 

To  Preserve  Gr lass  from  tlie  Sun. 

Reduce  a quantity  of  Gum  Tragacanth  to  fine  powder,  and  dissolve  it  for  24  hours 
in  white  of  egg  well  beat  up. 

Water-color  Drawings. 

Canada  Balsam,  1 part;  Oil  of  Turpentine,  2 parts. 

Mix  and  size  drawing  before  applying. 

Objects  of  Natural  History,  Sliells,  Risk,  etc. 

Mucilage  of  Gum  Tragacanth  and  of  Gum  Arabic,  each  1 oz. 

Mix,  and  add  spirit  with  Corrosive  Sublimate,  to  precipitate  the  more  stringy  por- 
tion of  the  Gum. 

Iron  and  Steel. 

Mercury,  120  parts;  Tin,  10  parts;  Green  Vitriol,  20  parts;  Hydrochloric  Acid  of 
1.2  sp.  gr.,  15  parts,  and  pure  Water,  120  parts. 

Blackboards. 

Shellac  Varnish,  5 gallons;  Lamp-black,  5 ozs.;  fine  Emery,  3 ozs.;  thin  with 
Alcohol,  and  lay  in  3 coats. 

Black. 

Heat,  to  boiling,  Linseed  Oil  Varnish,  10  parts,  with  Burnt  Umber,  2 parts,  and 
powdered  Asphaltum,  1 part. 

When  cooled,  dilute  with  Spirits  of  Turpentine  as  may  be  required. 

Balloon. 

Melt  India  Rubber  in  small  pieces  with  its  weight  of  boiled  Linseed  Oil. 

Thin  with  Oil  of  Turpentine. 

Transfer. 

Alcohol,  5 ozs. ; pure  Venice  Turpentine,  4 ozs. ; Mastic,  1 oz. 

To  render  Canvas  Waterproof'  and  Bliable. 

Yellow  Soap,  1 lb  , boiled  in  6 pints  of  Water,  add,  while  hot,  to  112  lbs.  of  oil  Paint. 

Waterproof  Bags. 

Pitch,  8 parts,  Wax  and  Tallow,  each  1 part. 

To  Clean  Varnish. 

Mix  a lye  of  Potash  or  Soda,  with  a little  powdered  Chalk. 

STAINING.  ? 

Wood  and  Ivory. 

Yellow. — Dilute  Nitric  Acid  will  produce  it  on  wood. 

Red. — An  infusion  of  Brazil  Wood  in  Stale  Urine,  in  the  proportion  of  1 lb.  to  a 
gallon,  for  wood,  to  be  laid  on  when  boiling  hot,  also  Alum  water  before  it  dries. 

Or,  a solution  of  Dragon’s-blood  in  Spirits  of  Wine. 

Black. — Strong  solution  of  Nitric  Acid. 

Blue. — For  Ivory:  soak  it  in  a solution  of  Verdigris  in  Nitric  Acid,  which  will  turn 
it  green ; then  dip  it  into  a solution  of  Pearlash  boiling  hot. 

Purple. — Soak  Ivory  in  a solution  of  Sal-ammoniac  into  four  times  its  weight  of 
Nitrous  Acid. 

Mahogany. — Brazil,  Madder,  and  Logwood,  dissolved  in  water  and  put  on  hot. 


MISCELLANEOUS  MIXTURES. 


8 77 


MISCELL  ANEOU  S. 

Blacking  for  Harness. 

Beeswax,  .5  lb. ; Ivory  Black,  a ozs.  - Spirits  of  Turpentine,  x oz. ; Pruss.an  Blue 
ground  in  oil,  i oz  ; Copal  „ g f mixture  is  quite  cold;  make  it 

harness,  then  polish  lightly 

with  silk. 

To  Clean  Brass  Ornaments. 

Brass  ornaments  that  have  not  b-n  gilt  or  lackered  “^n"’L?e,tn  S£ 

«>- wHh  swons  Tripoii- 
To  Harden  Brills,  Chisels,  etc. 

Temper  them  in  Mercury. 

To  Clean  Coral. 

-r^^SS^^e.  If  much  discolored,  let 
it  remain  in  solution  for  a few  hours. 

Blacking,  withovit  [Polishing. 

Molasses,  4 ozs. ; Lamp-black,  .5  oz. ; Yeast,  a table-spoonful;  Eggs,  2;  Olive  0,1, 
a teaspoonful ; Turpentine,  a teaspoonful.  Mix  well. 

To  be  applied  with  a sponge,  without  brushing. 

Dubhing. 

Resin,  2 lbs. ; Tallow,  1 lb. ; Train-oil,  1 gallon. 

Anti-friction  G-rease. 

Tallow  100  lbs  • Palm-oil,  70  lbs.  Boiled  together,  and  when  cooled  to  80°,  strain 
th" k'sTevefa'nd  mix  will  28  lbs.  of  Soda,  and  x.  5 gallons  of  Water, 
for  Winter,  take  25  lbs.  more  oil  in  place  of  the  Tallow. 

Or,  Black  Lead,  1 part;  Lard,  4 parts. 

To  Attach  Hair  Belt  to  Boilers. 

Red  Lead,  x lb. ; White  Lead,  3 lbs. ; and  Whiting,  8 lbs.  Mixed  with  boiled  Lin- 
seed  Oil  to  consistency  of  paint. 

[Pastils  for  Fumigating. 

Gum  Arabic  2 ozs. ; Charcoal  Powder,  5 ozs. ; Cascarilla  Bark,  powdered,  .75  oz. ; 
Saltpetre,  .25  drachm.  Mix  together  with  water,  and  make  into  shape. 

Bor  Writing  upon  Zinc  Labeis. -Horticultural 
Dissolve  too  grains  of  Chloride  of  Platinum  in  a pint  of  water;  add  a little  Mu- 

Cila0g“  Sabammon^^'dr. ; Verdigris, . dr. ; Lamp-black,  .5  dr. ; Water,  xo  drs.  Mix. 
To  [Remove  old  Ironmold. 

Remoisten  part  stained  with  ink,  remove  this  by  use  of  Muriatic  Acid  diluted  by 
5 or  6 times  ifs  weight  of  water,  when  old  and  new  stain  will  be  removed. 

To  Cut  India  Rubber. 

Keep  blade  of  knife  wTet  with  water  or  a strong  solution  of  Potash. 

Adhesive  for  Rubber  Belts. 

Coat  driving  surface  with  Boiled  Oil  or  Cold  Tallow,  and  then  apply  powdered 
Chalk. 

X-jiard. 

50  parts  of  finest  Rape-oil,  and  1 part  of  Caoutchouc,  cut  small.  Apply  heat  until 
it  is  nearly  all  dissolved. 

To  [Preserve  Heather  Belting  or  Hose. 

Apply  warm  Castor  Oil.  For  hose,  force  it  through  it. 

To  Oil  Heather  Belting. 

Apply  a solution  of  India  Rubber  and  Linseed  Oil. 

4 E 


miscellaneous  mixtures. 


*-^Mow  1 IPbT’  T warm. 

Beef  Tallow,  3 lbs. ; Beeswax,  , lb.  Heated  and  applied  warm  to  both  sidea 

0 s 

Lay  dull  files  in  diluted  Sulphuric  Acid  until  they  are  bitten  deep  enough 
Apply  Aqua.ammoniaRera°Ve  °U  fr°m  leather. 

Wash  with  a solution  of  Pearlash  in  water 

Or,  Extract  of  Litherium  diluted  with  from  200^3^ pa^f  witen' 

, To  Remove  IPaint 

Apply  hot^tmd ?et2 rentai'nnfor>i° day ’ 4 °ZS'  ’ boiling  Water,  with  Quicklime,  .5  lb 

r,  Extract  of  Litherium,  thinly  brushed  over  the  surface  ? or  3 times. 
r,  . To  Clean  ALarTble 

Mix  with  water. 

n ..  ' Paste  for  Cleaning  Metals 

Spirits  of  Turpentfne.’  Rottenstone>  6 Parts-  Mix  with  equal  parts  of  Train  Oil  and 
Watchmaker’s  Oil,  whieh 

Place  coils  of  thin  Sheet  Lead  in  a bottle  with  Olive  °r  Thicl3:eils* 

a few  weeks,  and  pour  off  the  clear  oil.  th  ° V 0lL  ExP°se  it  to  the  sun  for 

..  , durable  IPaste. 

wHh  a mtle  cold 

being  added  to  it);  add  a little  BrownSnrarm  r,rL  o1le  boding  water  is 
prevent  fermentation,  and  a few  drops  of  (hi  of  Lavender Th?u,b: which  will 
coming  moldy.  When  dried,  dissolve  in  water. Lavender>  whlch  will  prevent  it  be- 
lt will  keep  for  two  or  three  years  in  a covered  vessel. 

Stains. 

lo  Remove. — Stains  of  Iodine  are  removed  bv  reotififvi  «n,v(  T 7 
alic  or  Superoxalate  of  Potash;  Ironmolds  bv  wd-rSpi  *t;  Ink  stams  b^  Ox- 
W]th  Ink,  then  remove  them  in  the  usual  way  ’ bUt  lf  obstlDate>  moisten  them 

ot“SSS?  fr0m  'removed  by  Spirits  of  Hartshorn,  or 

Ch with  fresh  solution  of 

part  in  solution  of  Ammonia  or  of  Hyposulphite™?  Soda^T^r”6  White’  dip  the 
with  clean  water.  p 01  ^)OC‘a-  In  a few  minutes  wasli 

tu?eroS’Vhe  Stai“ed  Iinen  °Ver  a baSin  of  water,  and  wet  mark  with  Tine- 

Rod  T°a  rreserve  Bottoms  of  Iron  Steam-boilers 
r-5  parteby  w7e5igphtns;  Wtian  Red’  17  part3i  ™tin«>  &S  Parts;  and  LUharge, 

, . T . To  Preserve  Sails. 

Aatfae„VwUh%2lube“vitHo,,  and  il  "*«>  -°  gallons 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS.  879 


Whitewash. 


For  outside  exposure,  slack  Lime,  .5  bushel,  iu  a barrel;  add  common  Salt,  1 lb. ; 
Sulphate  of  Zinc,  .5  lb. ; and  Sweet  Milk,  1 gallon. 


Boiled  Oil  and  finely  powdered  Charcoal,  each  1 part;  mix  to  the  consistence  of 
paint.  Apply  2 or  3 coats. 

This  composition  is  well  adapted  for. casks,  water-spouts,  etc. 


Rub  surface  with  Pumice  Stone  am  iter  until  the  rising  of  the  grain  is  removed. 
Then,  with  powdered  Tripoli  and  boi]  Linseed  Oil,  polish  to  a bright  surface. 


Chrome  Green,  .25  oz. ; Sugar  of  Lead,  1 lb. ; ground  fine,  in  sufficient  Linseed  Oil 
to  moisten  it.  Mix  to  the  consistency  of  cream,  and  apply  with  a soft  brush. 

The  glass  should  be  well  cleansed  before  the  paint  is  applied.  The  above  quantity  is  sufficient  for 
about  200  feet  of  glass. 

To  Make  Drain  Tiles  Porous. 

Mix  sawdust  with  the  clay  before  burning. 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS. 

I# it  is  required  to  lay  out  a tract  of  land  in  form  of  a square,  to  be  en- 

closed with  a post  and  rail  fence,  5 rails  high,  and  each  rod  of  fence  to  con- 
tain 10  rails.  What  must  be  side  of  this  square  to  contain  just  as  many 
acres  as  there  are  rails  in  fence  ? 

Operation,  i mile  =-  320  rods.  Then  320  X 320  = 160,  sq.  rods  in  an  acre  = 640 
acres  y and  320  X 4 sides  and  X 10  rails  = 12  800  rails  per  mile. 

Then  as  646  acres  : 12800  rails  12800  acres  : 256000  rails,  which  will  enclose 
256000 ’acres,  and  -1/256000  X 69.5701  = number  of  yards  in  side  of  a sq.  acre , and 
-f- 1760,  yards  in  a mile  = 20  miles. 

2.— How  many  fifteens  can  be  counted  with  four  fives? 


3.  — What  are  the  chances  in  favor  of  throwing  one  point  with  three  dice? 

Operation. — Assume  a bet  to  be  upon  the  ace.  Then  there  will  be  6 X 6 X 6 = 216 

different  ways  which  the  dice  may  present  themselves , that  is , with  and  without  an  ace . 

Then,  if  the  ace  side  of  the  die  is  excluded,  there  will  he  5 sides  lefty  and  5X5X5 
= 125  ways  without  the  ace. 

Therefore,  there  will  remain  only  216  — 125  = 91  ways  in  which  there  could  he  an 
ace.  The  chance,  then,  in  favor  of  the  ace  is  as  91  to  125 ; that  is,  out  of  216  throws, 
the  probability  is  that  it  will  come  up  91  times , and  lose  125  times. 

4. — The  hour  and  minute  hand  of  a clock  are  exactly  together  at  12; 
when  are  they  next  together  ? 

Operation. — As  the  minute  hand  runs  11  times  faster  than  the  hour  hand,  then, 
as  11  : 60  ::  1 : 5 min.  27^  sec.  — time  past  1 o'clock. 

5. — Assume  a cube  inch  of  glass  to  weigh  1.49  ounces  troy,  the  same  of 
sea-water  .59,  and  of  brandy  .53.  A gallon  of  this  liquor  in  a glass  bottle, 
which  'weighs  3.84  lbs.,  is  thrown  into  sea-water.  It  is  proposed  to  deter- 
mine if  it  will  sink,  and,  if  so,  how  much  force  will  just  buoy  it  up? 

Operation.  3.84  X 12  -f- 1.49  — 30.92  cube  ins.  of  glass  in  bottle. 

231  cube  ins.  in  a gallon  X .53  = 122.43  ounces  of  brandy. 

Then,  bottle  and  brandy  weigh  3.84  X 12-1-122.43  = 168.51  ounces , and  contain 
261.92  cube  ins.,  which  X -59  = 154-53  ounces,  weight  of  an  equal  bulk  of  sea-water. 

And,  168.51  — 154. 53  = 13  98  ounces , weight  necessary  to  support  it  in  the  water. 


To  Preserve  Woodwork. 


To  IP  sh  Wood. 


Faint  for  Window  Glass. 


880  MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS. 


6.— A fountain  has  4 supply  cocks,  A,  B,  C,  and  D,  and  under  it  is  a ci«« 
tern,  which  can  be  filled  by  the  cock  A in  6 hours,  by  B in  8 hours,  by  C in 
10  hours,  and  by  D in  12  hours ; now,  the  cistern  has  4 holes,  designated  E 
F,  G,  and  II,  and  it  can  be  emptied  through  E in  6 hours,  F in  5 hours,  G in 
4 hours,  and  H in  3 hours.  Suppose  the  cistern  to  be  full  of  water,  and  that 
all  the  cocks  and  holes  were  opened  together,  in  what  time  would  the  cistern 
be  emntied? 


be  emptied? 

Operation.— Assume  the  cistern  to  hold  120  gallons. 

hrs.  gall. 

: 1 : 20  at  A. 

'.1  : 15  af  B. 

: : 1 : 12  at  C. 

1 : 10  at  D. 

[ hour , 57  gallons. 


hrs.  gall. 

If  6 : 120  : 
8 : 120  ; 
10  : 120  ; 
12  : 120  : 
Run  in  in  1 


hr3. 

If  6 
5 
4 
3 

Run  out  in  1 hour , 


gall. 

120  ::  1 
120  ::  1 
120  ::  1 
120  ::  1 


hrs.  gall. 

20  at  E. 

24  at  F. 

30  at  G. 

40  at  H. 

1 14  gallons. 

57 

Run  out  in  1 hour  more  than  run  in,  57  gallons. 

Then,  as  57  gallons  : 1 hour  ::  120  gallons  : 2.158-}-  hours. 

7.— A cistern,  containing  60  gallons  of  water,  has  3 cocks  for  discharging 
it ; one  will  empty  it  in  1 hour,  a second  in  2 hours,  and  a third  in  3 hours ; 
in  what  time  will  it  be  emptied  if  they  are  all  opened  together? 

Operation.— 1st,  .5  would  run  out  in  1 hour  by  the  2d  cock,  and  .333  by  the  3d- 
consequently,  by  the  3 would  the  reservoir  be  emptied  in  1 hour.  .5  -f-  .333  -}- 1 = 
§d~  § + f,  being  reduced  to  a common  denominator , the  sum  of  these  3 — JL1- . whence 
the  proportion,  1 1 : 60  6 : 32^-  minutes. 

A reservoir  has  2 cocks,  through  which  it  is  supplied ; by  one  of  them 
it  will  fill  in  40  minutes,  and  by  the  other  in  50  minutes ; it  has  also  a dis- 
charging cock,  by  which,  when  full,  it  may  be  emptied  in  25  minutes.  If 
the  3 cocks  are  left  open,  in  what  time  would  the  cistern  be  tilled,  assuming 
the  velocity  of  the  water  to  be  uniform  ? 

Operation.— The  least  common  multiple  of  40,  50,  and  25,  is  200. 

Then,  the  1st  cock  will  fill  it  5 times  in  200  minutes , and  the  2d,  4 times  in  200 
minutes , or  both,  9 times  in  200  minutes ; and,  as  the  discharge  cock  will  empty  it 
8 times  in  200  minutes , hence  9 — 8 = 1,  or  once  in  200  minutes  = 3. 2 hours. 

9-  The  time  of  the  day  is  between  4 and  5,  and  the  hour  and  minute 
hands  are  exactly  together ; what  is  the  time  ? 

Operation.— Difference  of  speed  of  the  hands  is  as  1 to  12  = 11. 

4 hours  X 60  = 240,  which  -4-  n = 21  min.  49.09  sec.,  which  is  to  be  added  to  4 hours. 

10.— Out  of  a pipe  of  wine  containing  84  gallons,  10  were  drawn  off,  and 
the  vessel  refilled  with  water,  after  which  10  gallons  of  the  mixture  were 
drawn  off,  and  then  10  more  of  water  were  poured  in,  and  so  on  for  a third 
and  fourth  time.  It  is  required  to  compute  how  much  pure  wine  remained 
in  the  vessel,  supposing  the  two  fluids  to  have  been  thoroughly  mixed. 

Operation.  84  — 10  = 74,  quantity  after  the  1st  draught. 

Then,  84  : 10  74  : 8.8095,  and  74  — 8.8095  = 65.1905,  quantity  after  2 d draught. 

84:10: : 65. 1905 : 7. 7608,  and  65. 1905  — 7. 7608  = 5 7. 4297,  quantity  after  3 d draught. 

84: 10:  ‘.57. 4297: 6. 8367,  and  57.4297  — 6.8367  = 50.593,  quantity  after  4th  draught , 
= result  required. 

I1’ — A.  reservoir  having  a capacity  of  10000  cube  feet,  has  an  influx  of 
750  and  a discharge  of  1000  cube  feet  per  day.  In  what  time  will  it  be 
emptied  ? 


Operation. 


— = 40  days. 


Contrariwise 
In  what  time  will  it  be  filled  ? 


1000  — 750 

The  discharge  being  1000  and  the  influx  1250  cube  feet  per  hour. 


1250  — 1000 


Operation. 


= 40  hours  = 1 day  16  hours. 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS.  88 1 


12.— A son  asked  his  father  how  old  he  was.  His  father  answered  him 
thus’:  If  you  take  away  5 from  my  years,  and  divide  the  remainder  by  8, 
the  quotient  will  be  one  third  of  your  age ; but  if  you  add  2 to  your  age,  and 
multiply  the  whole  bv  3,  and  then  subtract  7 from  the  product,  you  will  have 
the  number  of  years  of  my  age.  What  were  the  ages  of  father  and  son  i 

Operation.—  Assume  father’s  age  37. 

Then  - 5 = 32,  and  32  = 8 = 4,  and  4 X 3 = 12,  son’s  age.  Again : 12  + 2 = 14, 
and  14  X 3 = 42,  and  42  - 7 = 35-  Therefore  37  - 35  - 2,  error  too  little. 

Again:  Assume  father’s  age  45;  then  45  — 5 = 4°,  and  4°  = 8 = 5-  The™^e 
5 x 3 — 15,  son's  age.  Again:  15  + 2=  17,  and  17  X 3 = 5L  and  51  — 7 — 44-  There- 
fore 45  — 44  = 1,  error  too  little. 

Hence  (45  sup.  X 2 error)  — (37  sup.  X 1 error)  = 90  - 37  = 53,  and  2 — 1 = 1. 

Consequently,  53  is  father's  age.  Then  53  — 5 = 48,  and  48 -4- 8 = 6 = .333  of  son's 
age , and  6 X 3 = 18  years , son's  age. 

—Two  companions  have  a parcel  of  guineas.  Said  A to  B,  if  y 011  will 
give  me  one  of  your  guineas  I shall  have  as  many  as  you  have  left,  B re- 
plied, if  you  will  give  me  one  of  your  guineas  I shall  have  twice  as  many  as 
you  will  have  left.  How  many  guineas  had  each  of  them  ? 


Operation.— Assume  B had  6. 

Then  A would  have  had  4,  for  6 — 1 = 4 + 1 = 5-  Again : 4 (A’s  parcel)  — 1 = 3, 
and  6 + 1 = 7,  and  3 X 2 = 6.  Therefore  7 — 6 = 1,  error  too  tittle. 

Again : Assume  B had  8. 

Then  A would  have  6,  for  8 — 1 = 6+  1 =7.  Again : 6 (A’s  parcel)  — 1 = 5,  and 
8 -f- 1 = 9,  and  5X2  = 10.  Therefore  10  — 9 ==  1,  error  too  great. 

Hence  8X1  = 8,  and  6X1=6.  Then  8 + 6 = 14,  and  1 + 1 = 2.  Whence,  di- 
viding products  by  sum  of  errors,  14  = 2 = 7 = B’s  parcel , and  7 — 1 — 5 + 1— 6 
for  A when  he  had  received  x of  B ; also  5-iX2  = 7 + x=  8 = B’s  parcel  when  he 


had  received  1 of  A. 

1 4 —If  a traveller  leaves  New  York  at  8 o’clock  in  the  morning,  and  walks 
towards  New  London  at  the  rate  of  3 miles  per  hour,  without  intermission ; 
and  another  traveller  starts  from  New  London  at  4 o clock  in  the  evening, 
and  walks  towards  New  York  at  the  rate  of  4 miles  per  hour  continuously ; 
assuming  distance  between  the  two  cities  to  be  130  miles,  whereabouts  upon 
the  road  will  they  meet? 


Operation.  — From  8 to  4 o’clock  is  8 hours;  therefore,  8X3  = 24  miles,  per- 
formed by  A before  B set  out  from  New  London;  and,  consequently,  130  — 24=106 
are  the  miles  to  be  travelled  between  them  after  that. 


Hence,  as  (3  + 4)  7 : 3 ’.  I 106  : All  — 45a  more  miles  travelled  by  A at  the  meeting; 
consequently?  24  + 45  f-  = 69-f  miles  from  New  York  is  place  of  their  meeting. 

I5  _If  from  a cask  of  wine  a tenth  part  is  drawn  out  and  then  it  is  filled 
with  water ; after  which  a tenth  part  of  the  mixture  is  drawn  out;  again 
is  filled,  and  again  a tenth  part  of  the  mixture  is  drawn  out:  now,  assume 
the  fluids  to  mix  uniformly  at  each  time  the  cask  is  replenished,  what  frac- 
tional part  of  wine  will  remain  after  the  process  of  drawing  out  and  replen- 
ishing has  been  repeated  four  times  ? 

Operation.— Since  .1  of  the  wine  is  drawn  out  at  first  drawing,  there  must  remain 
q After  cask  is  filled  with  water,  .1  of  whole  being  drawn  out,  there  will  remain 
*.9  of  mixture;  but  .9  of  this  mixture  is  wine;  therefore,  after  second  drawing,  there 

will  remain  .9  of.  9 of  wine,  or  ; and  after  third  drawing,  there  will  remain  .9 


93 

Of.  9 of.  9 Of  wine,  or  — . 

Hence,  the  part  of  wine  remaining  is  expressed  by  the  ratio  .9,  raised  to  a power 
exponent  of  which  is  number  of  times  cask  has  been  drawn  from. 

q4 

Therefore,  fractional  part  of  wine  is  — .6561. 

4°E* 


882  MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS. 


16. — There  is  a fish,  the  head  of  which  is  9 ins.  long,  the  tail  as  long  as 
the  head  and  half  the  body,  and  the  body  as  long  as  both  the  head  and  tail. 
Required  the  length  of  the  fish. 

of°ta ^RATIOX'~Assume  b0(Jy  t0  be  24  ins-  in  length.  Then  24-4-  2 -j-  9 = 21,  length 

Hence  21  -f  9 = 30,  length  of  body,  which  is  6 ins.  too  great. 

Again : assume  the  body  to  be  26  ins.  in  length.  Then  26  -4-  2 -f  9 = 22  length  of 
tail.  Hence  22  -|-  9 = 31,  length  of  body,  which  is  5 ins.  too  great. 

Therefore,  by  Double  Position,  divide  difference  of  products  (see  rule,  page  99) 
by  difference  of  errors  (the  errors  being  alike),  26  X 6 — 24  X 5 = 36  = difference  of 
products , and  6 — 5 = 1 = difference  of  errors.  J 

Consequently,  36  = 1 = 36,  length  of  body,  and  36  -4-  2 -}-  9 = 27,  length  of  tail , and 
30 4-27 + 9 = 72  ins. , length  required. 

17. — A hare,  50  leaps  before  a greyhound,  takes  4 leaps  to  the  greyhound's 
3,  but  2 leaps  of  the  hound  are  equal  to  3 of  the  hare’s.  How  many  leaps 
must  the  greyhound  take  before  he  can  catch  the  hare? 

Operation.— As  2 leaps  of  the  greyhound  equal  3 of  the  hare,  it  follows  that  6 of 
the  greyhound  equal  9 of  the  hare. 

While  the  greyhound  takes  6 leaps,  the  hare  takes  8;  therefore,  while  the  hare 
takes  8,  the  greyhound  gains  upon  her  1. 

Hence,  to  gain  50  leaps,  she  must  take  50  X 8 = 400  leaps;  but,  while  hare  takes 
400  leaps,  greyhound  takes  300,  since  number  of  leaps  taken  by  them  are  as  4 to  3. 

18. — If  a basket  and  1000  eggs  were  laid  in  a right  line  6 feet  apart,  and 
10  men  (designated  from  A to  J)  were  to  start  from  basket  and  to  run  alter- 
nately, collect  the  eggs  singly,  and  place  them  in  basket  as  collected,  and 
each  man  to  collect  but  10  eggs  in  his  turn,  how  many  yards  would  each 
man  run  over,  and  what  would  be  entire  distance  run  over*? 

Operation.  — A’s  course  would  be  6X2  feet  {first  term)  -f- 10  X 6 X 2 feet  ( last 
term)  = 132  = sum  of  first  and  last  terms  of  progression. 

Then  1324-2X10=:  660  feet  = number  of  times  X half  sum  of  extremes  = sum  of 
all  the  terms,  or  the  distance  run  by  A in  his  first  turn. 

B’s  course  would  be  11X6X2  = 132  feet  { first  term)  4-20X6X2  ='  240  feet  {last 
term)  = 372  = sum  of  first  and  last  terms. 


Then  372  4-2X10  = i860  = sum  of  all  the  times,  or  B 's  first  turn. 

A’s  last  course  would  be  901  x 6 X 2 = 10812  feet  for  the  first  term  and 
= 10920  feet  for  the  last  term  of  his  last  turn. 


910X6X2 


Then  10  812  -f- 10  920  -4-  2 X 10  = 108  660  = sum  of  the  terms,  or  distance  run. 

B’s  last  course  would  be  911  x 6 X 2 = 10932  feet  for  the  first  term  and  920X6X2 
— 11 040  feet  for  the  last  term  of  his  last  turn. 


Then  10  932  -f-  1 1 040  -4-2X10  = 109  860  = sum  of  the  terms  or  distance  run. 

Therefore,  if  A’s  first  and  last  runs  — 660  and  108  660  feet,  and  the  number  of 
terms  10,  then,  by  Progression,  the  sum  of  all  the  terms  ==  546  600  feet. 

And  if  B’s  first  and  last  runs=  i860  and  109  860  feet,  and  the  number  of  terms  10, 
then  the  sum  of  all  the  terms  = 558600  feet. 

Consequently,  558  600  — 546  600  = 12  000  = common  difference  of  runs,  which,  be- 
ing added  to  each  man’s  run  = sum  of  all  runs,  or  entire  distance  run  over. 


A’s 

run,  546  600  = 

182  200  yds. 

F’s 

B’s 

“ 558600  = 

186200  “ 

G’s 

C’s 

“ 570600  = 

190200  u 

H’s 

D’s 

“ 582600  = 

194200  “ 

I’s 

E’s 

“ 594600  = 

198200  “ 

J’s 

run,  606  600  = 202  200  yds. 
“ 618600  = 206200  u 

“ 630600  = 210200  11 

u 642600  = 214200  u 
“ 654600  = 218200  “ 


6 006  000  feet,  which -4- 5280  = 1137.5  miles. 

I9-  i11  a pair  of  scales,  a body  weighs  90  lbs.  in  one  scale,  and  but  40 

lbs.  in  the  other,  what  is  the  true  weight? 

V (40  x 90)  = 60  lbs. 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS.  883 


20.  If  a steamboat,  running  uniformly  at  the  rate  of  15  miles  per  hour 

through  the  water,  were  to  run  for  1 hour  with  a current  of  5 miles  per  hour, 
then  to  return  against  that  current,  what  length  of  time  would  she  require 
to  reach  the  place  from  whence  she  started  ? 

Operation.  15  — j-  5 — 20  miles,  the  distance  vun  during  the  houi . 

Then  15  — 5 = 10  miles  is  her  effective  velocity  per  hour  when  returning , and 
20^-10  — 2 hours , the  time  of  returning , and  2 + 1 = 3 hours , or  the  whole  time  oc- 
cupied. 

Or,  Let  d represent  distance  in  one  direction , t and  t'  greater  and  less  times  of  run- 
ning in  hours , and  c current  or  tide. 

a , v X ? — d 

Then  = velocity  of  boat  through  the  water , and = c. 

’ t X t'  1 

21.  — Flood-tide  wave  in  a given  river  runs  20  miles  per  hour,  current  of 
it  is  3 miles  per  hour.  Assume  the  air  to  be  quiescent,  and  a floating  body 
set  free  at  commencement  of  flow  of  the  tide ; how  long  will  it  drift  in  one 
direction,  the  tide  flowing  for  6 hours  from  each  point  of  river  ? 

Operation. — Let  x be  the  time  required;  202:  = distance  the  tide  has  run  up,  to- 
gether with  the  distance  which  the  floating  body  has  moved;  33  — whole  distance 
which  the  body  has  floated. 

Then  20 x- 


-3x 


: 6 X 20,  or  the  length  in  miles  of  a tide. 

c = — x6  = 7 hours.  3 minutes , 31.765  seconds. 
20  — 3 


^ 3 

22. — A steamboat,  running  at  the  rate  of  10  miles  per  hour  through  the 
water,  descends  a river,  the  velocity  of  which  is  4 miles  per  hour,  and  re- 
turns in  10  hours  ; how  far  did  she  proceed? 

Operation.— Let  x = distance  required, 

returning.  Then, f-  --  — 10;  6x  + 143 1 

14  0 


— - — — time  of  going. = time  of 

10  + 4 10  — 4 

- 840 ; 20X  — 840 ; 840  -4-  20  ==  42  miles. 


23. — From  Caldwell’s  to  Newburgh  (Hudson  River)  is  18  miles ; the  cur- 
rent of  the  river  is  such  as  to  accelerate  a boat  descending,  or  letard  one 
ascending,  1.5  miles  per  hour.  Suppose  two  boats,  running  uniformly  at  the 
rate  of  15  miles  per  hour  through  the  water,  were  to  start  one  from  each 
place  at  the  same  time,  where  will  they  meet? 

Operation.— Let  3 = the  distance  from  N.  to  the  place  of  meeting;  its  distance 
from  C.,  then , will  be  18  — x. 

Speed  of  descending  boat,  15  + 1.5  = 16.5  miles  per  hour  ; of  ascending  boat,  15 

x 18  — x 

1.3  — 13.  ^ miles  per  hour.  — - — = time  of  boat  descending  to  point  of  meeting.  ^ ^ 

= time  of  boat  ascending  to  point  of  meeting. 


x 18 — x 

These  times  are  of  course  equal;  therefore,  -7—= — • 

16.5  i3-5 

16. 53,  and  13.  $x  -j- 16.  sx  = 297,  or  30X  = 297. 


Then,  13. 53  = 297  — 


Hence  x — — 9.9  miles,  the  distance  from  Newburgh. 

3° 

24. — There  is  an  island  73  miles  in  circumference;  3 men  start  together 
to  walk  around  it  and  in  the  same  direction  : A walks  5 miles  per  day,  B 8, 
and  C 10 ; when  will  they  all  come  aside  of  each  other  again  ? 

Operation.— It  is  evident  that  A and  C will  be  together  every  round  gone  by  A ; 
hence  it  remains  to  ascertain  when  A and  B will  be  in  conjunction  at  an  even  round, 
as  3 miles  are  gained  every  day  by  B.  Therefore,  as  3 : x 73  : 24.33+;  but,  as 
the  conjunction  is  a fractional  number,  it  is  necessary  to  ascertain  what  number  of 
a multiplier  will  make  the  division  a whole  number. 

7^-f-24.'n+  = s,  the  number  of  days  required  in  which  A will  go  round  5 times, 
B 8,  and  C 10  times. 


884  MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS. 


25.— Assume  a cow,  at  age  of  2 years,  to  bring  forth  a cow-calf,  and  then 
to  continue  yearly  to  do  the  same,  and  every  one  of  her  produce  to  bring 
forth  a cow-calf  at  age  of  2 years,  and  yearly  afterward  in  like  manner ; 
how  many  would  spring  from  the  cow  and  her  produce  in  40  years  ? 

Operation. — The  increase  in  1st  year  would  be  o,  in  2d  year  1,  in  3d  1,  in  4th  2 
in  5O1  3,  in  6th  5,  and  so  on  to  40  years  or  terms,  each  term  being  — sum  of  the  two 
preceding  ones.  The  last  term,  then,  will  be  165  580  141,  from  which  is  to  be  sub- 
tracted 1 for  the  parent  cow,  and  the  remainder,  165  580140,  will  represent  increase 
required. 

26— The  interior  dimensions  of  a box  are  required  to  be  in  the  propor- 
tions of  2,  3,  and  5,  and  to  contain  a volume  of  1000  cube  ins. ; what  should 
be  the  dimensions  ? 


Operation.  — 3 /IO°° X 2 3 = 6. 43 ; 3 00X38  J j,65;  and  3 A°ooX53  =l6  { 
v 2X3X5  V 2x3x5  y V 2x3X5 

And  what  for  a box  of  one  half  the  volume,  or  500  cube  ins.,  and  retaining 
same  proportionate  dimensions  ? 


Operation.— 2 x 3 X 5 = 30,  and  — = 15. 


Then 


/15X  6.433  /15  X 9-653  ^ / 

V 30  —S'1!  y ~ = 7. 66 ; and3y 


/15  X i63 


= 12  ms. 


30  * V 30 

27. — The  chances  of  events  or  games  being  equal,  what  are  the  odds  for 
or  against  the  following  results  ? 


Five  Events. 

Four  Events. 

Odds. 

Against. 

In  favor. 

Odds. 

Against. 

In  favor. 

31  to  1 

4.33  to  I 

All  the  5 
4 out  of  5 

1 out  of  5 

2 out  of  5 

15  to  1 
2.2  to  1 

All  the  4 
3 out  of  4 

1 out  of  4 

2 OUt  Of  4 

5 to  3 in  favor  of  the  5 events  result- 
ing 3 and  2. 

5 to  3 against  2 events 
the  4 events  do  not  result 

only,  or  that 
2 and  2. 

Three  Events. 

T'wo  Events. 

Odds. 

Against. 

In  favor. 

Odds. 

Against. 

In  favor. 

7 to  1 
Even 

All  the  3 
( 2 or  all  out 

t of  3 

1 out  of  3 
1 2 or  all  out 

l of  3 

3 to  1 
Even 

Both  events 
( 1 only  out 
\ of  2 

1 out  of  2 
( 1 only  out 
\ of  2 

3 to  1 in  favor  of  the  3 events  result- 
ing 2 and  1. 

Even  that  the  events  result  1 and  1. 

28. — Required  the  chances  or  probabilities  in  events  or  games,  when  the 
chances  or  probabilities  of  the  results,  or  the  players,  are  equal. 


Events 

Games. 

That  a 
named  event 
occurs  a 
majority  or 
more  of 
times. 

Against  a 
named  event 
occurring 
an  exact 
majority  of 
times. 

Against  each 
event  occur- 
ring an  equal 
number 
of  times. 

Events 

or 

Games. 

That  a 
named  event 
occurs  a 
majority  or 
more  of 
times. 

Against  a 
named  event 
occurring 
an  exact 
majority  of 
times. 

Against  each 
event  occur- 
ring an  equal 
number  of 
times. 

21 

Even 

5 to  1 

_ 

ii 

Even 

3.4  to  1 



20 

1.33  to  1 

— 

4.66  to  1 

10 

1.7  to  I 

3.06  to  I 

*9 

Even 

4.5  to  I 

— 

9 

Even 

3 to  1 

— 

18 

1.55  to  I 

— 

4. 4 to  1 

8 

1.75  to  I 

2.66  to  1 

17 

Even 

4.4  to  I 

— 

7 

Even 

2.7  to  1 

— 

16 

1.5  to  1 

— 

4. 1 to  1 

6 

2 to  I 

— 

2.2  to  I 

15 

Even 

4 to  i 

— 

5 

Even 

2.2  tO  I 

— 

14 

1.5  to  I 

— 

3.8  to  I 

4 

2.2  to  I 

— 

1.66  to  I 

13 

Even 

3.7  to  I 

— 

3 

Even 

1.66  to  1 

— 

12 

1.6  to  I 

3.44  to  I 

2 

3 to  I 

— ' 

Even. 

29. — The  chances  of  consecutive  events  or  results  are  as  follows  : 

11. — 2047  to  1.  | 10.— 1023  to  1.  | 9. — 5ntoi.  | 8. — 255  to  1.  | 7. — 127  to  1.  | 6. — 63  to  1. 

Hence  it  will  be  observed  that  the  chances  increase  with  the  number  of  events 
very  nearly  in  a duplicate  ratio. 

Illustration. — The  chances  of  n consecutive  events  compared  with  10,  are  as 
2047  to  1023,  or  2 to  1. 


MISCELLANEOUS  OPERATIONS  AND  ILLUSTRATIONS.  885 

30.— Required  the  chances  or  probabilities  of  events  or  results  in  a given 
number  of  times.  ... 

The  numerator  of  a fraction  expresses  the  chance  or  probability  either  for  the  re- 
sult or  event  to  occur  or  fail,  and  the  denominator  all  the  chances  or  probabilities 
both  for  it  to  occur  or  fail. 

Thus,  in  a given  number  of  events  or  games,  if  the  chances  are  even,  the  proba- 
bility of  any  particular  result  is  as  -±-  = ~ 5 , etc.,  being  1 out  of 

2,  2 out  of  4,  etc. , or  even. 

If  the  number  of  events  or  games  are  3,  then  the  probability  of  any  par- 
ticular result,  as  2 and  1,  or  1 and  2,  is  determined  as  follows : 

Number  of  permutations  of  3 events  are  1 X 2 X 3 = 6,  which  represents  number 
of  times  that  number  of  events  can  occur,  2 and  1,  or  1 and  2,  to  which  is  to  be 
added  the  2 times  or  chances  they  can  occur  all  in  one  way  or  the  reverse  thereto. 

HenCe  6 = 1=  . 3—  = l,or3to  1 in  favor  of  result;  and  probability  ot 

1 2 d-  6 4 4 3 1 . * 

one  Dartv  naming  or  winning  tw’O  precise  events  or  results,  as  winning  2 out  01  3, 
is  determined  as  follows:  Number  of  permutations  and  chances,  as  before  shown, 

3 ’ ’ ' 

' 8 


are  8.  Hence,  number  of  his  chances  being  3, 


3 — 3 ? — = — , or  3 to  5 in 

o 1 j ~ - 3 5 

favor  of  result;  and  probability  of  one  party  naming  or  winning  all  or  3 events 
or  results  is  determined  as  follows:  Number  of  permutations  and  chances  being 
also,  as  before  shown,  8.  Hence,  as  there  is  but  one  chance  of  such  a lesult, 


1 + 7" 


_ — A , or  1 to  7 in  favor  of  result. 

’ 8 8—1  7 


If  number  of  events,  etc.,  are  4,  then  probability  of  any  particular  result, 
as  2 and  2,  or  of  winning  2 or  more  of  them,  is  determined  as  follows : 
Number  of  permutations  and  chances  of  4 events  are  16.  Hence,  as  number  of 

chances  of  such  a result  are  : 


of  the  result,  and  that  the  results  do  not  occur  precisely  2 and  2.  The  number  of 


= — or  as  11  to  5 in  favor 

16  16  — 11  5 


chances  of  such  a result  being  10, 


6 -j-  10 


— , or  5 to  3 against  it. 
5 3 


If  number  of  events,  etc.,  are  5,  then  probability  of  any  particular  result, 
as  3 and  2,  is  determined  as  follows : 

Number  of  permutations  and  chances  being  32,  and  number  of  chances  of  such 
a result  being  20,  ^ ^ ^ f , or  as  5 to  3 in  favor  of  the 

result;  and  that  it  may  occur  precisely  3 out  of  5,  the  number  of  chances  are 


10 


JO-f-22  32  l6 


= — } or  11  to  5 against  it. 


3I. What  is  the  dilatation  of  the  iron  in  a railway  track  per  mile,  be- 

tween the  temperatures  of  — 20°  and  +130°? 

Operation. 200  -f  1300  = 1500.  The  dilatation  of  wrought  iron  (as  per  table, 

page  519)  is,  from  320  to  2120  = i8o°  = .ooi  257  5 times  its  length. 

1 °47  9.  of  5280  (feet  in  a mile)  = 


Hence,  as  180  : 150  ::  .001  257  5 : .001047  9 
5-53  feet  Per  m^e- 

32  — A steamer  having  an  immersed  amidship  section  of  125  sq.  feet,  has 
a speed  of  15  miles  per  hour  with  300  H>.  What  power  would  be  required 
for  one  of  like  model,  having  a section  of  150  sq.  feet  for  a speed  of  20  miles . 
As  power  required  for  like  models  is  as  cube  of  speeds, 


Then  — 1.2  relative  sections , and 
125 

Hence,  1 : 1.2  2.37  : 2.844  times  IP. 


20 a = 8000 
i53  = 3375 


— 2. 37  relative  powers. 


886 


MARINE  STEAMERS  AND  ENGINES. 


MARINE  STEAMERS  AND  ENGINES. 

Iron  Cruiser  (IPropeller). 

“Zabiaca,’’  I.  R N.—  Vertical  Direct  Engine  {Compound).— Length  between 
perpendwulai'S,  228  feet ; at  water-line  of  12  feet,  220  feet ; beam,  30  feet ; hold,  17.5 

Displacement  at  load  draught  of  12.58  and  14.58  feet,  1202  tons.  Per  inch  at  load- 
line,  11.58  tons.  Areas.— Of  Load-line,  4867  sq.  feet;  of  Sails,  12  312  sq.  feet. 

Coefficients.  Of  Total  Displacement,  . 5 ; of  Surface,  . 74 ; of  Cylindroid  from  cyl- 
inder, .61;  of  Cylindroid  from  parallelopipedon,  .475. 

Cylinders. — 34  and  59  ins.  in  diam.  by  36  ins.  stroke  of  piston. 

Pressure  of  Steam.— 78  lbs.  per  sq.  inch,  cut  off  at  23  ins.  full  throttle.  Revolu- 
tions  89.4  per  minute  IIP,  1400.  Pitch  of  Propeller,  19  feet.  Speed,  14  knots 
pei  hour.  Fuel. — Anthracite  coal,  1.6  lbs.  per  IIP  per  hour.  * 4 

Centres  of  Gravity.  —Forward  of  after  perpendicular,  100  feet;  below  meta-centre 
at  draught  of  10.46  and  12.21  feet,  2.81  feet,  and  at  load-line  3.12  feet.  Of  Buoyan- 
cy, below  load-line,  4. 97  feet.  Of  Engines , Boilers,  Water,  etc. , aft  of  centre  of  length 
25.25  feet;  do.  above  top  of  keel,  9.17  feet.  b ’ 

Meta-centre. — Above  centre  of  buoyancy  for  mean  draught  of  u.3  feet  = 4 feet. 

Iron  Freight  and  Passenger  (Propellers). 

“ Orient.  ’’-Vertical  Direct  Engine  {Compound).— Length  upon  deck , 460  feet : 
beam,  46. 35  feet ; depth  to  main  deck,  27.  i feet ; to  spar  deck,  35. 1 feet. 

Immersed  section  at  load-line , 1094  sq.feet.  Displacement  at  load  draught  of  26  q 
feet,  9500  tons  ; per  inch,  40  tons.  Tons,  3440-5380. 

Cylinders.— 1 of  60  ins.  in  diam.,  and  2 of  85  ins.,  by  5 feet  stroke  of  piston.  Con- 
denser.  — Surface,  12  000  sq.  feet.  Propeller.  — 4 blades,  22  feet  in  diam.  Pitch,  30 
teet.  Shaft,  20  ins.  in  diam.  ’ J 

Boilers.—  4 (cylindrical  tubular),  15.5  feet  in  diam.  by  17.5  feet  in  length;  6 fur- 
naces, 4 feet  in  diam.  by  6 feet  in  length.  Pressure  of  Steam,  75  lbs.  per  sq.  inch. 
Revolutions,  60  per  minute.  IIP,  5400.  Bulkheads,  12.  Decks',  3 of  iron. 

Capacity  — 30oo  tons  coal,  3600  tons  (measurement)  cargo,  120  1st  class  passen- 
gers, 130  2d,  and  300  3d  class,  or  3000  troops  and  406  horses. 

Water  Ballast.—  Aft,  82  feet  in  length.  Rig  4-masted  bark.  Passage , 35  days 
Plymouth  to  Australia.  Weights.- Hull,  Engines,  and  Boilers,  4940  tons. 

“Arizona.’’— Vertical  Direct  Engine  {Compound).  — Length  between  verven- 
diculars  and  for  tonnage,  450  feet;  breadth,  45.5  feet;  depth , 35.7  9 feet;  Tons,  514T55. 

Cylinders.—  1 of  62  ins.  and  2 of  90  ins.  in  diam.,  by  5.5  feet  stroke  of  Diston 
Condenser.—  Surface,  i2  54osq.  feet.  ’ J b 01  Plsl°n. 

Propeller  (Cast  Steel).— Diam.,  23  feet;  weight,  27  tons. 

Boilers.— 6 of  13.5  feet  in  diam.,  3 of  10  feet  in  length,  and  3 of  18  feet  Heatina 

RewZtioZ50-0,  pqer  m nufe  aiTTpVq6fee^  of  Steam,  86  lbs.  per  sf  inch, 

revolutions,  ^5  per  minute.  IIP,  6306.  Speed,  17  knots  per  hour. 

“Normandie  ’’—Vertical  Direct  Engines  ( Compound ) -Length  ,qo  feet  n 
ms. , beam,  49  feet  1 1 ms.  Hold , 37  feet  5 ins.  Mean  draught  at  trial,  20 %{t  % 
sf0aCement'  7656  tons • Irnnersed  Section  at  load -draught  of  24.25  feet,  io6o 

6,  mf"  ^Hn^f  l™3£t3-7i  inS”  and  3 of  74-875  ins.  in  diam. ; stroke  of  pistons, 
67  ins. , ratio  ot  low  to  high  pressure,  i to  4.46.  1 

strokeofiriston3;  T/f  V f -P  single  acting,  34  ins.  diam. ; 

11  by  11  inch  engines  Centrifugal  Pum2>*—3,  12.5  ms.  in  diam.,  driven  by  three 

.P,Sf^VCyIlfdriCal  tubular),  4 double  end,  13.5  feet  in  diam.,  18.5  feet  in 

XHefuna  wL  ’ I3'75  T 9-5  feet  in  length.  Grates,  808.5  sq.  feet. 
Heating  Surface,  21  405  sq.  feet.  Steam  Room , 3950  cube  feet. 

XrD,  €SS%hn?f  Ste™n\85  lbs- ; cut  off  at  • 75  stroke.  Revolutions,  59  per  minute.  IIP, 
8006.  Shaft,  23.625  ms.  in  diam.  Propeller , 22  feet  in  diam.  Pitch,  31  feet. 

complet e)  1 3,6  tons.S  ^ h°"r'  WeigM  of  Eng‘nes,  Boilers,  and  Water  in  boilers, 


MARINE  STEAMERS  AND  ENGINES. 


887 


“City  of  San  Francisco.”  — Vertical  Direct  Engine  [Compound).  — Length 
over  all , 352  feet;  for  tonnage , 339  feet;  beam,  40. 2 feet;  hold,  28  feet  10  ins.;  Load 
draught,  22  feet. 

Cylinders , 2.— 51  and  88  ins.  in  diam.  by  5 feet  stroke  of  piston.  Condenser. — 
Surface,  6425  sq.  feet. 

Pressure  of  Steam,  80  lbs.  per  sq.  inch.  Revolutions,  55.  Speed,  14  knots  per  hour. 
Propeller,  4 blades,  20  feet  in  diam.  by  25  feet  pitch. 

Boilers.—  6 (cylindrical  tubular),  13  feet  in  diam.  Heating  Surface , 10650  sq.  feet. 
Grates , 378  sq.  feet.  Ratio  of  Grate  to  heating  surface,  1 to  28,;  to  tube  area,  9 to  1 ; 
to  smoke-pipe  area,  6.66  to  1. 

Iron  -A.xixili.ary  Freight. 

Vertical  Direct  Engine  [Compound). — Length  on  deck,  135  feet ; beam,  22.5  feet; 
hold,  11  feet. 

Load-draught , 4 feet  10  ins.  and  10  feet  6 ins.  Free  board,  1.5  feet. 

Cylinders. — 21  and  40  ins.  in  diam.  by  27  ins.  stroke  of  piston.  Condenser. — Sur- 
face, 617  sq.  feet. 

Boiler  (cylindrical  tubular). — 12  feet  in  diam.  by  9.5  feet  in  length.  Heating 
surface,  1205  sq.  feet.  Grates,  38.5  sq.  feet. 

Pressure  of  Steam,  80  lbs.  per  sq.  inch.  Speed,  10.8  knots  per  hour.  IIP,  370. 
Consumption  of  coal,  8.5  tons  in  24  hours.  Rig. — Schooner. 

“Isle  of  Dursey.” — Vertical  Direct  Engines  [Compound  Triple  Expansion). 
— Length  on  deck,  210  feet ; beam,  31.25  feet ; hold,  14. 1 f$et.  Tons,  620.963. 

Cylinders.  — 2,  each  15.75  and  22  ins.,  and  44.33  ins.  in  diam. ; stroke  of  piston, 
2.75  feet.  Condenser. — 466  .75  inch  tubes,  No.  18  B W G.  Surface , 792  sq.  feet. 
Propeller. — 4 blades,  12.5  feet  in  diam.  Pitch,  14.5  feet.  Surface,  38.5  sq.  feet. 
Pressure  of  Steam,  150  lbs.  per  sq.  inch.  Revolutions , 73  per  minute. 

Boiler. — 1 (horizontal  tubular).  Heating  surface,  1650  sq.  feet.  Grate,  42  sq.  feet. 
IIP  per  sq  foot  of  grate,  12.3 ; of  heating  surface,  .374.  Total,  500. 

Fuel.—  Bituminous,  1.5  lbs.  per  IIP  per  hour.  Rig.—  Fore-topsail  schooner. 

Iron  Fire-boat. 

“Zophar  Mills.” — Vertical  Direct  Engine. — Length  on  load-line,  n 5 feet ; 
beam,  molded , 24  feet;  hold  at  side,  8 feet  8 ins. 

Immersed  section  at  load-line  of  7 feet,  150  sq.feet. 

Cylinder.  — 30  ins.  in  diam.  by  30  ins.  stroke  of  piston;  volume  of  piston  space, 

12.25  cube  feet.  Condenser. — Surface , 900  sq.  feet. 

Boilers  (return  tubular). — Two,  8 feet  in  width  by  14  feet  in  length.  Heating  sur- 
face, 2120  sq.  feet.  Grates,  80  sq.  feet. 

Pressure. — 70  lbs.  per  sq.  inch,  cut  offat . 5 stroke.  Revolutions , 84  at  45  lbs.  press- 
ure, cut  off  at  .5.  Speed.— 12.5  miles  per  hour. 

Propeller. — 4-bladed,  8 feet  9 ins.  in  diam. 

Pumps,  Vertical  Duplex.  Steam  cylinders , 4.  — 16  ins.  in  diam.  by  9 ins.  stroke. 
Pumps , 4.  — 7.5  ins.  in  diam.  by  9 ins.  stroke.  Receiving  pipes , 8.5  ins.  in  diam. 
Revolutions , no  per  minute. 

Discharge. — 2200  gallons  per  minute;  or,  8 streams,  2.5  ins.  to  3.25  ins.  hose, 
average  75  feet  in  length  of  hose  each,  and  1.5  ins.  nozzles,  160  feet.  Or,  4 streams, 

3.25  ins.  hose,  100  feet  in  length  of  hose  each,  connected  to  one  length  of  16  feet  of 
4-inch  hose,  and  3.25  ins.  nozzle,  280  feet. 

Steel  Launch. 

Inverted  Direct  Engine  [Non- condensing).— Length,  25  feet ; beam , 5 feet;  hold , 

2. 5 feet. 

Cylinder.—  5 ins.  in  diam.  by  5 ins.  stroke  of  piston. 

Hull,—  Frame,  .75  x .75  inch,  No.  12  W G.  Keel,  stem  and  stern-post,  each, 

1.5  X 1.25  ins. 


888  MARINE  STEAM  VESSELS  AND  ENGINES. 


Steel  Yaclits.  (Propellers). 

“Lady  Torfrida.”— Vertical  Direct  Engines  (Compound).  — Length,  200  feet 
8 ins.;  beam , 25  feet  7 ins. ; hold , 15  feet  7 ins.  Tons,  611. 

Cylinders.— 3,  one  of  24  ins.  in  diam. , and  two  of  34  ins.  by  30  ins.  stroke  of  piston. 

Condenser. — Surface , 1978  sq.  feet.  Circulating  Pump,  double,  12  by  17  ins.  Air- 
pump,  single,  20  by  17  ins. 

Boilers  (return  tubular).— 14. 5 feet  in  diam.  by  9 feet  in  length.  Heating  Sur- 
face, 1887  sq.  feet.  Grate,  77  sq.  feet. 

Pressure  of  Steam,  no  lbs.  Vacuum,  28.5  ins.  IIP,  1020.  Propeller,  Manganese 
bronze,  n feet  in  diam.  Pitch,  14.5  feet.  Speed.— is  knots  per  hour. 

Iron. 

“ Isa.”  — Vertical  Direct  Engines  (Compound).  — Length  of  keel,  118.66  feet; 
beam , 18.75  feet ; hold,  10  feet.  Tons,  248. 

Cylinders,  3.— 10, 15,  and  28  ins.  in  diam.  by  2 feet  stroke  of  piston.  Condenser.— 
Surface,  350  sq.  feet.  Circulating  Pump,  6 ins.  in  diam.  by  12  ins.  stroke. 

Pressure  of  Steam.— 120  lbs.  per  sq.  inch  full  stroke.  Revolutions,  112  per  minute. 
Speed,  12  knots  per  hour. 

Propeller.— 2 blades,  8.5  feet  in  diam.  Pitch,  12.25  feet. 

Composite. 

“Radha.”  — Vertical  Direct  Engines  (Compound).  — Length  for  tonnage,  142 
feet ; beam,  20  feet ; depth  of  hold,  8 feet  8.5  ins.  Tons,  77.04  and  149.15.  ’ 

Immersed  section  at  load-draught  of  8.25  feet,  115  sq.feet. 

Cylinders , 3. — 1 of  20  ins.  in  diam.,  and  2 of  26  by  2 feet  stroke  of  piston. 

Condenser. — Surface , 800  sq.  feet. 

Boiler  (flue  and  return  tubular). — 9 feet  8 ins.  wide,  and  15  feet  in  length.  Heat- 
ing surface,  1947  sq.  feet.  Grate,  48  sq.  feet. 

Propeller , 7.5  feet  in  diam.  Pitch,  12  feet.  Revolutions,  135  per  minute. 

Pressure  of  Steam. — 100  lbs.,  cut  off  at  .5.  Blast  draught. 

“Siesta.” — Vertical  Direct  Engine  (Compound),  Herreshoff. — Length  on  deck 
over  all,  98  feet ; at  water-line,  90.3  feet;  beam  at  deck,  1 7 feet ; at  water-line , 15.16 
feet;  depth  of  hull  from  rabbet  of  keel  to  top  of  shear  plank,  8.33  feet ; draught  of 
water  at  load-line,  5.66  feet. 

Immersed  section  at  load-line,  43  sq.  feet.  Displacement  at  load-draught,  63.83  tons. 

Area  of  water  section,  878.7  sq.feet,  and  of  immersed  surface  of  hull,  1438  sq.feet. 

Ratio  of  water  surface  to  its  circumscribing  parallelogram,  .64  ; of  immersed  trans- 
verse section  to  its  do.  do. , . 584  ; and  of  displacement  of  immersed  hull  above  lower 
edge  of  rabbet  of  keel  to  its  circumscribing  parallelopiped,  .5677. 

Cylinders,  2.— 10.5  and  18  ins.  in  diam.  by  18  ins.  stroke  of  piston.  Volume  of 
piston  space,  3.45  cube  feet.  Relative  volumes  of  displacement  of  cylinders,  1 to 
2.96.  Air-pump , single  acting,  6 ins.  in  diam.  by  6.25  ins.  stroke. 

Circulating  and  Feed  Pumps,  single  acting,  1.125  ins.  in  diam.  by  18  ins.  stroke. 

Condenser,  External.— Surface,  731,  5 ins.  by  29.5  ins.  tubes;  condensing  surface, 
235  sq.  feet. 

Propeller,  4 blades,  4 feet  7 ins.  in  diam.  Pitch , 8 feet.  Helicoidal  area  of  blades, 
9.46  sq.  feet.  Transverse  area,  6.59  sq.  feet. 

Shaft. — Journal,  3.875  X 8 ins. ; stress,  3.75  ins.  Engine  space , 3 feet  by  5.5  feet 
in  length. 

Boiler  (vertical  double  coil). — Diam.  outside  of  casing,  6.66  feet;  height,  8 feet 
10  ins.  Heating  surface,  558  sq.  feet.  Grates , 26  sq.  feet. 

Smoke-pipe,  23.5  ins.  in  diam.  by  25  feet  above  grates.  Steam  room,  5.7  cube  feet. 

Heating  surface  to  Grate,  21.5  to  1. 

Pressure  of  Steam.  60.7  lbs.  per  sq.  inch,  cut  off  in  small  cylinder  at  .88  of  stroke, 
and  in  large  at  .3  stroke.  In  small  cylinder  at  end  of  stroke,  55.2  lbs. ; in  large 
cylinder  at  commencement  of  stroke,  50.6;  and  at  end  of  stroke,  15.6  lbs.  Mean 
back  pressure  in  small  cylinder,  47.36  lbs. ; and  in  large,  5.77. 

Revolutions , 193  per  minute.  Speed',  12.75  miles  (11.06  knots)  per  hour.  Slip  of 
Propeller,  27.3  per  cent. 


MARINE  STEAM  VESSELS  AND  ENGINES.  889 

Herreshoff.— Vertical  Direct  Engine  {Compound).— Length  on  deck,  100 feet; 
beam,  12.5  feet. 

Cylinder.— 1 of  12.5  and  21.5  ins.  in  diam.  by  16  ins.  stroke  of  piston. 

Pressure  of  steam , 120  lbs.  per  sq.  inch.  Revolutions,  480  per  minute.  Speed , 
22.5  knots  per  hour. 

Thrust  of  Propeller  at  15.73  knots,  4080  lbs. 

Torpedo  Boats.  (^Propellers.) 

Iron. 

Vertical  Direct  Engine  (Compound).— Length,  no  feet ; beam,  12.5  feet. 

Displacement , 52  tons. 

Cylinders,  1.— 12.5  ins.  and  21.5  ins. ; stroke  of  piston,  16  ins. 

Boiler. — (Horizontal  tubular.)  Diam.,  4.75  feet.  Tubes,  125  of  2 ins.  in  diam. 
Heating  surface,  1016  sq.  feet.  Speed,  20.3  knots  per  hour. 

Steel.  Composite. 

“Torpedo  Boat,”  R.  N.  — Vertical  Direct  Engine  (Compound),  Herreshoff 
Mfg.  Co.— Length,  59.5  feet;  beam,  ;.^feet. 

Cylinders. — 6 and  10.5  ins.  in  diam.  by  10  ins.  stroke  of  piston. 

Condenser , External.— Surface.  Boiler  (vertical  coil).— Tubes  2 ins.  in  diam.  and 
300  feet  in  length. 

Propeller.  — 4 blades,  38  ins.  in  diam.  by  5 feet  pitch. 

Weight  at  load-draught  of  hull  of  1.5  feet;  armament  and  stores,  8 tons. 

Iron.  Side  Wheels. 

“Princess  Marie  and  Elizabeth. ’’—Oscillating  Engine  (Compound).— Length 
on  load-line,  274.8  ins.;  beam,  34.75  feet;  hold,  24.25  feet.  Tons,  1606. 

Cylinders,  2. — 60  and  104  ins.  in  diam.  by  3.5  feet  stroke  of  piston. 

Pressure  of  Steam.—  70  lbs.  per  sq.  inch,  cut  off  at  .6  stroke.  Revolutions,  32.75 
per  minute.  Speed,  17. 12  knots  per  hour.  IIP,  3543. 

Cjnsumption  of  fuel,  1.92  lbs.  per  IIP  per  hour.  Cost,  £54900  sterling. 

Cutter  (Corrugated). 

“La  Bonita.” — Inclined  Engine  (Non-condensing).— Length  upon  deck,  42  feet ; 
beam,  9 feet ; hold , 3 feet. 

Immersed  section  at  load-line , 8.75  sq.  feet.  Displacement  at  load-draught  of  1.3 
feet,  8386  lbs.  Tons,  9.65,  0.  M. 

Cylinder. — 8 ins.  in  diam.;  stroke  of  piston,  1 foot;  volume  of  piston  space,  .35  cube 
foot.  Water-wheels. — Diam.  5.66  feet.  Blades,  7 ; breadth,  2.3  feet;  depth,  7 ins. 

Boiler.— (Horizontal  tubular).  Heating  surface , 95  sq.  feet.  Grates,  6 do.  Fuel , 
coal  or  wood.  Exhaust  draught. 

Pressure  of  steam. — 65  lbs.  per  sq.  inch.  Revolutions,  54  per  minute.  IIP,  9. 

Hull.—  Corrugated  and  galvanized  plates,  .0625  inch  thick.  Weights.—  Hull,  2876 
lbs. ; Engine  and  wheels,  2400  lbs. ; Boiler,  2260  lbs. ; pipes,  grates,  etc.,  750  lbs. 

Steel. 

Ferry  Boat.— Inclined  Engine  (Surface  Condensing). — Length,  78  feet;  beam, 
15  feet;  hold,  8 feet  amidships  and  5 feet  at  ends. 

Load-draught,  2. 25  feet. 

Cylinder , 15.5  ins.  by  24  ins.  stroke  of  piston.  Boiler  (cylindrical  tubular).— Steel; 
heating  surface,  220  sq.  feet. 

Pressure  of  Steam,  80  lbs.  per  sq.  inch.  Plates,  .125,  .1875,  and  .25  ins. 

Light  Draught. 

“Ho-nam.” — Vertical  Beam  Engine  (Compound). — Length  upon  deck,  280  feet; 
beam,  Tsfeet ; depth  from  hold  to  upper  deck,  30  feet.  Tons,  2364. 

Cylinders , 1 — 40  and  72  ins.  in  diam. ; stroke  of  piston,  10  feet.  IIP,  3000. 

Speed , 16.14  knots  per  hour.  Passengers,  2000. 

Decks , 3:  Main,  Saloon,  and  Promenade.  Rig.—  Schooner. 

F 


89O  MARINE  STEAM  VESSELS  AND  ENGINES, 


Wood  Side  "Wheels.  Passenger  and  Deck  Cargo. 

“City  of  Fall  River,”  New  York  to  Fall  River,  Mass.,  Vertical  Beam  En- 
gine (Compound). — From  notes  of  James  E.  Sague  and  John  B.  Adger , Jr.  Length 
on  water-line , 260  feet;  over  all , 273  feet;  beam,  42  feet;  over  guards , 73  feet;  hold 

18  feet ; 1723  tons  N.  M. 

Immersed  section  at  load-line  of  9 feet  3 ins.  (1750  tons),  365  sq.feet;  and  at  load- 
draught  of  12  feet,  480  sq.feet.  Displacement  at  load-draught , 2350  tons. 

Cylinders , 2. — 1 of  44  ins.  in  diam.  by  8 feet  stroke  of  piston,  and  1 of  68  ins.  in 
diam.  by  12  feet  stroke.  Clearance  at  each  end  of  high  pressure,  4.6  per  cent,  and 
at  low  pressure,  3 per  cent.  Volumes,  85  and  303  cube  feet. 

Receiver,  89.13  cube  feet.  Air-pump , 37  ins.  in  diam.  by  4.75  feet  stroke  of  pis- 
ton. Condenser. — Surface,  4067  sq.  feet. 

Water-wheels , 25.5  feet  in  diam.  Blades , feathering,  12  of  40  ins.  in  depth  by  10 
feet  in  length.  Centre  of  Pressure  on  Blades,  n.  22  feet  from  axis  of  shaft. 

Boilers.—  2 (flue  and  return  tubular),  17.5  feet  in  width  by  15  feet  in  length,  220 
tubes  3.5  ins.  in  diam.  and  12  feet  in  length.  Grates,  230  sq.  feet. 

Fuel.  —Anthracite.  Natural  Draught.  Consumption,  1463  lbs.  per  hour;  refuse, 
281  lbs.  = 19-23  per  cent,  per  sq.  foot  of  grate,  and  12.73  lbs. 

Pressure  of  Steam.—  High-pressure  cylinder,  throttle  open  and  cut  off  at  .445, 
mean  in  boiler  per  guage,  70  lbs. ; in  receiver,  n lbs. ; mean  effective  pressure,  41.8 
lbs.  per  sq.  inch. 

Low-pressure  cylinder,  at  point  of  cutting  off  of  .45,  17.42  lbs.  above  zero;  mean 
effective  pressure,  12.4  lbs.  per  sq.  inch.  Expansion  of  steam,  6.99  times.  Vacuum, 

28.4  ins. 

IP.— High-pressure  cylinder,  783;  low-pressure,  840. 

Revolutions,  25.8  per  minute. 

Feed  Water , 27  854  lbs.  per  hour;  per  IEP,  17.17  lbs.  Temperatures. — Feed  water, 
970;  sea  water,  49. 40;  water  of  condensation,  900;  heat  units  per  hour  per  IIP, 

19  090. 

Stress  of  wheels,  20.4  per  cent. 

Condensing  water,  per  IIP  per  hour,  407  lbs. 

Consumption.—  Compound  engine,  2.03  lbs.  per  IIP;  and  as  a simple  condensing 
engine,  without  high-pressure  cylinder,  2.84  lbs. 

Evaporation  per  hour , 1208  lbs.  water;  per  lb.  of  combustible  from  2120,  11.75 
lbs.;  from  temperature  of  feed  (970),  10.22  lbs. ; from  feed  per  lb.  of  coal,  826  lbs. 
Temperature  of  gases  in  chimney  485°. 

Heating  Surface , 29  sq.  feet  to  1 of  grate. 

Weights,  Engine,  and  Frame,  250  tons;  boilers  complete,  120  tons;  water,  50  tons. 
Speed,  14.14  knots  per  hour;  and  IIP,  1623. 

Draft  of  water , 10.65  feet;  Displacement,  1938  tons. 

“City  of  Boston,”  New  York  and  Norwich. —Vertical  Beam  Engine  (Con- 
densing).— Length  upon  load-line,  320  feet ; beam,  39  feet ; hold,  12.6  feet. 

Immersed  section  at  load-line,  288  sq.feet.  Displacement  1450  tons,  at  load-draught 
of  8. 25  feet. 

Cylinder. — 80  ins.  in  diam.  by  12  feet  stroke  of  piston.  Volume,  419  cube  feet. 
Water-wheels.  — Diam.  37  feet  8 ins.  Arms , 36.  Blades,  37;  breadth  of  do.,  10 
feet;  depth  of  do.,  30.5  ins.  Dip  at  load-line,  4.25  feet. 

Boilers.— 2 (flue  and  return  tubular),  Shells,  12.5  feet  in  diam.,  and  in  length 

26.5  feet.  Heating  Surface,  10 120  sq.  feet.  Grates,  184  sq.  feet. 

Pressure  of  Steam. — 35  lbs.  per  sq.  inch,  cut  off  at  .5  stroke.  Revolutions  (maxi- 
mum), 19.75  per  minute.  IIP,  2500. 

Fuel. — Anthracite;  Blast.  Consumption,  at  ordinary  speed,  5200  lbs.  per  hour. 
Weights  of  Engine,  Boilers,  etc.,  263  tons. 

Hull.—^  Weight,  800  tons.  Light  draught  of  hull  without  fuel,  water,  or  furni- 
ture, 7 feet. 


MARINE  STEAM  VESSELS  AND  ENGINES.  89 1 


Wood  Propellers. 

Herreshoff,  R.  N.,  Vertical  Direct  Engine  [Compound).— Length  on  deck , 46 
feet ; over  all,  48  feet;  beam,  gfeet;  hold , 5 feet. 

Displacement  at  load-line,  7-44  tons.  Area  of  section  at  load-lime , 217.8  sq.feet. 
Area  of  wetted  surface,  365. 5 sq.  feet.  Coefficient  of  fineness, . 396. 

Cylinder.—  8 and  14  ins.  in  diam.  by  9 ins.  stroke  of  piston. 

Condenser,  External. — Surface. 

Propeller.— 4 blades,  3 feet  in  diam.  by  4 feet  1 inch  pitch. 

Blower,  42  ins.  in  diam. 

Boiler  (vertical  coil).  Heating  surface , 174  sq.  feet.  Grates , 12.  5 sq.  feet. 

Pressure  of  Steam,  53  lbs.  per  sq.  inch.  Revolutions,  333  per  minute.  IIP,  68.4. 
Speed  10  18  knots  per  hour.  With  129  lbs.  and  466  revolutions,  14.26  knots.  IIP, 
169.5.’  Weight  of  Engines,  Boiler , and  Water,  5300  lbs. 

Herreshoff,  Vertical  Direct  Engine  [Compound).  — Length  over  all , 8 6 feet; 
beam , n feet.  Displacement,  27  tons. 

Cylinder.— 13  and  22  ins.  in  diam.  by  12  ins.  stroke  of  piston. 

Surface  Condensing. 

Pressure , 130  lbs.  per  sq.  inch. 

Revolutions,  460  per  minute.  Speed,  20  knots  per  hour.  IIP,  425. 

Propeller , 3 blades.  Pitch,  5 feet. 

Herreshoff  R I N.— Vertical  Direct  Engine  [Compound).— Length  over  all, 
60  feet ; beam , 7 feet ; hold , 5. 5 feet.  Displacement  at  load-draught  of  32  ms. , 7 tows 
(2240  lbs. ). 

Cylinders.—  8 and  14  ins.  in  diam.  by  9 ins.  stroke  of  piston.  Surface  condenser. 
Pressure  of  Steam.— 140  lbs.  per  sq.  inch,  cut  off  at  .5. 

Revolutions , 600  per  minute.  Speed,  19.875  knots  per  hour. 

Cable  or  Rope  Towing. 

“ Nyitra.  Horizontal  Direct  Engines  [Condensing).— Length  of  boat,  xf&feet; 
beam,  24. 5 feet ; hold,  7. 5 feet. 

Immersed  section,  74.4  sq.feet.  Displacement,  200  tons  at  load-line  of  3.7  5 feet. 
Immersed  section , 263.7  sq.  feet.  Displacement,  949  tons.  Tow.  3 barges. 

Cylinders. — 2 of  14.18  ins.  in  diam.  by  23.625  ins.  stroke  of  piston. 

IIP,  net  effective , 100.  Speed,  7.73  miles  per  hour. 

Propellers.  — Twin,  4 feet  2 ins.  in  diam. 

Stress. — Cable,  7485  lbs.  Per  ton  of  displacement,  6.5  lbs. ; per  sq.  foot  of  im- 
mersed section,  22  lbs. 

Fuel.  — Per  mile  and  ton  of  displacement  (1149),  .078  lbs. 


Towing.  Wood  Side  VWlieels. 
uWm  h.  Webb.”— Harbor  and  Coast.— Vertical  Beam  Engines  [Condensing). 
—Length  upon  deck,  185.5  feet;  beam,  30. 25  feet;  hold,  10.8  feet. 

Immersed  Section  at  load-line,  194  sq.feet.  Displacement  498.25  tons,  at  load- 
draught  of  7. 25  feet. 

Cylinders.— 2,  of  44  ins.  in  diam.  by  10  feet  stroke  of  piston ; volume,  21 1 cube  feet. 
Condensers.— Jet,  2,  volume  105  cube  feet.  Air-pumps.—  2,  volume  45  cube  feet. 

Water-wheels. — Diam.,  30  feet.  Blades  (divided),  21;  breadth  of  do.,  4.6  feet; 
depth  of  do.,  2.33  feet.  Dip  at  load-line,  3.75  feet. 

Boilers.  — 2 (return  flue).  Heating  surface,  3280  sq.  feet.  Grates,  147.5  sq.  feet. 
Smolce-pipe.— Area,  11.6  sq.  feet,  and  35  feet  in  height  above  the  grate  level. 
Pressure  of  Steam.— 35  lbs.  per  sq.  inch,  cut  off  at  .5  stroke.  Revolutions,  22  pel 
minute.  IIP,  1500. 

Fuel. — Anthracite  or  Bituminous.  Consumption,  1680  lbs.  per  hour. 

Speed.— 20  miles  per  hour. 

Weights.  — Engines,  Wheels,  Frame,  and  Boilers,  310579  lbs. 


892 


RIVER  STEAMBOATS  AND  ENGINES. 


Wood  Side  ‘W'h.eels.  Passenger. 

“Daniel  Drew,”  New  York  to  Albany.— Vertical  Beam  Engine  ( Condensing ). 
— Length  upon  deck , 251.66  feet ; at  load-line , 244  feet ; beam , 31  feet ; hold , 9. 25  feet. 

Immersed  section  at  load-line , 136  sq.feet.  Displacement  380  tons , load-draught 
of  4.83  feet. 

Cylinder.  — 60  ins.  in  diam.  by  10  feet  stroke  of  piston;  volume , 196  cube  feet. 
Condenser. — «/e£,  volume  68  cube  feet.  Air-pump,  volume  26  cube  feet. 

Water-wheels.  — Diam.  29  feet.  Arms , 24.  Blades , 24;  breadth  of  do.,  9 feet; 
depth  of  do.,  26  ins.  Dip  at  load-line,  2.33  feet. 

Boilers. — 2 (return  flue),  29  feet  in  length  by  9 feet  in  width  at  furnace.  Shell, 
diam.  8 feet.  Heating  surface , 3350  sq.  feet.  Grates,  105  sq.  feet.  Cross  area  of 
lower  flues,  15.5  sq.  feet;  of  upper,  13  sq.  feet.  Weight , 80650  lbs. 

Smoke-pipes. — 2,  area  25.13  sq.  feet,  and  32  feet  in  height  above  the  grate  level. 

Pressure  of  Steam. — 35  lbs.  per  sq.  inch,  cut  off  at  .5  stroke.  Revolutions  ( maxi- 
mum),  26  per  minute.  IIP,  1720. 

Fuel. — Anthracite;  Blast.  Consumption , 3800  lbs.  per  hour. 

Speed , 22.3  miles  per  hour.  Slip  of  Wheels  from  Centre  of  Pressure,  12.5  per  cent. 

Frames. — Molded,  15.75  ins. ; sided,  4 ins. ; and  20  ins.  apart  at  centres. 

“ Mary  Powell,”  Hudson  River.— Vertical  Beam  Engine  {Condensing).— Length 
on  water-line,  286  feet ; over  all , 294  feet ; beam , 34  feet  3 ins.  ; over  all,  64  feet ; hold , 

9 feet.  Deck  to  promenade  deck,  10  feet. 

Immersed  section  at  load  - line  of  6 feet , 200  sq.  feet.  Displacement , 800  tons  at 
mean  load-draught  of  6 feet. 

Area  of  transverse  head  surface  of  hull  above  water,  2000  sq.feet. 

Cylinder. — 72  ins.  in  diam.  by  12  feet  stroke  of  piston;  volume,  338  cube  feet. 
Clearance  at  each  end,  12.5  cube  feet. 

Steam  and  Exhaust  Valves , 14.75  ins.  in  diam.  Air-pump,  40  ins.  in  diam.  by  5 
feet  2 ins.  stroke  of  piston.  Condenser. — Jet , 128  cube  feet.  Crank-pin , 8.75  ins.  in 
diam.  x 10.75  ins. 

Beam,  22.5  feet  in  length;  centre,  9.75  in  diam. 

Water-wheels — Diam.  31  feet;  blades  (divided),  26;  breadth  of  do.,  10  feet  6 ins. ; 
width,  1 foot  6 ins. ; immersion,  3 feet  6 ins.  Shafts.—  Journal,  15.625  ins.  by  17  ins.’ 

Boilers. — 2 (flue  and  return  tubular),  of  steel,  n feet  front  by  26  feet  in  length; 
shell,  10  feet  in  diam.  and  16  feet  1 inch  in  length.  Furnaces,  2 in  each,  of  4 feet 

10  ins.  by  8 feet  in  length.  Heating  Surface,  2660  sq.  feet;  and  Superheating,  340 
sq.  feet  in  each.  Grates , 152  sq.  feet.  Flues,  10  in  each,  transverse  area,  n feet 
7 ins.  Tubes,  80  in  each,  4.5  ins.  in  diam.,  6 feet  6 ins.  in  length,  and  8 feet  7 ins. 
in  transverse  area. 

Steam  Chimneys . 8 feet  in  diam.  x 12  feet  in  height.  Smoke-pipe , 4 feet  6 ins.  in 
diam.  and  68  feet  in  height  from  grates. 

Combustion , Blast.  Blowers , 4 feet  in  diam.  and  3 feet  in  width.  Revolutions,  78 
per  minute.  Fuel  (anthracite),  6280  lbs.  per  hour,  or  40  lbs.  per  sq.  foot  of  grate 
per  hour.  Per  sq.  foot  of  heating  surface,  2.25  lbs. 

Speed , 23.65  miles  per  hour. 

Pressure  of  Steam,  28  lbs.  per  sq.  inch,  cut  ofT  at  .47  stroke;  terminal  pressure, 
16.4  lbs. ; throttle,  .625  open.  Vacuum,  25  ins.  Revolutions , 22.75  per  minute. 

Temperatures.— Reservoir,  120°.  Feed  water,  1200.  Chimney,  7400.  IP.— Total, 
1900.  IIP,  1560.  Net,  1450. 

Evaporation.— Water  per  lb.  of  coal,  from  1200,  7 lbs. ; per  lb.  of  combustible, 
from  1200,  8. 2 lbs.  Steam  per  total  IP  per  hour,  21. 1 lbs.  Coal  per  do.  do.,  3. 14  lbs. 

Weights.  Engine.  — Frame,  keelson,  out -board  wheel -frames  donkey  engine, 
and  boiler,  blower  engines  and  blowers,  all  complete,  360000  lbs.  Boilers.—  Iron 
return  flue.  120000  lbs.  Steel  return  tubular,  116000  lbs.  Water,  128000  lbs. 

Capacity.—  2000  passengers  and  their  baggage. 

Memoranda.  — Th i s vessel  was  originally  but  266  feet  in  length,  and  when  length- 
ened the  cylinder  of  62  ins.  in  diam.  was  removed  and  replaced  with  one  of  72  ins. 
Engine  designed  throughout  for  original  cylinder  and  a pressure  of  from  50  to  55 
lbs.,  cutting  olf  at  .625  of  stroke,  with  throttle  wide  open. 

Engines  and  Boilers  built  by  Fletcher,  Harrison,  & Co.,  New  York,  1861  and  1875. 


RIVER  STEAMBOATS  AND  ENGINES. 


893 

‘‘Solano  ” Ferry  Boat.— Vertical  Beam  Engines  [Condensing).— Length  over 
all , 424  feet;  on  keel,  406  feet;  beam  [molded),  64  feet;  hold  at  18.5  feet;  at  ends, 

15  feet  10  ins.  ; width  over  guards,  116  feet. 

Light  draught,  5 feet;  loaded,  6.5  feet.  Tons , 3541. 

Cylinders.—  2 of  60  ins.  diam.  by  11  feet  stroke  of  piston. 

Wheels , 34  feet  in  diam.  by  17  feet  face.  Blades,  24. 

Boilers,  8.— Steel;  7 feet  in  diam.  by  28  feet  in  length.  Heating  surface,  19640 
sq.  feet.  ’ Grates,  288  sq.  feet.  IIP,  4000. 

Passenger  and.  Liglit  IPreiglrt. 

“Seth  Grosvenor.”— Steeple  Engine  [Condensing).— Length  upon  deck,  9 $feet ; 
beam , 17.2  feet;  hold,  5 feet. 

Immersed  section  at  load-line , 43  sq.feet.  Displacement  73  tons,  at  load-draught 

of  3-25  fat. 

Cylinder.— 28  ins.  in  diam.  by  3 feet  stroke  of  piston;  volume,  12.8  cube  feet. 
Water-wheels.  — Diam.  13.5  feet.  Blades,  14;  breadth  of  do.,  3 feet;  depth  of  do., 

1.25  feet. 

Boiler  (flue  and  return  tubular). — Heating  surface,  540  sq.  feet.  Grates,  22.5  sq. 
feet.  Area  of  tubes,  367  sq.  ins.  IIP,  90. 

Weights.—  Engine,  Wheels,  Frame,  and  Boiler,  61  556  lbs.  = 27-4  tons. 

The  operation  of  this  vessel  was  in  every  way  successful,  being:  very  fast,  economical  in  fuel,  etc., 
and  she  would  have  been  improved  if  the  hull  had  had  15  feet  additional  length,  all  other  dimensions 
and  capacities  remaining  the  same. 

"Wood.  Stern  Wlieels. 

Passenger  and  Deck  Dreiglit. 

“Montana.” — Horizontal  Engines  [Non-condensing). — Length  upon  deck  [over 
all),  248  feet;  at  water-line , 245  feet;  beam,  48  feet  8 ins.  [over  all,  50  feet  4 ins.); 
hold,  6 feet;  draught  of  water  at  load-line , 5.5  feet. 

Immersed  section  at  load-line,  244  sq.  feet.  Displacement  at  mean  light  draught 
of  22  ins. , 594  tons  (2000  lbs. ) 

Cylinders.— Two,  18  ins.  in  diam.  by  7 feet  stroke  of  piston. 

Valves,  4.5  and  5 ins.  in  diam.  Piston-rod , 4 ins.  Steam-pipe,  4.5  ins.  Connect- 
ing-rod, 30  feet  in  length. 

Water-wheel , 19  feet  in  diam.  by  35  feet  face;  blades,  3 feet  in  depth.  Shaft, 

10.25  ins.  in  diam. 

Boilers. Four  (horizontal  tubular),  42  ins.  in  diam.  by  26  feet  in  length.  Two 

flues  in  each,  15  ins.  in  diam.  Heating  surface,  0 ffective,  1023,  total  1431  sq.  feet. 
Furnace,  6.5  X 17  feet.  Grates , 4.16  X 17  feet;  surface,  70.8  sq.  feet.  Smoke-pipes. 

Two,  3 feet  in  diam.  by  55  feet  3 ins.  in  height.  Exhaust  or  Blower  draught. 

Calorimeter.  — Of  Bridge,  15.27;  of  Flues,  9.82;  and  of  Chimneys,  14.14  sq.  feet. 
Areas  of  grate,  compared  to  calorimeter  of  flues,  7.2;  to  ditto,  of  chimneys,  5;  and 
of  bridge,  4.6  sq.  feet. 

Steam-room,  562 ; and  water  space,  294  cube  feet. 

Hull.— Frames,  4X6  ins.  and  15  ins.  apart  at  centres.  Intermediate  do.,  4X6 
ins.,  and  running  for  7.5  feet  each  side  of  keelson.  Planking. — Bottom,  oak,  4 ins. ; 
side  do.,  2.5  to  4 ins.  Deck  beams,  pine,  3X6  ins.  Deck  plank,  2.5  ins.  Keelson, 
oak;  side  do.,  eight  each  side,  one  each  7,  8.75,  and  9 ins.,  and  five  6.75  ins.  Wales, 
one  each  side,  9 and  7 ins.  by  3,  and  one  10  X 2.5  ins.  Deck  posts,  3.5  X 3 ins.  ar*d  4 
feet  apart.  Deck  beams,  5. 5 X 3 ins.  Knuckles , oak,  6 X 12  ins.  Bulkheads,  one 
longitudinal  and  one  athwartship  at  shear  of  stern.  Sheathing  of  wrought  iron, 
.0625  to  .125  inch  from  just  below  light  to  load-line. 

Hog  Posts.— White  pine,  8.5  and  u ins.  square.  Chains , 1.5  ins.  in  diam. 

Weights. — Boilers,  29  264;  water,  18  351 ; and  boilers,  chimneys,  grates,  and  water, 
55672  lbs.  Hull,  oak,  520560;  Pine,  91  437;  Bolts,  spikes,  etc.,  8000,  and  Deck  and 
guards,  76000  lbs. ; Hull  alone,  310  tons. 

Weight  of  hull  compared  to  one  of  iron  as  8 to  5,  effecting  a difference  of  about 
100  tons. 


894 


RIVER  STEAMBOATS. SAILING  VESSELS. 


Passenger  and.  Peck  PreigLt. 

“Pittsburgh.” — Horizontal  Engines  {Non- condensing).— Length  on  deck , 252 
feet;  beam,  39  feet ; hold,  6 feet;  draught  of  water  at  load-line,  2 feet. 

Immersed  section  at  load-line,  75  sq.feet.  Displacement  at  load-draught  of  2 feet, 
380  tons  (2000  lbs.). 

Cylinders.— Two,  21  ins.  in  diam.  by  7 feet  stroke  of  piston. 

Water-wheel.— 21  feet  in  diam.  by  28  feet  face. 

Boilers.—  2 (horizontal  tubular),  47  ins.  in  diam.  by  28  feet  in  length.  Two  fires 
in  each. 

Iron.  Stern  Wheels. 

Horizontal  Engines  ( Non-condensing ).  — Length  upon  deck , no  feet ; beam,  14 
feet  ( deck  projecting  over,  4 feet ) ; hold,  3.5  feet. 

Immersed  section  at  load-line , 10.25  sq.feet.  Displacement  at  load-draught  of  1.1 
feet,  33  tons. 

Cylinders.— Two,  of  10  ins.  in  diam.  by  3 feet  stroke  of  piston;  volume  of  piston 
space,  1.6  cube  feet. 

Wheel. — Diam.  13  feet.  Blades , 13;  breadth  of  do.,  8.5  feet;  depth  of  do.,  8 ins. 

Revolutions,  33  per  minute.  Boiler.— One  (horizontal  tubular).  Tubes,  100  of  2 
ins.  in  diam. 

Fuel. — Bituminous  coal.  Consumption , 4480  lbs.  in  24  hours. 

Hull.— Plates,  keel,  No.  3;  bilges,  No.  4;  bottom,  No.  5;  sides,  Nos.  6 and  7. 
Frames , 2.5  X - 5 ins.,  and  20  ins.  apart  from  centres. 

Steel. 

“Chattahoochee.”— Inclined  Engines  {Non- condensing).— Length  on  deck , 157 
feet;  beam,  31. 5 feet;  hold,  5 feet. 

Immersed  section  at  load-line,  153  sq.feet.  Freight  capacity , 400  tons  (2000  lbs.). 

Cylinders.— Two,  15  ins.  in  diam.  by  5 feet  stroke;  volume  of  piston  space,  12.26 
cube  feet. 

Wheel.—  One,  18  feet  in  diam. ; blades,  2 feet  in  depth. 

Boilers. — Three  (cylindrical  flued).  Diam.  42  ins. ; length,  22  feet;  2 flues  of  10 
ins.  in  each.  Heating  surface,  690  sq.  feet.  Grates , 48  sq.  feet. 

Pressure  of  Steam,  160  lbs.  per  sq.  inch,  cut  off  at  .375.  Revolutions,  22  per  min. 

Consumption  of  Fuel,  12  tons  (2000  lbs.)  in  24  hours.  Plating  of  Hull , .1875  to 
.25  inch.  Light  draught , 21  ins. 

Iron  Propellers. 

Vertical  Direct  Engines  {Non-condensing). — Length  on  deck , 70 feet;  beam,  10.5 
feet ; draught,  12  ins. 

Propellers,  2.-2  blades,  16.  ins.  in  diam.,  set  n ins.  below  water-line. 

Boiler  (tubular  coil).  Revolutions , 480  per  minute. 

Speed,  10.49  miles  Per  hour. 

Water  led  to  propellers  through  tunnels  in  bottom  at  sides. 

“Louise.”— Vertical  Tandem  Engines  {Compound).— Length,  60  feet;  beam , 12 
feet ; hold,  4.25  feet. 

Displacement  at  load-draught  of  2. 5 feet , 8 tons. 

Cylinders , 5 and  10  ins.  in  diam.  by  8 ins.  stroke  of  piston. 

Surface  Condenser.— Boiler  (vertical  tubular),  4 feet  in  diam.  by  8.5  in  length. 

Iron  Sailing  "Vessels. 

Passenger  and  Preiglxt. 

English. — Ship. — Length  upon  deck , 178  feet ; do.  at  mean  load-line  of  19. 16  feet. ,177 
feet ; keel,  171  feet;  beam,  32.88  feet ; depth  of  hold,  21.75  feet ; keel  (mean),  i.'j^feet. 

Immersed  section  at  load-line,  387  sq.  feet.  Displacement  at  load-draught  of  19. 16 
feet,  1385  tons;  at  deep  load-draught  of  20  feet,  1495  tons;  and,  in  proportion  to  its 
circumscribing  parallelopipedon, . 524. 

Load-line.— Area  at  load-draught,  4557  sq.  feet.  Angle  of  entrance  57°;  of  clear- 
ance, 64°.  Area  in  proportion  to  its  eircumscrib  ng  parallelogram,  .784. 


YACHTS. CUTTERS. PILOT  BOAT. 


895 

rprttre  of  Gravity , 6.416  feet  below  mean  load-line.  Centre  of  Displacement  (grav- 
ity of  b 6-  25  feet  below  load-line;  and  4.33  feet  before  middle  of  length  of  load-line. 
Immersed  Surface.— Bottom,  7370  sq.  feet.  Keel , 1130  sq.  feet.  Sails , 13  282  sq.feet. 
Meta-centre , 6.66  feet  above  centre  of  gravity  of  displacement  Centre  of  Effort 
before  centre  of  displacement,  3. 5 feet ; height  of  do.  above  mean  load-line,  55. 5 ^et. 

Launch..  'Wood. 

Steam  Launch  “ Herreshoff. ’’—Vertical  Engine  {Compound).— Length,  33  feet 
1 inch  ; beam,  8.7  5 feet. 

Displacement  at  mean  load-draught  of  {to  rabbet  of  keel)  19  ins.,  8929  lbs. 
Weights.—  Hull  and  Machinery,  6555  lbs.  Coal,  1120  lbs. 

Yachts.  Wood. 

“America  ” Schooner.—  Length  over  all,  98  feet ; upon  deck,  94  feet;  at  load-line, 
qo5 feet;  beam,  22.5  feet ; at  load-line,  22  feet ; depth  of  hold  9. 25 feet.  Height  at 
side  from  under  side  of  garboard  sir  alee,  1 1 feet.  Sheer , forward,  3 feet ; aft,  1 . 5 feet, 
Tmmersed  section  at  load-line , 121.8  sq.  feet.  Displacement  at  load-draught  of  8. 5 
feet  from  under  side  of  garboard  strake  and  of  11  feet  aft,  191  tons;  and,  m pro- 
portion to  Volume  of  circumscribing  par allelopipedon,  .375. 

Displacement  at  4 feet  {from  garboard  strake ),  43  tons  ; at  5 feet,  66  tons ; at  6 
feet,  93  tons;  at  7 feet,  127  tons;  and  at  8 feet , 167  tons. 

Centre  of  Gravity.— Longitudinally,  1.75  feet  aft  of  centre  of  length  upon  load- 
line  Sectional,  2.58  feet  below  load-line.  Of  Fore  body,  14.25  feet  forward;  and 
of  After  body,  19  feet  aft.  Meta-centre , 6.72  feet  above  centre  of  gravity. 

Centre  of  Effort , 31  17  feet  from  load-line.  Centre  of  Lateral  Resistance , 6.33  feet 
abaft  of  centre  of  gravity.  Area  of  Load-line , 1280  sq.  feet.  Mean  girths  of  im- 
mersed section  to  load-line,  25  feet. 

Load-draught.—  Forward,  4.91  feet;  aft,  11.5  feet.  Rake  of  Stem,  17  feet. 

Soars  — Mainmast , 81  feet  in  length  by  22  ins.  in  diam.  Foremast , 79.5  feet  in 
lenath  by  24  ins.  in  diam.  Main  boom , 58  feet  in  length.  Main  gaff , 28  feet.  Fore 
gaff ; 24  feet.  Rake,  2.7  ins.  per  foot.  Drag  of  Keel,  3 feet.  Tons,  170.56. 

“Julia,”  Sloop  .—Length  for  tonnage , 72.25  feet;  on  water-line,  qo  feet  7 ins.; 
beam,  19  feet  8 ins.;  hold,  6 feet  8 ins.  Tons,  0.  M.  83.4;  N.  M.  43.98. 

Load-draught,  6.25  feet. 

Sails.— Mainsail,  hoist,  49.75  feet,  foot  54.25,  and  gaff  27.66;  Jib , hoist,  49.75  feet, 
foot  39. 5,  and  stay  63. 5.  Gaff  topsail,  hoist,  24. 5 feet. 

Areas. — Mainsail,  2322  sq.  feet.  Jib,  986,  and  Topsail,  454. 

Cutters. 

“Tara”  {English)  Sloop.—  Length  on  load-line,  66  feet;  beam,  11.5  feet. 

Immersed  section  at  load-line,  11.5  sq.feet.  Displacement,  75  tons. 

Spars.-Mast,  deck  to  hounds,  42  feet.  Boom,  58  feet.  Gaff,  39  feet  Bowsprit 
outside  of  stem,  30  feet.  Mast  to  stem,  26  feet.  Topmast,  foot  to  hounds,  25  feet. 
Balloon  topsail  yard,  46  feet.  Canvas,  area,  3450  sq.  feet.  Tons,  t.  H.,  90. 
Ballast.— At  Keel , 38.5  tons.  Hull,  1.5  tons. 

“Mischief”  {English),  Sloop.— Length  on  load-line,  61  feet ; beam,  19.9  feet. 
Immersed  section  at  load-line , 60  sq.  feet.  Displacement , 55  tons. 

Pilot  Boat. 

“Wm.  H.  Aspinwall,”  Schooner.  —Length  of  keel , 74  feet;  upon  deck,  80  feet; 
beam,  19  feet ; hold , 7.6  feet.  Draught  of  water.  6 feet  forward ; aft , 9.  $ feet. 

Keel,  22  ins.  in  depth.  False  keel,  12  ins.  in  depth  at  centre. 

Spars.— Mainmast.  77  feet  in  length.  Foremast,  76  feet.  Main  boom,  46  feet 
Main  gaff,  21  feet.  Fore  gaff,  20  feet. 

Tons.—  N.  M.,  46.32. 


896  PASSAGES  OF  STEAMBOATS. ICE-BOATS. 


PASSAGES  OF  STEAMBOATS. 

Distances  in  Statute  Miles. 


1807,  Clermont , of  N.  Y.,  New  York  to 
per  hour,  neglecting  effect  of  the  tide. 


Albany,  145  miles,  in  32  hours  = 4. 53  miles 


1811,  New  Orleans , of  Pittsburgh,  Penn,  (non-condensing  and  stern-wheel)  Pitts- 
burgh to  Louisville,  Ky.,  650  miles,  in  2 days  22  hours. 

1849,  Alida,  of  N.  Y.,  Caldwell’s,  N.  Y.,  to  Pier  1,  North  River,  43.25  miles,  in  1 
hour  42  min. , ebb  tide  = 2. 75  miles  per  hour.  Speed  = 22. 19  miles  per  hour  i860 
30th  Street,  N.  Y.,  to  Cozzens’s  Pier,  West  Point,  50.5  miles,  in  2 hours  4 min.,  and 
to  Poughkeepsie,  74.25  miles,  in  3 hours  27  min.,  5 landings,  flood  tide.  And  1853 
Robinson  Street  to  Kingston  Light,  90.375  miles,  in  4 hours,  making  6 landings' 

tiond  t.idp  ' ° ’ 


1850,  Buckeye  State,  of  Pittsburgh,  Penn,  (non  - condensing),  Cincinnati  to  Pitts- 
burgh, 500  miles  (200  passengers),  53  landings,  in  1 day  19  hours;  4 miles  per  hour 
adverse  current.  Speed  ==  15.63  miles  and  1.23  landings  per  hour.  Average  depth 
of  water  in  channel  7 feet. 

1852,  Reindeer , of  N.  Y.,  New  York  to  Hudson,  116.5  miles,  in  4 hours  57  min 

making  5 landings.  Flood  tide.  ^ 5/  ’ 

1853,  Shotwell , of  Louisville,  Ky.  (non-condensing),  New  Orleans  to  Louisville 
1450  miles,  8 landings,  in  4 days  9 hours;  4.5  to  5.5  miles  per  hour  adverse  cur- 
rent. Speed  = 18.81  miles  per  hour. 

Note.— In  1817-18  the  average  duration  of  a passage  from  New  Orleans  to  Louisville  was  27  days. 
12  hours;  the  shortest,  25  days.  * 

1855,  New  Princess,  of  New  Orleans  (non-condensing),  New  Orleans,  La.,  to  Natchez 
Miss.,  310  miles,  in  17  hours  30  min.;  3.5  to  4 miles  per  hour  adverse  current! 
Speed  ==  20.98  miles  per  hour. 

1864,  Daniel  Drew,  of  N.  Y.,  Jay  Street.  N.  Y.,  to  Albany,  148  miles,  in  6 hours  51 
min. , 9 landings.  Flood  tide.  Speed  of  boat  = 2?.  6 miles  per  hour. 

1867,  Mary  Powell,  of  N.  Y.,  Desbrosses  Street,  N.  Y.,  to  Newburgh,  60.5  miles,  in 
2 hours  50  min. , 3 landings;  from  Poughkeepsie  to  Rondout  Light,  15.375  miles,  in 
39  min.,  flood  tide.  1873,  Milton  to  Poughkeepsie,  light  draught  and  flood  tide,  4 
miles,  in  9 min. ; and  1874,  Desbrosses  Street  to  Piermont,  24  miles,  in  1 hour;  to 
Caldwell’s,  43.25  miles,  in  1 hour  50  min.  Speed  = 22. 77  to  23  miles  per  hour. 


Runs  from  New  York  to  Albany,  146  miles,  by  different  Boats. 


1826,  Sun. 12  hours  16  min. 

182 6,  North  America* . 10  “ 20  “ 

1841,  Troy  f 8 “ 10  “ 

1841,  South  America  t.  7 “ 28  “ 


1852,  Fr.  Skiddy  § 6 hours  24  min. 

i860,  Armenia  n 7 “ 22  “ 

1864,  Daniel  Drew } 6 “ 51  “ 

1864 , Ch’ncey  Vibbard  X.  6 “ 42  “ 


* 7 landings.  + 4 landings.  % 9 landings.  § 6 landings.  ||  11  landings. 

Timing  Distance.— From  14th  St.,  Hudson  River,  N.  Y.,  to  College  at  Mount  St.  Vincent,  13  miles. 


Note. — Where  landings  have  been  made,  and  the  river  crossed,  the  distance  between  the  points 
given  is  correspondingly  increased. 

1870,  R.  E.  Lee , of  St.  Louis  (non-condensing),  New  Orleans  to  St.  Louis,  Mo.,  1180 
miles  (without  passengers  or  freight),  4 to  5 miles  per  hour  adverse  current;  Vicks- 
burg, 1 day  38  min.;  Memphis,  2 days  6 hours  9 min.;  Cairo,  3 days  1 hour.;  and  to 
St.  Louis,  3 days  18  hours  14  min.,  inclusive  of  all  stoppages. 

1870,  Natchez,  of  Cincinnati,  Ohio,  from  New  Orleans  to  Baton  Rouge,  120  miles, 
in  7 hours  40  min.  42  sec. 


Runs  from  Neio  Orleans  to  Natchez,  295  miles , by  different  Boats. 

1814,  Orleans,  6 days  6 hours  40  min.  I 1856,  New  Princess , 17  hours  30  min. 

1840,  Edward  Shippen,  1 day  8 hours.  | 1870,  R.  E.  Lee , 16  hours  36  min.  47  sec. 

Ice-L>oats. 

Distances  in  Statute  Miles. 

1872,  Haze,  of  Poughkeepsie,  N.  Y.,  to  buoy  off  Milton,  4 miles,  in  4 min. 

1872,  Whiz,  of  Poughkeepsie,  N.  Y.,  to  New  Hamburg,  8.375  miles,  in  8 min. 


i 

i 


PASSAGES  OF  STEAMERS  AND  SAILING  VESSELS.  897 


PASSAGES  OF  STEAMERS  AND  SAILING  VESSELS. 

Distances  in  Geographical  Miles  or  Knots. 

Steamers.  Side-wlxeels. 

1807.  Phoenix , of  Hoboken,  N.  J.  (John  Stevens),  New  York,  N.  Y.,  to  Philadelphia 
Penn.  First  passage  of  a steam  vessel  at  sea. 

1814,  Morning  Star , of  Eng.,  River  Clyde  to  London,  Eng.  First  passage  of  an 
English  steamer  at  sea. 

1817,  Caledonia , of  Eng. , Margate,  Eng.,  to  Cassel,  Germ.,  180  miles,  in  24  hours. 

1810  Savannah , of  N.Y.,  about  340  tons  0.  M.,  Tybee  Light,  Savannah  River,  Ga 
to  Rock  Light,  Liverpool,  Eng.,  3640  miles,  in  25  dags  14  hours;  6 days  21  hours  of 
which  were  under  steam. 

1825  Enterprise , of  Eng.,  500  tons,  Falmouth,  Eng.,  to  Table  Ray,  Africa,  in  57 
days ; and  to  Calcutta,  India,  in  113  days.  First  passage  of  a steamer  to  India. 

1830,  Hugh  Lindsay , 41 1 tons,  80  HP,  Bombay,  India,  to  Suez,  Egypt,  3103  miles, 
in  31  days  running  time. 

1837,  Atlanta , of  Eng.,  650  tons,  Falmouth,  Eng.,  to  Calcutta,  in  91  days. 

1839,  Great  Western , of  Eng.,  Liverpool  to  New  York,  N.  Y.,  3017  miles,  in  12 
days  18  hours. 

1870  Scotia , of  Eng.,  Queenstown,  Ireland,  to  Sandy  Hook,  N.  J.,  2780  miles,  in 
8 days  7 hours  31  min.  1866,  New  York  to  Queenstown,  2798  miles,  in  8 days  2 
hours  48  win.;  thence  to  Liverpool,  Eng.,  270  miles,  in  14  hours  59  min.;  total,  8 
days  17  hours  47  min. 

Screw. 

1874.  India  Government  Boat , Steel,  length  87  feet,  beam  12  feet,  draught  of  water 
3.75  feet,  mean  speed  for  one  mile  20.77  miles  per  hour,  and  maintained  a speed  of 
18.92  miles  in  1 hour. 

1877,  Lusitania,  of  Eng.,  London  to  Melbourne,  Australia,  via  Cape,  11  445  miles, 
in  38  days  23  hours  40  min. 

Sailing  Vessels. 

1851,  Chrysolite  (clipper  ship),  of  Eng.,  Liverpool,  Eng.,  to  Anjer,  Java,  13000 
miles,  in  88  days.  The  Oriental , of  N.  Y.,  ran  the  same  course  in  89  days. 

1853,  Trade  Wind  (clipper  ship),  of  N.  Y.,  San  Francisco,  Cal.,  to  New  York,  N.  Y., 
13610  miles,  in  75  days. 

1854,  Lightning  (clipper  ship),  of  Boston,  Mass.,  Melbourne,  Australia,  to  Liver- 
pool, Eng.,  12  190  miles,  in  64  days. 

1854,  Comet  (clipper  ship),  of  N.  Y.,  Liverpool,  Eng.,  to  Hong  Kong,  China,  13040 
miles,  in  84  days. 

1854,  Sierra  Nevada  (schooner),  of  N.  H.,  Hong  Kong,  China,  to  San  Francisco, 
Cal.,  6000  miles,  in  34. days. 

1854,  Red  Jacket  (clipper  ship),  of  N.  Y.,  Sandy  Hook,  N.  J.,  to  Melbourne,  Aus- 
tralia, 12720  miles,  in  69  days  11  hours  1 min. 

1855,  Euterpe  (half-clipper  ship)  of  Rockland,  Me.,  New  York  to  Calcutta,  India, 
12  500  miles,  in  78  days. 

i860,  Andrew  Jackson  (clipper  ship),  of  Boston,  New  York,  N.  Y.,  to  San  Fran- 
cisco, Cal.,  13  610  miles,  in  80  days  4 hours. 

1865,  Dreadnought  (clipper  ship),  of  Boston,  Honolulu,  Sandwich  Islands,  to  New 
Bedford,  Mass.,  13470  miles,  in  82  days ; and  1859,  Sandy  Hook,  N.  J.,  to  Rock 
Light,  Liverpool,  Eng.,  3000  miles,  in  13  days  8 hours. 

1865,  Sovereign  of  the  Seas  (medium  ship),  of  Boston,  Mass.,  in  22  days  sailed 
5391  miles  = 245  miles  per  day.  For  4 days  sailed  341.78  miles  per  day,  and  for  1 
day  375  miles. 

1866,  Henrietta  (schooner  yacht),  of  N.  Y.,  Sandy  Hook,  N.  J.,  to  the  Needles, 
Eng.,  3053  miles,  in  13  days  21  hours  55  min.  16  sec. 

1866,  Ariel  and  Serica  (clipper  ships),  of  England,  Foo-chou-foo  Bar,  China,  to 
the  Downs,  Eng.,  13  500  miles,  in  98  days. 

1869,  Sappho  (schooner  yacht),  of  N.  Y.,  Light-ship  off  Sandy  Hook,  N.  J.,  to 
Queenstown,  Ireland,  2857  nvles,  in  12  days  9 hours  34  min. 


898  ELEMENTS  OP  MACHINES  AND  ENGINES, 


ELEMENTS  OF  MACHINES  AND  ENGINES. 

BLOWING  ENGINES. 

Furnaces. — Two.  Fineries. — Two.  {England.) 

240  Tons  Forge  Pig  Iron  per  Week . 

Engine  (non-condensing).—  Cylinder,  20  ins.  in  diam.  by  8 feet  stroke  of  pistoa 
Boilers. — Six  (plain  cylindrical),  36  ins.  in  diam.  and  28  feet*  in  length.  Grates , 
100  sq.  feet. 

Blowing  Cylinders. — Two,  62  ins.  in  diam.  by  8 feet  stroke  of  piston.  Pressure , 
2.17  lbs.  per  sq.  inch.  Revolutions , 22  per  minute. 

Pipes , 3 feet  in  diam.=  168  area  of  cylinder. 

Tuyeres. — Each  Furnace,  2 of  3 ins.  in  diam. ; 1 of  3.25  ins. ; and  1,  3 of  3 ins. 
Each  Finery,  6 of  1.33  ins. ; and  1,  4 of  1. 125  ins. 

Temperature  of  Blast , 6oo°.  Ore,  40  to  45  per  cent,  of  iron. 

Furnaces. — Eight , diam.  16  to  18  feet.  Dowlais  Iron  Works  {England). 
1300  Tons  Forge  Iron  per  Week;  discharging  44000  Cube  Feet  of  Air  per 
Minute. 

Engine  (non-condensing). — Cylinder , 55  ins.  in  diam.  by  13  feet  stroke  of  piston. 
Pressure  of  Steam.— 60  lbs.  per  sq.  inch,  cut  off  at  .33  the  stroke  of  piston.  Valves , 
120  ins.  in  area. 

Boilers. — Eight  (cylindrical  flued,  internal  furnace),  7 feet  in  diam.  and  42  feet  in 
length ; one  flue  4 feet  in  diam.  Grates , 288  sq.  feet. 

Fly  Wheel. — Diam.,  22  feet;  weight,  25  tons. 

Blowing  Cylinder , 144  ins.  in  diam.  by  12  feet  stroke  of  piston. 

Revolutions , 20  per  minute.  Blast , 3.25  lbs.  per  sq.  inch.  Discharge  pipe,  diam. 
5 feet,  and  420  feet  in  length.  Valves. — Exhaust,  56  sq.  feet;  Delivery,  16  sq.  feet. 

Furnaces. — Lackenby  {England). 

800  Tons  Iron  per  Week. 

Engine  (horizontal,  compound  condensing). — 32  and  60  ins.  in  diam.  by  4.5 
feet  stroke  of  piston. 

Blowing  Cylinders. — Two,  80  ins.  in  diam.  by  4.5  feet  stroke  of  piston.  Pressure, 
4. 5 lbs.  per  sq.  inch.  Revolutions , 24  per  minute. 

Pipe,  30  ins.  in  diam. ; volume,  12.25  times  that  of  blowing  cylinders. 

IP. — Engine,  290  lbs. ; Blowing  cylinders,  258;  efficiency,  89  per  cent. 

Valves. — Area  of  admission,  .16  of  area  of  piston;  of  exit,  .125. 

Volume. — 190000  cube  feet  of  air  are  supplied  per  ton  of  air. 


Blower  and.  Exhausting  Fan.  (Sturtevant’s.) 


Blower. 

Grate 

Surface. 

Inlet. 

Outlet. 

Diam.  of 
Pulley. 

Face  of 
Pulley. 

Revolu- 

tions. 

Air  per 
Minute. 

IP. 

No. 

Sq.  Feet. 

Diam.  Ins. 

Diam.  Ins. 

Ins. 

Ins. 

Per  Min. 

Cube  Feet. 

00 

5 

5 

4 

2.75 

2 

3000 

500 

— 

0 

6 

5-75 

4-75 

3 

2.25 

2600 

6oo 

— 

X 

8 

6-5 

5-75 

3-5 

3 

2200 

764 

.6 

2 

10 

7-5 

7-5 

3-75 

3-5 

1928 

1 019 

•79 

3 

14 

9 

9 

4-25 

4 

1638 

1427 

1 .11 

4 

20 

10.5 

10.5 

5 

5 

1410 

1936 

x-5i 

5 

27 

12 

12 

6 

5- 25 

1194 

2 701 

2.1 

6 

36 

14 

14 

7 

6-5 

1018 

3669 

2.86 

7 

48 

16 

16 

9 

7-5 

878 

4847 

3-77 

8 

62 

18 

18 

10 

8.5 

7 66 

6115 

4.76 

9 

80 

21 

21 

12 

10.5 

671 

8i54 

6.35 

10 

100 

24 

24 

14 

12 

598 

10702 

8-34 

* 40  feet  would  have  afforded  economy  in  fuel. 


ELEMENTS  OF  MACHINES  AND  ENGINES.  899 


COTTON  FACTORIES.  (English. ) 

For  driving  22060  Hand-mule  Spindles , with  Preparation,  and  260  Looms , 
with  common  Sizing. 

Engine  (condensing)—  Cylinder,  37  ins.  in  diam.  by  7 feet  stroke  of  piston; 

volume  of  piston  space,  53.6  cube  feet. 

Pressure  of  Steam.— ( Indicated  average)  16.73  lbs.  per  sq.  inch.  Revolutions , 17 
per  minute.  , . _ . . 

Friction  of  Engine  and  Shafting.-(  Indicated)  4.75  lbs.  per  sq.  inch  of  piston. 

IIP  125.  Total  power  = i.  Available,  deducting  friction  717. 

f 305  hand-mule  spindles,  with  preparation , 

, . or  230  self-acting  “ “ 

Notes.— Each  IIP  will  drive  j or  £ throstle  “ “ 

t or  10. 5 looms,  with  common  sizing. 


Including^ ==  3 hand-mule,  or  2.25  self-acting  spindles. 
1 self-acting  spindle  = 1.2  hand-mule  spindles. 


DREDGING  MACHINES. 

Dr  edging  20  Feet  from  Water-line,  or  180  Tons  of  Mud  or  Silt  per  Hour 
11  Feet  from  Water-line. 

Length  upon  deck , 123  feet;  beam,  26  feet  Breadth  over  all , 41  feet. 

Immersed  section  at  load-line , 60  sq.feet.  Displacement , 141  tons,  at  load-draught 
0/2.83  feet. 

Engine  (non-condensing ).-Cylinders,  two,  12.125  ins.  in  diam.  by  4 feet  stroke 
of  piston. 

Boilers,— Two  (cylindrical  flue),  diam.  40.5  ins.,  and  length,  20  feet  3 ins.;  two 
flues,  14.625  ins.  in  diam.  Heating  surface , 617  sq.  feet.  Grates , 37  sq.  teet. 

Pressure  of  Steam,  25  lbs.  per  sq.  inch;  throttle  .25  open,  cut  off  at  .5  the  stroke 
of  piston.  Revolutions , 42  per  minute. 

Buckets.  — Two  sets  of  12,  2.5  feet  in  length  by  15  ins.  at  top  and  2 feet  deep;  vol- 
ume, 6.25  cube  feet.  Chain  Links , 8 ins.  in  length  by  .5  inch  diam. 

Scows  or  Camels. — Four,  of  40  tons  capacity  each. 

STEAM  HOPPER  DREDGER*  ( Wm . Simons  Sf  Co.) 


Iron. 

“Neptune”  (English). — Length,  150  feet;  breadth,  yifeet. 

Dredge  from  6 Ins.  to  2 5 Feet.  Capacity  of  Hopper,  500  to  600  Tons. 
Engines. — Two  (compound),  375  IP,  for  dredging  and  propulsion,  and  one  for 
raising  bucket-frame  and  anchor-posts. 

A like  designed  dredger  of  1000  tons  capacity  has  dredged  10000  tons  silt  per 
week  and  transported  it  7.5  miles. 


Dredging  400  Tons  of  Mud  or  Silt  per  Hour , 5 to  35  Feet  in  Depth 
Capacity  of  Hopper,  1300  Tons. 

Engines.—  Two  (compound),  IP  700.  Speed.— 7.5  knots  per  hour. 

Steam  Dredging  Crane.  (English.) 

Lift , 30  Feet  per  Hour. 


Lbs. 
21 280 
24  640 


Lifting 

Power. 

Volume 

of 

Bucket. 

Mud  or 
Silt. 

Coal  and 
Sand. 

Excava- 

tion 

Ground. 

Weight 
of  Crane. 

Lifting 

Power. 

Volume 

of 

Bucket. 

Mud  or 
Silt. 

Coal  and 
Sand. 

Excava- 

tion 

Ground. 

Tons. 

Lbs. 

Tons. 

Tons. 

C.  Yds. 

Lbs. 

Tons. 

Lbs. 

Ton®. 

Tons. 

C.  Yds. 

2.  5 

1x20 

25 

20 

20 

18000 

5 

2240 

50 

40 

3° 

3 

1680 

37-5 

32 

25 

33480 

7 

3360 

60 

54 

40 

gOO  ELEMENTS  OF  MACHINES  AND  ENGINES, 


Iron. 

Dredger  and  Hopper  Barge  (Compound;  English). — Length,  extreme  120  feet' 
beam,  32  feet ; hold,  10. 5 feet.  Breadth  of  bucket  well,  6. 75  feet.  Load-draught,  6 feet. 

Cylinders.—  21.5  and  40  ins.  in  diam.  by  2.5  feet  stroke  of  piston.  Condenser .— 
Surface , 600  sq.  feet.  Circulating  Pump , single  acting.— 15  ins.  in  diam.  by  xc  ins 
stroke  of  piston. 

Boiler.  — Heating  surface,  1150  sq.  feet.  Grates , 40  sq.  feet.  Steam -room  300 
cube  feet.  ’ 0 

Shaft. — 7.75  ins.  in  diam. 

Bucket  ladder. — Of  wrought  iron,  74  feet  in  length,  5 feet  in  depth  at  centre,  and 
2 feet  2 ins.  at  ends.  Buckets. — 34;  volume,  15  cube  feet  each. 

Excavation  and  Delivery.— For  a transit  of  7.5  miles,  3000  tons  per  day. 

Hopper  Barge. — Length  between  perpendiculars,  115  feet;  beam,  32  feet; 
hold,  9 feet  11  ins.  Load-line  with  400  Tons  dredge,  8 feet. 

Hoppers — Length,  50  feet ; breadth  at  top , 22  feet;  at  bottom,  9 feet. 

Cost. — Dredge , $ 90000;  Hoppers,  $ 18  000  each. 

Maintenance.— Dredger,  1.75  cents;  Hopper,  1.7  cents;  Towing,  1.2  cents  per  ton 
of  dredge  excavated  and  delivered. 

“Hercules,”  Panama  Canal.  — Length  on  deck , 100  feet ; beams,  40,  60,  and  4 c 
feet ; depth  of  hold,  12  feet.  Slot , 36  feet  in  length  by  6 feet  7 ins.  in  width. 

Ways.— Two,  one  40  feet  and  one  60  feet,  by  5 feet  in  width. 

Buckets.  — 38 ; volume,  1.33  cube  yards.  Spuds,  2 feet  in  diam.  and  60  in  length. 

Engines.— Two  of  100  IP  each,  and  two  of  40  IP  each. 

Boilers.— Three  (horizontal  tubular),  16  feet  in  length. 

Elevator  and  Discharge.—  Maximum,  24  cube  yards  per  minute. 

Crane.  (“W  ood.) 

Hull. — Length  on  deck,  100  feet ; beam,  ^\feet ; load-draught,  4.5  feet. 

Radius  of  crane,  46  feet;  height,  70  feet;  counter -balance,  70  tons. 

Boiler. — Heating  surface,  500  sq.  feet.  Pressure  of  Steam,  80  lbs.  per  sq.  inch. 
HP,  150. 

Propellers. — Two,  4.25  feet  in  diam.  Speed,  5 miles  per  hour. 

Engine  to  operate  crane.  Cylinder. — 10  ms.  in  diam.  by  12  ins.  stroke  of  piston. 

FLOUE  MILLS. 

30  Barrels  of  Flour  per  Hour. 

Water-wheels,  Overshot. — 5,  diam.  18  feet  by  14.5  feet  face.  Buckets , 15 
;ns.  ih  depth.  Water. — Head,  2.5  feet.  Opening,  2.5  ins.  by  14  feet  in  length  over 
each  wheel. 

5 Barrels  of  Flour  per  Hour , and  Elevating  400  Bushels  of  Grain  36  Feet. 

Water-wheel,  Overshot. — Diam.  22  feet  by  8 feet  face.  Buckets,  52  of  1 
foot  in  depth.  Water. — Head , from  centre  of  opening,  25  ins.  Opening,  1.75  ins. 
by  80  ins.  in  length. 

Revolutions,  3.5  per  minute.  Stones,  three  of  4.5  feet;  revolutions,  130, 

Three  Run  of  Stones , Diameter  4 Feet. 

Water-wheel,  Overshot. — Diam.  19  feet  by  8 feet  face.  Buckets , 14  ins.  in 
depth. 

Or, 

Steam-engine  (non-condensing). — Cylinder,  13  ins.  in  diam.  by  4 feet  stroke. 

Boiler  (cylindrical  flued).— Diam.  5 feet  by  30  in  length;  two  flues  20  ins.  in  diam. 


ELEMENTS  OF  MACHINES  AND  ENGINES.  QOI 


IP 


No. 

4 

6 

io 

15 

20 

25 


HOISTING  ENGINES. 

Eor  3?ile  Driving,  Hoisting,  NTining,  etc. 
Lidgerwood  Manuf’g  Co.,  New  York. 


Single  Cylinders. 


Cylinder. 


Ins. 

5X5 

6X8 

7 X 10 

8 X 10 

9 X 12 
10  X 12 


Capacity. 


Lbs. 

1000 

1250 

1800 

2800 

4000 

5000 


Cost,  with 
Boiler.* 


$ 

600 

675 

825 

1050 

1275 

1375 


12 
20 
30 
40 
50 
Complete. 


Double  Cylinders. 


Cylinder. 


Ins. 

5 X 8 
6X8 

7 X 10 

8 X 10 

9 X 12 
10  X 12 


Capacity. 


Lbs. 

2000 

2500 

3500 

6000 

8000 

9000 


Cost,  with 
Boiler.* 


950 

1050 

1350 

1550 

2000 

2350 


Details  and  Operation. 


Engine. 

Dram. 

Be 

Dimen- 

sions. 

>iler. 

Tubes. 

Ram. 

Leaders. 

Hoist. 

Lift. 

Ram. 

Blows 

per 

Minute. 

Piles 
per  10 
Hours. 

Fuel 

per 

Hour. 

H? 

10* 

20 

Ins. 

12  X 24 
14  X 26 

Ins. 

32  X 75 
40  X 84 

No. 

48  of  2 in. 
80  of  2 in. 

Lbs. 

1953 

2700 

Feet. 

40 

75 

Feet. 

8 to  12 
8 to  ii 

No. 
25 
1 29 

No. 

50 

IOO 

Lbs. 

70 

80 

* Weight  complete,  8500  lbs. 


Mining  Engines  and.  Boilers.  {Various  Capacities.) 

Engine , Boiler , etc.,  as  given  for  Pile  Driving,  page  902. 

Operation.  — 250  to  300  tons  of  coal  in  10  hours.  Fuel , 40  lbs.  coal  per  hour. 
Water,  20  gallons  per  hour. 

Weight  of  Engine  and  Boiler,  4500  lbs. 


Hancock  Inspirator.  F01 ' a Lift  of  Water  of  25  Feet. 


No. 

Diam 

Steam-pipe. 

ieter. 

Suction. 

Discharge 
at  Pressure 
of  60  Lbs. 

No. 

Diam 

Steam-pipe. 

eter. 

Suction. 

Discharge 
at  Pressure 
of  60  Lbs. 

Ins. 

Ins. 

G’lls.per  h’r. 

Ins. 

Ins. 

G’lls.per  h’r. 

10 

•375 

•5 

120 

30 

1.25 

i-5 

1260 

12.5 

•5 

•75 

220 

35 

1.25 

i-5 

1740 

15 

•5 

•75 

300 

40 

1-5 

2 

2230 

20 

•75 

1 

* 540 

45 

i-5 

2 

2820 

25 

1 

1.25 

900 

50 

2 

2-5 

3480 

Temperature  of  water  not  over  1450  for  a low  lift,  and  ioo°  for  a high  lift. 


HYDROSTATIC  PRESS.  ( Cotton) 

30  Bales  of  Cotton  per  Hour. 

Engine  (non-condensing). — Cylinder,  10  ins.  in  diam.  by  3 feet  stroke  of  piston. 
Pressure  of  Steam,,  50  lbs.  per  sq.  inch,  full  stroke.  Revolutions,  45  to  60  per 
minute. 

Presses. — Two,  with  12-inch  rams;  stroke,  4.5  feet. 

Pumps. — Two,  diam.  2 ins. ; stroke,  6 ins. 


For  83  Bales  per  Hour. 

Engine  (non-condensing).— Cylinder,  14  ins.  in  diam.  by  4 feet  stroke  of  piston. 
Boilers. — Three  (plain  cylindrical),  30  ins.  in  diam.  and  26  feet  in  length.  Grates , 

32  sq.  feet.  Pressure  of  Steam , 40  lbs.  per  sq.  inch.  Revolutions , 60  per  minute. 

Presses. — Four,  geared  6 to  1,  with  two  screws,  each  of  7.5  ins.  in  diam.  by  1.625 
in  pitch. 

Shaft  (wrought  iron).  — Journal,  8.5  ins.  Fly  Wheel,  16  feet  in  diam. ; weight, 
8960  lbs. 


902 


ELEMENTS  OF  MACHINES  AND  ENGINES. 


LOCOMOTIVE. 

“Experiment”  [Compound).— Cylinders,  one  each,  12  and  26  ins.  in  diam.,  and 
one  26  ins.  by  2 feet  stroke  of  piston. 

Boiler.— Heating  surface , 1083.5  sq.  feet.  Grate , 17. 1 sq.  feet.  Pressure  of  Steam , 

150  lbs.  per  sq.  inch,  cut  off  at  .35.  Speed , 50  miles  per  hour.  Weight.—  Empty, 

34-75  tons. 

Street  Railroad  or  Tramway  Engine. 

Cylinder , 7 ins.  in  diam.  by  n ins.  stroke  of  piston. 

Boiler  78  tubes  1.75  ins.  in  diam.  by  4 feet  in  length.  Heating  surface , 160  sq. 
feet.  Grate , 4.25  sq.  feet.  Wheels,  2.33  feet  in  diam.  Base , 4.5  feet.  Gauge , 4 
feet  8. 5 ins. 

— Average  per  mile  in  England,  2.52  pence  sterling  = 4.48  cents. 

PILE-DRIVING. 

Driving  One  Pile. 

Engine  (non-condensing).—  Cylinder,  6 ins.  in  diam.  by  1 foot  stroke  of  piston. 

Boiler  (vertical  tubular).— 32  ins.  in  diam.,  and  6.166  feet  in  height.  Grates,  3.7 
sq.  feet.  Furnace,  20  ins.  in  height.  Tubes,  35,  2 ins.  in  diam.,  4.5  feet  in  length. 

Revolutions,  150  per  minute.  Drum,  12  ins.  in  diam.,  geared  4 to  1.  Leader , 40 
feet  in  height.  Ram.—  2000  lbs.,  2 blows  per  minute.  Fuel,  30  lbs.  coal  per  hour. 

Driving  Two  Piles. 

Engine  (non-condensing).— Cylinders,  two,  6 ins.  in  diam.  by  18  ins.  stroke  of 
piston. 

Boiler  (horizontal  tubular).— Shell,  diam.  3 feet,  and  6 feet  in  length.  Furnace 
end  3.75  feet  in  width,  3.5  feet  in  length,  and  6 feet  in  height. 

Pressure  of  Steam,  60  lbs.  per  sq.  inch.  Revolutions,  60  to  80  per  minute. 

Frame  8.5  feet  in  width  by  26  feet  in  length.  Leaders , 3 feet  in  width  by  24  feet  • 
in  height.  Rams.—  Two,  1000  lbs.  each,  5 blows  per  minute. 

PUMPING  ENGINES.  ' 

Corliss  Steam-engine  Co.,  Providence,  R.  I. — Vertical -Beam  Engine  (Com-  , 
pound).—  Cylinders.— 18  and  36  ins.  in  diam.  by  6 feet  stroke  of  piston. 

Pumps. — Four  plunger,  19  ins.  in  diam.  by  3 feet  stroke  of  piston.  Displacement 
per  revolution  of  engine,  84.96  cube  feet. 

Boilers.  — Three,  vertical  fire  tubular.  Grate.— 93  sq.  feet.  Heating  surface,  1680 
sn  feet.  Pressure  of  Steam.  125  lbs.  per  sq.  inch,. cut  off  at  .22  feet.  Revolutions,  / , 
36  per  minute.  IIP  313.  Fly-wheel.— 25  feet  in  diam.,  weight  62000  lbs. 

Fuel  —Cumberland  coal,  486  lbs.  per  hour,  inclusive  of  kindling  and  raising  steam,  i 
Ash  and  Clinkers.  9.4  per  cent.  Duty  for  one  week,  113  271 000  foot-lbs. 

Water  delivered,  17621  gallons  per  minute,  against  head  of  180  feet. 

Duty,  average  for  1883,  per  100  lbs.  anthracite  coal,  106  048  000  foot-lbs. 


For  Elevating  200000  Gallons  of  Water  per  Hour. 

Lynn  Mass. — Engine  (Compound). — Cylinders,  17.5  and  36  ins.  in  diam.  by  7 feet 
stroke  of  piston;  volume  of  piston  space,  61.2  cube  feet.  Air  Pump  (double  act- 
ing), 11.25  ins-  diam.  by  49.5  ins.  stroke  of  piston. 

Pump  Plunger , 18.5  ins.  in  diam.  by  7 feet  stroke. 

Boilers.— Two  (return  Sued),  horizontal  tubular;  diam.  of  shell,  5 feet;  drum,  3 
feet;  tubes,  3 ins.  Length  of  shell,  16  feet.  Grates,  27.5  sq.  feet. 

Pressure  of  Steam,  90.5  lbs. ; average  in  high-pressure  cylinder  86  lbs.,  cut  off  at 
1 foot,  or  to  an  average  of  44.5  lbs. ; average  in  low-pressure  cylinder,  27  lbs.,  cut 
off  at  6 ins.,  or  to  an  average  of  10.8  lbs. 

Revolutions,  18.3  per  minute.  Fly  Wheel. — Weight,  24000  lbs. 

Evaporation  of  Water , 4644  lbs.  per  hour.  Loss  of  action  by  Pump,  4 per  cent. 

Consumption  of  Coal—  Lackawanna,  291  lbs.  per  hour. 

Duty  205772  gallons  of  water  per  hour,  under  a load  and  frictional  resistance  of 
73.41  ibs.  per  square  inch,  equal  to  103  923  217  foot-lbs.  for  each  100  lbs.  of  coal. 


ELEMENTS  OF  MACHINES,  MILLS,  ETC. 


903 


“ Gashill at  Saratoga , N.  Y. 

Engine  ( Horizontal  Compound).  Cylinders.—  High  pressure,  2 of  21  ins.  diam. 
Low  pressure,  2 of  42  ins.  diam.,  all  3 feet  stroke  of  piston.  Pumps.—  Two  of  20  ins. 
diam.  by  3 feet  stroke  of  piston. 

Fly  Wheel,  12.33  feet  in  diam. ; weight,  12000  lbs. 

Boilers  (horizontal  tubular).— Two  of  5.5  feet  in  diam.  by  18  feet  in  length.  Heat- 
ing surface,  2957  sq.  feet.  Grates , 51  sq.  feet  of  grate;  to  heating  surface,  1 to  58, 
and  to  transverse  section  of  tubes,  1 to  7.  Chimneys , 75  feet. 

Pressure  of  Steam.— Mean  of  20  hours,  74.25  lbs.  per  sq.  inch.  Revolutions , 17.87 
per  minute.  IIP. High-pressure  cylinders,  109.2;  low-pressure,  76.65.  Total,  185.8. 

Fuel. Anthracite,  6.9  lbs.  per  sq.  foot  of  grate  per  hour.  Evaporation , per  sq. 

foot  of  heating  surface  per  hour,  1.175  lbs. ; per  lb.  of  coal,  9.25  lbs. ; per  cent,  of 
non-combustible,  3.2. 

Duty,  112899993  foot-lbs.  per  100  lbs.  coal.  Heating  surface  per  IIP,  14.9. 

Steam  per  sq.  foot  of  surface  per  hour,  1. 19  lbs. ; per  sq.  foot  of  surface  per  lb.  of 
coal  per  hour  from  212°,  11.28  lbs. 


Ericsson’s  Caloric.  For  an  Elevation  of  50  Feet. 


Dimen- 

sions. 

Space 

occupied. 

Floor.  | Height. 

Volume 

per 

Hour. 

Pipes, 
Suction 
and 
Dis-  1 
charge,  j 

Fuel 

per  Hour. 
Nut 

Anthr.  I Gas. 

Furnace. 
Gas.  | Coal. 

COST 

Deep 

Pump. 

Well  Pi 
Extra. 
Pipes  pe 
Plain. 

imp. 

sr  Foot. 
Galvan 

Ins. 

Ins. 

Ins. 

Gall. 

Ins. 

Lbs. 

Cub.  ft. 

% 

$ 

$ 

$ 

% 

5 

34Xi8 

48 

150 

•75 

— 

15 

150 

— ' 

— 

— 

— 

6 

39X20 

51 

200 

•75 

2-5 

18 

200 

210 

— 

— 

— 

8 

48X21 

63 

350 

1 

3-3 

25 

235 

250 

10 

.64 

.86 

12 

54X27 

63 

800 

i-5 

6 

— 

— 

320 

15 

.80 

*•*5 

12* 

42X52 

65 

1600 

2 

12 

— 

— ‘ 

450 

25t 

.92 

1.25 

* Over  90  feet,  92  cents.  t Duplex. 


Including  engine  and  pump,  oil-can  and  wrench,  complete  in  all  but  suction  and 
discharge-pipe. 

SUGAR  MILLS. 

Expressing  40000  lbs.  Cane-juice  per  day,  or  for  a Crop  of  5000  Boxes  of 
450  lbs.  each  in  four  Months ’ Grinding. 

Engine  (non-condensing).— Cylinder,  18  ins.  in  diam.  by  4 feet  stroke  of  piston. 
Boiler  (cylindrical  flued).— 64  ins.  in  diam.  and  36  feet  in  length;  two  return  dues, 
20  ins.  in  diam.  Heating  surface,  660  sq.  feet.  Grates,  30  sq.  feet. 

Pressure  of  Steam,  60  lbs.  per  sq.  inch,  cut  off  at  . 5 the  stroke  of  piston.  Revolu- 
tions, 40  per  minute. 

Rolls.  — One  set  of  3,  28  ins.  in  diam.  by  6 feet  in  length;  geared  1 to  14.  Shafts, 
11  and  12  ins.  in  diam.  Spur  Wheel , 20  feet  in  diam.  by  1 foot  in  width.  Fly 
Wheel.  18  feet  in  diam. ; weight,  17  400  lbs. 

Weights.—  Engine,  61  460  lbs. ; Sugar  Mill,  65  730  lbs. ; Spur  Wheel  and  Connect- 
ing Machinery  to  Mill,  28680  lbs. ; Boiler,  18  520  lbs. ; Appendages,  6730  lbs.  Total, 
181 120  lbs. 

STONE  AND  ORE  BREAKERS.  (Blake’s.) 


No. 

Re- 

ceiver. 

Pul 

D’m. 

ley. 

Face. 

45 

> % 

Power 

re- 

quired. 

Weight. 

No. 

Re- 

ceiver. 

Pul 

D’m. 

ley. 

Face. 

| V’locitj 
per 

1 Minute, 

Power 

quired. 

Weight. 

Ins. 

Feet. 

Ins. 

Feet. 

PP. 

Lbs. 

Ins. 

Feet. 

Ins. 

Feet. 

HP. 

Lbs. 

A 

4X10 

1.66 

6 

250 

4 

4 000 

5 

9Xi5 

2-5 

9 

250 

9 

13360 

1 

5XlO 

2-75 

6 

180 

5 

6 700 

6 

11X15 

2-33 

6 

180 

9 

11  6co 

2 

7X10 

2 

7-5 

250 

6 

8000 

7 

13X15 

2-33 

8 

180 

9 

11  760 

3 

5Xi5 

2-33 

8 

180 

9 

9 100 

8 

15X20 

3-5 

10 

150 

12 

32  600 

4 

7X15 

! 2.33 

9 

180 

9 

10490 

9 

18X24 

6 

12 

125 

12 

37  5oo 

Note. — Amount  of  product  depends  on  distance  jaws  are  set  apart,  and  speed. 
Product  given  in  Table  is  due  when  jaws  are  set  1.5  ins.  open  at  bottom,  and  ma- 
chine is  run  at  its  proper  speed  and  diligently  fed.  It  will  also  vary  somewhat  with 
character  of  stone.  Hard  stone  or  ore  will  crush  faster  than  sandstone. 

A cube  yard  of  stone  is  about  one  and  one  third  tons. 


904 


ELEMENTS  OF  MACHINES. CHIMNEYS. 


STEAM  FIRE-ENGINE. 

Ainoskeag,  IN'.  FI . ls^t  Class. 

Steam  Cylinder.— Two  of  7.625  ins.  in  diam.  by  8 ins.  stroke  of  piston. 

Water  Cylinder.— Two  of  4.5  ins.  in  diam. 

Boiler  (vertical  tubular).— Heating  surface , 175  sq.  feet.  Grates , 4.75  sq.  feet. 
Pressure  of  Steam.— 100  lbs.  per  sq.  inch.  Revolutions , 200  per  minute. 
Discharges.—  Two  gates  of  2.5  ins.,  through  hose,  one  of  1.25  ins.  and  two  of  1 inch. 
Projection.  — Horizontal,  1.25  ins.  stream,  311  feet;  two  1 inch  streams,  256  feet. 
Vertical,  1.25  ins.  stream,  200  feet.  Water  Pressure. — With  1.125  ins.  nozzle,  200  lbs. 
Time  of  Raising  Steam,— From  cold  water,  25  lbs.,  4 min.  45  sec. 

Weights.  — Engine  complete,  6000  lbs. ; water,  300  lbs. 

SAW-MILL. 

Two  Vertical  Saws,  34  Ins.  Stroke,  Lathes,  etc. 

Engine  (non-condensing).  Cylinder. — 10  ins.  in  diam.  by  4 feet  stroke  of  piston. 
Boilers. — Three  (plain  cylindrical),  30  ins.  in  diam.  by  20  feet  in  length. 

Pressure  of  Steam.— go  lbs.  per  sq.  inch.  Revolutions , 35  per  minute. 

Note.— This  engine  has  cut,  of  yellow-pine  timber,  30  feet  by  18  ins.  in  1 minute. 

STONE  SAWING. 

Emerson  Stone  Saw  Co.  (Diamond  Stone  Saw,  Pittsburgh,  Penn.).— 
20  IP,  150  sq.  feet  of  Berea  sandstone,  inclusive  of  both  sides  of  cut,  in  1 hour. 

CHIMNEYS. 

Lawrence,  Mass . Octagonal,  222  Feet  above  Ground,  and  19  Feet  below. 
Foundation,  35  Feet  square  and  of  Concrete  7 Feet  deep.  ( Hiram  F.  Mills.) 

Shaft.—  234  feet  in  height,  20  feet  at  base,  and  11.5  at  top;  28  ins.  thick  at  base 
and  8 at  top.  Core.—  2 feet  thick  for  27  feet,  and  1 foot  for  154. 

Horizontal  Flues.  — 7.5  feet  square,  and  Vertical  flue  or  cylinder  of  8.5  feet,  234 
high,  with  walls  20  ins.  thick  for  20  feet,  16  for  17  feet,  12  for  52  feet,  and  8 for  145  feet. 
Purpose.  — For  700  sq.  feet  grate  surface.  Weight.  — 2250  tons.  Bricks , 550000. 

New  York  Steam  Heating  Co.  Quadrilateral,  220  Feet  above  Ground 
and  1 Foot  below.  {Chas.  E.  Emery , Ph.D .) 

Shaft. 220  feet  in  height,  and  27  feet  10  ins.  by  8 feet  4 ins.  in  the  clear  inside. 

Foundation.— 1 foot  below  high  water.  Capacity.—  Boilers  of  16000  IP. 


Cost  of  Steam-Engines  and  Boilers  complete,  and.  of 
Operation  per  Bay  of  IO  Honrs,  inclusive  of  Labor, 
Enel,  and  llepairs.  {Chas.  E Emery,  Ph.D.) 


HP. 

Engine. 

Water 
orated 
IIP  per 
Hour. 

Evap- 
i per 
Lb.  of 
Coal. 

Coa 

IIP. 

1 per 
Day. 

Labor. 

Sup- 
plies 
and  Re- 
pairs. 

Cost 

of 

Coal.* 

Total 
Cost  of 
Operat’n, 
including 
Coal. 

6.25 
12.5 
29 
1 1 2 
276 
552 

Portable  Vertical^ 

Horizontal ( ^ 8 

Single  Condensing. . . 

Lbs. 

42 

38 

32 

23 

22.2 

22.2 

Lbs. 

7-5 

7-5 

8 

8.8 

8.8 

8.8 

Lbs. 

56 

51 

40 

26. 1 

25.2 
25.2 

Lbs. 

394 

7i7 

1 308 
3 3oo 
7 83i 
15  663 

$ 

i-75 

i-75 

2.25 

3- 75 

4- 25 
6 

$ 

•33 
.41 
.60 
1.17 
2. 12 
4.02 

•73 

1- 33 

2- 43 
6.14 

14.58 
29. 16 

2.86 
3-56 
5-45 
11.66 
22.27 
41-  S2 

* $ 4.42  per  ton  (2240  lbs.),  including  cartage. 


GRAPHIC  OPERATION. 


905 


GRAPHIC  OPERATION. 


Solutions  of  Questions  toy-  a G-rapliic  Operation. 
1.  If  a man  walks  5 miles  in  1 hour,  how  far  will  he  walk  in  4 hours? 


continue  the  time  to 


Operation.  — Draw  horizontal  line,  divide  it  into  equal  parts, 
as  1,  2,  3,  and  4,  representing  hours.  From  each  of  these 
points  let  fall  vertical  lines  A C,  1 1,  etc.,  and  divide  A C into 
miles,  as  5,  10,  15,  and  20,  and  from  these  points  draw  equi- 
distant lines  parallel  to  the  horizontal. 

Hence,  the  horizontal  lines  represent  time  or  hours,  and 
the  vertical,  distance  or  miles. 

Therefore,  as  any  inclined  line  in  diagram  represents  both 
time  and  distance,  course  of  man  walking  5 miles  in  an  hour 
is  represented  by  diagonal  Ac;  and  if  he  walks  for  4 hours, 
4,  and  read  off  from  vertical  line  A C the  distance  = 20  miles. 


2.  How  far  will  a man  walk  in  2 hours  at  rate  of  10  miles  in  1 hour  ? 


His  course  is  shown  by  the  line  A o,  representing  20  miles. 

3.  If  two  men  start  from  a point  at  the  same  time,  one  walking  at  the 
rate  of  5 miles  in  an  hour  and  the  other  at  10  miles,  how  far  apart  will  they 
be  at  the  end  of  2 hours  ? 

Their  courses  being  shown  by  the  lines  A r and  A 0,  the  distance  r 0 represents 
the  difference  of  their  distances,  10  20  = 10  miles. 

4.  How  long  have  they  been  walking? 

Their  courses  are  now  shown  by  the  lines  A 0 and  A 4,  the  distance  2 4 represents 
the  difference  of  their  times,  or  2 ^ 4 = 2 hours. 

5.  When  they  are  10  miles  apart,  how  long  have  they  been  walking? 

Their  courses  are  again  shown  by  the  lines  A r and  A 0,  the  distance  r o repre- 
sents the  difference  of  their  distances  of  10  miles,  and  A 2,  2 hours. 

6.  If  a man  walks  a given  distance  at  rate  of  3.5  miles  per  hour,  and  then 
runs  part  of  distance  back  at  rate  of  7 miles,  and  walks  remainder  of  dis- 
tance in  5 minutes,  occupying  25  minutes  of  time  in  all,  how  far  did  he  run? 

Operation.  — Draw  horizontal  line,  as  A C, 
representing  whole  time  of  25  minutes;  set 
off  point  e representing  a convenient  fraction 
of  an  hour  (as  10  minutes),  and  a i equal  to 
corresponding  fraction  of  3.5  miles  (or  .5833); 
draw  diagonal  A n,  produced  indefinitely  to  0, 
and  it  will  represent  the  rate  of  3.5  miles  per 
hour. 

Set  off  C r equal  to  5 minutes,  upon  same 
scale  as  that  of  A C;  let  fall  vertical  r s , and 
draw  diagonal  C u at  same  angle  of  inclination 
as  that  of  An;  then  from  point  u draw  diagonal  u 0,  inclined  at  such  a rate  as  to 
represent  7 miles  per  hour;  thus,  if  i n represents  rate  of  3.5  miles,  s 0,  being  one 
half  of  the  distance,  will  represent  7 miles. 

The  whole  distance  between  the  two  points  is  thus  determined  by  C x,  and  dis- 
tance ran  by  u s,  measured  by  scale  of  miles  employed. 

Verification.—  The  distances  A e and  A i are  respectively  10  minutes  = .166  of  an 
hour,  and  .5833  mile  = .166  of  3.5  miles.  Hence,  C x~. 875  mile,  and  u s = . 5833 
mile.  Consequently,  the  man  walked  A0  = .875  mile  = 15  minutes,  ran  Om  = 
.5833  mile  = 5 minutes,  and  walked  u C = .2916  mile. 

7.  If  a second  man  were  to  set  out  from  C at  same  time  the  man  referred 
to  in  preceding  question  started  from  A,  and  to  walk  to  A and  return  to  C, 
at  a uniform  rate  of  speed  and  occupying  same  time  of  25  minutes,  at  which 
points  and  times  will  he  meet  the  first  man? 

Operation. — As  A C represents  whole  time,  and  C x distance  between  the  two 
points,  v z and  0 x will  represent  course  of  second  man  walking  at  a uniform  rate, 
and  he  will  meet  the  first  man,  on  his  outward  course,  at  a distance  from  his  start- 
ing-point of  A,  represented  by  A o,  and  at  the  time  A a;  and  on  his  return  course 
at  distance  Av.xm.  and  at  the  time  A c. 

4g* 


go6 


MISCELLANEOUS. 


MISCELLANEOUS. 


NTo.,  Diameter,  and.  Number  of  Sliot.  {American  Standard.) 
Compressed  Back  Shot. 


No. 

Diam. 

Shot 
per  Lb. 

No. 

Diam. 

Shot 
per  Lb. 

No. 

Diam. 

Shot 
per  Lb. 

Inch. 

NO. 

Inch. 

No. 

Inch. 

No. 

3 

•25 

284 

1 

•3 

173 

00 

•34 

115 

2 

.27 

232  . 

0 

•32 

140 

000 

•36 

98 

Balls,  .38  Inch,  85  No.  per  lb. ; .44  Inch,  50  No.  per  lb. 


Chilled  Shot. 


No. 

Diam, 

Shot 
per  Oz. 

No. 

Diam. 

Shot 
per  Oz. 

No. 

Diam. 

Shot 
per  Oz. 

No. 

Diam. 

Shot 
per  Oz. 

Inch. 

No, 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

12 

•05 

2385, 

9 

.08 

585 

6 

.11 

223 

1 

.16 

73 

11 

.06 

, 1380 

8 

Trap 

495 

5 

.12 

172 

B 

•17 

61 

10 

Trap 

1130 

8 

.09 

4°9 

4 

•13 

136 

BB 

.18 

52 

10 

•°7 

868 

7 

Trap 

345 

3 

.14 

109 

BBB 

.19 

43 

9 

Trap 

I 7i6 

7 

.1 

299 

2 

15 

88 

Drop  Shot. 


No, 

Diam. 

Pellets 
per,  Oz. 

No. 

Diam. 

Pellets 
per  Oz. 

No. 

Diam. 

Pellets 
per  Oz. 

No. 

Diam . 

Pellets 

perOz. 

Inch.. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Extra  Fine  Dust 

•OI5. 

84021 

9 

Trap 

688 

5 

. 12 

168 

BBB 

.19 

42 

Fine  Dust 

•03 

10784 

9 

.08 

568 

4 

•13 

132 

T 

.2 

36 

Dust 

.04 

4 565  ' 

8 

Trap 

472 

3 

.14 

106 

12 

-05 

2 326 

8 

.09 

399 

2 

•15 

86 

TT 

.21 

31 

11 

.06 

I 346 

7 

Trap 

338 

1 

. 16 

71 

F 

.22 

27 

10 

Trap 

I 056 

7 

. 1 

291 

B 

•17 

59 

so 

.07 

848 

6 

. 1 1 

218 

BB 

. 18 

50 

FF 

•23 

24 

The  scale  of  the  Le  Roy  standard  (adopted  by  the  Sportsman’s  Convention)  com- 
mences with  .21  inch  for  TT  shot,  and  reduces  .01  inch  for  each  size  to  .05  inch  for 
No.  12.  The  number  of  pellets  per  oz.  being  the  actual  number  in  perfect  shot. 

The  number  of  pellets  by  this  standard  is  nearly  identical  with  that  of  the  Amer- 
ican Standard. 

Tatham’s  scale  is  same  as  Le- Boy’s;  but  number  of  pellets  is  deduced  mathemat- 
ically,, by  computing  them  from  the  specific  gravity  of  the  lead. 


Drains.  Diameter  and  G-rade  of,  to..  Discharge  Rainfall. 


Diam. 

Grade  ' 
1 Inch. 

, Acres. 

Diam. 

Grade 
1 Inch. 

Acres. 

Ins. 

Ins. 

4 

3° 

•5 

40 

1.2 

20 

.6 

20 

1-5 

5 

80 

•5 

7 

20 

1.2 

60 

.6 

60 

*■5 

20 

1. 

8 

120 

. L5 

6 

60 

IL 

8q 

i.8, 

Diam. 

Grade 
1 Inch. 

Acres. 

Diam. 

Grade 
1 Inch. 

Acres. 

Ins. 

Ins. 

5-8 

60 

2. 1 

80 

9 

120 

2. 1 

15 

240 

7- f 

80 

2.5 

120 

7.8 

60 

2-75 

80 

9 

12 

, 120 

4-5 

60 

10 

80 

5-3 

18 

240 

10 

British  and  Nletrio  Measures,  Commercial  Equivalents 

of.  (tr.  Johnstons  Stones , F.  R.  S.) 


Length.  Millimeters. 

Inch 9T4-4 

Foot 3°4-^ 

Yard.. 25.4 


Weight.  Grammes. 

Pound 453-6 

Ounce 28.35 

Grain .0648 


Volume.  Cube  Centimeter. 

Gallon.. 4554 

Quart 1136 

Ounce 28.4 


MEMORANDA, 


907 


MEMORANDA. 

Kiysical  and.  MecTianical  Elements,  Constructions, 
and  Results. 

Belting.  Double.  —600  HP  (to  be  transmitted)  -f-  velocity  of  belt  in  feet  per 
minute,  or  191  IP  4-  number  of  revolutions  per  minute  X diameter  of  pulley  in  feet 
— width  in  ins.  Machine  Belts. — 1500  to  2000  BP  -r-  velocity  of  belt  in  feet  per 
minute  = width  in  ins.  ( Edward  Sawyer.) 

Blast  Pipe  of  a Locomotive.  Best  height  is  from  6 to  8 diameters 
of  pipe,  and  best  effect  when  expanded  to  full  diam.  of  pipe  at  2 diameters  from  base. 

Boiler  Riveting.  A riveting  gang  (2  riveters  and  1 boy)  will  drive  in  shell, 
furnace,  etc.,  a mean  of  12.5  rivets  per  hour. 

Brick  or  Compressed  Fuel  is  composed  of  coal  dust  agglomerated 
by  pitchy  matter,  compressed  in  molds,  and  subjected  to  a high  temperature  in  an 
oven,  in  order  to  expel  the  moisture  or  volatile  portion  of  the  pitch  and  any  fire- 
damp that  may  exist  in  the  cells  of  the  coal. 

Bridge,  Highest.  At  Garabil,  France,  413  feet  from  floor  to  surface  of  water, 
and  1800  feet  in  length. 

Bronze,  JVTalleakle.  P.  Dronier,  in  Paris,  makes  alloys  of  copper  and 
tin  malleable  by  adding  from  .5  per  cent,  to  2 per  cent,  quicksilver. 

Building  Department,  Requirements  of.  (New  Yorlc.) 

Furnace  Flues  of  Dwelling  Houses  hereafter  constructed  at  least  8-inch  walls  on 
each  side.  The  inner  4 ins.  of  which,  from  bottom  of  flue  to  a point  two  feet  above 
2d  story  floor,  built  of  fire-brick  laid  with  fire-clay  mortar;  and  least  dimensions  of 
furnace  flue  8 ins.  square,  or  4 ins.  wide  and  16  ins.  long,  inside  measure;  and  when 
furnace  flues  are  located  in  the  usual  stacks,  side  of  flue  inside  of  house  to  which  it 
belongs  may  be  4 ins.  thick.  If  preferred,  furnace  flues  may  be  made  of  fire-clay 
pipe  of  proper  size,  built  in  the  walls,  with  an  air  space  of  1 inch  between  them, 
and  4 ins.  of  brick  wall  on  outside. 

Boiler  Flues  to  be  lined  with  fire-brick  at  least  25  feet  in  height  from  bottom, 
and  in  no  case  walls  of  said  flues  to  be  less  than  8 ins.  thick. 

All  flues  not  built  for  furnaces  or  boilers  must  be  altered  to  conform  to  the  above 
requirements  before  they  are  used  as  such. 

Buildings,  Proteetion  of,  from  Lightning.  A wire  rope  of 
4 lbs.  per  yard  is  held  to  be  the  most  efficient. 

Single  Conductors,  weighing  8 lbs.  per  yard  and  4 lbs.  for  duplicated  and  all  others, 
may  be  located  50  feet  apart,  thus  bringing  every  portion  of  the  building  to  which 
they  are  applied  within  25  feet  of  their  protection. 

Iron  is  the  be^st  material  for  a conductor;  it  should  be  continuous,  and  all  joints 
soldered.  Several  points  are  preferable  to  one.  and  greater  surface  should  be  given 
to  connections  w’ith  the  earth  than  usually  practised.  (Sir  W.  Thompson.) 

For  othe^  information,  see  Van  Nostrand's  Magazine , 1 V.  F,  Aug.  1882 ,page  154. 

Cement.  Iron  to  Stone.—  Fine  iron  filings,  20  parts,  Plaster  of  Paris,  60,  and 
Sal  Ammoniac,  1 ; mixed  fluid  w’ith  vinegar,  and  applied  forthwith. 


Chimney  Braviglit.  W — w h = D.  W and  w representing  weights  of  a 
cube  foot  of  air  at  external  and  internal  temperatures , h height  of  chimney  or  pipe  in 
feet , and  D value  of  draught.  See  Weight  of  Air,  page  521. 

Chinese  or  India  Ink  improves  with  age,  should  be  kept  in  dry  air, 
and  in  rubbing  it  down,  the  movement  should  be  in  a right  line  and  with  very  little 
pressure. 


MEMORANDA. 


908 


Coal,  Effective  Value  off  Theoretical  quantity  of  heat  per  IP  is 
2564  units  per  hour,  and  average  quantity  of  heat  in  a lb.  of  coal  that  is  utilized 
in  the  generation  of  steam  in  a boiler  is  8500  units;  hencej  theoretical  quantity 

of  coal  required  per  IP  per  hour=  = .3  lbs.,  after  the  water  has  been  heated 

into  atmospheric  steam,  being  theoretically  nearly  7.5  per  cent,  of  total  heat  re- 
quired to  change  30  lbs.  water  at  6o°  into  steam  of  60  lbs.  effective  pressure. 

The  total  heat  developed  by  the  combustion  of  coal,  when  utilized  evaporatively, 
ranges  from  .55  to  .8,  but  in  practice  it  does  not  exceed  65  per  cent. 


Coast  and.  Bay  Service.  A velocity  of  current  of  2.5  feet  per  second 
will  scour  and  transport  silt,  and  5 to  6.5  feet  sand.  For  river  scour  the  velocities 
are  very  much  less. 

Cold,  Greatest.  — 2200,  produced  by  a bath  of  Carbon,  Bisulphide,  and 
liquid  Nitrous  Acid. 

Corrosion  of*  Iron,  and  Steel.  The  corrosion  of  steel  over  iron  is, 
as  a mean,  fully  one  third  greater. 

Cost  of  Family  of  Mechanics  in  Erance  ranges  from  $220 
to  $600  per  annum,  of  which  clothing  costs  16  parts,  food  61,  rent  15,  and  mis- 
cellaneous 8. 


Crushing  Resistance  of  Brich.  A pressed  brick  of  Philadelphia 
clay  withstood  a pressure  of  500000  lbs.  for  a period  of  5 minutes. 

Earthwork.  Shovelling.  — Horizontal,  12  feet.  Vertical,  6 feet.  When 
thrown  horizontal,  12  to  20  feet,  1 stage  is  required,  and  from  20  to  30,  2 stages. 
When  vertical,  6 to  10  feet,  1 stage  is  required. 

Wheelbarrow.  — Proper  distance  up  to  200  feet. 


Number  of  Loads  and  Volume  of  Earth,  per  Lay. 
One  Laborer . (C.  Herschell , C.  E.) 


Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume. 

Feet. 

No. 

Cub.  Yds. 

Feet. 

No. 

Cub.  Yds. 

Feet. 

No. 

Cub.  Yds. 

20 

120 

23-5 

150 

96 

13-3 

350 

88 

11. 6 

5° 

no 

16.9 

200 

94 

12.8 

400 

86 

11. 2 

7° 

100 

14.4 

250 

92 

12.4 

450 

84 

10.9 

100 

98 

13.8 

300 

90 

12 

500 

82 

10.5 

Volume  of  a barrow  load,  2.5  cube  feet. 


Portable  Railroad  and  Hand  Cars.—  For  a distance  of  550  feet,  60  cube  yards  can 
be  transported  per  day. 

Horse  Cart.—'V olume  of  Earth  transported  per  Lay. 

One  Laborer. 


Distance. 

Trips. 

Volume. 

Distance. 

Trips. 

Volume.  I 

| Distance. 

Trips. 

Volume. 

Feet. 

No. 

Cub.  Yds. 

Feet. 

No. 

Cub.  Yds. 

Feet. 

No. 

Cub.  Yds. 

300 

86 

17. 1 

1000 

43 

8.6 

2000 

25 

5 

500 

67 

13.6 

I5°°  ; 

3i 

6.4  1 

2500 

21 

4*3 

Volume  of  each  load,  8 cube  feet. 

Ox  Cart  is  less  in  cost  at  expense  of  time.  / 


Electinc  Light,  Candle  Lower  off  Maxim  Incandescent  Lamp.— 
Current  with  30  Faure  cells,  74  volts,  1.81  Amperes,  16  standard  candles.  With  50 
like  cells,  124  volts,  and  3.2  Amperes,  333  candles.  ( Paget  Hills , LL.  D.) 

The  elavated  electric  lights  at  Los  Angeles,  Cal.,  are  distinctly  visible  at  sea  for  a 
distance  of  80  miles. 

Engine  and  Sugar  Mill,  Weights  off  Engine  (non- condensing). 
—Cylinder.— 30  ins.  in  diam.  by  5 feet  stroke  of  piston.  Boilers  (cylindrical  flue).— 
70  ins  in  diam.  by  40  feet  in  length.  Weights.  — Engine,  105000  lbs. ; Boilers , com- 
plete, 75  000  lbs. ; Sugar-mill , 40  ins.  by  8 feet,  220050  lbs. ; Connecting  Machinery , 
137  1 79  lbs.  Cane  carriers,  etc.,  46  787  lbs. 


MEMORANDA. 


909 


Filtering  Stone.  Artificial—  Clay,  15  parts;  Levigated  Chalk,  1.5;  and 
Glass  Sand,  coarse,  83.5.  Mixed  in  water,  molded,  and  hard  burned. 

Fire-engine,  Steam.  Relative  effect  for  equal  cost  compared  with  a 
hand  engine,  as  1 to  1 13.  Each  IH>  requires  about  112  weight  of  engine. 


Floating  Bodies,  Velocities  of.  At  low  speeds  resistance  increases 
somewhat  less  than  square  of  velocity.  In  a Canal,  at  a speed  of  5 miles  per  hour, 
a large  wave  is  raised,  which  at  a speed  of  9 miles  disappears,  and  when  speed  is 
superior  to  that  of  the  wave,  resistance  of  boat  is  less  in  proportion  to  velocity,  and 
immersion  is  reduced. 

Length  of  Vessel. — The  proper  length  for  a vessel  in  feet  (upon  the  wave-line 
theory)  is  fifteen  sixteenths  of  square  of  her  speed  in  knots  per  hour. 


Flow  of  ^Air.  67  y/h  = Velocity  per  second  X C.  h representing  column 
of  water  in  ins.,  and  C a coefficient  ranging  from  56  to  100. 


Circular  orifices,  thin  plate 5^  to  -79 

Cylindrical  mouth-pieces,  short 81  “ .84 

do.  do.  rounded  at  inner  end 92  ‘ .93 

Conical  converging  mouth  pieces 9 “ 1 

Conoidal  mouth  piece,  alike  to  contracted  vein 97  ‘ 1 


Fines,  Corrugated.  ( Wm.  Parker.)  ~ - — Working  stress  in 

lbs.  per  sq.  inch.  T representing  thickness  in  16 ths  of  an  inch , and  D diameter  in  ins. 

Steel,  corrugations  1.5  ins.  deep.  Experiments  upon  a furnace  31.875  ins.  in 
diarn.,  6.75  feet  in  length,  and  with  13  corrugations. 

Foundation  Files.  When  piles  are  driven  to  a solid  foundation,  they  act 
as  columns  of  support,  and  are  designated  Columns , and  when  they  deriv°  their 
supporting  power  from  the  friction  of  the  soil  alone,  they  are  termed  Piles. 

Authorities  differ  greatly  as  to  the  factor  of  safety  for  Piles,  varying  .1  to  .01  of 
impact  of  ram.  ( Weisbach. ) 

As  columns,  their  safe  load  may  be  taken  at  from  750  to  900  lbs.  per  sq.  inch. 
Authorities  give  a higher  value  (Rankine  and  Mahon,  1000);  but  it  is  to  be  borne 
in  mind  that  when  piles  are  driven  to  a solid  resistance,  they  are  frequently  split, 
and  consequently  their  resistance  is  much  decreased. 

As  a rule,  the  following  coefficients  for  ordinary  structures  are  submitted: 

When  the  piles  are  wholly  free  from  vibration  consequent  upon  external  impulse, 
.35  to  .4.  and  when  the  structures  are  heavy  and  exposed  to  irregular  loading,  as 
storehouses,  etc. , .15  to  .2. 

Ordinarily,  the  bearing  of  a properly  driven  pile  not  less  than  10  ins.  in  diam.  may 
be  taken  at  10  tons. 

Friction  of  Bottoms  of  Vessels.  At  a velocity  of  7 knots  per 
hour,  a foul  bottom  requires  2.42  LP  over  that  for  a clean  bottom. 

Friction  of  Planed  Brass  Surfaces  in  muddy  water  is  .4  pressure. 


Gras,  Steam,  and  Hot-air  Engines.  Relative  costs  of  gas,  steam, 
and  air  engines  per  IP:  Otto  Gas  engine,  8.75;  Steam  engine,  3.5;  and  Hot-air 
engine,  4. 


Heat.  Available  heat)  *6431535 

upended  per  IIP  per  hour)  Total  heat  of  combustion  X Coefficient  for  fuel 
consumption  of  coal  per  IIP. 

Coal  14000X772  units  = 10808000.  Theoretical  evaporative  power  — 15  lbs. 

16  431  535 

water.  Efficiency  of  furnace  = .5;  then  10  808  000  X • 5 = 5 404  000,  and  — 

5 404  000 


= 3.04  lbs.  per  IIP  per  hour. 


Ice  Boats,  Speed  of.  Maj.-Gen.  Z.  B.  Tower.  U.  S.  A.,  assigns  the  speed  of 
Ice  boats  at  twice  that  of  the  wind,  and  the  angle  of  sail,  to  attain  greatest  speed, 
to  be  less  than  900. 

Japan  Coal.  Analysis  of  Bituminous. — Specific  Gravity,  1.231.  Carbon, 
77.59.  Hydrogen,  5.28.  Oxygen,  3.26.  Nitrogen,  2.75.  Sulphur,  1.65  Ash,  8.49. 
and  loss,  .98. 

Its  evaporative  effect  = 4. 16  lbs.  water  per  lb.  of  coal. 


910 


MEMORANDA. 


Lee-way.  A full  modelled  vessel,  with  an  immersed  section  of  i to  6 of  her 
longitudinal  section,  and  with  an  area  of  36  sq.  feet  of  sails  to  1 of  immersed  sec- 
tion, will  drift  to  leeward  1 mile  in  6.  A medium  modelled  vessel,  with  an  im- 
mersed section  of  1 to  8,  and  with  like  areas  of  sail  and  section,  will  drift  1 in  9. 

LigRt,  Standard,  of.  Photometric , English. — Spermaceti  candles,  6 per 
lb. ; 120  grains  per  hour.  Carcel  burner  = 9.5  candles. 

Locomotive  .A.xles,  Friction  of.  .016  of  weight.  Hence,  if  radius  of 
wheel  = .1,  axle  friction  at  periphery  -~ 10  ==  3.73  at  periphery. 

Mercurial  Grange.  To  prevent  freezing,  apply  or  introduce  Glycerine  on 
top  of  column. 

Metal  Products  of  TJ.  S.,  18SS.  Value,  $222000000. 

Mississippi  Diver,  Silt  in.  Near  St,  Charles  the  volume  of  silt 
borne  per  day  in  1879  was  475457  cube  yards,  and  on  one  day,  July  3.  it  was 
4 113600.  At  times  the  volume  equals  3 ozs.  per  cube  foot  of  water. 

Motive  Power.  A sailing  vessel  having  a length  6 times  that  of  her 
breadth,  requires,  for  a speed  of  10  knots  per  hour,  an  impelling  force  of  48  lbs.  per 
sq.  foot  of  immersed  section. 

Mowing  Machine.  Kirby's  (Auburn,  N.  Y.) — 670  lbs.,  2 horses,  1 acre 
heavy  clover  in  46  min. 

Ordnance,  Energy  of.  In  a competitive  test  ot  a 9-incli  Woolwich 
gun,  and  a 5.75-inch  Krupp,  the  energy  per  inch  of  circumference  of  bore  was  re- 
spectively 1 18  and  123  foot-tons;  their  penetration  therefore  by  the  wrought-iron 
standard  being  about  the  same,  but  their  total  energies  were  respectively  16400 
and  5800  foot-tons. 

At  Mepper  a shot  of  no  lbs.,  with  a velocity  of  1749  feet  per  second,  and  a strik- 
ing energy  of  2300  foot-tons,  passed  through  a target  composed  of  two  plates  of  soft 
wrought  iron  7 ins.  thick,  with  10  ins.  of  wood  between  them,  and  passed  800  yards 
beyond. 

Petroleum.  One  lb.  crude  oil  heated  1 lb.  water  315. 750  = 28.21  lbs.  water 
at  6o°  converted  to  steam  at  2120.  Relative  evaporative  effects  of  Oil  and  Anthra- 
cite coal  as  1 to  3.45. 

Dopnlation,  Comparative  Density  of,  and  jNT umber  of 
Persons  living  in  a House  in  different  Cities. 

Chicago,  4;  Baltimore  and  Naples,  4.5;  Philadelphia,  6;  London,  Boston,  and 
Cairo,  8;  Marseilles,  9;  Pekin,  10;  Amsterdam,  n ; New  York,  13.5;  Hamburg, 
17.07;  Rome  and  Munich,  27;  Paris,  29;  Buda  Pesth,  34.2;  Madrid,  40;  St.  Peters- 
burg, 43.9;  Vienna,  60.5;  and  in  Berlin,  63. 


Dower  of  a Volcano.  An  eruption  of  that  of  Cotopaxi  has  projected 
a mass  of  rock  of  a volume  of  100  cube  yards  a distance  of  9 miles. 

Dower  Required  to  Dx*aw  a "Vessel  or  Load  up  an  In- 
clined Hydrostatic  Dail  or  Slip  "Way.  (Wm.  Boyd,  Eng.) 

WI  = R;  Cc?W-f-D  = F;  and  P d'  c =f  W representing  weight  of  vessel , or 
load  and  cradle , I inclination  of  ways , as  length  rise , R resistance  of  vessel  or  load , 
F friction  of  cradle  and  rollers , and  f friction  of  plunger  in  stuffing-box , all  in  tons, 
C and  c coefficients  of  friction  of  cradle  and  stuffing-box , d diameter  of  axle  of  rollers, 
d'  product  of  circumference  of  plunger  and  depth  of  collar  or  stuffing , all  in  ins.,  and 
P pressure  per  sq.  inch  on  plunger , in  lbs. 

Hence,  W = I,  and  R 4-  F -4-/ = power  in  tons. 

’ length  ’ 


Illustration.  — Assume  weight  of  a vessel  and  cradle  2000  tons,  pressure  on 
plunger  2500  lbs.  per  sq.  inch,  inclination  of  w*ays  1 in  20,  diameters  of  axle  of  roll- 
ers and  of  rollers  3 and  10  ins.,  depth  of  collar  2 ins.,  and  circumference  of  plunger 
50;  what  would  be  the  power  required?  C = .2,  and  c = .6. 


2000  . .2  X 3 X 2000  . 2500  X 2 X 5°  X - 6 

Then = 100  tons  ; - = 120  tons  ; - = 67  tons  ; 

20  10  2240 

and  100 -j- i2o-j- 67  = 287  tons. 


2240 


MEMORANDA. 


91 1 


Qt„„TTlfir  Ordinary  OistriDntion  of  Power 
If  PP0wer  developed  by  engine,  88  IH> ; Power  expended  in  its  operation, 

Friction  of  load  .^ys"*'  I Power  expended  by  slip  of  propeller. ...  14 

Friction  ; ” | “ “ m propulsion 71 

-C3  _ Contrifuaal  has  lifted  water  28  to  29  feet,  drawn  it  horizontally  800 

feeYanTt Also  drawn  it  24  feet,  and  projected  it  50  feet. 

’ Trains.  Power  and  Resistance. — A railway  train  running  at 

rat^f^omi^s  per  hour  = 88  feet  per  second,  and  velocity  a body  would  acquire 

• * qq  fppt  — 88  — 8 o22=  120.3  feet.  Consequently,  in  addition  to  power 

m m resistance  to  train,  as  much  power  must  be 

expended  to  "n  motion  at  lids  speed,  as  would  lift  it  in  mass  to  a beigbt  of 
1 21  feet  in  a second. 

Tf  thP  train  weiehed  100  tons  = 224000  lbs.,  then  224000  X 120.3  = 26747200 
foot  lbs  and  if  this  result  was  obtained  in  a period  of  5 minutes,  it  would  require 
1^  3 -J  5 X224000-  33 000  = 163. 3 H>  in  addition  to  that  required  for  frictional 

rGTo  raise  the  speed  of  a train  from  40  (58.66  feet  per  second)  to  45  (66  feet  per  sec- 
ond^ miles  per  hour,  the  power  required^in  addition  to  that  of  friction  would  be  as 

c8  66^8^02  = 53-44  feet  is  to  66 -=-8.02  = 67. 57  feet  = 67.57  — 53-44  = 1 4- 1 3 feet. 
Assume  a train  of  100  tons,  running  at  rate  of  60  miles  per  hour,  and  total  retard- 

in  a nower  at  1 its  weight  100-4-10  = 10.  Then  224000  X 10  X 120.3  — 26947  200  . 

22^400  = 1203  feet,  which  train  would  run  before  stopping.  If,  however,  tram  was 
ascending  a grade  of  i in  ioo,  the  retarding  force  — .11  (n  • IO°)  ® 

^ 640  distance  in  which  tram  would  come  to  rest  would  be  26947200^24640  = 
1093.6  feet. 

Relative  Non-condixctibility  of  Materials. 


Material.. 

Per  cent. 

Material. 

Per  cent. 

Material.  Percent. 

Hair  fait.  

100. 

83.2 

7i-5 

68 

Mineral  wool, No.  1 
Charcon  1 

67-5 

63.2 

55-3 

55 

Lime,  slacked 

Asbestos 

48 

36*3 

34-5 

13.6 

Mineral  Wool,  No.  2 
“ “ and  tar 

Sawdust 

Pine  wood 

Coal  ashes 

Loam 

Air  space,  2 ins. . . 

Resistance  to  a S team -vessel  in  au-  mm  w mer.  In  air 
10  per  cent,  of  IIP,  and  in  waiter,  at  a speed  of  20  miles  per  hour,  90  per  cent.,  or  8 
IIP  per  sq.  foot  of  immersed  amidship  section. 

Saws,  Circular.  30  ins.  in  diameter,  are  run  at  2000  revolutions  per  minute 
= 3-57  rail’es- 

SpUr  Gf  ear  has  been  driven  at  a velocity  of  1 mile  per  minute. 

Sugar  Mill  Rollers.  5 feet  by  28  ins.,  at  2.5  revolutions  per  minute, 
requires  20  IP,  and  18  feet  per  minute  is  proper  speed  of  such  rolls. 

Surface  Condensation,  Experiments  on.  (B.  G.  Nichol.) 
Tube  of  Brass , .75  Inch  External  Diameter.  No.  18  B W G,  Surface  = 1.0656 
sa.  feet.  Duration  o f Experiment , 20  Minutes. 

Horizontal. 


Steam. 

Vertical. 

Temperature 

255° 

256° 

Pressure  per  sq.  inch  per  gauge. . . 

17.75  lbs. 

18.25  lbs. 

Condensation  by  tube  surface 

18.5835  “ 

29.9585  “ 

“ per  sq.  ft.  of  “ per  hour 

52-32  “ 

84.34 

Condensed  during  experiment 

19.0625  “ 

1 30-4375  “ 

253° 

254° 

16.75  lbs. 

17.25  lbs. 

24.0835  “ 

43.0835  “ 

67.8 

121.29  “ 

24.5625  “ 

43-5625“ 

lbs. 


Steamers^  icngmes,  w eignts  ox.  cmymx,,  m 

Fittings  ready  for  Service  per  IIP. 

Mercantile  steamer 480  lbs.  I Light  draught 280 

English  Naval  “ 360  “ | Torpedoes 60 

Ordinary  Marine  Boiler  with  Water 196  lbs. 

"Wind,  Pressure  of.  Estimate  of  upon  Structures.  — 30  lbs.  per  sq.  foot. 
Per  lineal  foot  of  a locomotive  train  = 10  feet  in  height,  300  lbs.  per  sq.  foot, 

A Tornado  has  developed  a pressure  of  93  lbs.  per  sq.  foot. 


912 


MEMORANDA. 


"Via,  Suez  Canal.  Passages  by  Steamers.— 1882,  “ Stirling  Castle,”  Shang 
hai  to  Gravesend,  in  29  days  22  hours  and  15  min.,  including  1 day  22  hours  and  ~>q 
min.  in  coaling  and  detentions.  ' 0 

“ Glenare ,”  Amoy  to  New  York,  N.  Y.,  in  44  days  and  12  hours , including  deten- 
tion at  Suez.  From  Gibraltar  in  n days. 

Zinc  Foil  in  Steam-boilers.  Zinc  in  an  iron  steam-boiler  consti- 
tutes a voltaic  element,  which  decomposes  the  water,  liberating  oxygen  and  hydro- 
gen. The  oxygen  combines  with  fatty  acids  and  makes  soap,  which,  coating  the 
tubes,  prevents  the  adhesion  of  the  salts  left  by  evaporation.  The  mealy  deposit 
can  then  be  readily  removed. 

Files.  To  Compute  Extreme  Load  a Foundation  Pile  will  Sustain. 

R2  h 

p „ “ - = L.  R representing  weight  of  ram , P weight  of  pile , and  L extreme 

* T~  tv  X S 

load , all  in  lbs.;  h height  of  fall  of  ram,  and  s distance  of  depression  of  pile  with  last 
blows , both  in  feet. 

Illustration. — Assume  a ram  1000  lbs.  to  fall  20  feet  upon  a pile  of  400  lbs., 
what  resistance  will  the  earth  bear,  or  what  weight  will  the  pile  sustain  when 
driven  by  the  last  blow,  from  a height  of  20  feet,  .5  inch? 

s ==.5  of  12  ins.  =.0416. 


Then 


10002  X 20 
400  -J-  1000  X .0416 


20  000  000 
58.24 


= 343  406  lbs. 


Perimeter.  The  limits  or  bounds  of  a figure,  or  sum  of  all  its  sides. 

Of  a canal  it  is  the  length  of  the  bottom  and  wet  sides  of  its  transverse  section. 


Flood  Wave.  The  flood  wave  of  the  Ohio  River  in  March  (1884)  was  71 
feet  1 inch  at  Cincinnati,  being  higher  than  that  of  any  previous  record. 


Ice.  Crushing  Strength  of,  as  determined  by  U.  S.  testing  machine,  ranged 
from  327  to  1000  lbs.  per  sq.  inch. 

Atmosphere.  If  pure  air  is  exhausted  of  2.5  per  cent,  of  its  oxygen,  it  will 
not  support  the  combustion  of  a candle. 

Blasting  Paper.  Unsized  paper  coated  with  a hot  mixture  of  yellow 
prussiate  of  potash  and  charcoal,  each  17  parts;  refined  saltpetre,  35;  potassium 
chlorate,  70;  wheat  starch,  10,  and  water,  1500. 

Dry,  cut  into  strips,  and  roll  into  cartridges. 

Circular  Saws.  Speed,  9000  feet  per  minute.  Thus,  for  an  8 ins.,  4500 
revolutions,  and  progressively  up  to  a 72  ins.,  500  revolutions.  (Emerson.) 


Foods. 


Relative  Value  of,  compared  witL.  IOO  Lbs.  of 
Good  Bay. 

Additional  to  page  203. 


Lbs. 


Lbs. 


Lbs. 


Acorns 68 

Barley  and  Rye,  mix’d  179 

Barley  straw 180 

Buckwheat 64 

Buckwheat  straw 200 


Linseed 59 

Mangel-wurzel 339 

Pease  and  Beans 45 

Pea-straw 153 

Potatoes 175 


Rye 54 

Turnips 504 

Wheat 46 

Wheat,  Pea,  and  Oat- 
chaff 167 


Depth  of  tlie  Ooean.  Mean  depth  is  estimated  by  Dr.  Krummel  at 
1877  fathoms  = .4624  geographical  mile. 


Gas-engine.  A gas-engine  1.5  actual  IP  will  cost,  with  gas  at  8 cents  per 
hour,  10  cents  per  hour  for  10  hours.  (Am.  Engineer.) 


Locomotive.  Average  daily  run  100  miles  at  a cost  of  $ 12.80  for  driver, 
fireman,  fuel,  and  repairs.  (A'.  J.  Central  R.  R.  Co.) 

Consumption  of  Fuel  per  Mile.  Passenger , 25  to  30  lbs.  coal.  Freight , 45  to  55 
Ibs.3  or  one  cord  wood  per  40  miles. 


MEMORANDA. 


9 13 


Masonry.  In  laying  stones  in  mortar  or  cement,  they  should  rest  upon  the 
course  beneath  them,  more  than  upon  the  material  of  joint. 

Steel  Gr  un  (Krupp’s).  Bore,  15.75  ins.;  length  of  bore,  28.5  feet-  of 
gun  32.66  feet.  Weight,  72  tons.  Charge,  385  lbs.  prismatic  powder;  projectile 
chilled  iron,  1660  lbs.,  with  an  explosive  charge  of  22  lbs.  of  powder.  ’ 

Moment  of  shot  at  muzzle,  estimated  at  31  000  foot-tons,  and  range  15  miles. 

Saw-Mill.  7722  feet  of  1 inch  Poplar  boards  in  One  Hour. 

Engine  (Non-condensing).  Cylinder. — 12  by  24  ins.  stroke  of  piston. 

Boilers.  Two  (cylindrical  flued),  38  ins.  in  diam.  by  26  feet  in  length,  two  14  ins 
return  flues  m each  Heating  Surface.— 7 80  sq.  feet.  Grates .-42. 5 sq.  feet. 

Pressure  of  Steam.— 125  lbs.  per  sq.  inch,  cut  off  at  16.5  ins. 

^Revolutions.— 2 so  to  350  per  minute.  Saws.  — Two  circular,  60  and  66  ins.  in 

Note.—  Grates  set  28  ins.  from  under  side  of  boilers,  without  bridge-wall  and  a 
combustion  chamber  under  boilers,  4 feet  in  depth.  Fuel,  sawdust.  ’ 

Steam  Heating.  62  500  cube  feet  of  space  requires  6000  sq.  feet  of  heat- 
ing surface  to  attain  a temperature  of  700  in  the  vicinity  of  the  city  of  New  York 
in  its  coldest  weather.  J u 

Or,  One  sq.  foot  of  iron  pipe  will  heat  10. 5 cube  feet  of  space  in  an  ordinary  build- 
ing, temperature  of  exterior  air  70°.  (Felix  Campbell.)  * 

o °r  Steam  at  a Pressure  of  60  lbs.  + atmosphere  has 

a \ elocity  of  efflux  of  890  feet  per  second,  and  as  expanded,  a velocity  of  1445  feet. 

Blasting.  In  small  blasts  1 lb.  powder  will  detach  4.5  tons  material  and  in 
large  blasts  2.75  tons.  (See  page  443. ) dl>  na  m 


Delta,  Metal  (Iron  and  Bronze).  Specific  gravity  8 4 
(See  page  384).  b J 4 


Jarrah  Wood  of  Australia. 
A avails. 


Melting  point  1800°. 
Impervious  to  insects  and  the  Teredo 


Relative  water  evaporat- 


G as  an<i  Bituminous  Coal, 
ing  powers.  Gas,  20  to  21  lbs.  Coal,  9 lbs. 

,yessel.s*  For  each  f00t  of  depth  of  hold  (from  ceiling 
der  side  of  main  deck),  ,1  inch  added  to  1.5  ins.  for  a depth  of  8 feet.  Thus 
for  24  feet  depth  1.5,-f- . 1 X 8 co  24  = 3. 1 ins.  ( American . ) 

Or,  2 ins.  for  8 feet  depth  and  .1  for  each  foot  in  addition  thereto.  (Lloyd's.) 

Colors  for  NVorking  Drawings. 


Brass Gamboge. 

Bricks Carmine. 

Clay Burnt  Umber. 

Concrete Sepia  with  dark  markings. 

Copper Lake  and  Burnt  Sienna. 

Granite India  Ink,  light. 

Iron,  cast . . .Neutral  tint. 

“ wrought. Prussian  Blue. 

Lead Ind.  Ink  tinged  with  P.Blue. 


Steel Neutral  tint,  light* 

Water Cobalt. 

Wood Burnt  Sienna. 


f Burnt  Umber. 
Ye 


Stones  I bellow  Ochre. 

and  -l  and  Black. 

Earths  . . | “ and  B’t  Umber. 

Red  and  Indigo. 

(.  Burnt  Sienna  and  Indigo. 


Cllairi  Cable-  Square  of  diameter  of  chain  in  ins.  mul- 
tiplied  by  .35  will  give  volume  of  sp^ce  required  to  stow  1 fathom. 

; pat  aS  re ’in  ot^part  BitU“Cn  1 Part'  P°WdCTed  aSphalt  ™ P"*.  •»»«  -6 
Melt  bitumen,  add  asphalt  broken  small,  than  resin  oil  and  sand. 

Asphalt  Concrete.  Asphalt  mortar  u parts  and  broken  stone  9 parts. 

ni±^eStOS  iS  a flb:ous,  variety  of  Actinolite  or  Tremolite,  composed  of  silica 
aciS.  ’ magnesia’  oxlcle  of  iron-  water.  It  resists  heat,  ’moisture,  and  many 


memoranda. 


914 

TP r>r><l  of*  an  Esquimau.  Flesh  ot  a sea-horse  8.5  and 
BrSd  I M l£.%%  -5,  Spirits  »,  -d  Water  ,9  pint.  (Sir  W.  * Parry. ) 

r-oio-net’s  Concrete.  For  walls  that  resist  moisture.—  Sand,  Gravel,  and 
Pebbles^  parts;  Argillaceous  Earth  3 parts,  and  Quicklime  1 part. 

Hard  and  quick  setting.—  Sand,  Gravel,  and  Pebbles,  8 parts;  Earth,  burned  and 
powdered  Cinders,  each  1 part,  and  Unslacked  hydraulic  Lime  1.5  parts.  For  a very 
hard  mixture,  add  cement  1 part. 

Transmission  or  Conductivity  of  Temperature 
TTartli  At  Edinburgh  thermometers  set  at  a depth  of  t6  feet  in  the  earth  at- 
tained their  maximum  and  minimum  at  about  six  months  after  the  corresponding 
maximum  and  minimum  of  the  surface,  being  lowest  or  coldest  in  July. 

The  average  rate  of  transmission  of  heat,  as  observed  at  Schenectady,  N Y. , was, 
downwards,  2.9  feet  per  month,  and  upwards  3.4  feet.  (Ohn  H.  Landreth.) 

^Tiafts  When  loaded  transversely,  the  diameters  of  the  journal  should  first 
be^etermined  its  dimensions  then  at  any  other  point  can  be  deduced  from  those 
dLmeteS  it’being  observed  that  the  diameters  at  any  two  points  should  be  pro- 
portional to  the  cube  roots  of  the  stress  at  those  points. 

TnumaU  For  operation  at  high  speed  a greater  length  is  required  than  for  low 
th?  less  may  Se  its  diameter  for  a given  stress,  and 

consequently  the  friction  will  be  less, 
lengthen. 

,,=£ SSSHSSS 

as  1. 25  of  above  value.  ( W.  C.  Unwin. ) 

Ordway^o^New^ Meansi he  ^ernaine^th'e 
following  materials,  compared  with  a naked  pipe,  to  be. 


Hair-felt,  burlap 

Asbestos  paper,  hair-felt,  duck. 

Pine  charcoal * 

Air  space  . 


Cork  in  strips 

Rice- chaff. 

Clay  and  vegetable  fibre  . . 
Naked  pipe 


( Engineering , vol.  39,  page  206.) 

_ . ..  „ t oona  Height  between  decks 

jfc 

space  of  6 by  2 feet. 

Ham.moclcs.-To  compute  number  that  can  bewung  under  a deck. 

;~3  xl-n.  I representing  length  under  deck  in  feet,  and  b breadth  in  ins. 
6 16 

[Sir  G.  Wolseley.) 

~ _oo  lbs  water  evaporated  into  dry 

( Centennial  Exhibition , 1876. ) 34.  5 lbs.  water  as  abov  e trom  ieeu  at 

at  2120.  [Am.  Soc.  Mechanical  Engineers.) 


MEMORANDA. 


9*5 


of  Liebt  in  Water.  Mediterranean,  clear  sunlight. 
In  March,  at  a depth  of  1200  feet;  in  winter,  600  feet.  [M.  M.  Fol  and  Sarasm.) 

Railroad.  Horse.  First  in  operation  in  1826-7. 

Needles.  First  in  use  in  1545. 

Iron  Steamers.  First  build  in  1830. 

Lucifer  Nlatcli.  First  made  in  1829. 

Watches.  First  constructed  in  1476. 

Load  on  Stone  per  sq.  foot  Church  of  All-Saints  at  Angers,  86000  lbs. 
P<#theon  at  Rome,  60000  lbs. 


Flexible  Faint  for  Canvas.  Yellow  soap  1.66  parts, 
water  1.  Grind  while  hot  with  .83  parts  oil  paint. 


Boiling 


Fuel.  Evaporation  of  9 lbs.  water  from  2120 : 


lb.  good  coal. 

2 lbs.  dry  peat. 

3.25  “ cotton  stalks. 
3.75  “ wheat  straw. 


.75  lb.  petroleum. 

2. 5 lbs.  dry  wood. 

3.5  ‘‘  brushwood. 

4 “ megass,  or  cane  refuse. 


Tramways  ox*  Steel  !Raili*oad.s. 

Resistance  on  straight  and  level  tracks  15  to  40  lbs.  per  ton,  or  an  average  of 

30 Power  required  on  a good  track  to  start  a car,  as  determined  by  A.  W.  Wright, 
M W S E iVfi  s lbs.,  and  to  maintain  it  in  motion  17.2  lbs.  C.  E Emery  Ph.  D„ 
made^t  ij  lbs  5 On  a bad  track,  the  power  is  134.6  lbs.  to  start,  and  35  lbs.  to  main- 

‘“jpower  required,  as  determined  by  Mr.  Wright,  to  start  a car  with  an  average  load 
and  day’s  work  is  33.53  H>,  and  to  maintain  it  in  motion  133.22  H>. 

Average  work  of  a car-horse  5. 75  hours  per  day  for  a term  of  service  of  6 years. 
Strong  (fraught-horses  will  exert  a power  of  143  lb&  @ 2.75  miles  per  hour  for  22 
milesfand  an  ordinary  one  121  lbs.  per  25  miles.  (Gayffier.) 

Cable  Railway.  Mr.  Wright  gives  the  power  required  per  ton*  at  1.92  IP. 

* All  tons  here  and  elsewhere  are  given  at  2240  lbs. 


'T 


gi  6 


APPENDIX. 


APPENDIX. 

River  Steamboat.  Wood  Side  "Wheels. 
ITreiglit  and  Passenger. 

“ Bostona.  ” — Horizontal  Lever  Engines  {Non-condensing).— Length  on  deck 
302  feet  10  ins. ; beam , 43  feet  4 ins.;  hold , 6 feet.  Tons , 993.52. 

Immersed  section  of  light  draught  of  26  ws.,  83  sg./ee*.  Capacity  for  freight  1200 
tons  (2000  lbs.).  ’ 

Cylinders.— Two  of  25  ins.  in  diam.  by  8 feet  stroke  of  piston. 

Boilers.  — Four  of  steel,  47  ins.  in  diam.  by  30  feet  in  length,  6 flues  in  each 
Heating  surface , 903  sq.  feet.  Grate  surface , 98  sq.  feet.  £ 

Pressure  of  Steam,  154  lbs.  per  sq.  inch,  cut  off  at  .625. 

Revolutions , per  minute.  Speed , 10  miles  per  hour  against  current  of  upper 

Ohio,  3 to  5 miles. 


To  Compute  N£eta-cen  tre  of  Hall  of  a Vessel. 
Operation  of  Formula  in  Naval  Architecture,  page  660. 

Assume  a sharp-modelled  yacht,  45  feet  in  length,  13.5  feet  beam,  and  9.5  feet 
hold,  with  an  immersed  amidship  section  of  42  sq.  feet,  and  a displacement  of  goo 
cube  feet  at  a mean  draught  of  water  of  6 feet. 

2 /» y3  d x 

- f - — — — = Meta-centre.  See  pages  650,  659. 

Ordinates  (dec)  taken  at  intervals  of  2.5  feet  are  as  follows: 


.216 

2-x97 


y = o = 
y 1 3 = .63  = 

y23  = i.33  = 

y 3 3 — 2 3 = 8 

y 4 3 = 2.8s  = 21.952 
y53  — 3.63  _ 46.656 

y63—  53  =125 

y7  = 5 83  ~ 195.  II2 


y8  =6.53  = 287.496 
y 9 3 = 6.7s  = 300.763 
yio3  — 6.75  = 307.547 
y11  =6.5  =287.496 
y12  = 6.25  = 244.14 
yI33  = 5.8  =195.112 

y,433=  5 = 125 

y15  =4.2  = 74.088 


yl6  = 3*25  = 34. 328 

yI73  = 2>4  =13.824 
yl8g  — 1,5  = 3-375 
y19  = .8  = .512 

ya°3=  o = .0 

2272.814 

2-5 


5682.035 

Summation  of  function  of  cubes  of  ordinates  for  value  of  / y3  dx  = 5682.035. 

And  - 0f  5682.035  0f  6. 3i  = 4.21  feet. 

3 9°°  3 

Note. — The  other  elements  of  this  vessel  are: 

Area  of  load-line , 401.12  sq.  feet ; Displacement  in  weight , 27.974  tons ; do.  at  load- 
draught,  .955  tons  per  inch;  Depth  of  centre  of  gravity  of  displacement  below  load- 
line,  1.49  feet;  Volume  of  displacement,  to  volume  of  immersed  dimensions,  26.8 
per  cent. 

To  Compute  Height  of  Jet  in  a Ooxidnit  IPipe  from  a 
Constant  Head.  (Weisbach.) 
h v2  h' 

TV : — tv  . . = ■ — ‘ — h,  and  — = h".  h,  h',  and  h"  representing  heights 

I + (C+C'd)(v)  2" 

due  to  velocity  of  efflux,  loss  of  head  and  of  ascent,  l length  o f pipe  or  conduit,  and  d 
and  d'  diameters  of  pipe  and  jet,  all  in  feet,  v velocity  of  efflux  in  feet  per  second,  C 
and  C'  coefficients  of  friction  of  inlet  of  pipe  and  outlet , and  z a divisor  determined 
by  experiment  with  diameters  of . 5 to  1.25  ins. , ranging  from  1.06  to  1.08. 

Illustration. — If  conduit  pipe  for  a fountain  is  350  feet  in  length,  and  2 ins.  in 
diameter,  to  what  height  will  a jet  of  .5  inch  ascend  under  a head  of  40  feet? 
Assume  C andC"  .8  and  .5,  h — 25  feet,  d — 2 ins.  = .166,  and  .5  = .5-4-12  = .0416. 

Then 25 


1 -f  ^.8-f  .5 


--4.9  feet. 


66/ 


APPENDIX. 


917 


To  Compute  Head  and  Discharge  of  Water  in  Dipes  of 
Grreat  Length. 

It  becomes  necessary  first  to  determine  the  velocity  of  the  flow,  which  is  = 
4 v —v  — !.27o  — independent  of  friction.  V representing  volume  of  water 
3. 1416  d2  0 d~  ... 

in  cube  feet,  and  d diameter  of  pipe  in  ms. 


When  head,  length,  and  diameter  of  pipe  are  given 


V2  g h 


J 


i+C  + c- 


Coefficients  of  friction  C,  for  velocity  of  flow,  range  from  .0234  to  .0191  for  veloci- 
ties from  3 to  13  feet  per  second,  and  c that  for  the  pipe  as  a mean  at . 5.  See  Weis- 
bach’s  Mechanics,  Vol.  i.,  page  431. 

Illustration.— What  head  must  be  given  to  a pipe  150  feet  in  length  and  5 ins 
in  diameter,  to  discharge  25  cube  feet  of  water  per  minute,  and  what  velocity  will 
it  attain  at  that  head?  - - J 


C^.024  and  c — . 5* 


Then  1.273  ^ = *-273  X 2.4  = 3.055  feet  velocity  per  second , and 

(1+.5  + .024  150  X--2)  |^  = i.5  + 8;64X.  14  = 142/^  head. 

\ 5 / 64.33  _ 

•x/ds  c !l  V2  _ . . 

Or,  4. 72  — - — V in  cube  feet  per  minute , and  . 538  P / — r—  — “ 171  ins- 

y/ 1 -r-h  v 

Illustration.— Assume  elements  of  preceding  case. 

Then  4.72  ^3I25—  = 4 ?2  * = 25'67  cubefeet> aDd  -538  $JlS°  * **— 

V 150  -‘r-  1.42  10.25 

= . 538  X ^69  607  = . 538  X 9-  301  — 5 ins- 

To  Compute  Fall  of  a Canal  or  Open  ConcUxit  to  Con- 
duct and  Discharge  a.  Given  Volume  of  Water  per 

Coefficient  of  friction  in  such  case  is  assumed  by  Du  Buat  and  others  at 
.007  565. 

C lJL  x — = h.  h representing  height  of  fall,  l length  of  canal,  and  p net  perime - 
ter,  all  in  % ; A area  of  section  of  canal  in  sq.  feet,  and  v velocity  of  flow  in  feet 
per  second. 

Illustration  i -What  fall  should  he  given  to  a canal  with  a section  of  3 feet  at 
botlom  7 at  top,  and  3 in  depth,  and  a length  of  2600  feet,  to  conduct  40  cube  feet 

of  water  per  second  ? 

C = .oo76,  p = 3 -f  ( V32  + 22  X 2)  = 10. 21  feet,  A = 7 + ^ ~ = 1 5 sq.  feet,  and 

v — — 2. 66  feet. 

Then  .00,6  -600XKX2Z  x 2A6=  = I3  45  x = M8/eef. 

' 15  64.33 

2.— What  is  volume  of  water  conducted  by  a canal,  with  a section  of  4 feet  at 
bottom,  12  at  top,  and  5 in  depth,  with  a fall  of  3 feet,  and  a length  of  5800  feet? 

Jf—X^gh-v.  A = U+UX5  _ 40  sq.  feet,  and  p = 4 + (V52  + 4"'  X 2}  = 

16.8  feet. 


f 


40 


^3  X 64.33  x 3 = J X >93  = 3-3  feet,  and 


V .0076  X 5800  X 16.  i 
40  X 3. 23  feet  velocity  = 129.  2 cube  feet. 

For  Dimensions  of  transverse  profile  of  a canal,  see  Weisbach,  page  492,  vol.  i. 


918 


APPENDIX. 


STEAM,  VACUUM,  AND  HYDROSTATIC  GAUGES.  (Crosby’s.) 


INCHES. 


Diameter  of  Dial. . 

12 

10 

8.5 

6-75 

6 

5.5 

4-5 

3-5 

2-5 

Brass  No 

00 

0 

1 

2 

3 

4 

5 

6 

7 

Iron  No 

— 

_ 

i-5 

2.5 

3-5 

4-5 

5-5 

6-5 

7-3 

ADJUSTABLE-POP 

SAFETY-VALVES. 

(Crosby’s.) 

INCHES. 

Diameter  of  Valve. 

1 

| 1-25 

1 1,5 

1 2 

! 2-5 

I 3 

| 3-5 

1 4 

1 5 

Capacity  in  IP 

10 

1 20 

1 3° 

1 50 

1 .&> 

100 

150 

| 200 

1 3°° 

STEAM  SIPHON.  An  Independent  Lifting  Pump. 
Capacity  for  a Discharge  Pipe  2 Ins . in  Diameter,  per  Minute. 


Water  raised. 

j Pressure. 

Discharge. 

Water  raised. 

Pressure. 

Discharge. 

Feet. 

Ins. 

Lbs. 

Gallons. 

Feet. 

Ins. 

Lbs. 

Gallons. 

14 

6 

30 

63.54 

13 

2 

60 

119.68 

13 

2 

40 

85-71 

13 

2 

70 

138.44  f 

13 

2 

50 

IOO 

13 

2 

80 

157-57 

DISTANCES,  VELOCITIES,  AND  ACCELERATION. 

To  Compute  Velocities  of  an  Accelerated  Body. 


fv2  -j-  (2  v'  S),  Or,  v -f  t v'  = V.  v and  v'  representing  original  and  accelerated 
velocities,  and  V final  velocity , all  in  feet  per  second  ; S distance  or  space  passed  over 

in  feet , and  t time  in  seconds . ~ — = v'-  v'  representing  average  velocity  in 

feet  per  second.  V' t ==  S,  arid  2 V'  — V = v. 

Illustration  i. A body  moving  with  a velocity  of  10  feet  per  second,  is  acceler- 

ated at  rate  of  4 feet  per  second,  per  second,  for  a period  of  6 seconds;  what  are  its 
different  velocities? 
v = 10,  v'  = 4,  t — 6. 

Then,  10  -f  6l<4  = 34  feet  final  velocity.  10  = 22  feel  average  velocity. 


22X6  = 132  feet  distance  passed  over.  Vio2  -f  (2  X 4 X 132)  = VIJ&  — 34  feet> 
and  2 X 22  — 34  = 10  feet  original  velocity. 


k , V — v 
And,  — = v , 


V + t> 


X « = s, 


-Xt  = v'  S,  r2  -j-  2 v’  S = V2, 


V — v 


— — t,  and  \/V2  — 2 V S = v. 


2.- A body  is  projected  vertically  with  a velocity  of  200  fee' t per  se ^d,  and  is 
retarded  at  the  rate  of  30  feet  per  second,  per  second;  what t J 
passed  through  when  its  velocity  is  reduced  to  80  feet  per  second,  and  in  what  time . 

v — 200,  v'  = 30,  and  V = 80. 


v = 30 , 
Then 


200  — 80 

30 


= 4 seconds. 


80  4-  200  , 

— — X 4 = 560  feet. 


A vehicle  being  drawn  with  a velocity  of  25  feet  per  second  is  accelerated 
5 feet  per  second,  per  second;  what  is  its  velocity  and  time  of  operation  at  the  end 
of  100  feet? 


v = 25, 


Then 


5,  and  V = 100. 

25 


— iK  seconds. 


3 + 25  , 


5 


APPENDIX. 


. A stream  of  water  after  flowing  a distance  of  120  feet,  is  ascertained  to  have 
a vetocIty  ofTo  fteT  per’second,  with  an  accelerating  velocity  of  a feet  per  second, 
per  second;  what  was  its  primitive  velocity  and  time  of  flow  . 

S = X20,  V = 4P,  ^ = 2.  40-  33-47 

Then  V402  — 2 X 2 X 120  = 33.4 7 feet- 


- — 3. 26  seconds. 


j 


Delivery"  and  Friction  in  Dose. 

(R.  F.  Hartford , Am.  Soc.  C.  E.) 

Hose  2. 5 ins.  in  diameter.  Nozzles  not  exceeding  1. 5 ins. 

J2^.$T-  G 

Rubber  or  Leather . .0408  v d2  and  .497  c d2  Vp .=  G ) 

4.0484  G2 

d 


'2.012  G_d.  12>i8  cy/P  and  = 


and 

P;  .012  857  b G2  ?; 


c /p  v a*  ■ c2  d*  * ' 

.003 175  6 c2  d4  P Z and  .000021  4 6 l v*  d*  =J>;  P-p  = P';  P*  = r 

» = . , 3I4-96  <t~»)  d 467SO-8a  P (!-»)  , 

..306  (P-p)  and  - jnQ 


P 

1 — .003  175  & c2  d*  i and  — 


6c2rf4  6v2d4 

p — x.  G representing  gallons  dis- 


rharaed  ver  second  v velocity  in  feet  per  second , P pressure  of  stream  at  hydrant  or 
source  ofsuvplv  V ’ pressure  lost  in  hose , and  P'  pressure  at  nozzle  all  in  lbs.  per  sq 
foot  d diameter  of  nozzle  in  ins.,  H Acad  of  supply  at  hydrant , A Aead  at  nozzle  and 
l length  of  hose,  all  in  feet,  x fraction  of  P at  node,  & coefficient  of  material  of  hose, 
and  c for  nozzle. 

b = 1 for  rubber  hose  and  1.167  for  leather. 

c = .82  for  smooth  nozzle  and  .64  for  ring. 

Iliustration.—  Assume  length  of  a rubber  hose  200  feet,  pressure  at  hydrant  100 
lbs.,  diameter  of  ring  nozzle  1.25  ins.,  and  volume  of  discharge  4.97  gallons  per 
second ; what  are  the  other  elements  to  be  obtained  by  preceding  formulas  i 


.497  X -64  X 1.252  X a/  100  — 4-97  gallons. 

/^Si.X±9Z  and  = ins. 

V v V -64  v 100 

.012  857  X I X 4-97 2 x 200  = 63. 52  lbs. 

100  X • 3648  = 36. 48  lbs.  2. 306  (100  — 63. 52)  = 84. 12  feet. 


24.5.X4-97  =77  96/eet 

4.0484  x 4-97  2 _ £22 
.64s  X i-254  — x 
100  — 63.52  = 36.48  lbs. 


100  i 


77. q62 


9X2  g 

1— .003 175  X 1 X -64s  X 1.25*  X 200  = 1 — .6352  = .3648  = x. 
314.96(1  — 3648)  _ 2oo  and  46  750.82  X 100  (r  -t-3648)  - ^fret 
1 X .64=  X 1.25*  1 i X 77-96“  X 1-25*  , 

84.  12  X 63.  52 


= 84.12  feet. 


-63.52  = .3648  = x. 


36.48 


- = 146.47/eef. 


For  Vertical  Jets,  see  page  549. 

Gauging  of  Weirs. 

When  there  is  an  Initial  Velocity.  ( H -f-  A, i — h 2)  3 = H'.  H and  H represent- 
ing depth  of  water  on  weir , and  when  corrected  to  include  effect  of  initial  velocity  of 
approaching  water,  and  h head  to  which  this  velocity  is  due,  all  in  feet. 

Velocity  in  Pipes  C VrT  = V.  r representing  mean  radius  or  hydraulic  mean 
depth*  1 sine  of  angle  of  inclination  equal  to  loss  of  head  per  unit  of  length,  V velocity 
in  feet  per  second , and  *C  a mean  coefficient  of  142. 

In  small  Channels.  C = 30  to  50. 

Noth  —Sectional  area  of  a pipe  or  conduit,  divided  by  perimeter.  Is  termed  mean  radius,  and.  WTien 
the  pipe,  conduit,  or  channel  is  but  partially  filled,  the  area  is  termed  hydraulic  mean  depth. 


See  also  page  552. 


c 


920 


APPENDIX. 


Metric  Factors.  In  addition  to  pp.  27-37. 

By  Act  of  Congress , July,  1866.  By  French  Metric  Computation, 

IVIeasnres. 

1 Liter  per  cube  meter  = .007  48  gallons  per  cube  foot . . . j .007  48  gallons. 

Weiglits  and.  Pressures. 

1 Centimeter  of  mercury  per  sq.  inch  = .19291  lb.  per) 
sq.  inch. 


1 Atmosphere  (14.7  lbs.)  = 6.6679  kilograms 

1 Inch  of  mercury  per  sq.  inch  = 2.54  centimeters. . 

1 Pound  per  sq.  inch  = 453.6029  grams 

1 Cube  foot  per  ton  = .027^  cube  meter 


Heat. 

1 Caloric  per  Kilogram  = 1.8  heat  units  per  lb 

V elocity'. 

1 Meter  per  second  = 3.280  833  feet  per  second 


.192  911  7 lb. 
6.6678  kilograms. 
2.54  centimetres. 
453-  5926  grammes. 
.0279  cubic  metre. 

,|  1.8  heat  units. 

. | 3.280869  feet. 


Power  and  ‘Work. 

1 Kilogrammeter  {k  x m)  — 2.2046  X 3-28083 

1 Foot-pound  = .13826  kilogrammeters 

1 Kilogram  per  cheval  = 2.2352  lbs.  per  IP 

1 Sq.  foot  per  BP  = .091  63  sq.  meter  per  cheval 


7.233  foot-lbs. 

.13825  kilogrametre. 
2.2353  pounds. 

.091  63  sq.  metre. 


IVLiscellaneons. 


1 Avoirs  Lb.=  -4536  kilogram. 

1 Ton  = 1.016057  tonne. 

1 Sq.  Inch  = 645.161  29  sq.  miWrs. 
1 Mile  per  hour 


Sq.  Foot 


= .092903  sq.  meter. 

1 Cube  Foot  =.028317  cube  meter. 
1 Cube  Yard  = .764559  cube  meter. 
= 26.8225  meters  per  minute, 
x Knot  “ “ (6086.44  feet)  = 30.9192  “ “ “ 

1 Cube  Meter  per  minute  = 7.848  cube  yards  per  hour, 

x “ Yard  “ u = 45.8718  “ meters  u “ 


Locomotive  Brakes. 


v2  5 V* 

7 7 and  — - = distance  in  which  a train  is 

&4-4/  30/ 


stopped,  v and  V representing  velocity  in  feet  per  second , and  miles  per  hour , and 
f proportion  of  resistance  of  brakes  to  weight  of  train. 

Brakes,  self-acting,  on  all  wheels, /=  .14.  Ordinary  hand,/=  .023  to  031  As- 
cending 1 in  .5  resistance  is/+  2 ; descending  1 in  .5  /—  2r. 

Hydraulic  Hams.  Efficiency  decreases  rapidly  as  height  to  which  water 

is  to  be  raised  increases  above  the  fall  or  head. 

Number  of  times  the  height  to  which  the  water  is  raised  exceeds  that  of  the  head 
of  the  supply  and  efficiency  per  cent.  {Walter  S.  Hutton , C.  and  M.  E.) 

Number  ...  4 567  8 9 10  n 12  13  14  15  16  18  19  20  25 

Efficiency..  75  72  68  62  57  53  48  43  38  35  32  28  23  17  15  X2  o 

Speed  of  water  in  pumps,  200  feet  per  minute. 


Xo  Compute  Weight  of*  Water  at  any  Temperature. 

2 w 


500 


- = W.  W and  w representing  weights  of  water  per  cube 


= 62.425  lbs.,  and  461.2° 


T-f-46i.2c 

500  T T -f-  461. 2° 

foot  at  tempei'ature  T,  and  at  maximum  density  of  39. 2 
equal  absolute  temperature. 

Illustration. — Required  weight  of  a cube  foot  of  water  at  temperature  of  6o°. 

62.425  X 2 r 

- = 62.37  lbs. 


60 -f- 461. 2 
500 


+ 


500 

60  -j-  461.2 


appendix. 


921 


..  - Experiments  or  Performances  of 

Results  and  Boilers. 

_r  nn/1  Diameters  in  Inches , Revolutions  per  Minute , 

“krf  in  ita,  ^tr/aces  and  ^reas  m Ng.  J«. 


Elements  of  Engine. 


Harris. 

Non-con- 

densing. 


Corliss. 


Cylinder 

Revolutions 

Pressure  in  Pipe . . • 

Cut  off 

Mean  effective  Pressure 

IIP 

Friction  IP 

Net  IP y • • • ‘ 

Water  per  net  IP  I 

per  hour ) 

Coal  per  do 

Coal  per  IIP  per  I 
hour ) 

Vacuum 

Combustible  per  1 
IIP  per  hour  . ) 

Relative  efficiency  . . 

* Weight  of  engine,  40  cco  lbs. 


Con- 

densing. 

18X42 

73*6 
76*37 
7*94 
29.47 
115*43 
13. °7 
102.36 


Con- 

densing. 

24X60^ 

59*62 

92.88 

18.02 

89.38 

270.58 

12.55 


Boilers. 


Number 

Diameter. 

Length 

Tubes  50 

Heating  Surface. 

Grate  “ 

Calorimeter 

Heating  to  Grate 

Grate  to  Calorimeter. . 
Temperature  of  Feed. 
Steam  per  Lb.  of ) 
Combustiblet . . ) 
Steam  per  Lb.  of  ) 

Coal ) 

Coal  per  Sq.  Foot! 

of  Grate  per  hour ) " 
Steam  per  Temp.  2120 


60 

12 

4 

[536.92 

51*75 

1256.64 

29.7 


*/jji  1 . 

t Steam  per  lb.  of  coal  8.2.  lb,.,  and  evaporaUon  9 to 


Diameter 

of 

Wheel. 

Feet. 


windmills.  (Andrew  J.  Corcoran,  New  York.) 
Volume  of  Water  Pumped  per  Minute. 
From  10  to  200  Feet . 

V ERTICAI.  tKOM  W»  » 


Gallons. 

15.242 

48.262 

86.708 

111.665 

155.982 

249*93 

309.604 

532-5I7 

1080.112 


i5 

Gallons. 
10.162 
32-I75 
57*805 
74*443 
103.988 
159*954 
206 . 403 
355*oi2 


25 

Gal  Ions. 
6.162 
i9*x79 
33*941 
45* I39 

64.6 

97.682 

124.95 

212.38 


50 


728.828  I 430.848 


Gallons. 

3.016 

9.563 

17*952 

22.569 

3I-654 

52.165 

63.75 

106.964 

216.172 


75 


Gallons. 

6.638 
11.851 
i 5 • 3°4 
19.542 

32-513 

40.8 
7 1 . 604 


100 

Gallons. 

4*25 

8.485 

11.246 

16.15 

24.421 

31.248 

49.725 


Gallons.  I Gallons. 


[46.608  j 107. 712 


5.68 
7.807 
9*77 1 
i7*485 
19.284 
37*349 

74.8 


4*998 

8.075 

12.211 

I5-938 

26.741 

54*°43 


■Velocity  of  Wind. 

TTn:tGd  states  as  determined  by  the  Signal  Service  of  the 
”0  TwSn^ere  is  required  an 

av4age  velocity  of  wind  of  six  miles  per  hour. 

37- S _ pressure  oj  wind  per  sq.foot  of  surface  m lbs. 

4^7^Q><32^6 

Ot*  — V2  and  — u'2*  v representing  velocity  of  air  m feet  per  second , and 
’ 400  200 

in  miles  per  hour.  Wolff  .1.  Wiley 

X„te. For  u.ofal  table,  and  formula,  aee  “ Windmill,  a,  aPnme  Mover,  by  A.  R.  M , 

& Sons,  New  York,  1885. 


922 


APPENDIX. 


To  Compute  Head  in.  JL,L>s.  per  Sq.  Incli  to  Resist  Fric- 
tion of  An*  in  Long  and  Rectilineal  Pipes,  etc. 


V 1728  V2  L 

a 60"  V)  (3. 7 d) 5 83.  ] 


= H : 


H (3.7  d) 5 83. 

ya 


r-*  7 


^(3.7^5  83.3 


==  V; 


5 / L ^ a 60"  v P-{-H 

y 83.1  H ' 3‘7~  a’  ana  I2"  x 33  000  ~ ^ V representing  volume  discharged 

^ cube  feet  per  minute,  L length  of  pipe  in  feet,  d diameter  of  pipe  in  ins.,  H head 

and  n nllnUr/ri  ™ and  Per  *?•  inchi  v velocity  of  discharge  in  feet  per  sec- 
ond, a area  of  discharge  m sq.  ms.,  and  HP  horse-power  of  friction  of  air  alone. 

h Jmo3RA?I^)N'T'AsSUme  volume  of  air  discharged  44000  cube  feet  per  minute 
diameter  of  discharge  pipe  40.5+  ms.  (say  1280  sq.  ins.  net),  length  of  pipe  1000 
feet,  and  pressure  at  discharge  3.5  lbs.  per  sq.  inch. 


Then  44°°°Xr728 


1280  X 60 
44  000  2 X 1000 
6 310406  250000 


74i- 5 IP. 


ri  936090  000  X 1000 
83.1  X .3068 


99°  feet,  and  (3.7  x 4°-  5~H5  X 83.1  =6310406250000. 
3068  lbs.  ; cu^feet. 

-4-  3-7  = 40.5  ins.,  and 


1280  x 60  X 990  X 3. 5 + -3068 
12  X 33000 


Ice  or  Cold  Producing  Machines  witliont  Rower  or 
Pressure.  Dry- Agent  System.  (E.  Gillet,  New  York.) 

The  Operation  of  this  Machine  is  Automatic. 

Ice  produced  in  1 hour;  water  or  other  liquids  rapidly  cooled,  from  ioo°  and 
above  to  freezing-point  and  below.  ’ 

produce  simultaneously  35  lbs.  water  per  hour  at  a tempera 

peraturJ06^ -maohines  produce  in  like  manner  70  lbs.  water  per  hour  at  a like  tem- 

Ihe  capacity  of  these  machines  can  be  increased  to  any  size  as  required,  and  they 
ar£u  P ™ e to  t^ie  Pr.°duction  of  ice,  refrigerating  of  brine,  meats,  etc. 

the  Refrigerator,  which  is  or  can  be  applied  to  the  machine,  is  cooled  by  the  ex- 
cess  of  low  temperature  of  the  solution,  after  making  the  ice,  cooling  the  water  etc. 

Ihe  chemical  element  cannot  crystalize  while  in  process  of  evaporation5  but 
when  allowed  to  coo  it  will  crystalize  at  once,  and  be  readily  granulated  if  allowed 
to  drop  from  a low  elevation. 

Owing  to  the  use  of  one  or  more  Economizers  embraced  in  the  system,  1. 5 to  2 
lbs.  of  water  is  required  per  lb.  of  ice.  7 5 

This  system  consists  simply  in  mixing  a special  salt  with  water,  and  reducing  it 
by  evaporation,  or  evaporating  a few  hundredweight  of  water  per  hour,  the  water 
decreasing  in  temperature  while  passing  from  one  Economizer  to  another,  and  in- 
versely on  returning  to  Evaporator.  ’ 

Friction,  of  Water  in  Pipes.  (Weisbach.) 

. 1865  l V2  fi  V 

c = h.  I representing  length  of  pipes  in  feet , v = , or  velocity 

in  feet  per  second,  V volume  of  water  in  cube  feet  elevated  per  second,  d diameter  of 
pipes  m ms.,  and  C a coefficient,  ranging  from  .069  when  velocity  — .1  foot,  .0387  for 
’ -°375  f°r  l f™t,  .0265 for  2 feet,  .023  for  4 feet,  .0214  for  6 feet,  .0205  fir  8 
feet,  .0193 /or  12  feet,  and  .01 82 /or  20  feet. 

Illustration.— Assume  volume  125  cube  feet,  raised  25  feet  per  hour,  through  a 
pipe  2 ins.  in  diameter  and  500  feet  in  length;  how  many  feet  of  vertical  head  will 
the  friction  m the  pipe  be  equal  to  ? 

Then  ^36^  x 22 " = 1,59  velocit>r>  and  c = -°28- 
Hence>  " ^ 5g  X 1 59  X .028  = 3.3  feet,  and  25-1-3.3  = 28.3  feet. 


ORTHOGRAPHY  OF  TECHNICAL  AVORDS  AND  TERMS.  923 
ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 

a uniformity  of  expression. 

Abut.  To  meet,  to  adjoin  to  at  the  end  to  border  upon.  Abut  end  of  a log,  etc. , 
is  that  having  the  greatest  diameter  or  side. 

But  and  Butt  end,  when  applied  in  this  manner,  are  corruptions. 

Adit  In  Mining , the  opening  into  a mine. 

‘ • Tho  middle  or  centre  of  a vessel,  either  fore  and  aft  or  athwartships. 

Tht:“p  tomemof  ad  veSat  0,  and  is  termed  dead  flat. 

rod  tn  nainted  and  carved  or  sculptured  ornaments  of  imaginary 
foliage* a^d^ani mals, 'In  which  there  are  no  perfect  figures  of  either.  Synonymous 
with  Moresque.  . 

Arbor  The  principal  axis  or  spindle  of  a machine  of  revolution. 

facesTof  a bodyl"  forming^n'e^terhir  angle^me^ea^o^er.^T^e  edges  ofVa  bodyj 
as  a brick,  are  arrises. 

Ashlar  In  Masonry , stones  roughly  squared,  or  when  faced. 

Hart.  Across,  from  side  to  side,  transverse,  across  the  line  of  a vessel's 
course. 

Athwartships , reaching  across  a vessel,  from  side  to  si  e. 

Bagasse.  Sugar-cane  in  its  crushed  state,  as  delivered  from  the  rollers  of  a mill. 
Balk.  In  Carpentry,  a piece  of  timber  from  4 to  10  ins.  square. 

Baluster.  A small  column  or  pilaster;  a collection  of  them,  joined  by  a rail,  forms 
a balustrade. 

Banister  is  a corruption  of  balustrade. 

Bark  A ship  without  a mizzen-topsail,  and  formerly  a small  ship. 

Bateau.  A light  boat,  with  great  length  proportionate  to  its  beam,  and  wider  at 
its  centre  than  at  its  ends. 

Batten  In  Carpentry , a piece  of  wood  from  i to  2 5 ins.  thick  ^ [rom  1 7 

infCbread?h  When  less  than  6 feet  in  length,  it  is  termed  a deal-end. 

Berme.  In  Fortifications  and  Engineering , a space  embaPnk- 

and  a moat  or  fosse,  to  arrest  the  ruins  of  a rampart  The  level  top  oi 
ment  of  a canal,  opposite  to  and  alike  to  the  towpath. 

Bevel.  A term  for  a plane  having  any  other  angle  than  45°  or  9°  • 

Binnacle.  The  case  in  which  the  compass,  or  compasses  (when  two  are  used), 
set  on  board  of  a vessel.  a 

Bit.  The  part  of  a bridle  which  is  put  into  an  animal’s  mouth.  In  Caipentry,  a 
boring  instrument. 

Bitter  End.  The  inboard  end  of  a vessel’s  cable  abaft  the  bitts. 

Bills.  A vertical  frame  upon  a deck  of  a vessel,  around  or  upon  which  is  secured 
cables,  hawsers,  sheets,  etc. 

Bogie.  Pivoted  truck,  to  ease  the  running  of  an  engine  or  car  around  a curve^ 
Boomkin.  A short  spar  projecting  from  the  bow  or  quarter  of  a vessel,  to  exten 
the  tack  of  a sail  to  windward. 

Bowlder.  A stone  rounded  by  natural  attrition;  a rounded  mass  of  rock 
ported  from  its  original  bed. 

Breast-summer.  A lintel  beam  in  the  exterior  wall  of  a building. 

Buhr-stone.  A stone  which  is  nearly  pure  silex,  full  of  pores  and  cavities,  a 
used  for  Mills. 

Bunting.  Woolen  texture  of  which  colors  and  flags  are  made. 

Burden.  A load.  The  quantity  that  a ship  will  carry.  Hence  burdensome. 

Cag.  A small  cask,  differing  from  a barrel  only  in  size.  Commonly  written  Keg. 


924  ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 


Caliber.  An  instrument  with  semi-circular  legs,  to  measure  diameters  of  snheres 
or  exterior  and  interior  diameters  of  cylinders,  bores,  etc.  ’ 

A pair  of  Calibers  is  superfluous  and  improper. 

Calk.  To  stop  seams  and  pay  them  with  pitch,  etc.  To  point  an  iron  shoe  so  as 
to  prevent  its  slipping.  0 

Cam  An  irregular  curved  instrument,  having  its  axis  eccentric  to  the  shaft 
upon  which  it  is  fixed.  0 “u 

Camber.  To  camber  is  to  cut  a beam  or  mold  a structure  archwise  as  deck 
beams  of  a vessel.  » ueLK' 


Camboose.  The  stove  or  range  in  which  the  cooking  in  a vessel  is  effected  Tho 
cooking-room  of  a vessel;  this  term  is  usually  confined  to  merchant  vessels  n 
vessels  of  war  it  is  termed  Galley.  in 

CW  In  Engineering,  a decked  vessel,  having  great  stability,  designed  for  use 
weight  or  bufk  SUn  V6SSelS  °r  structures-  A|s°  to  transport  loads  of  great 

A Scow  is  open  decked. 

Cantle.  A fragment;  a piece;  the  raised  portion  of  the  hind  part  of  a saddle. 
Cantime.  The  space  between  the  sides  of  two  casks  stowed  aside  of  each  other 
TantUne.  “ “ th°  CaDtline  °f  tW°  0tliers>  il  is  said  to  be  stowed  bilge  and 

Capstan.  A vertical  windlass. 

inferring  fisheries!'  VCSSel  (°f  25  °r  3°  t0nS’  bllrdeD)  used  uPon  the  coast  of  Prance 

reeelveX  2^?.^  VnJber  se‘ fore.  and  aft  from  the  deck  beams  of  a vessel,  to 
leceive  the  ends  of  the  ledges  in  framing  a deck. 

Carvel  built.— A term  applied  to  the  manner  of  construction  of  small  boats  to 
signify  that  the  edges  of  their  bottom  planks  are  laid  to  each  other  like  to  the  man- 
ner of  planking  vessels.  Opposed  to  the  term  Clincher. 


Caster.  A small  phial  or  bottle  for  the  table.  Casters.  Small  wheels  placed 
upon  the  legs  of  tables,  etc.,  to  allow  them  to  be  moved  with  facility. 

Catamaran.  A small  raft  of  logs,  usually  consisting  of  three,  the  centre  one  be- 
ing longer  and  wider  than  the  others,  and  designed  for  use  in  an  open  roadstead 
and  upon  a sea-coast. 

Chamfer.  A slope,  groove,  or  small  gutter  cut  in  wood,  metal,  or  stone. 

Chapelling.  Wearing  a ship  around  without  bracing  her  fore  yards. 

Chimney.  The  flue  of  a fireplace  or  furnace,  constructed  of  masonry  in  houses 
and  furnaces,  and  of  metal,  as  in  a steam  boiler.  See  Pipe. 


Chinse.  To  chinse  is  to  calk  slightly  with  a knife  or  chisel. 

Chock.  In  Naval  Architecture , small  pieces  of  wood  used  to  make  good  any  de- 
ficiency in  a piece  of  timber,  frame,  etc.  See  Furrings. 

Choke.  To  stop,  to  obstruct,  to  block  up,  to  hinder,  etc. 

Cleats.  Pieces  of  wood  or  metal  of  various  shapes,  according  to  their  uses,  either 
to  belay  ropes  upon,  to  resist  or  support  weights  or  strains,  as  sheet,  shoar,  beam 
cleats,  etc. 

Clincher  built.  A term  applied  to  the  construction  of  vessels’  bottoms,  when 
the  lower  edges  of  the  planks  overlay  the  next  under  them. 

Coak.  A cylinder,  cube,  or  triangle  of  hard  wood  let  into  the  ends  or  faces  of  two 
pieces  of  timber  to  be  secured  together.  The  metallic  eyes  in  a sheave  through 
which  the  pin  runs.  In  Naval  Architecture , the  oblong  ridges  banded  on  the  masts 
of  ships. 

Coamings.  Raised  borders  around  the  edges  of  hatches. 

Coble.  A small  fishing-boat. 

Cocoon.  The  case  which  certain  insects  make  for  a covering  during  the  period 
of  their  metamorphosis  to  the  pupa  state. 


ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS.  925 

Cog.  In  Mechanics,  a short  piece  of  wood  or  other  material  let  into  the  faces  of 
a bodv  to  impart  motion  to  another.  A term  applied  to  a tooth  in  a wheel  when  it 
is  made  of  a different  material  than  that  of  the  wheel.  In  Mining,  an  intrusion  of 
matter  into  fissures  of  rocks,  as  when  a mass  of  unstratifled  rocks  appears  to  be  in- 
jected  into  a rent  in  the  stratified  rocks. 

Coaaina  In  Carpentry , the  cutting  of  a piece  of  timber  so  as  to  leave  a part 
alike^to  a^cog,  and  the  notching  of  the  upper  piece  so  as  to  conform  to  and  receive 
it.  Alike  to  indenting  or  tabling. 

Colter.  The  fore  iron  of  a plough  that  cuts  earth  or  sod. 

Compass.  In  Geometry , an  instrument  for  describing  circles,  measuring  figures,  etc. 

A pair  of  Compasses  is  superfluous  and  improper. 

Connectina  Rod  In  Mechanics , the  connection  between  a prime  and  secondary 
mover,  as  between  the  piston-rod  of  a steam-engine  and  the  crank  of  a water-wheel 
or  fly-wheel  shaft. 

The  term  Pitman  is  local,  and  altogether  inapplicable. 

Contrariwise.  Conversely,  opposite.  Crossways  is  a corruption. 

Corridor.  A gallery  or  passage  in  or  around  a building,  connected  with  various 
departments,  sometimes  running  within  a quadrangle ; it  may  be  opened  or  enclosed. 
In  Fortifications , a covert  way. 

Cyma.  A molding  in  a cornice. 

Damasquinerie.  Inlaying  in  metal. 

Davit.  A short  boom  fitted  to  hoist  an  anchor  or  boat. 

Deals.  In  Carpentry , the  pieces  of  timber  into  which  a log  is  cut  or  sawed  up. 
Their  usual  thickness  is  3 by  9 ins.  and  exceeding  6 feet  in  length. 

Improperly  restricted  to  the  wood  of  fir-trees. 

Dike.  In  Engineering , an  embankment  of  greater  length  than  breadth,  imper- 
vious to  water,  and  designed  as  a wall  to  a reservoir,  a drain,  or  to  resist  the  influx 
of  a river  or  sea. 

Dingay  (Nautical).  A ship  or  vessel’s  small  boat. 

Dock.  In  Marine  Architecture , an.  enclosure  in  a harbor  or  shore  of  a river,  for 
the  reception,  repair,  or  security  of  vessels  or  timber.  It  may  be  wholly  or  only 
partially  enclosed.  See  Pier. 

When  applied  to  a single  pier  or  jetty,  it  is  a misapplication. 

Dowel.  A pin  of  wood  or  metal  inserted  in  the  edge  or  face  of  two  boards  or 
pieces,  so  as  to  secure  them  together. 

This  is  very  similar  to  coaking,  but  is  used  in  a diminutive  sense.  An  illustration  of  it  is  had  in  the 
manner  a cooper  secures  two  or  more  pieces  in  the  head  of  a cask. 

Draught.  A representation  by  delineation.  The  depth  which  a vessel  or  any 
floating  body  sinks  into  water.  The  act  of  drawing.  A detachment  of  men  from 
the  main  body,  etc. 

Ordinarily  written  draft. 

Dutchman.  In  Mechanics , a piece  of  like  material  with  the  structure,  let  into  a 
slack  place,  to  cover  slack  or  bad  work.  See  Shim. 

Edgewise.  An  edge  put  into  a particular  direction.  Hence  endwise  and  sidewise 
have  similar  significations  with  reference  to  an  end  and  a side. 

Edgeways  is  a corruption. 

Euphroe.  A piece  of  wood  by  which  the  crowfoot  of  an  awning  is  extended. 
Fault.  In  Mining , a break  of  strata,  with  displacement,  which  interrupts  opera- 
tions. Also,  fissures  traversing  the  strata. 

Felloe , Felloes.  The  pieces  of  wood  which  form  the  rim  of  a wheel. 

Fetch.  Length  of  a reservoir,  pond,  etc.,  along  which  the  wind  may  blow  towards 
the  embankment  or  dam. 

Flange.  A projection  from  an  end  or  from  the  body  of  an  instrument,  or  any 
part  composing  it,  for  the  purpose  of  receiving,  confining,  or  of  securing  it  to  a sup- 
port or  to  a second  piece. 

Flier.  In  Carpentry , a straight  line  of  steps  in  a stairway. 

Frap.  To  bind  together  with  a rope,  as  to  frap  a fall,  etc. 


926  ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 


Frieze  In  Architecture,  the  part  of  the  entablature  of  a column  which  is  between  f 
""  TeVt  “id  next  the  base,  left  by  the  removal  of  the  top  or 

« *».  “ — » 

brfn.,  tbeir  faces  to  the  required  shape  or  level. 

Gliding.  Putting  galets  into  pointing-mortar  or  cement. 

Galete  Pieces  of  stone  chipped  off  by  the  stroke  of  a chisel.  See  Spall 

: Galiot.  A small  galley  built  for  speed,  having  one  mast,  and  from  .6  to  =o  thwarts 

for  rowers  A Dutch-constructed  brigantine, 
j Gate.  In  Mechanics,  the  hole  through  which  molten  metal  is  poured  into  a mold 

, for  casting.  " “"^^^.wheete  for  transmlttitigmbtlon.  To  gear* 

) by  passins 
around  his  belly.  In  Printing,  the  bands  ol  a press. 

Gnarled.  Knotty. 

ii  nvnvtfi  To  clean  a vessel’s  bottom  by  burning. 

Tig  Z tin,  off  grass,  shells,  etc,  from  a ship’s  bottom.  Synonymous 
r<  with  Breaming. 

u,  and  aSSenSlotheSem0  Cat  harping*; ropes  which  brace  in  the  shrouds  of 

in  to  the  huU  of  a vessel  when  ter  ends  drop  below  h4r 
aD  C®«i»r  InNaral  Architecture,  calking  with  a large  maul  or  beetle. 

"to  press,  to  crowd,  to  wedge  in.  In  Nautical  language,  to  squeeze  tight. 

",  Pe^d-  /o  from  one  tack  to  another  ; hence  JiUng,  the  shifting 

of  a boom. 

Jigging.  Washing  minerals  in  a sieve  of  tbe  floor  timbers, 

anf  eeSy  W M l"  When  Rcated  on  & floors  or  at  the  sides,  it  is  termed  a 
sisters  or  a side  keelson. 

Si 

S«,.  'wm>  «*»»"•»“  »•"■«*■ 

“CSX  * ™ » .> « — »'  - — « ■»  " 

will  not  float  or  sit  upright.  ^ rin^  or  securing  her  to  a 

SI  < ^-'anting  edge  of  a sail  when 

not  secured  to  a spar  or  stay. 


( 

fici 

( 

( 

to  1 
clei 

C 

the 

C 
pie( 
whi 
of  s 

C 
C( 
Cl 
of  tt 


ORTHOGRAPHY  OP  TECHNICAL  WORDS  AND  TERMS.  g2f 

Luf  The  fullest  part  of  the  bow  of  a vessel. 

Mall.  A large  double-headed  wooden  hammer. 

Mantle.  To  expand,  to  spread.  Mantelpiece.  The  shelf  ovef  a fireplace  in  front 
of  a chimney. 

Marquetry.  Checkered  or  inlaid  work  in  wood. 

Matrass.  A chemical  vessel  with  a body  alike  to  an  egg,  and  a tapering  neck. 
Mattress  A quilted  bed ; a bed  stuffed  with  hair,  moss,  etc.,  and  quilted. 

Mitred.  In  Mechanics,  cut  to  an  angle  of  45°,  or  two  pieces  joined  so  as  to  make 
a right  angle. 

Mizzen-mast.  The  aftermost  mast  in  a three-masted  vessel. 

Mold  In  Mechanics , a matrix  in  which  a casting  is  formed.  A number  of  pieces 
of  vellum  or  like  substance,  between  which  gold  and  silver  are  laid  for ^the  P^rpos q 
of  bein°-  beaten.  Thin  pieces  of  materials  cut  to  curves  or  any  required  figure  In 
Naval  'Architecture  pieces  of  thin  board  cut  to  the  lines  of  a vessel  s timbers  etc. 

Fine  earth,  such  as  constitutes  soil.  A substance  which  forms  upon  bodies  in 
warm  and  confined  damp  air. 

This  orthography  is  by  analogy,  as  gold,  sold,  old,  bold,  cold,  fold,  etc. 

Molding.  In  Architecture,  a projection  beyond  a wall,  from  a column,  wainscot,  etc 
Moresque.  See  Arabesque. 

Mortise.  A hole  cut  in  any  material  to  receive  the  end  or  tenon  of  another  piece. 
Muck.  A mass  of  dung  in  a moist  state,  or  of  dung  and  putrefied  vegetable  matter. 
Mullion.  A vertical  bar  dividing  the  lights  in  a window  ; the  horizontal  are 
termed  transoms. 

Net.  Clear  of  deductions,  as  net  weight. 

Newel.  An  upright  post,  around  which  winding  stairs  turn. 

Nigged.  Stone  hewed  with  a pick  or  pointed  hammer  instead  of  a chisel. 

Ogee.  A molding  with  a concave  and  convex  outline,  like  to  an  S.  See  Cyma 
and  Talon. 

Paillasse.  Masonry  raised  upon  a floor.  A bed. 

Pargeting.  In  Architecture , rough  plastering,  alike  to  that  upon  chimneys. 
Parquetry.  Inlaying  of  wood  in  figures.  See  Marquetry. 

Parral.  The  rope  by  which  a yard  is  secured  to  a mast  at  its  centre. 

Pawl.  The  catch  which  stops,  or  holds,  or  falls  on  to  a ratchet  wheel. 

Peek  The  upper  or  pointed  corner  of  a sail  extended  by  a gaff,  or  a yard  set  ob- 
liquely to  a mast  To  peek  a yard  is  to  point  it  perpendicularly  to  a mast. 

Pendant.  A short  rope  over  the  head  of  a mast  for  the  attachment  of  tackles 
thereto ; a tackle,  etc. 

Pennant.  A small  pointed  flag. 

Pier.  In  Marine  Architecture,  a mole  or  jetty,  projecting  into  a river  or  sea,  to 
protect  vessels  from  the  sea,  or  for  convenience  of  their  lading.  See  Dock. 

Erroneously  termed  a Dock. 

Pile.  In  Engineering , spars  pointed  at  one  end  and  driven  into  soil  to  support  a 
superstructure  or  holdfast.  Spile  is  a corruption. 

Pipe.  In  Mechanics , a metallic  tube.  The  flue  of  a fireplace  or  furnace  when 
constructed  of  metal;  usually  of  a cylindrical  form. 

The  term  or  application  of  Stack  (which  refers  solely  to  masonry)  to  a metallic  pipe  is  a misappli- 
cation. 

Piragua.  A small  vessel  with  two  masts  and  two  boom -sails. 

Commonly  termed  Perry-augur. 

Pirogue.  A canoe  formed  from  a single  log,  propelled  by  paddles  or  by  a sail, 
with  the  aid  of  an  outrigger. 


Plastering. 


Jl  dill  uuu  *65^*  • 

In  Architecture , covering  with  plaster  cement  or  mortar  upon  walls  , 
England,  termed  laying , if  in  one  or  two  coat  work;  and  priclcing  up, 

a t 


or  laths.  In  — 0 

if  in  three-coat  work. 

Plumber  block.  A bearing  to  receive  and  support  the  journal  of  a shaft. 
Polctcre.  Masts  of  one  piece,  without  tops. 


928  ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND  TERMS. 


Poppets.  In  Naval  Architecture , pieces  of  timber  set  perpendicular  to  a vessel’s 
bilge-ways,  and  extending  to  her  bottom,  to  support  her  in  launching. 

Porch.  An  arched  vestibule  at  the  entrance  of  a building.  A vestibule  supported 
by  columns.  A portico. 

Portico.  A gallery  near  to  the  ground,  the  sides  being  open.  A piazza  encom- 
passed with  arches  supported  by  columns,  where  persons  may  walk;  the  roof  may 
be  flat  or  vaulted.  J 

Pozzuolana.  A loose,  porous,  volcanic  substance,  composed  of  silicious  argilla- 
ceous, and  calcareous  earths  and  iron. 

Prize.  In  Mechanics , to  raise  with  a lever.  To  pry  and  a pry  are  corruptions. 

Proa,  Flying.  A narrow  canoe,  the  outer  or  lee  side  being  nearly  flat.  A frame- 
work, projecting  several  feet  to  the  windward  side,  supports  a solid  bearing  in  the 
form  of  a canoe.  Used  in  the  Ladrcne  Islands. 

Purlin.  In  Carpentry , a piece  of  timber  laid  horizontal  upon  the  rafters  of  a 
roof,  to  support  the  covering. 

Ramp.  In  Architecture , a flight  of  steps  on  a line  tangential  to  the  steps.  A 
concave  sweep  connecting  a higher  and  lower  portion  of  a railing  wall  etc.  A 
sloping  line  of  a surface,  as  an  inclined  platform. 

Rarefaction.  The  act  or  process  of  distending  bodies,  by  separating  their  parts 
and  rendering  them  more  rare  or  porous.  It  is  opposed  to  Condensation. 

Rebate.  In  Mechanics , to  pare  down  an  edge  of  a board  or  a plate  for  the  purpose 
of  receiving  another  board  or  plate  by  lapping.  To  lap  and  unite  edges  of  boards 
and  plates.  In  Naval  Architecture , the  grooves  in  the  side  of  the  keel  tor  receiving 
the  garboard  strake  of  plank. 

Commonly  written  Rabbet. 

Remou.  Eddy  water  without  progressive  action,  in  bed  of  a river*  a return  of 
water  against  direction  of  flow  of  a river. 

Rendering.  In  Architecture , laying  plaster  or  mortar  upon  mortar  or  walls. 
Rendered  and  Set  refers  to  two  coats  or  layers,  and  Rendered , Floated,  and  Set  to 
three  coats  or  layers. 

Reniform.  Kidney-shaped. 

Resin.  The  residuum  of  the  distillation  of  turpentine.  R0Sin  is  a corruption. 

Riband.  In  Naval  Architecture , a long,  narrow,  flexible  piece  of  timber. 

Rimer.  A bit  or  boring  tool  for  making  a tapering  hole.  In  Mechanics , to  Rim 
is  to  bevel  out  a hole.  Riming.  The  opening  of  the  seams  between  the  planks  of 
vessel  for  the  purpose  of  calking  them. 

Rotary.  Turning  upon  an  axis,  as  a wheel. 

Rynd.  The  metallic  collar  in  the  upper  mill-stone  by  which  it  is  connected 
the  spindle. 

Sagging.  A term  applied  to  the  hull  of  a vessel  when  her  centre  drops  below 
ends.  The  converse  of  Hogging. 

Scallop.  To  mark  or  cut  an  edge  into  segments  of  circles. 

Scarcement.  A set  back  in  the  face  of  a wall  or  in  a bank  of  earth.  A footii 

Scarf.  To  join ; to  piece ; to  unite  two  pieces  of  timber  at  their  ends  by  rui 
the  end  of  one  over  and  upon  the  other,  and  bolting  or  securing  them  togethei 

Scend.  The  settling  of  a vessel  below  the  level  of  her  keel. 

Selvagee.  A strap  made  of  rope-yarns,  without  being  twisted  or  laid  up,  a 
tained  in  form  by  knotting  it  at  intervals. 

Sennit.  Braided  cordage. 

Sewage.  The  matter  borne  off  by  a sewer. 

Sewed.  In  nautical  language , the  condition  of  a vessel  aground ; she  is  s 
sewed  by  as  much  as  the  difference  in  depth  of  water  around  her  and  her 
depth. 

Sewerage.  The  system  of  sewers. 

Shaky.  Cracked  or  split,  or  as  timber  loosely  put  together. 

Shammy.  Leather  prepared  from  the  skin  of  a chamois  goat. 


ORTHOGRAPHY  OF  TECHNICAL  WORDS  AND 


sheer  In  Naval  Architecture,  the  curve  or  bend  of  a ship's  deck  To 

at  the  upper  cuts,  and  used  to  elevate  heavy 
bodies,  as  masts,  e“.  piece  of  wood  or  iron  lot  into  a slack  place  in  a 

fra»a h 

Shoal,  A great  multitude;  a crowd;  a multitude  of  fish. 

77'ar.  ‘ IToblique  brace,  the  upper  end  resting  against  the  substance  to  be  sup- 

P Shales.  Pieces  of  plank  under The  heels  *^n  which  wood;  coal,  etc. , are 

thfown' or  tlTToKw  Aural  contraction  of  a river.  A young  p,g. 
Sidewise.  See  Edgewise. 

Sianalled.  Communicated  by  signals. 

Zi‘ the  horizontal  piece  of  tim‘ 

be^stoAnf  at  the  bottom  ^ t0  draw  duids  out  of  vessels. 

sfr'Tht  extreme  after-part  of  the  keel  of  a vessel;  the  portion  that  supports 
the  rudder-post. 

Slantwise.  Oblique ; not  perpendicular. 

Sleek  To  make  smooth.  Refuse;  small  coal.  . , 

JZker.  A spherical-shaped,  curved,  or  plane-surfaced  instrument  with  which  to 
smooth  surfaces. 

Slue  The  turning  of  a substance  upon  an  axis  within  its  figu  . 

J , 1 term  applied  to  planks  when  their  edges  at  their  ends  are  curved  or 

rofS'upwa^wsSe  .{  the  ends  of  a full-modelled  vessel. 

r ctene  etc  chipped  off  by  the  stroke  of  a hammer  or  the  force 

Spall.  A p.ece  of  stone,  etc.  chippea  pjeces 

of  a blow.  Spoiling,  break  g P triangnlar  spaCe  between  the  outer  lines 

or^SLf  an  IrcCa  horizontal  line  drawn  from  its  apex,  and  a vertical  hue 

i)0<=e  of  shielding  the  deck-beams  from  the  shock  of  a sea. 

, alike  to  alme,  Me,r  with  leafhgo. 
■vulcanized  rubber,  used  to  facilitate  the  drying  of  wet  floors;  0r  “f S ° ber  Th( 
Stack  In  Masonry,  a number  of  chimneys  or  p.pes  standing  together. 

^The^ppli^  the  smoke-pipe  of  a steam-boiler  is  wholly  erroneous. 

Stape.  Ill  Engineering,  the  interval  or  distance  between  two  e.evalions,  in  sho 
ling,  throwing,  or  lifting. 

Sleeving.  The  elevation  of  a vessel’s  bowsprit,  cathead,  etc. 

Stroke.  A breadth  of  plank. 

Strut.  An  oblique  brace  to  support  a rafter. 

Style.  The  gnomon  of  a sun-dial.  o ^ ,rArt 

s-T;r“r„=rrr,- ...  • 

blow  from  a hammer. 


930  ORTHOGRAPHY  OP  TECHNICAL  WORDS  AND  TERMS. 

Talus.  In  Architecture , tlie  slope  or  batter  of  a wall  naraoet  etc  Tn 
a sloping  heap  of  rubble  at  foot  of  a cliff.  ’ parapet>  etc-  Iu  Geology, 

distffints  weight*^®''’  “ W°°deB  beariB®  t0  receive  the  end  «f  a girder  to 
Templet.  A mold  cut  to  an  exact  section  of  any  piece  or  structure 

izB>  - 

Terring.  The  earth  overlying  a quarry. 

Tester.  The  top  covering  of  a bedstead. 

Tioles.  The  pins  in  the  gunwale  of  a boat  which  are  used  as  rowlocks 
Thwarts.  The  athwartship  seats  in  a boat. 

eurrenUnstead  of  tbe^wind.  °f  & VeSSe‘  ^ aBCh°r’ Wben  shc  rides  in  dire<=tiott  of  the 
Tire.  The  metal  hoop  that  binds  the  felloes  of  a wheel. 

shoS^ecureA  St0PPCr  °f  * PieC°  °f  ordBanco'  The  iron  bottom  which  grape- 

framesMai’k'  W°°deB  PiDS  CmpIoyed  secure  the  planking  of  a vessel  to  the 

boring  at  greaVfepff  **  instr,!ment  in  the  comminution  of  rock  in  earth- 

piecesof  timbe/set'horLomAti6 ’ a movabIe  form  °f  support.  In  Mast-making , two 
pieces  oi  timber  set  horizontally  upon  opposite  sides  of  a mast-head.  u 

2 nee.  In  Seamanship,  to  haul  or  tie  up  by  means  of  a rope  or  tricing-line. 

furaace  °n  °r  Tuyere’  The  nozzle  of  a bellows  or  blast-pipe  in  a forge  or  smelting- 

Vice-  In  Mechanics,  a press  to  hold  fast  anything  to  be  worked  upon. 

Voyal.  In  Seamanship,  a purchase  applied  to  the  weighing  of  an  anchor  lead 
to  a capstan. 

Wagon.  An  open  or  partially  enclosed  four-wheeled  vehicle,  adapted  for 
transportation  of  persons,  goods,  etc. 

Wear.  In  nautical  language,  to  put  a vessel  upon  a contrary  tack  bv  turi 

her  around  stem  to  the  wind.  y 

Weir.  A dam  across  a river  or  stream  to  arrest  the  water;  a fence  of  twiff 
stakes  in  a stream  to  divert  the  run  offish. 

Whipple-tree.  The  bar  to  which  the  traces  of  harness  are  fastened. 

Wind-rode.  The  situation  of  a vessel  at  anchor,  when  she  rides  in  directio- 
the  wind  instead  of  the  current. 

Windrow.  A row  or  line  of  hay,  etc.,  raked  together. 

TFtMe  An  instrument  fitted  to  the  end  of  a boom  or  mast,  with  a ring  thr  1 
which  a boom  is  rigged  out  or  mast  set  up. 

Woold.  To  wind ; particularly  to  bind  a rope  around  a spar,  etc. 

Roil.  To  render  turbid,  to  stir  or  mix. 


THE  EJSTB. 


I 

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